No title

Letters

to

the

Editor

The Mathematical InteUigencer encourages comments about the material in this isxue. Letters to the editor should be sent to the editor-in-chief, Chandler Davis.

One-Way Link Leonard Gillman, in his review of Emblems of Mind, alludes to w h a t I always think of as the one-way correlation b e t w e e n mathematics and mus i c - t h a t while mathematicians often have an affinity for music, m u s i c i a n s are m u c h less apt to have one for mathematics. He says this is "presumably b e c a u s e although a person with n o understanding of music can nevertheless enjoy a m u s i c a l performance, it is unlikely that a n y o n e can curl up with a m a t h e m a t i c s b o o k and enjoy it w i t h o u t u n d e r s t a n d i n g it."

My o w n e x p l a n a t i o n is quite different. It is simply that if one has an aptitude for b o t h m u s i c and mathematics, the practical consideration of having to make a living will often dictate the direction one takes. Jacob E. Goodman Department of Mathematics City College, CUNY New York, NY 10031 USA e-mail: [email protected]

9 1999 SPRINGER VERLAG NEW YORK, VOLUME 21, NUMBER 1, 1999

3

Letters

to

the

Editor

The Mathematical InteUigencer encourages comments about the material in this isxue. Letters to the editor should be sent to the editor-in-chief, Chandler Davis.

One-Way Link Leonard Gillman, in his review of Emblems of Mind, alludes to w h a t I always think of as the one-way correlation b e t w e e n mathematics and mus i c - t h a t while mathematicians often have an affinity for music, m u s i c i a n s are m u c h less apt to have one for mathematics. He says this is "presumably b e c a u s e although a person with n o understanding of music can nevertheless enjoy a m u s i c a l performance, it is unlikely that a n y o n e can curl up with a m a t h e m a t i c s b o o k and enjoy it w i t h o u t u n d e r s t a n d i n g it."

My o w n e x p l a n a t i o n is quite different. It is simply that if one has an aptitude for b o t h m u s i c and mathematics, the practical consideration of having to make a living will often dictate the direction one takes. Jacob E. Goodman Department of Mathematics City College, CUNY New York, NY 10031 USA e-mail: [email protected]

9 1999 SPRINGER VERLAG NEW YORK, VOLUME 21, NUMBER 1, 1999

3

)pinior

On Blindness Lemme B. Bourbaki

The Opinion column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editor-chief endorses or accepts responsibility for them. An Opinion should be submitted to the editor-inchief, Chandler Davis.

4

Vine, when it is judged at competitions, is judged blindly. Musicians auditioning for coveted orchestra spots are judged blindly. The old Coke-versus-Pepsi taste test was blind. Even love is blind. But, alas, mathematics manuscripts, when judged for acceptance by journals, are subject to a bias: the author's name and school affiliation are attached to the submitted manuscript for no apparent reason other than to influence the referee. Some referees blanch at the idea of blind refereeing. Their usual defense of this position--author name and institution affiliation somehow help the referee render a judgment about the quality of the manuscript--at once undermines their stance, by acknowledging precisely that this information does influence the referee. But this is not surprising. Reading a mathematics manuscript carefully is demanding and time-consuming. The typical referee must work hard to find the time--between writing his own papers, teaching his classes, serving on committees, ad nauseam--just to glad-hand all of the manuscripts that editors send his way, to say nothing of actually reading them thoroughly and carefully. The temptation to cut corners is overwhelming. Imagine the typical time-pinched referee with two manuscripts on his desk, one from, say, Alotta Reputation at The Hugh G. Goes University of the Rather Impressive, the other from Joe Feeblepuss at Southeast State University of Agronomy. The poor referee has little time to devote to these manuscripts. The editor has reminded him (twice!) that the reports on both manuscripts are long overdue. Is it a stretch to imagine this frazzled referee will be inclined to give a pro forma scanning of the A. Reputation manuscript before rubberstamping a positive review, while reserving the full powers of his mordant scrutiny for the feckless Feeblepuss manuscript? Or, perhaps, also give a

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THE MATHEMATICAL INTELLIGENCER 9 1999 SPRINGER-VERLAG NEW YORK

pro forma scanning of Feeblepuss's manuscript before rubber-stamping a negative review? ("The mathematics in this manuscript seems to be correct, but it is probably too specialized to be of interest to a wide audience.") The occasional delusional referee may seduce himself into thinking that he can judge manuscripts this way without bias (yea, right, and lobbyists' money doesn't influence politicians); the rest of us know that, of course, referees are influenced by the manuscript author's name and school affiliation. Now it's true that if feckless Feeblepuss submits a rare manuscript of exceptionally high, award-winning quality, it will probably be accepted by a reputable journal. Eventually. But not many mathematicians, not even "big name" mathematicians, often do awardwinning research. Aside from a tiny handful of mathematicians from each epoch, most of us do mostly good competent work that is not particularly monumental. Most of us are worker bees, quietly going about our business of filling in small gaps in the theresa, making computations that support or refute conjectures, etc. There are only a few queens. But the publishing process is blind to this fact. It treats far too many workers like queens, and treats the rest of the workers cavalierly. It doesn't even get the partit i o n - i n t o queens and workers--right: Galois's manuscript was rejected by a referee suitably unimpressed with the young Frenchman's name and pedigree. The American Mathematical Society experimentally used blind refereeing for its Proceedings once for a y e a r - the experiment was discontinued [2]-but other fields often use blind refereeing, e.g., The Journal of the History of Ideas asks its authors to omit their identity from submitted manuscripts [4]. Not only is this obviously fairer (Shaugnessy [8] notes that "some papers are published because of the rep-

utation of the a u t h o r s or institutions 9 editors or reviewers let inferior pap e r s 'slide' if they are s u b m i t t e d from a prestigious r e s e a r c h e r or institution"; s e e also [1] and [6]), it also i m p r o v e s the quality of the p a p e r s that are published. F o r instance, in their l a n d m a r k study, P e t e r s and Ceci [7] evaluated 12 p s y c h o l o g y j o u r n a l s that u s e d nonblind review by r e s u b m i t t i n g manus c r i p t s that had p r e v i o u s l y b e e n published in the same j o u r n a l two y e a r s before, changing only the n a m e s of the a u t h o r s a n d their institutions 9 Only 2 o u t of 16 reviewers felt that previously p u b l i s h e d but u n r e c o g n i z e d p a p e r s w e r e suitable for publication. Witness also the conclusions of Fisher, et al. [3]: "Blinded reviewers a n d editors in this study, but not n o n b l i n d e d reviewers, gave better scores to authors with m o r e previous articles. These results suggest that blinded reviewers m a y provide m o r e unbiased reviews and that nonblinded reviewers m a y be affected b y various types of bias." And fmaUy, c o n s i d e r the results of Labland's and Piette's massive study [5]: "Articles published in journals using blinded p e e r review were cited significantly m o r e than articles published in j o u r n a l s using nonblinded p e e r r e v i e w . . . Journals using nonblinded p e e r review publish a larger fraction of p a p e r s that should not have b e e n published than do j o u r n a l s using blinded p e e r review. When reviewers

k n o w the identity of the author(s) of an article, they are able to (and evidently do) substitute particularistic criteria for tmiversalistic criteria in their evaluative process." H e r e w i t h then, a m o d e s t p r o p o s a l to realign the m a t h e m a t i c a l manuscript s u b m i s s i o n ritual with b o t h fairness and excellence: 1. A u t h o r selects j o u r n a l a n d s e n d s m a n u s c r i p t to editor 9 2. E d i t o r f o r w a r d s manuscript, s a n s a u t h o r ' s n a m e and school affiliation, to referee. 3. Referee carefully reviews manuscript a n d s e n d s r e c o m m e n d a t i o n to editor. 4. E d i t o r u s e s referee's r e p o r t to inform his d e c i s i o n about w h e t h e r o r n o t to a c c e p t manuscript. I suspect, though, that m a t h e m a t i c s m a n u s c r i p t s will be r e v i e w e d with flagrant bias for s o m e time to come. The p e o p l e w h o have the p o w e r to i m p r o v e the p r o c e s s - - j o u r n a l e d i t o r s - - a r e t h e m s e l v e s "name-recognizable" a n d a m o n g t h o s e w h o have the m o s t to lose b y m a k i n g the p r o c e s s fair and increasing the quality of their journals. I imagine it w o u l d be difficult for t h e m to relinquish their prerogative to exercise their o w n shallow bias. W h e n Oedipus, King of Thebes, found o u t h e ' d m a r r i e d his mother, and

(probably) killed his father, the only logical action for this "blind" man to t a k e was to gouge his eyes out. I'm n o t suggesting that t h e editors of mathematics journals, b l i n d though they a r e to their o w n bias (and its c o n c o m i t a n t a d v o c a c y for less t h a n the b e s t p a p e r s in the p a g e s of t h e i r journals), gouge their own eyes out. I am suggesting that the rest of us help t h e m s e e - - r e move their b l i n d n e s s - - r e n d e r i n g selfmutilation unnecessary. Towards t h a t end, and in s u m m a r y , a simple argument:

i f attaching the author's name and school affiliation to the manuscript influences the referee, this is obviously unfair bias and should be avoided; i f attaching the author's name and school affiliation to the manuscript does not influence the referee, then there should be no objection to removing them. REFERENCES

1. Banner, J.M., Preserving the integrity of peer-review, Scholarly Publishing 19 (1988), no. 2, 109-115. 2. Notices of the American Mathematical Society 26 (1979), 119. 3. Fisher, M., et al., The effects of blinding on

4.

MOVING? W e n e e d y o u r n e w a d d r e s s so that y o u d o n o t miss a n y issues of

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6.

THE MATHEMATICAL INTELLIGENCER. Please s e n d y o u r o l d a d d r e s s (or label) a n d n e w a d d r e s s to:

Springer-Verlag N e w Y o r k Inc., Journal Fulfillment Services P.O. Box 2485, Secaucus, NJ 07096-2485 U.S.A.

7.

8.

acceptance of research papers by peer review, Journal of the American Medical Society 272 (1994), no. 2, 143-146. Journal of the History of Ideas 58 (1997), no. 1. Labland, D.N., and Piette, M.J., A citation analysis of the impact of blinded peer review, Journal of the American Medical Society 272 (1994), no. 2, 147-151. McGiffert, M., Is justice blind? An inquiry into peer-review, Scholarly Publishing 20 (1988), no. 1, 43-48. Peters, D.P., and Ceci, S.J., Peer-review practices of psychological journals: the fate of published articles, submitted again, Behav. Brain 5 (1982), 187-195. Shaughnessy, A.F., Comment; Blind peer review of journal articles, Drug Intelligenceand Clinical Pharmacy 22 (1988), no. 12, 1006.

Please give us six weeks notice. Lemme B. Bourbaki Southeast State University of Agronomy

VOLUME 21, NUMBER 1, 1999

5

PAULUS GERDES

Molecular Modeling of I--ullerenos with I ',exastrips*

~

ecently [1] Cuccia, Lennox, and Ow showed how origami, the ancient Japanese art of paper folding, can be used for the modeling of fuUerenes. They chose modular origami, wherein simple modules are interlocked to form larger and more elaborate structures. In this paper another, and relatively easy, way will

be presented to build models of fullerenes and related molecules using hexastrips (Fig. 1). It will be shown that these hexastrip models correspond to a particularly stabilizing Kekul~ structure which may render them useful in narrowing down the search for possible fullerene isomers. Hexastrips were introduced by the author in the early 1980's when he was exploring possibilities of incorporating a hexagonal basket weaving technique into the teaching of geometry in Mozambique [2]. In the north of Mozambique, Makhuwa craftsmen weave their light transportation baskets (litenga) and their fish traps (lema) with a pattern of regular hexagonal holes (Fig. 2). The strands are woven over-and-under in three directions leading to a very stable fabric. This structure consti-

tutes a model for a layer of graphite: Imagine the carbon atoms arranged at the vertices of the hexagonal holes; the edges of these holes represent single bonds between the

Figure 1. Hexastrip. The dotted line segments indicate the folds of the cardboard paper.

*Reprinted with permission from The Chemical Intelligencer Vol. 4 (1).

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THE MATHEMATICAL INTELLIGENCER 9 1999 SPRINGER-VERLAG NEW YORK

Figure 2. Plane part of a hexagonally woven basket.

Figure 4. Pentagonal hole surrounded by hexagonal holes.

Figure 3. (a) Pattern of hexagonal holes; (b) Model for a layer of graphite.

carbon atoms, and the crossings of two strands between two neighboring vertices of two neighboring hexagonal holes represent the double bonds (Fig. 3). The same hexagonal basket-weaving technique has been used in several other regions of Africa and the world [3]. In Madagascar fish traps and transport baskets are made using it. In Kenya it is used for making cooking plates, and among the Pygmies (Zalre) for carrying baskets, as well as among various Amerindian peoples in Brazil (Ticuna, Omagua, etc.), Ecuador (Huarani), and Guyana (Yekuana). The Micmac-Algonkin Indians of Canada use it for their large eastern snowshoes, as do Eskimos in Alaska. In Asia the use of the hexagonal basket-weaving technique is well spread, from the Munda in India, the Kha-Ko in Laos, to Malayasia, Indonesia, China, Japan, and the Philippines. Artisans all over the world discovered that if they use this open hexagonal weave to produce a basket, they have to "curve" the faces at the basket's "comers." They found that this can only be done by reducing the number of strands at the comers, and so they weave comers with pentagonal holes [4]. Figure 4 displays such a pentagonal hole surrounded by five hexagonal holes. The extreme situation would be a "basket" consisting of pentagonal holes only. This happens with the Malaysian "sepak raga" ball, which has twelve pentagonal holes (Fig. 5). Various variations of the "sepak raga" game are played in other parts of Southeast Asia, including Burma, Thailand, the Philippines,

and Indonesia. It is a game with a long tradition. Dunsmore refers to a legend about a 14th-century Malay ruler who held his audience spellbound by kicking the ball more than 200 times without letting it touch the ground [5]. The structure of the "sepak raga" ball is very similar to that of the m o d e m soccer ball (since the end of the 1960s), and constitutes a model for buckminsterfullerene C60 (Fig. 6): Imagine once more the carbon atoms arranged at the vertices of the holes of the "sepak raga" ball (this time, 12 pentagonal holes, leading to 60 atoms); the (rectified) edges of these holes represent the single bonds between the carbon atoms, and the crossings of two strands between neighboring vertices of neighboring pentagonal

Figure 5. Malaysian "sepak raga" ball.

VOLUME 21, NUMBER 1, 1999

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Figure 6. (a) Schematic representation (front view) of the woven "sepak raga" ball; (b) Schematic representation o f the modern s o c c e r ball structure; (c) The " s e p a k r a g a " - - s o c c e r ball structure o f buckminsterfullerene.

holes represent the double bonds. The hexagonal tings of carbon atoms are held together by alternately single and double bonds. There are 20 hexagonal rings. Both the hexagonal rings and the global icosahedral structure of C60 may be more easily visible if we weave the ball using hexastrips (Fig. 7). Hexastrips are cardboard strips in which a series of folds have been introduced in such a way that they facilitate the weaving together of the strips in three directions. Figure 8 shows how the fLrst folds may be produced to make a hexastrip, and Figure 9 shows how to join three hexastrips---over-and-under. The strips may be held together using paper clips or gluing their overlapping rhombi. Curl and Smalley (USA), and Kroto (UK) were awarded the 1996 Nobel Prize in Chemistry for their 1985 discovery of C60, observed in the mass spectrometer, and their conjecture that it would have the symmetrical structure of a truncated icosahedron [6]. The possible existence of a such structured, stable carbon molecule had been conceived in 1970 by Osawa in Japan [7]. Curl, Kroto, and Smalley named the molecule buckminsterfullerene after the designer/

Figure 7. Hexastrip model of C6o.

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THE MATHEMATICALINTELLIGENCER

Figure 8. (a) Wrapping one strip around the other to obtain the first folding line (b) The second folding line is m a r k e d by folding the upp e r part of the second strip in such a w a y that it b e c o m e s a d j a c e n t

to the first strip. This process is repeated to produce the various folds,

Figure 9, Joining three hexastrips.

Figure 13. Hexastrip model of the tetrahedral isomer of C168. Figure 10, Hexastrip model of C72.

Figure 11. Hexastrip model of a nanotubule isomer of C~2o.

Figure 12. Hexastrip model of the isomer of C12o with tetrahedral symmetry.

inventor o f t h e geodesic domes, a n d i n d i c a t e d s o c c e r e n e as a p o s s i b l e alternative name. Looking at the structure o f the "sepak raga" ball, it b e c o m e s c l e a r that sepak-raga-ene could also have b e e n a p o s s i b l e n a m e for C60. It m a y be interesting to n o t e that the a r c h i t e c t R. B u c k m i n s t e r Fuller (1895-1983) h a d such a "sepak raga" b a l l - - ( f o r inspirat i o n ? ) - - o n t h e shelf of a b o o k c a s e in his d o m e home [8]. Prior to the discovery of buckminsterfullerene only t w o forms of crystalline carbon w e r e known: graphite and diamond. Since the 1990 success of Kr~ttschmer (Germany) a n d Huffman (USA) in synthesizing m e a s u r a b l e quantities of C60, m a n y o t h e r fullerenes and related m o l e c u l e s have b e e n studied. F u n e r e n e s are defined as c l o s e d cage molecules c o m p r i s e d entirely of sp2-hybridized c a r b o n s arranged in hexagons a n d p e n t a g o n s [9]. As a c o n s e q u e n c e of Euler's t h e o r e m a b o u t the relationship b e t w e e n the n u m b e r of vertices (V), the n u m b e r of edges (E), a n d the n u m b e r of faces (F) o f a c o n v e x polyhedron, V - E + F = 2, the total number of p e n t a g o n a l rings in a fullerene m u s t always be 12. Fig. 10 s h o w s a hexastrip m o d e l o f C72 with two p o l a r h e x a g o n a l holes. It is the s m a l l e s t e x a m p l e o f a carbon~ b a s e d nanotubule, a cylindrical fullerene tube, and has a sixfold r o t a t i o n a l axis. Fig. 11 s h o w s a h e x a s t r i p m o d e l o f a n o t h e r nanotubule, this time c o m p o s e d of two d i a m e t r i c a l l y o p p o s e d hemispherical C60 caps, j o i n e d b y a fivefold cylindrical wall o f two r o w s o f h e x a g o n a l holes. H a ~ n g 12 p e n t a g o n a l holes (12 x 5 = 60 vertices) and 10 h e x a g o n a l holes (10 x 6 = 60), the m o d e l r e p r e s e n t s an i s o m e r o f Ct20. A h e x a s t r i p m o d e l of a n o t h e r i s o m e r of Ct20 is s h o w n in Fig. 12. It has global t e t r a h e d r a l symmetry: the twelve p e n t a g o n a l holes are c l u s t e r e d in four groups o f three at the c o r n e r s of a t r u n c a t e d t e t r a h e d r o n a n d are s u r r o u n d e d by single b a n d s of h e x a g o n a l holes. The smallest possible tetrahedral hexastrip model is one for Cs4: in the middle of each of the four faces there is a hexagonal hole. The tetrahedral structure b e c o m e s clearly visible in the hexastrip model of an i s o m e r of C~6s shown in Fig. 13: t h r e e hexagonal holes on each o f the four "faces" and one hexagonal hole on each of the six "edges." A n o t h e r possibility consists of the twelve p e n t a g o n a l

VOLUME 21, NUMBER 1, 1999

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Figure 14. Hexastrip model of the octahedral isomer of C276.

holes being distributed over the fullerene in six groups of two. If the groups of two pentagonal holes are c o m p o s e d of neighboring pentagonal holes, and regularly distributed over the closed surface, then the structure has a global octahedral form, as the hexastrip model of an isomer of C276 in Fig. 14 displays. At the corners of the woven truncated octahedron there are two opposite pentagonal holes surrounded by a layer of six hexagonal holes. To learn to make hexastrip models, one might start by weaving models for some small quasi-fullerenes. Closed carbon cages containing other than 5- and 6-membered rings are k n o w n as quasi-fullerenes. Fig. 16 shows a model of C24 woven of four hexastrips (each with only six folds): instead of twelve pentagonal holes, there are six square holes; it has both the form of a truncated cube and of a truncated regular octahedron. Still smaller is the model of C12, woven with three hexastrips (each with only four folds), which has tetrahedral symmetry (Fig. 17). Models of quasi-funerenes with, for instance, heptagonal rings [10] m a y also be built using hexastrips. Fig. 18 shows a hexastrip model of a quasi-fullerene C576 in the form of a torus; it has 12 pentagonal and 12 heptagonal holes. The heptagonal holes produce the concave regions. Hexastrip models of several non-cage carbon molecules

Figure 16. Hexastrip model of a C24 cluster [14].

Figure 17. Hexastrip model of C12 with tetrahedral symmetry.

Figure 18. Hexastrip model of the isomer of C576 with the form of a Figure 15. Hexastrip model of the icosahedral isomer of C240.

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THE MATHEMATICAL INTELLIGENCER

torus.

Figure 21. Hexastrip model of C120 with the numbering scheme of Fig. 20.

Figure 19. Structural motifs and hexastrip models of (top) crysene; (middle) coronene; (bottom) corannulene, like carbon clusters.

m a y also b e built. Fig. 19 s h o w s m o d e l s of crysene-, c o r o n e n e - and corannulene-like c a r b o n clusters. H e x a s t r i p m o d e l s o f fullerenes and related c a r b o n struct u r e s are n o t only beautiful a n d relatively e a s y to m a k e o f c h e a p m a t e r i a l s - - a n d m a y as s u c h b e attractive f r o m a didactic p o i n t of v i e w - - b u t t h e y also give a c l e a r p i c t u r e of the b o n d i n g situation: the e d g e s of the holes r e p r e s e n t the

single bonds, a n d the folds (that is, w h e r e w o v e n h e x a g o n s are a d j a c e n t ) the double bonds. Given a h e x a s t r i p model, it is n o t difficult to d e t e r m i n e the n u m b e r of c a r b o n a t o m s implied, as t h e r e are no p r o b l e m s w i t h a p o s s i b l e double counting of vertices. Conversely, if a n u m b e r n is equal to 60 + 6m, w h e r e m is zero or an integer greater than 1, then it is p o s s i b l e to c o n s t r u c t a h e x a s t r i p m o d e l of Cn. F o r example, in t h e c a s e o f n = 120, w e have 120 = 60 + 6 • 10, and s e v e r a l n u m b e r i n g s c h e m e s m a y b e w o r k e d out to see which h e x a s t r i p i s o m e r s are possible. Fig. 20 displays a possible n u m b e r i n g s c h e m e for the top half of t h e C120 i s o m e r s h o w n in Fig. 21. This i s o m e r is different from the ones p r e s e n t e d in Fig. 11 and 12. When it is not possible to write n in the form of 60 + 6m, as in the c a s e n = 70, t h e n t h e r e m a y exist a variation of a h e x a s t r i p model. In fact, for t h e relatively stable C70 it is p o s s i b l e to w e a v e two s e m i C60 m o d e l s (see Fig. 22), and j o i n them: On the adjacent central h e x a s t r i p s there are ten vertices, r e p r e s e n t i n g the e x t r a 10 c a r b o n atoms. H e x a s t r i p m o d e l s p r e s e n t o t h e r a d v a n t a g e s as well which m a y t u r n out to be useful in t h e analysis of the possible e x i s t e n c e of certain i s o m e r s of fullerenes. Schmalz, et al. p o i n t e d out in 1986 that c a r b o n cage

Figure 20. Hexastrip numbering scheme for the top half of a C12o isomer. The numbers 5 and 6 represent pentagonal and hexagonal holes.

Figure 22. Hexastrip model of C70 composed of two woven halves.

VOLUME 21, NUMBER 1, 1999

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a carbon cylinder. One might pose the question: do hexastrip weavable fullerenes provide another, and maybe powerful, way of reducing the number of candidate isomers. As woven structures they are very stable. What could in the molecules' microcosm correspond to the stable hexagonal basket weave technique? Or, formulated in another manner, do hexastrips with their zigzagging folds (double b o n d s . . . ) correspond to something strain-reducing or stability-reinforcing in fullerenes? [13]

Acknowledgments Maurice Bazin, Arnout Brombacher, and Marcos Cherinda are thanked for stimulating conversations. The Research Department of the Swedish International Development Agency is thanked for financial support, and the University of Georgia (Athens, USA) for the conditions created for doing research during the author's 1996-1997 sabbatical leave from Mozambique's Universidade Pedag6gica. REFERENCES AND NOTES

structures in which the pentagons are isolated are likely to be more stable than structures in which they abut. Subsequently the prescription that abutting pentagons are to be avoided has become known as the "isolated-pentagon-rule" (IPR). It appears to be obeyed by all fullerenes found and characterized so far [11]. The hexastrip models satisfy the "isolated-pentagon-rule": If two pentagonal holes would abut, at that place there would not be any pentagonal holes any more, but only a nonagonal hole. In their study on competing factors in fullerene stability, Fowler et al. note that the isolated pentagon rule is compatible with considerations of 7r electronic stability, but that pentagon isolation in itself does not guarantee it [12]. As a technique for reducing the number of candidate isomers, the isolated pentagon rule is initially very successful. At C7s there are over twenty thousand general fullerenes, but only five IPR isomers. However, for n = 100, there are 450 IPR isomers, and for n = 120 there are 10774 IPR isomers. Fowler et al. analyze three possible ways to reduce the number of candidate isomers. For n = 120, there are 4 isomers with optimal "hexagon neighbor indices," forty have the form of a "leapfrog" cage, and one is

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1. Cuccia, L.A.; Lennox, R.B.; Ow, F.M. The Chemical Intelligencer 1996, 2(2), 26-31 2. See, for example, Gerdes, P. Educational Studies in Mathematics 1988, 19, 137-162; Gerdes, P. Ethnogeometrie: Kulturanthropolo gische Beitrage zur Genese und Didaktik der Geometrie; Franzbecker Verlag: Bad Salzdethfurth, 1990; pp 282-287 3. For examples and references, see Gerdes, P. Ethnogeometrie: Kulturanthropologische Beitr~ge zur Genese und Didaktik der Geometrie; Franzbecker Verlag: Bad Salzdethfurth, 1990; pp 52-53. Further examples may be found in Faubl6e,J. Ethnographie de Madagascar, Musee de I'Homme: Paris, 1946; pp 19, 28, 38; Somjee, S. Material Culture of Kenya, East African Educational Publishers: Nairobi, 1993, 96; Meurant, G.; Thompson, R.F. Mbuti Design--Paintings by Pygmy Women of the Ituri Forest, Thames and Hudson: London, 1995, 162; Guss, D.M. To Weave and Sing--Art, Symbol, and Narrative in the South American rain Forest, University of California: Berkeley, 1989, 73; Lane, R.F. Philippine Basketry: an Appreciation, Bookmark Inc.: Manila, pp 14, 44, 152, 170, 213; Ranjan, M.P., Bamboo and Cane crafts of North East India, National Institute of Design, 1986 4. Cf. Gerdes, P. In Fivefold Symmetry; Hargittai, I., Ed.; World Scientific: Singapore, 1992; pp 245-261 5. Dunsmore, S. Sepak Raga (Takraw)--The Southeast Asian Ball Game, Sarawak Museum: Kuching, 1983, 2 6. Kroto, H.W.; Heath, J.R.; O'Brian, S.C.; Curl, R.F.; Smalley, R.E. Nature, 1985, 318, 162 (Reproduced in Aldersey-Williams, H. The most beautiful molecule: The discovery of the Buckyball; John Wiley & Sons: New York, 1995) 7. Cf. e.g. Kroto, H.; Fischer, J.; Cox, D., Eds., The Fullerenes, Pergamon Press: Oxford, 1993, pp 1, 11 ; Hirsch, A. The Chemistry of Fullerenes, GeorgeThieme Verlag: Stuttgart, 1994, 5; Dresselhaus, MS.; Dresselhaus, G.; Ecklund, P.C. Science of Fullerenes and Carbon Nanotubes, Academic Press, San Diego, 1996, 2 8. As can be see in a photograph in Snyder, R., Ed. Buckminster Fuller: Autobiographical Monologue/Scenario, St. Martin's Press: New York, 1980, p 151 9. Cf., e.g., Taylor, R. The Chemistry of Fullerenes, World Scientific: (continued on page 27) Singapore, 1995

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Parallel Worlds: Escher and Mathematics, Revisited This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of "mathematical community" is the broadest. We include "schools" of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from

l[=-z.--]a M a r j o r i e

Senechal,

Editor

he popularity--and u b i q u i t y - - o f the graphic work of the Dutch artist M.C. Escher (1898-1974) continues unabated: books on his w o r k remain in print, the public never seems to tire of Escher posters, mugs, Tshirts, calendars, and other paraphernalia, and exhibitions of his w o r k are packed. Over 300,000 visitors attended the six-month "M. C. Escher: A Centennial Tribute" at the National Gallery of Art in Washington last spring; exhibitions have recently been held, or soon will be held, in Brazil, Mexico, the Czech Republic, Hong Kong, Great Britain, China, Greece, Italy, Argentina, and Peru. "People are attracted like magnets to these works. They come closer and closer and closer, and they stay there an incredible amount of time," says Jean-Francois I~ger of the National Gallery of Canada. "Studies have shown that the average length of time that a gallery visitor will stay in front of a work of art is 17 seconds. But they stay minutes in front of Escher's, and discuss, and comment, and say, 'Do you see this, have you seen that?'." What is the magnet, what is the attraction? ls it profound, or is it superficial? It has b e c o m e rather fashionable to affect weariness with these questions. Although Escher was "discovered" by

T

I

research mathematicians (and other scientists) in the 1960's, t h e i r - - o u r - enthusiasm for his work has waned as (or because?) the public's has waxed. "Of course, the article contains the inevitable reference to Escher, the philistine mathsman's favorite artist," sniffed an a n o n y m o u s referee for the interdisciplinary journal Leonardo a few years ago [1]. Art critics have been disdainful all along, insisting that Escher will be, at most, a footnote in the history of twentieth-century art. But while this assessment may be correct, is it fair? E s c h e r never claimed to be either a mathematician or an artist. "My uncle floated between art and m a t h e m a t i c s - - t h o s e are his words," says his n e p h e w Nol Escher [2]. He was not at home in either world, yet he perhaps illuminates a profound relation between them. M.C. Escher's hundredth birthday provides an occasion for the mathematical commtmity to revisit his w o r k and come to terms with it. The Escher Centennial Congress, held in Rome and Ravello, Italy, June 24-28, 1998, brought together a diverse group of mathematicians, scientists, artists, designers, m u s e u m educators, and others to consider the entire range of Escher's work, "from landscapes to mindscapes," from m a n y different per-

mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.

Please send all submissionsto the Mathematical Communities Editor, Marjorie Senechal, Department of Mathematics, Smith College, Northampton, MA 01063, USA; e-mail: [email protected]

Ravello: M.C. E s c h e r ' s h o m e in 1923. P h o t o g r a p h by M a j o r i e Senechal.

9 1999 SPRINGER-VERLAGNEWYORK, VOLUME21, NUMBER1, 1999

13

M. C. Escher's Print Gallery (1956 lithograph), 9 1998 Cordon Art, B.V.--Baarn--Holland. All rights reserved.

spectives [3]. During that congress I asked a small subset of the invited speakers to explore the reasons for Escher's enduring popularity with the general public in general, and in particular whether his appeal is in any sense "mathematical." The following comments splice together excerpts from two wide-ranging discussions. The participants were George Escher, a retired aeronautical engineer and oldest son of M.C. Escher; I s t v ~ Hargittai, Professor of Chemistry, Hungarian Academy of Sciences, author of numerous books on symmetry; Douglas Hofstader, Center for Research on Concepts and Cognition, Indiana University, author of G6del, Escher, Bach; Claude Lamontagne, Professor of Psychology at the University of Ottawa; Jean-Francois Leg~r, Education Director of the National Gallery of Canada in Ottawa; Arthur Loeb, Professor of Design Science at Harvard University; Istv~in Orosz,

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THE MATHEMATICAL INTELLIGENCER

Budapest, artist (considered by some to be Escher's "successor"); and Doris Schattschneider, Professor of Mathematics, Moravian College, author of Visions of Symmetry.

Senechal: It is a truism that art critics dislike Escher's work but the public loves it. Many people have speculated on possible reasons f o r the first, but f e w seem to have seriously considered the second. Today, let's forget about the critics, and consider the public instead. And let's begin in a skeptical vein. I don't know of any other artist's work that has been so commercialized, not even Picasso 's. To what extent is Escher's popularity due to the commercialization? Or is the commercial success due to Escher's appeal? I-Iofstadter: You can't just say, well, we're going to make all those ties! People aren't necessarily going to buy them.

Esther: Yes, but there was a very organized sales campaign of the Escher concept which was invented after father died or maybe even before, by people around him who said, "If we let it go, it will just fall apart." Because of the character of the people involved then and the people involved now, that's what you have: marketing specialists. Senechal: Does that explain w h y other artists, such as Vasarely and Magritte, whose work challenges the imagination in ways somewhat analogous to Escher's, don't have the same mass following? Or is it, at least in the case of Vasarely, because his geometrical illusions are j u s t abstract figures, not embedded in fanciful worlds? Lamontagne: Maybe it is partly because they are not marketed the way Escher is, but also there is an immediacy in Escher. Magritte is not as easy to interpret. Escher chose simple things, waterfalls, monks walking. It looks understandable at first--but then you find surprises in it. L~ger: I'm not sure that everybody likes Escher. When we were working on our public programming, we tried to identify who would be most interested in him. We concluded that it would be young adults, who were interested in mind games and things like that. It may be that people become interested in Escher at a certain age, and then their interest fades a bit. Maybe Escher appeals to this group because his work is immediate: what you see is what you get. Orosz: The most terrible experience for us artists is when a viewer at an exhibition stops for a half of a second in front of our work and then walks on to the next one. This is impossible in front of a print of Escher. And usually I feel, when I see his work in an exhibition or in a book, that after some minutes the picture is not important anymore, the important thing is the thinking, the mysterium. Over time, it will be even more important than the picture. This may be why the publishers use his works in calendars, because people have to live with them for a month at least, and they see it every day.

Hofstadter: I don't r e m e m b e r w h e r e I first saw "Metamorphosis," it was probably in s o m e book, but I r e m e m b e r the fascination of the changing forms. I was n e v e r as attracted to the tessellatious as I was to the metamorphoses, the idea that here is something that is tessellating, but it's changing into something else. And then, on top of that, it changes from being a two-dimensional thing to a three-dimensional thing, and then b a c k into a two-dimensional thing, and then into a n o t h e r t h r e e - d i m e n s i o n a l thing, and then it w i n d s up being a village that plunges into the s e a with chess pieces, and words! There were so many ideas tangled together there in such an elegant and graceful and, again, startling and astonishing m a n n e r - - t h a t ' s what g r a b b e d me. It was a two-dimensionai, three-dimensional constant interplay and then bringing in these other worlds, like medieval villages, chess, the world of the intellect, the world of the p a s t a medieval village connotes m o r e than just the past, it again connotes a kind of mystical quality, something that's gone but that radiates a kind of c h a r m that I

M. C. Escher's Day and

can't put m y fmger on very well. And that, to me, was also marvelous. Orosz: It's not the visual image that is m o s t important, it's something in the mind. Still, it is very easy to s p e a k a b o u t the w o r k of Escher, m u c h easier t h a n to s p e a k about a b s t r a c t o r o t h e r k i n d s of art. S o m e h o w it is v e r y close to c o m m u n i c a t i o n - - y e t it is n o t visual c o m m u n i c a t i o n , n o r is it v e r b a l communication. Lamontagne: With Escher, the revealing t h a t h a p p e n s in the graphics is always a c c o m p a n i e d by a concealing w h i c h u n c o v e r s itself through t i m e as the visual s y s t e m seeks interpretations. E s c h e r was an incredible visual engineer; he e x p e r i m e n t e d with j u s t a b o u t all t h e w a y s in which you c a n int e r v e n e in the visual p r o c e s s to fool the system. I see t h r e e directions in his work. One I call "two-D," the tilings; ano t h e r is "three-D," the i m p o s s i b l e figures; and the third is w h a t I call "through-D," like the Print Gallery, in w h i c h s u b j e c t a n d object toggle with one another. The guy looking at the print is an o b j e c t but w h e n y o u go b a c k

he b e c o m e s the s u b j e c t and then he turns into an o b j e c t again. Loeb: We've h e a r d that it's the young p e o p l e w h o t a k e to E s c h e r ' s work. That m a y b e true; as a natural scientist I tend to question these things, but I think it's p r o b a b l y true. At this particular conference w e have several m u c h o l d e r people, b u t I think t h a t they bec a m e i n t e r e s t e d in E s c h e r when t h e y w e r e younger. It s h o u l d be possible to find out. Sehattschneider: I think you're right that p r o b a b l y the p r i m a r y audience is high school and college. I think part of it has to do w i t h the irreverence o f s o m e o f E s c h e r ' s a r t - - t h e y say it is "cool," "awesome." But p e o p l e w h o like to solve p r o b l e m s , w h o like to try to figure things out, are i m m e d i a t e l y att r a c t e d r e g a r d l e s s o f age. I think t h a t ' s w h y scientists a n d m a t h e m a t i c i a n s a r e so a t t r a c t e d - - i t ' s not so much t h a t there is m a t h e m a t i c s in it. Lamontagne: Like m o s t young undergraduate s t u d e n t s in the 60's, I got a kick out of Escher, a n d I had m y p o s t e r s - - i t didn't turn into mania,

Night (1938 woodcut), 9 1998 Cordon Art, B.V.--Baarn--Holland. All rights reserved.

VOLUME21, NUMBER1, 1999 15

Up and Down (1947 lithograph), 9 1998 Cordon Art, B.V.--Baarn--Holland. All rights reserved, M. C. Escher's

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THE MATHEMATICALINTELLIGENCER

though. I really enjoyed it for a period, but t h e n I m o v e d on to something else, and I forgot a b o u t Escher. Then a few years ago the National Gallery o f C a n a d a in O t t a w a d e c i d e d to p u t up an exhibition a n d they were looking for s o m e o n e for the c o m m i t t e e w h o h a d s o m e k n o w l e d g e and expertise. Someone in the m u s e u m h a d been one of m y s t u d e n t s of p e r c e p t i o n and r e m e m b e r e d t h a t I h a d an interest in knowledge a n d illusions a n d vision a n d t h a t I was at the University of Ottawa, so soon I was b a c k in the Escherian world. I was v e r y h a p p y to be in it: in fact, I found in E s c h e r ' s w o r k the w h o l e p r o b l e m s p a c e in which I h a d b e e n playing o v e r the previous 20 years! It has to do with knowledge, and with the. fragility o f knowledge, with the unavoidable h y p o t h e t i c a l n a t u r e of knowledge. I s t a r t e d looking at the prints from that perspective, trying to see if I c o u l d fit t h e m into a unity. I don't have c l o s u r e on it, but I'm p r e t t y e x c i t e d a b o u t the w a y it is shaping up. H a r g i t t a i : This kind of discussion inevitably p r o m p t s m e to a s k m y s e l f w h a t I like in E s c h e r most, a n d w h a t I use E s c h e r m o s t for. I use him m o s t for his p e r i o d i c drawings, b u t I d o n ' t think I like t h e m most. After a while they b e c o m e very m u c h the same, boring a n d simplistic. If I c o u l d j u s t c h o o s e the one thing that I like most, it w o u l d b e his wild flowers. I s t a r t e d wondering, w h y do I like his wild flowers so m u c h ? It is p r o b a b l y b e c a u s e of m y s c i e n c e background: his wild flowers a r e v e r y geometric, t h e y a r e s t r i p p e d o f m a n y things, and t h e y s e e m to m e to give a fantastic m o d e l of nature. S o m e t h i n g is there, it is v e r y important, b u t m a n y o t h e r things are j u s t ignored, as in any very good model. His p e r i o d i c d r a w i n g s are e x t r e m e l y useful for me, b u t in this case "what y o u see is w h a t y o u get": after a while y o u get v e r y u s e d to it. I always get an uneasy feeling w h e n I see that m a t h t e a c h e r s are s p e n d i n g I d o n ' t k n o w h o w m a n y class hours on Escher. I think it's a v e r y g o o d w a y to m a k e children h a t e him and that kind of work. In fact, he is a unique artist for t h e conn e c t i o n b e t w e e n art and science. Lamontagne: E v e r y b o d y has s e e n illusions in p s y c h o l o g y b o o k s o r even

M. C. Escher's Castrovalva (1930 lithograph), 9 1998 Cordon Art, B.V.--Baarn--Holland. All

rights reserved.

m o r e widely available literature, b u t t h e y are crude. E s c h e r p u t t h e m into a w o r l d that has s o m e cogency, s o m e consistency. He u s e s a variety of them, s o m e of w h i c h d o n ' t strike us as being illusions, for i n s t a n c e the w a y in which he uses the v a r i o u s w o r l d s that point to one another, to m a k e p e o p l e realize that k n o w l e d g e c a n n o t be trusted, b u t at the s a m e time, it can be trusted. It can be t r u s t e d locally, but there's alw a y s a globality t h a t might s h o w that it does n o t m a k e sense. This local/ global relationship is fundamental in cognitive s c i e n c e as well as in mathematics. H o f s t a d t e r : T h e r e is, as I'm sure everyone knows, a b r a n d o f literature that may have s t a r t e d in South A m e r i c a called magic, or magical, realism, in w h i c h t h e r e is a mixture of reality and p a r a n o r m a l events. I haven't r e a d much of it; the only time I att e m p t e d to r e a d s o m e - - G a b r i e l Garcia Marquez's One Hundred Years of Solitude--I f o u n d I j u s t couldn't t a k e it, I couldn't s t a n d it. And yet,what is the difference b e t w e e n that kind of literature, w h i c h m i x e s reality with mystical, u n e x p l a i n a b l e events that violate the laws o f physics, and "High and Low," E s c h e r ' s p r i n t in which the scene is r e p e a t e d twice, with the b o y sitting on the staircase, with the p a l m trees in the courtyard, the t o w e r that is both going up and down, the windows right side up on one side and upside down on the other side, gravity obviously flowing in t w o different directions in the s a m e building. In s o m e sense that's magical realism, yet I love that! I don't understand w h a t it is in myself that fends Garcia Marquez uninteresting and silly, yet finds E s c h e r captivating and mesmerizing. There's a sense of m y s t i c i s m in it: I think the w o r d mysticism isn't wrong. I ' m not a mystic, b u t t h e r e ' s an a p p e a l to a sense of marvelous mystery, w h i c h is also w h a t caught m e so m u c h in "Day and Night," the first E s c h e r p r i n t I ever saw. The birds, not only intersecting and forming their o w n b a c k g r o u n d , b u t also becoming fields a n d then day and night in the s a m e p l a c e at the s a m e time, all o f that w a s overwhelming. It was so strange and c o m p l e x and weird. E s e h e r : Maybe this is b e c a u s e you can

VOLUME21, NUMBER1, 1999 17

l o o k at an E s c h e r print again and again and again, a n d think a b o u t it. S e n e c h a l : Yes, I think the difference between magical realism in literature and the magical sense in Escher is that as you look at Escher more and more, you begin to understand it. You don't see how he could possibly have thought of it, but you do see how it was actually executed. You begin to see, for example, why this seems to be convex when you look at it one way, but concave another, instead of just being baffled by it. You become intellectually engaged in trying to understand Escher, while with Garcia Marquez and the other magical realist writers that I have read, no understanding is possible because there's nothing there to understand. It's j u s t magic. Schattschneider: I agree. I don't think that "what y o u see is w h a t y o u get" with Escher. I gave John Conway a c o p y of m y b o o k [4], and he later told me that it t o o k him six m o n t h s to read. I said, "John, if I tell p e o p l e it t o o k y o u six m o n t h s to r e a d m y book, no one will open it!" He replied that at first he b e g a n to d e v o u r it, b u t then he d e c i d e d to put it on t h e p i a n o and only allow himself to t u r n one page a day, b e c a u s e he really w a n t e d to s t u d y it. When he s l o w e d down, he s a w things he h a d never seen b e f o r e although he h a d l o o k e d at t h e s e prints m a n y times. L ~ g e r : My u n d e r s t a n d i n g of the exp r e s s i o n "what y o u see is w h a t you get" is that it is immediate, in the s e n s e that the m e s s a g e is all included in the work: you c a n c o m e to it knowing nothing a b o u t art, a n d still you will get something out o f it. You don't have to k n o w w h a t w a s p r i o r to that, or after that, it d o e s n ' t cite s o m e b o d y else, y o u can engage in it with no prior knowledge of it. t I o f s t a d t e r : A n d yet, w h e n one k n o w s s o m e of E s c h e r ' s o t h e r works, one r e a d s his l a n d s c a p e s in a w a y that one might not have r e a d t h e m without that context. One h a s the s e n s e that this is s o m e b o d y w h o a p p r e c i a t e s magic. You feel it in that landscape, even though it's not directly there, and even though it was done m a y b e 20 y e a r s before something like "Day and Night." You feel that s a m e s e n s e o f the magical, a

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THE MATHEMATICALINTELLIGENCER

s e n s e of engagement, depth, power, space, and space b e t w e e n lines. L ~ g e r : The more I l o o k at Escher, the m o r e I a m interested in his landscapes. Even art critics will agree with that. I w o u l d wish that m o r e p e o p l e w o u l d focus on the Italian l a n d s c a p e s . E s t h e r : F a t h e r thought t h a t a m o n g all the artists who d e p i c t e d landscapes, he w a s nothing special. They w e r e all d e d i c a t e d p e o p l e with g o o d eyes, wanting to s h o w w h a t t h e y saw; he w a s one of thousands. It's only b e c a u s e he s w i t c h e d out of that field that his w o r k in it b e c o m e s visible; that's the strangest thing about the whole phenomenon. F a t h e r never c o n s i d e r e d himself an artist: b e c a u s e he h a d a certain prec o n c e i v e d i d e a of w h a t an artist was, he thought he wasn't one, a n d he couldn't d r a w anyway. But c a n you see, t h r o u g h his prints, t h a t he w a s looking at the w o r l d so intensely, with such interest, that it c o m e s through, it resonates, within you: "Oh, so t h a t ' s w h a t t h e w o r l d can look like!" Loeb: Maurits e x p r e s s e d surprise to m e m a n y times that p e o p l e w e r e seeing mystical things in his prints. He did n o t e x p e c t that at all. George, did you have any e x p e r i e n c e with this? Escher: Well, yes. It w a s r a t h e r funny, the reactions that father got to many of his prints. People saw their own imaginings in them, not w h a t he h a d m e a n t to say. What he m e a n t to say is what's there, and nothing more, according to him. These other p e o p l e s a w reincarnations and mystical things. Loeb: Maybe that is the "magic mirror" of M. C. Escher. Maybe t h a t ' s w h a t we all see: w e see ourselves m i r r o r e d in his work. L a m o n t a g n e : The question of interp r e t a t i o n is a very subtle one. The attitude that we should n o t interpret, t h a t w e should be cautious, is very naive: if you have to be c a u t i o u s w h e n y o u i n t e r p r e t then y o u have to be cautious w h e n you think, b e c a u s e thinking is interpreting. To an extent, I'm a Popperian. That is, I agree with the p h i l o s o p h e r Karl P o p p e r that all forms o f knowledge, including p e r c e p t u a l knowledge, are c o n j e c t u r a l o r hypothetical, and the only w a y in w h i c h w e can h o p e to p r o g r e s s in o u r under-

standing is to formulate our k n o w l e d g e in a falsifiable w a y [5]. F o r example, a p e r s o n c o m i n g into a different culture r e a d s it in a w a y that is refutable t h r o u g h further experience. That is, I think, a b e t t e r reading t h a n a naive reading of it which is not refutable. Seneehal: Is there anything mathematical in the appeal of Escher, or is that completely beside the point? Lamontagne: Perhaps mathematics was to E s c h e r as g r a m m a r w a s to S h a k e s p e a r e . Mathematics is form. I - I o f s t a d t e r : At the time I w a s writing my book, which b e c a m e k n o w n as GSdel, Escher, Bach [6], it w a s n o t called t h a t at all: the w o r k i n g title w a s s o m e t h i n g like "GSdel's T h e o r e m a n d the H u m a n Brain." As I was writing a n d writing, I realized that for m a n y of the c o n c e p t s that I h a d called "strange loops" o r "tangled hierarchies," i m a g e s that I k n e w from E s c h e r w e r e a p p e a r ing in m y head, over and o v e r again. F o r a while t h o s e images j u s t h e l p e d m e e x p r e s s myself m o r e clearly; t h e y h e l p e d m e get m o r e sharply into w o r d s w h a t I w a s trying to get across. But then eventually it o c c u r r e d to me, m y gosh, I s h o u l d be showing m y r e a d e r s this stuff, I s h o u l d not be s i m p l y having it in m y h e a d as a crutch o r an aid, I s h o u l d b e sharing this. If it's useful to m e as a w r i t e r to have an E s c h e r picture in m y head, it will be useful for m y r e a d e r s to have it in theirs. At that p o i n t E s c h e r b e c a m e an integral p a r t of the book, and it w a s a b o u t the s a m e time that Bach w a s entering, for v e r y different reasons. F o r me, m a n y o f the c o n c e p t s that I w a s trying to get across, particularly this notion of strange loop, were extremely well represented in Escher's pictures, and they were deeply connected, as I said, with GSdel's theorem and certain things in mathematical logic. I d o u b t that Escher had those notions in mind explicitly, but the abstraction that underlies GSdel's p r o o f and the abstraction that underlies "The Print G a l l e r y " - - t h e i d e a of a s y s t e m folding a r o u n d and engulfing itself, is the s a m e c o n c e p t as in a s y s t e m that can r e p r e s e n t its o w n predicates, a syst e m that c a n talk a b o u t itself. L a m o n t a g n e : I've b e e n r a i s e d in a c o n t e x t of Piagetian thought, in the

Piaget world, w h i c h is still quite valid. Piaget talks a b o u t a d o l e s c e n c e as the p e r i o d w h e n cognition o p e n s up to the r e a l m of possibilities. Before t h a t - - h e calls it the c o n c r e t e o p e r a t i o n a l per i o d - t h e mind is reactive a n d it can do very fancy things, b u t on the basis o f actual things, c o n c r e t e objects. But w h e n you reach the formal o p e r a t i o n a l stage, which starts a r o u n d 12, as you r e a c h 12, 13, 14, the m i n d o p e n s up. It realizes that there is not only actuality, b u t that actuality can l e a d to potentiality. A n d so the y o u n g m i n d s open up to the fact that t h e y are w h a t they are, b u t within the c o n t e x t of a huge c o m b i n a t o r i c s that is at the s a m e time physical, social, a n d psychological. Do y o u k n o w the test that Piaget did with m a t h e m a t i c s - - w i t h p e r m u t a t i o n s and combinations, showing h o w kids at the p r e - o p e r a t i o n a l level, a n d c o n c r e t e operational level, and formal o p e r a t i o n a l level handle c o m b i n a t o r i a l t a s k s [7]? Before adolescence, a child can comp u t e the n u m b e r of a r r a n g e m e n t s of

any given n u m b e r of objects, b u t cannot even u n d e r s t a n d the question if y o u a s k for the n u m b e r of arrangem e n t s o f N objects. A d o l e s c e n t s can think a b o u t N. In addition to explaining m a t h understanding, Piaget's i d e a s are beautiful m e t a p h o r s for the m i n d in general. When you get to the formal o p e r a t i o n a l stage, that is, a d o l e s c e n c e , then y o u o p e n up to the p o s s i b l e and y o u realize that you are one a m o n g s t an infinite set of possibilities. Now, at that age, t h e r e is at the s a m e t i m e the fear of losing w h a t you are b u t t h e exc i t e m e n t o f discovering w h a t y o u might be, a n d w h a t the w o r l d might be. I think t h a t it is in this a r e a that w e can locate the great fascination for Escher. This is o u r stopping point, but it is not the end. This particular c o n v e r s a t i o n w a s one a m o n g an infinite set o f possible c o n v e r s a t i o n s a b o u t the w o r k of M. C. E s c h e r and, m o r e generally, a b o u t t h e d e e p relations b e t w e e n art a n d m a t h e m a t i c s and the h u m a n mind.

Like E s c h e r ' s visual puzzles, it l o o p s b a c k on itself, leading us through n e w l a n d s c a p e s that s o m e h o w are familiar. REFERENCES

[1] John Rigby, private communication. [2] Nol Escher, private communication. [3] "M. C. Escher: From Landscapes to Mindscapes" was the title of an exhibition held at the National Gallery of Canada in Ottawa in 1998. The proceedings of the Escher Centennial Congress will be published shortly by Springer-Verlag. [4] Doris Schattschneider, Visions of Symmetry: notebooks, periodic drawings, and related work of M. C. Escher, New York, W.H. Freeman, 1990. [5] Karl Popper, Conjectures and Refutations: the growth of scientific knowledge, London, Routledge & Kegan Paul, 1963. [6] Douglas Hofstader, GSdel, Escher, Bach: an eternal golden braid, New York, Basic Books, 1979. [7] Jean Piaget and Barbel Infelder, La genese de I'idee de hasard chez /'enfant, Paris, Presses Universitaires de France, 1951.

VOLUME 21, NUMBER 1, 1999

19

GIAN-CARLO ROTA

Two Turning Points in Invariant Theory* 0

nvariant theory is the great Romantic story of mathematics. For 150 years, f r o m its beginnings with Boole to the time, around the middle of this century, when it branched off into several independent disciplines, mathematicians of all countries were brought together by their common faith in invariants: in England, Cayley, MacMahon,

Sylvester, and Salmon, and later, Alfred Young, Aitken, Littlewood, and Turnbull; in Germany, Clebsch, Gordan, Grassmann, Sophus Lie, Study; in France, Hermite, Jordan, and Laguerre; in Italy, Capelli, Brioschi, Trudi, and Corrado Segres, d'Ouidio; in America, Glenn, Dickson, Carus (of the Carus Monographs), Eric Temple Bell, and, later, Hermann Weyl. Seldom in history has an international community of scholars felt so united by a common scientific ideal for so long a stretch of time. In our century, Lie theory and algebraic geometry, differential algebra, and algebraic combinatorics are offsprings of invariant theory. No other mathematical theory, with the exception of the theory of functions of a complex variable, has had as deep and lasting an influence on the development of mathematics. Eventually, invariant theory was to become a victim of its own success: The very term "invariant theory" is nowadays applied to diverse offspring theories, so that it has become all but meaningless. It is no wonder if you are baf-

fled by the title of this lecture. Which invariant theory is it about? It is about classical invariant theory. The old treatises are being dusted off the shelves of library basements and reread, reinterpreted, and presented in a language that meets the standard of rigor of our day. The program of classical invariant theory, that had for some time been given up as hopeless, is again being pursued, and success may at last be within reach. I will review two turning points in the history of invariant theory. The first, the "new" one, happened around the turn of the century, and its effects are still being felt all over mathematics. The second, the "old" one, happened very early in the game and led to a serious misunderstanding that lasts to this day. A pedestrian definition of invariant theory might go as follows: invariant theory is the study of orbits of group actions. Such a definition is correct, but it must be supple-

*Slightly edited second Colloquium Lecture delivered at the Annual Meeting of the American Mathematical Society, Baltimore, January 8, 1998.

~0

THE MATHEMATICALINTELLIGENCER9 1999 SPRINGER-VERLAGNEWYORK

m e n t e d b y a p r o g r a m m a t i c statement. H e r m a n n Weyl, in t h e introduction to his b o o k The Classical Groups, summ a r i z e d the p r o g r a m in t w o assertions. The first s t a t e s that "all geometric facts are e x p r e s s e d by the vanishing of invariants," and the s e c o n d s t a t e s that "all invariants are invariants o f tensors." Let m e briefly c o m m e n t on t h e s e lofty s t a t e m e n t s . What is a g e o m e t r i c fact? It is a fact a b o u t s p a c e that is indep e n d e n t of the choice of a c o o r d i n a t e system. G e o m e t r i c facts are d e s c r i b e d b y m e a n s of equations w h i c h require a c h o i c e of coordinates. In a v e c t o r space V of d i m e n s i o n n, one c h o o s e s a c o o r d i n a t e s y s t e m Xl, x 2 , . . . , Xn. Since Descartes, w e have l e a r n e d to e x p r e s s g e o m e t r i c facts b y equations in the c o o r d i n a t e s Xl, x2, 99 9 Xn. However, a b o u t 100 y e a r s ago, m a t h e m a t i c i a n s and physicists m a d e the shoctdng discovery that the usual equations (i.e., equations in the commutative ring generated by the variables Xl, x2, 9 9 xn) are inadequate for the description of a lot o f geometric and physical facts. Motivated b y this discovery, they introd u c e d a m o r e general ring. This is the ring of non-commutative polynomials in the coordinates Xl, x 2 , . . . , xn. Homogeneous elements of this ring (i.e., h o m o g e n e o u s noncommutative polynomials in the variables Xl, x 2 , . . . , Xn) are called tensors. If we believe Hermann Weyl's philosophy, then w e will be satisfied that equations in the t e n s o r algebra suffice for the description of any geometric fact w e will ever meet. Furthermore, ff these equations are to express geometric properties, then they must hold no m a t t e r what coordinate system is chosen; in other words, equations that describe geometric facts m u s t be invariant under changes of coordinates. The p r o g r a m o f invariant theory, from Boole to our day, is precisely the translation of geometric facts into invariant algebraic equations expressed in terms o f tensors. This p r o g r a m of t r a n s l a t i o n of g e o m e t r y into a l g e b r a w a s to b e carried out in t w o steps. The first s t e p c o n s i s t e d in d e c o m p o s i n g t e n s o r a l g e b r a into irreducible c o m p o n e n t s u n d e r changes o f coordinates. The s e c o n d s t e p cons i s t e d in devising an efficient notation for t h e e x p r e s s i o n of invariants for each i r r e d u c i b l e component. The first step w a s successfully c a r r i e d o u t in this century; the s e c o n d w a s a b a n d o n e d s o m e t i m e in the twenties, a n d only recently has it resurfaced. The d e c o m p o s i t i o n o f t e n s o r algebra into i r r e d u c i b l e c o m p o n e n t s was d i s c o v e r e d a r o u n d the turn o f the cent u r y a l m o s t s i m u l t a n e o u s l y b y Issal Schur a n d Affred Young. The gist of this d e c o m p o s i t i o n is one of t h e great a d v a n c e s in m a t h e m a t i c s o f all times, and it m a y b e worthwhile to p r e s e n t it in a form that can be m a d e available to undergraduates. Let us c o n s i d e r functions of three variables, s u c h a s f ( x l , x2, x3). Two well-known c l a s s e s of functions o f t h r e e varia b l e s are s y m m e t r i c functions, t h o s e that satisfy the equations

fs(Xb x2, X3) = fs(Xil, xi2, xi3) for every p e r m u t a t i o n sending the indices (i, 2, 3) to (il, i2, i3), and s k e w - s y m m e t r i c functions, defined by the equations

fa(Xl, X2, X3) = -4-fa(Xil, Xi 2, Xi3), w h e r e the sign is § 1 or - 1 a c c o r d i n g as the p e r m u t a t i o n sending the indices (1, 2, 3) to (il, i2, i3) is even or odd. It is n o t true that a function of t h r e e variables is the s u m of a s y m m e t r i c function and a s k e w - s y m m e t r i c function. A third type o f function is required, w h i c h is called a cyclic function, c h a r a c t e r i z e d by the equation

fc(Xl, x2, x3) + fc(x3, xt, x2) + fc(x2, x3, Xl) = O. Every function of t h r e e variables c a n b e uniquely written as the s u m o f a symmetric function, a s k e w - s y m m e t r i c function, a n d a cyclic function, in symbols,

f(Xl, X2, X3) = fs(Xl, X2, X3) + fa(Xl, X2, X3) + fc(Xl, X2, x3). E a c h o f t h e three s y m m e t r y c l a s s e s is invariant u n d e r permutations; this is obvious for s y m m e t r i c and skew-symmetric functions b u t not quite so o b v i o u s for cyclic functions. These t h r e e invariant s u b s p a c e s play for the group of p e r m u t a t i o n s of a set of t h r e e e l e m e n t s a role analogous to the role o f the eigenvectors of a s y m m e t r i c matrix. F o r f u n c t i o n s f ( x l , x2, x3, x4) of f o u r variables, there a r e five s y m m e t r y classes, which are defined as follows: 1. S y m m e t r i c functions 2. S k e w - s y m m e t r i c functions 3. Cyclic-symmetric functions, satisfying the four equations

f(xb f(Xl, f(xl, f(xl,

x2, x3, x4) X2, X3, X4) x2, x3, x4) x2, x3, x4)

+ f ( x l , x4, x2, x3) § f(x4, x2, Xl, x3) + f(x4, Xl, x3, x2) + f(x3, xl, x2, x4)

+ + + +

f ( x t , x3, x4, x2) = 0, f(x3, x2, x4, x l ) = 0,

f(x2, x4, x3, xl) = O, f(x2, x3, Xl, x4) = 0

4. F u n c t i o n s satisfying the f o u r equations f(Xl, X2, X3, X4) § f(x2, Xl, X3, X4) § f ( x l , x2, x4, x3) + f(x2, Xl, x4, x3) = 0, f ( x l , x2, x3, x4) + f(x3, x2, xl, x4) + f ( x l , x4, x3, x2) + f(x3, x4, xl, x2) = 0, f(Xl, X2, X3, X4) § f(Xl, X3, X2, X4) § f(x4, x2, x3, Xl) § f(x4, x3, x2, xl) = O, sign(q) f(x~l, x~2, x~3, xw4) ----0 5. F u n c t i o n s satisfying the equations

f(xt, x2, x3, x4) - f ( x 2 , Xl, x3, x4) f(xl, x2, X4, X3) § f(x2, Xl, X4, X3) ----0, f ( X l , X2, X3, X4) --f(x3, X2, Xl, X4) --

f(xl, x4, x3, x2) + f(x3, x4, Xl, x2) = 0, f ( x l , x2, x3, x4) - f ( x l , x3, x2, x4) f(x4, x2, x3, Xl) + f(x4, x3, x2, x l ) = 0, Z f(xol, x~2, x~3, x~4) = 0 Every function of four variables is uniquely e x p r e s s i b l e a s the s u m of five functions, e a c h one belonging to one o f these s y m m e t r y classes. E a c h s y m m e t r y class is invariant under permutations. More generally, every function o f n variables f ( x l , x2, 99 9 xn) c a n be uniquely w r i t t e n as the sum of Pn functions, e a c h one belonging to a different s y m m e t r y class. Here, Pn equals the n u m b e r o f p a r t i t i o n s of the integer n.

VOLUME 21, NUMBER 1, 1999

21

E a c h s y m m e t r y class is defined by equations w h i c h are not difficult to find. This d e c o m p o s i t i o n holds for tensors as well, after some cosmetic changes of notation. To this day, only two symmetry classes o f t e n s o r s have been s t u d i e d in any detail. Symmetric t e n s o r s are ordinary c o m m u t a t i v e polynomials such as we l e a r n e d to use in analytic geometry. Skew-symmetric tensors are polynomials in the c o o r d i n a t e s xl, x 2 , . . 9 x~ p r o v i d e d that the variables are a s s u m e d to satisfy the equations xixj = -xjxi. Tensors belonging to symmetry classes o t h e r than the classes of s y m m e t r i c and skewsymmetric t e n s o r s also o c c u r in g e o m e t r y a n d physics. However, these s y m m e t r y classes have b e e n studied very little, and they are a long w a y from being understood. So m u c h for the w o r d "new" in the i n t r o d u c t i o n of this lecture; let us n e x t do s o m e justice to the w o r d "old." I will describe the m o s t p e c u l i a r feature of classical invariant theory, n a m e l y the s y m b o l i c or u m b r a l notation, to which Eric Temple Bell d e d i c a t e d his Colloquium Lectures in 1927. I will c o n s i d e r the simplest group, n a m e l y the group of translations o f the line. The unusual f e a t u r e s of the symbolic m e t h o d will a l r e a d y be a p p a r e n t in this special case. Let p(x) and q(x) b e m o n i c p o l y n o m i a l s in the variable x. I write t h e m in the following quaint notation:

p(x)=xn§247247

""§

an-lX§

and

" " + ( k k_ 1) bk-lX+bk 9 I a s s u m e that t h e p o l y n o m i a l q(x) is o f l o w e r degree than the p o l y n o m i a l p(x); that is, that k -< n. Define the t r a n s l a t i o n o p e r a t o r Tv on a p o l y n o m i a l p(x) as follows:

TCp(x) = p(x + c). Let us write

+

(2)P2(c)xn-2+'"+(nn_ The f l h coefficient c o m p u t e d to b e

1) Pn l(C)X+pn(C).

pj(c) of the p o l y n o m i a l p(x + c) is

Pj(o) = aj T (J) aj lC § (~) aj-2 c2 + ''' + cj. A p o l y n o m i a l I(al, a2, 9 9 9 , an, bl, b2, 9 9 9 , bk) in the variables al, a 2 , . . . , an, bl, b 2 , . . . , bk is said to be an invariant of the two p o l y n o m i a l s p(x) and q(x) w h e n

I(al, a2,. . . , an, bl, b 2 , . . . , b k ) =

I(pl(c), p 2 ( c ) , . . . , pn(c), ql(c), q 2 ( c ) , . . . , qk(C))

22

THE MATHEMATICALINTELLIGENCER

I(TCp(x), TCq(x)) = I(p(x), q(x)) for all c o n s t a n t s c. Invariant t h e o r y is c o n c e r n e d with the p r o b l e m of finding all invariants of a given set o f polynomials, as well as their significance. What is m e a n t by the "significance" of an invariant? I a p p e a l to H e r m a n n Weyl. "Every" p r o p e r t y of p o l y n o m i a l s w h i c h is invariant u n d e r the group of translations is exp r e s s e d b y the vanishing o f a set of invariants. In o t h e r words, "any" set of p o l y n o m i a l s w h i c h is invariant u n d e r t r a n s l a t i o n s is the same set as a set of p o l y n o m i a l s o b t a i n e d b y setting to zero a set of invariants of such polynomials. It is i m p o s s i b l e to u n d e r s t a n d this s t a t e m e n t w i t h o u t ex~ amples. Let us c o n s i d e r the s i m p l e s t a n d oldest example. The p r o p e r t y of a quadratic p o l y n o m i a l

q(x) = x 2 + 2blx + b2 of having a double r o o t is invariant u n d e r translations; in o t h e r words, if the p o l y n o m i a l q(x) has a double root, so d o e s the p o l y n o m i a l q(x + c) for any c o n s t a n t c. F o l l o w i n g H e r m a n n Weyl, w e look for an invariant w h o s e vanishing e x p r e s s e s this property. Sure enough, it is e a s y to c h e c k t h a t the discriminant

D(bl, b2) = b2 - b2

q ( x ) = x k + ( ~ ) b l X k - l + ( ~ ) b 2 xk-2+

p ( x + c) = xn T ( 1 ) P i ( C ) x n - i

for all c o m p l e x n u m b e r s c. By a b u s e of notation, w e w r i t e

I(p(x), q(x)) and we s p e a k o f / a s being an invariant of the p o l y n o m i a l s p(x) and q(x). In this abusive notation, a polyn o m i a l I is said to be an invariant of the p o l y n o m i a l s p(x) and q(x) w h e n e v e r

is the d e s i r e d invariant. This example, due to Boole, w a s the s p a r k that led to the birth o f invariant theory. One often h e a r s the s e n t e n c e "Hilbert killed invariant theory," referring to w h a t w e call the Hilbert b a s i s theorem. It is n o t true. Hilbert loved invariant theory, a n d he w e n t on publishing striking p a p e r s in invariant t h e o r y aft e r he p r o v e d the basis t h e o r e m . Some of the m o s t fascinating results in invariant t h e o r y w e r e d i s c o v e r e d in the first 20 y e a r s of this century, a long time after Hilbert p r o v e d his basis theorem. What then is the r e a s o n for the subsequent t e m p o r a r y d e m i s e of invariant theory? One r e a s o n is the e n d e m i c use of the s y m b o l i c o r u m b r a l notation. Dieudonn6 w r o t e that half the s u c c e s s of a p i e c e of m a t h e m a t i c s d e p e n d s on a p r o p e r choice of notation. It w o u l d b e interesting to m a k e a list of unfortunate notations that killed various c h a p t e r s of mathematics, as well as a list of felicitous n o t a t i o n s t h a t p r o m o t e d the developm e n t of o t h e r b r a n c h e s o f m a t h e m a t i c s . The s y m b o l i c o r u m b r a l n o t a t i o n was a c a t a s t r o p h e . A n u m b e r of m a t h e m a t i c i a n s tried to m a k e s e n s e of the symbolic m e t h o d without success, the three m o s t n o t a b l e ones being H e r m a n n Weyl, Eric Temple Bell, and E d w a r d Hegeler C a m s . Bell failed to p r o p e r l y define u m b r a l notation, and his Algebraic Arithmetic r e m a i n s to this d a y the b o o k of seven seals. If Weyl a n d Bell had lived 50 y e a r s longer, so as to benefit

from the d e v e l o p m e n t o f w h a t w a s in their t i m e called "modern" algebra, t h e y w o u l d u n d o u b t e d l y have s u c c e e d e d in p r o p e r l y defining u m b r a l notation. In o u r day, as I will s h o w you, this is easy: it will only t a k e a few minutes. Before I start spouting definitions, let m e say w h a t I a m not going to say. Umbral n o t a t i o n can be s h o w n to be equivalent (or "cryptomorphic," to use a t e r m invented by m y late friend Garrett Birkhoff) to ano t h e r notation that has gained great notoriety in o u r day: t h e n o t a t i o n of H o p f algebras. I will not justify this sibylline p r o n o u n c e m e n t , n o t b e c a u s e it is difficult to do so, but bec a u s e it is not needed. Let us go on to the definition of u m b r a l notation. Side b y side with the p o l y n o m i a l s p ( x ) and q(x), w e c o n s i d e r a n o t h e r p o l y n o m i a l a l g e b r a C[x, a, fi] in t h r e e variables x, a, and/3, together with a linear functional E defined on the underlying v e c t o r s p a c e C[x, a, /3]. The definition of the linear functional E is the k e y point. It is carried out in steps:

The u m b r a l or symbolic m e t h o d consists o f replacing all o c c u r r e n c e s of the coefficients of t h e polynomials p ( x ) and q(x) b y u m b r a e and equivalences. F o r example,

p ( x ) -~ (x + a) n and q(x) ~ (x + fi)k. Let us carefully c h e c k the first equivalence. By definition, the equivalence m e a n s the s a m e as

E ( p ( x ) ) = E ( ( x + a)n). Because E ( x j ) = x j for all non-negative integers j, this identity can be r e w r i t t e n as

p(x) = E((x + 00% E x p a n d i n g the right-hand side by the binomial theorem, w e obtain

S t e p 1. Set

E ( x j) = x j for all non-negative integers j , in particular E(1) = 1. Thus, the range o f the linear functional E is C[x]. S t e p 2. Set

By linearity, this equals

E(aJ) = aj; in particular, we have E(oJ) = 0 if j > n. S t e p 3. Set

E(fi j) = bj; in particular, we have E(fiJ) = 0 i f j > k. S t e p 4. This is the m a i n step. Set

E ( ,~i l ~ x ~) = E ( , ~ ) E ( f ~ ) x ~. Following Sylvester, the variables a and fi are called umbrae. In o t h e r words, the linear functional E is multiplicat i r e on distinct umbrae. S t e p 5. E x t e n d b y linearity. This c o m p l e t e s the definition of the linear functional E. We n e x t c o m e to the m o s t disquieting feature of u m b r a l notation. Let f ( a , fl, x) and g(a, fi, x) be t w o p o l y n o m i a l s in the variables a, fi, and x. We write f ( a , fi, x) ~ g(a, fi, x)

Evaluating the linear functional E, w e see that this, in turn, equals

xn+(nl)a,xn-'+(2)a2xn-2+"'+(nnl)an ,X+an, as desired. The e x p r e s s i o n

( x + ~)~ is c a l l e d a n u m b r a l r e p r e s e n t a t i o n o f t h e p o l y n o m i a l p(x). In u m b r a l notation, a c o m p l e x n u m b e r r is a r o o t of the p o l y n o m i a l equation p ( x ) = 0 if and only if ( r + o0 n --~ 0. Similarly, in u m b r a l notation, the p o l y n o m i a l TCp(x)= p ( x + c) m a y be r e p r e s e n t e d as follows:

to m e a n

p ( x + c) = (x + ~ + c) n,

E(f(~,/3, x)) = E(g(~,/3, x)). Read = as "equivalent to." The "classics" w e n t a bit t o o far; they w r o t e

and this yields the u m b r a l e x p r e s s i o n for the coefficients pj(c) o f the p o l y n o m i a l p ( x + c), n a m e l y

pj(C) ~-- (0~ + C) j. f ( a , fi, x ) = g(a, fl, x); t h a t is, t h e y r e p l a c e d the s y m b o l ~ by o r d i n a r y equality. This was an excessive a b u s e of notation. The "classics" w e r e a w a r e of the error, a n d while they a v o i d e d c o m p u t a tional errors by clever artistry, they were unable to get a w a y from the abuse.

Let us n e x t see h o w umbra] n o t a t i o n is related to invariants. Let us a s s u m e that the t w o p o l y n o m i a l s p ( x ) and q(x) have the s a m e degree n. Then, an invariant A of the p o l y n o m i a l s p ( x ) and q(x) m a y be d e f i n e d as follows:

A(q(x), p ( x ) ) -~ (fl - oOn.

VOLUME 21, NUMBER 1, 1999

23

The evaluation of the invariant A in t e r m s of the coefficients o f p ( x ) a n d q ( x ) proceeds as follows: A ( q ( x ) , p ( x ) ) = E(([3 - c~)~)

+

i)

Thus, we see that apolarity gives a trivial a n s w e r to the following question: w h e n c a n a polynomial p ( x ) be w r i t t e n as a linear c o m b i n a t i o n of polynomials of the form (x r l ) n, (x - r 2 ) n , . . . , (x - rn)n? A beautiful theorem on apolarity was proved by the British mathematician John Hilton Grace. I state it without proof: G r a c e ' s T h e o r e m . If two polynomials p(x) and q(x) of degree n are apolar, t h e n e v e r y d i s k i n the c o m p l e x p l a n e c o n t a i n i n g e v e r y zero o f p ( x ) also c o n t a i n s at least one

+

z e r o o f q(x).

§ (--1)n-1

n -- 1

=bn-(1)

bn_lal§

....

§ "'" § ( - 1 ) n - 1 ( n n- 1) b l a n - l § Why is A a n invariant? This is best seen in umbral notation: A(TCq(x), TCp(x)) ~-- (fi + c - a - c) n = (fl - a)n.

The invariant A is called the apolar invariant; two polynomials p ( x ) and q ( x ) having the property that A ( q ( x ) , p ( x ) ) = 0 are said to be apolar. In u m b r a l notation, two polynomials are apolar w h e n e v e r

([3 - a) n --- O. The concept of apolarity has a distinguished pedigree going all the way b a c k to Apollonius. What is the "significance" of the apolar invariant? What does it m e a n for two polynomials to be apolar? This question is a n s w e r e d by the following theorem: T h e o r e m 1. S u p p o s e that r is a root o f the p o l y n o m i a l q(x), that is, that q(r) = O. Then, the p o l y n o m i a l s q ( x ) a n d p ( x ) = (X -- r) n are apolar.

Grace's Theorem is an i n s t a n c e of what might be called a sturdy theorem. For almost 100 years, it has resisted all attempts at generalization. Almost all k n o w n results a b o u t the distribution of zeros of polynomials in the c o m p l e x p l a n e are corollaries of Grace's theorem. I will next generalize the apolar invariant to the case of two polynomials p ( x ) and q ( x ) of different degrees n a~d k, with k -< n. To this end, let us slightly generalize the defmition of invariant, as follows. A polynomial I(al, a 2 , . . . , an, bl, b 2 , . . . , bk, X) i n t h e v a r i a b l e s at, a2, 9 9 9 an, bl, b2, 9

bk, x i s s a i d t o b e a n i n -

v a r i a n t of the polynomials p ( x ) and q ( x ) when I(al, a2, 9 9 9 an, bl, b2, . 9 bk, x ) = I(pt(c), P2(C), 9 9 9 pn(c), ql(c), q2(c), 9 9 9 qk(C), X + C)

for all complex n u m b e r s c. Sometimes, these more general invariants are called covariants. Now define a more general apolar invariant as follows: A ( q ( x ) , p ( x ) ) -~ ([3 - a ) k ( x -- a) n-k.

Again, we say that two polynomials p ( x ) and q ( x ) are apolar w h e n A ( q ( x ) , p ( x ) ) is identically zero; that is, zero for all x. T h e o r e m 1 remains valid as stated; that is, if q(r) = 0, t h e n the polynomial p ( x ) = ( x - r) n is apolar to q(x). Let us consider a special case. Suppose that q ( x ) is a quadratic polynomial a n d p ( x ) is a cubic polynomial: q(x) = x 2 + 2 b l x + b2

PROOF. For p ( x ) = ( x - r) n, we have oLj ~ ( - - r ) j, a n d hence A ( q ( x ) , p ( x ) ) = (fl - ( - r ) ) n = ([3 + r) n ~-- 0

p ( x ) = x 3 + 3 a l x 2 + 3a2x + a3.

as desired.

Then, we have, in u m b r a l n o t a t i o n

COROLLARY. I f the p o l y n o m i a l q(x) h a s n d i s t i n c t zeros rl, r2, 9 9 9 , rn, a n d i f the p o l y n o m i a l p ( x ) i s a p o l a r to q(x), then there e x i s t c o n s t a n t s Cl, c2, 9 9 9 , C n f o r w h i c h p ( x ) = c l ( x - r O n + c2(x - r2) n §

"'"

§ Cn(X

--

rn) n.

PROOF. The d i m e n s i o n of the affine s u b s p a c e of all (not necessarily monic) polynomials p ( x ) which are apolar to q(x) equals n. But if the polynomial q(x) has simple roots, then by the above t h e o r e m the polynomials ( x - r l ) n , ( x r2) n, . . . , ( x - rn) n are linearly i n d e p e n d e n t a n d apolar to q(x). Hence, the polynomial p ( x ) is a linear c o m b i n a t i o n of these polynomials. This completes the proof.

24

THE MATHEMATICALINTELUGENCER

and

A ( q ( x ) , p ( x ) ) ~- (fi - a)2(x - a) = ([32 _ 2a[3 + 02)x - a[3 2 § 202[3 -- 02.

Evaluating the linear f u n c t i o n a l E, we obtain the following explicit expression for the apolar invariant: A ( q ( x ) , p ( x ) ) = E((fi 2 - 2a[3 + 02)x - a[32 + 2a213 - 02) = (b2 - 2 a l b l § a 2 ) x -- a l b 2 + 2a2bl - a3.

Thus, a quadratic polynomial q ( x ) and a cubic polynomial p ( x ) are apolar if and only if their coefficients satisfy the two equations b2 - 2 a l b l + a2 = O, - a i b 2 + 2a2bl - a3 = O.

Using t h e s e equations, w e can prove two i m p o r t a n t theorems: T h e o r e m 2. There is, i n general, one ( m o n i c ) quadratic p o l y n o m i a l w h i c h is apolar to a given cubic polynomial. PROOF. Indeed, the a b o v e equations m a y b e r e w r i t t e n as b2 - 2 a t b l = --a2, - a l b 2 + 2a2bl = a3. The solutions bt a n d b2 for given a b a2, and a3 are, in general, unique. T h e o r e m 3. There is a l w a y s a one-dimensional space o f ( m o n i c ) cubic p o l y n o m i a l s w h i c h are apolar to a given quadratic polynomial. PROOF. Indeed, given bt a n d b2, w e m a y solve for at, a2, and a3 from the equations

- 2 a t b l + a2 = -b2, - a l b 2 + 2a2bt = a3. These equations always have a single infinity of solutions, a s t h e y used to s a y in the old days. T h e o r e m s 2 a n d 3 p r o v i d e a simple and explicit m e t h o d for solving a cubic equation. It goes as follows. Given the cubic polynomial

p(x) = x 3

+ 3alx 2 +

3a2x + a3,

first, b y T h e o r e m 2 w e fend a unique quadratic p o l y n o m i a l q(x) w h i c h is a p o l a r to p(x). In general, such a quadratic p o l y n o m i a l q(x) has t w o distinct zeros r l a n d r2. By T h e o r e m 1, the cubic p o l y n o m i a l s (x - r t ) 3 and (x - r2) 3 are a p o l a r to q(x). Second, b y T h e o r e m 3, the affine linear s p a c e o f cubic p o l y n o m i a l s a p o l a r to q(x) h a s d i m e n s i o n 2. As p ( x ) is a p o l a r to q(x), w e conclude that p ( x ) is a line a r c o m b i n a t i o n of (x - r l ) 3 a n d (x - r2) 3. In symbols,

T h e o r e m 4. The d i m e n s i o n of the space o f all ( m o n i c ) p o l y n o m i a l s o f degree k that are apolar to a p o l y n o m i a l of degree n equals 2k - n, i n general, w h e n k -< n. T h e o r e m 5. The d i m e n s i o n o f the space o f all ( m o n i c ) p o l y n o m i a l s of degree n that are apolar to a p o l y n o m i a l of degree k equals k, i f k <-n. Let us try to solve an equation of degree 5 in much t h e s a m e w a y a s w e solved a cubic equation. Given the quintic p o l y n o m i a l

p(x) = x 5

+

5al x4 + 10a2x 3 + 10a3x 2 + 5a4x + a5 = 0,

T h e o r e m 4 a s s u r e s us that t h e r e is, in general, a unique cubic p o l y n o m i a l q(x) which is a p o l a r to p(x). In general, this cubic p o l y n o m i a l has three distinct z e r o s rb r2, and r3. By T h e o r e m 1, the polynomials (x - r l ) 5, (x - r2) 5, and (x r3) 5 are linearly i n d e p e n d e n t a n d a p o l a r to q(x). By T h e o r e m 5, the dimension of the s p a c e of all p o l y n o m i a l s a p o l a r to q(x) equals 3. But the p o l y n o m i a l p ( x ) is a p o l a r to q(x). Hence, p ( x ) can be w r i t t e n in the form

p ( x ) = Cl(X

--

r t ) 5 + c2(x - r2) 5 + c3(x - r3) 5

for suitable c o n s t a n t s ci. Thus, w e s e e that a generic polynomial o f d e g r e e 5 can be w r i t t e n a s a linear c o m b i n a t i o n o f fifth p o w e r s o f t h r e e linear polynomials. These are comp u t e d b y solving linear, quadratic, a n d cubic equations. This r e d u c t i o n to canonical form o f t h e quintic is as close as w e can c o m e to solving a quintic equation b y radicals. At this point, s o m e o n e in the a u d i e n c e will raise his o r h e r h a n d a n d say: "Excuse me, but t h e u m b r a l m e t h o d y o u have i n t r o d u c e d is not even g o o d e n o u g h to e x p r e s s the discriminant o f a quadratic equation!" Quite right. The definitions of u m b r a e a n d of the linear functional E have an obvious generalization to any a r r a y of polynomials, s a y p l ( x ) , p2(x), 9 9 9 pe(x). One simply considers the s p a c e of p o l y n o m i a l s

C[x, ~1, ~ 2 , . . - , ae], and one sets

p ( x ) = c(x - r t ) 3 + (1 - c)(x - r2) 3

E(d)

for s o m e c o n s t a n t c. O b s e r v e that c, rl, a n d r2 are comp u t e d b y solving linear and quadratic equations. In this way, the solution of the cubic equation p ( x ) = 0 is r e d u c e d to the solution of the equation

to equal the f l h coefficient of the p o l y n o m i a l pt(x). What is crucial, t h e linear functional E is again multiplicative on distinct umbrae: E (ala2a3""x i j k s ) = E(a[)E(oJ)E(o~k)'"X e.

c(x - r t ) 3 = - ( 1 - c)(x - r2) 3, a n d this is easily solved b y taking a cube root. This m e t h o d o f solving a cubic equation is the only one I c a n r e m e m ber. Let m e digress with a p e r s o n a l anecdote. A few y e a r s ago, I w a s lecturing on this m a t e r i a l at a s y m p o s i u m in comb i n a t o r i c s that t o o k p l a c e at the University at Minnesota. Persi Diaconis was sitting in the front row, and I c o u l d tell a s I s t a r t e d to lecture that he w a s falling asleep; he eventually b e g a n to doze off. But the m o m e n t I m e n t i o n e d the magic w o r d s "solving a cubic equation," he w o k e up with a s t a r t and said: "Really! How?" The p r e c e d i n g t w o t h e o r e m s are easily generalized.

Now c o m e s the catch: the polynomiais---saypt(x),p2(x), 9 9 p e ( x ) - - n e e d not be distinct. In fact, the most important case occurs w h e n each of the polynomials pl(x), p2(x), 9. . , Pe(X) is equal to the s a m e p(x). In this case, the definition of the linear functional E may be simplified as follows: 1.

E ( a j) = aj for every i, and 2.

E ( a~ o j a k" "X e) = aiajak" " x e for all non-negative integers i, j , k , . . . ,

~.

VOLUME 21, NUMBER 1, 1999

25

Umbrae 0/1, 0/2,.. 9 , 0/~ satisfying points 1 and 2 are said to b e exchangeable. Thus, for exchangeable umbrae, w e have (X "4- 0/1) n ~'-- (X A- 0/2) n.

Eric Temple Bell, w h o w r o t e = in p l a c e of = , was baffled by the fact t h a t t w o u m b r a e could be e x c h a n g e a b l e without being equal. I can n o w state the main t h e o r e m o f invariant theory, considering a single polynomial. T h e o r e m 6. E v e r y i n v a r i a n t o f a p o l y n o m i a l p ( x ) is obtained by evaluating some p o l y n o m i a l i n the differences ai - 0/j and 0/i - x, where c~r and 0/j are exchangeable umbrae. Conversely, every p o l y n o m i a l i n such differences is equivalent to a n i n v a r i a n t o f the p o l y n o m i a l p ( x ) . The p r o o f is e x t r e m e l y simple, but will be omitted. Let us r e v i e w s o m e classical examples. The discriminant of a quadratic p o l y n o m i a l p ( x ) = x 2 + 2 a , x + a2 m a y be umbrally r e p r e s e n t e d as follows: D(p(x))

~ -

(0/, - ot2)2/2,

w h e r e a~ a n d 0/2 a r e e x c h a n g e a b l e umbrae. Indeed, E((0/,

- 0/2) 2) -- E ( 0 / 2 )

- 2 E ( 0 / l O l 2 ) A- E ( a 2)

= a2 - 2a 2 + a2 = 2(a2 - a2), as desired. Next, c o n s i d e r a cubic polynomial p ( x ) = x 3 + 3alx 2 + 3a2x + a3. The d i s c r i m i n a n t o f this polynomial, let us call it D(p(x)), equals ( e x c e p t for a multiplication c o n s t a n t ) the expression 2 2 D(p(x)) = 6ala2 - 8a 3 - 8a3a3 + 12a,a2a3

2a23.

The umbral e x p r e s s i o n o f the discriminant is e a s i e r to remember: D(p(x)) --~ (0/,

-

0L2)2(0~3 -

0/4)2(0/1 -

0/4)((~2 -

0/3).

As you know, the discriminant vanishes if a n d only if the cubic equation p ( x ) = 0 h a s a double root. The Hessian of a cubic polynomial c a n b e elegantly written in n m b r a l n o t a t i o n as follows: H ( p ( x ) ) ~- (0/1 - 0 / 2 ) 2 ( a l - x ) ( 0 / 2 - x ) .

The Hessian v a n i s h e s if and only if the cubic p o l y n o m i a l is the third p o w e r o f a polynomial o f d e g r e e 1. Allow me a n o t h e r digression. On hearing a b o u t the vanishing of the H e s s i a n as the condition t h a t a cubic polynomial be a p e r f e c t cube, it c o m e s naturally to a s k the general question: w h i c h invariant o f a p o l y n o m i a l of degree n vanishes if a n d only if the polynomial is the k-th p o w e r of s o m e p o l y n o m i a l o f degree n/k? Here, k is a divisor of n. F o r a long time, I thought the a n s w e r to this question to be b e y o n d reach, until one day, while leafing t h r o u g h the s e c o n d v o l u m e of Hilbert's collected p a p e r s , I accidentally discovered that Hilbert h a d c o m p l e t e l y s o l v e d it. The solution can be elegantly e x p r e s s e d in u m b r a l notation. This is only one of several striking results of Hilbert's on invariant t h e o r y that have b e e n forgotten. Let us c o n s i d e r n e x t an invariant o f the quintic. T h e o r e m

26

THE MATHEMATICAL INTELLIGENCER

3 tells us that a quintic p ( x ) = x 5 + 5 a l x 4 + 10a2x 3 + 10a3x 2 + 5a4x + a5 h a s a unique a p o l a r cubic p o l y n o m i a l q(x). The polynomial q(x) is an invariant of p(x). Does it have a simple e x p r e s s i o n in u m b r a l notation? I n d e e d it does. The e x p r e s s i o n is the following: q(x) -~ (a2

-

~3)2(0/3

-

~1)~(0/,

-

0/2)~(~1

-

x)(~2

-

x)(0/3

-

x).

In the classical literature, this invariant is d e n o t e d b y the l e t t e r j. What p r o p e r t y will the quintic polynomial p ( x ) have w h e n the invariant j vanishes? The a n s w e r to this question is pleasing. The invariant j o f a quintic polynomial is identically equal to zero if a n d only if the quintic is a p o l a r to s o m e non-trivial p o l y n o m i a l o f d e g r e e 2. But then, T h e o r e m 5 tells us that the quintic m a y b e written in the form p ( x ) = c(x - r , ) 5 + (1 - c)(x - r2) 5,

w h e r e r l and rz are the r o o t s of a quadratic equation. Thus, t h e vanishing of the i n v a r i a n t j is a n e c e s s a r y and sufficient c o n d i t i o n that the quintic p o l y n o m i a l p ( x ) m a y be w r i t t e n as t h e s u m o f two, rather t h a n three, fifth p o w e r s o f linear polynomials. When this is the case, the fifth-degree equation p ( x ) = 0 can be s o l v e d b y radicals. By similar arguments, one can c o m p u t e all invariants w h o s e vanishing implies t h a t the equation of d e g r e e 5 is algorithmically solvable b y radicals. Twenty-three invariants p l a y a relevant role, as Cayley was first to show. Hilbert's t h e o r e m on finite generation of the ring o f invariants can be recast in the language of u m b r a e and c a n b e given a simple c o m b i n a t o r i a l p r o o f that d i s p e n s e s with t h e Hilbert basis theorem. In closing, let me t o u c h u p o n a n o t h e r r e a s o n for the a b a n d o n m e n t of the s y m b o l i c m e t h o d in invariant theory. In m a t h e m a t i c s , it is e x t r e m e l y difficult to tell the truth. The formal exposition of a m a t h e m a t i c a l t h e o r y d o e s n o t tell the w h o l e truth. The truth o f a m a t h e m a t i c a l t h e o r y is m o r e likely to be g r a s p e d while we listen to a c a s u a l rem a r k m a d e by s o m e e x p e r t t h a t gives a w a y s o m e h i d d e n motivation, w h e n w e finally pin d o w n the typical e x a m p l e s , o r w h e n w e discover w h a t t h e real p r o b l e m s are that w e r e s t o r e d behind the s h o w c a s e p r o b l e m s . P h i l o s o p h e r s and p s y c h i a t r i s t s should explain w h y it is that we m a t h e m a t i cians are in the habit of s y s t e m a t i c a l l y erasing o u r footsteps. Scientists have always l o o k e d a s k a n c e at this strange h a b i t of mathematicians, w h i c h has changed little f r o m P y t h a g o r a s to our day. The h i d d e n p u r p o s e o f t h e s y m b o l i c m e t h o d in inv a r i a n t t h e o r y w a s n o t s i m p l y t h a t o f finding e a s y exp r e s s i o n s for invariants. A d e e p e r faith w a s guiding this m e t h o d . It w a s t h e e x p e c t a t i o n t h a t the e x p r e s s i o n o f inv a r i a n t s b y the s y m b o l i c m e t h o d w o u l d e v e n t u a l l y g u i d e u s to single o u t the "relevant" o r "important" i n v a r i a n t s a m o n g a n infinite variety. It w a s the h o p e that t h e sign i f i c a n c e of the v a n i s h i n g o f an invariant c o u l d b e g l e a n e d f r o m its u m b r a l e x p r e s s i o n . The vanishing o f this faith w a s t h e m a i n r e a s o n for t h e d e m i s e of c l a s s i c a l in-

v a r i a n t theory, a n d the revival of this faith is the r e a s o n for its p r e s e n t rebirth. Whether or not we will s u c c e e d this second time, where the classics failed, is a cliffhanger. Wait a few years. I would n o t be giving this lecture if I did not believe in success. T h a n k you for your attention.

BIBLIOGRAPHY

Di Crescenzo, Antonio, and Rota, Gian-Carlo, Sul calcolo umbrale, Ricerche Matematica XLIII (1994), 129-162. Ehrenborg, Richard, and Rota, Gian-Carlo, Apolarity and canonical forms for homogeneous polynomials, Eur. J. Combinatorics 14 (1993), 157-181. Grosshans, Frank D., Rota, Gian-Carlo, and Stein, Joel A., Invariant Theory and Superalgebras, CBMS Regional Conferences in Mathematics Vol. 69, Providence, RI: American Mathematical Society (1987). Kung, J. P. S., and Rota, Gian-Carlo, The invariant theory of binary forms, Buff. Am. Math. Soc. (2) 10 (1984), 27-85. Metropolis, N., and Rota, Gian-Carlo, Symmetry classes: functions of three variables, Am. Math. Monthly 98 (1991), 328-332. Metropolis, N., Rota, G.-C., and Stein, Joel A., Theory of symmetry classes, Proc. Nat. Acad. Sci. 88 (1991), 8415-8419. Metropolis, N., Rota, Gian-Carlo, and Stein, Joel A., Symmetry classes of functions, J. AIg. 171 (1995), 845-866. Rota, Gian-Carlo, and Taylor, B. D. The classical umbral calculus, SIAM J. Math. Anal. 25 (1994), 694-711. Department of Mathematics Massachusetts Institute of Technology Cambridge MA 02139-4307, USA e-mail: [email protected]

GERDES (continued from page 12) 10. Cf. the study on the geometry of hypothetical curved graphite structures by Terrones, H.; Mackay A.L. In The Fullerenes, Kroto, H.; Fischer, J.; Cox, D., Eds. Pergamon Press: Oxford, 1993, pp 113-122; and Dresselhaus, M.S.; Dresselhaus, G.; Ecklund, P.C. Science of Fullerenes and Carbon Nanotubes, Academic Press: San Diego, 1996, chapter 19 11. Cf., e.g., Fowler, P.W.; Austin, S.J.; Manolopoulos, D.E. In Physics and Chemistry of the Fullerenes, Prassides, K., Ed.; Kluwer Academic Publishers: Dordrecht, 1994; pp 41-62 12. See [11] and Fowler, P.W.; Manolopoulos, D.E.; Ryan, R.P. In The Fullerenes, Kroto, H.; Fischer, J.; Cox, D., Eds. Pergamon Press: Oxford, 1993, pp 97-112 13. Hexastrip weavable isomers are what Schmalz and Klein call Clar Sextet isomers, having in common a special sort of Kekule structure possessed by no other fullerenes. In this structure every pentagon has five exo double bonds and every double bond is seen to take part in two conjugated 6-circuits, what represents the maximum possible stabilizing contribution from a Kekule structure. Schmalz and Klein show that such Kekule structures are only possible for buckminsterfullerene and fullerenes having 60 + 6m atoms for m > l . See Schmalz, T.G.; Klein, D.J. In Buckminsterfullerenes, Billups, W.E.; Ciufolini, M.A., Eds. VCH: New York, 1993, pp 83-101. The leapfrog fullerenes belong to the Clar Sextet isomers. Cf. Fowler, P.W.; Manolopoulos, D.E.; Ryan, R.P. In The Fullerenes, Kroto, H.; Fischer, J.; Cox, D., Eds. Pergamon Press: Oxford, 1993, pp 102-103 14. This cluster appears among the possible C24 clusters analyzed by Cioslowski, J. Electronic Structure Calculations on Fullerenes and Their Derivatives, Oxford University Press, New York, 1995, pp 162-164. Cf. also Terrones, H.; Mackay A.L. In The Fullerenes, Kroto, H.; Fischer, J.; Cox, D., Eds. Pergamon Press: Oxford, 1993, p 115

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Circles and Parabolas Sergey Markelov

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Alexander

Shen,

Editor

everal years ago, reading the Problems section of the A m e r i c a n Mathematical Monthly, I c a m e a c r o s s the following problem:

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C o n s i d e r two p a r a b o l a s : y = a x 2 + bx + c and y = d x 2 + ex + f, intersecting in two points. Let 1 be their c o m m o n chord, a n d m b e the tangent to b o t h p a r a b o l a s that t o u c h e s t h e m at X and Y. Then 1 intersects m in the point Z, w h i c h is the midp o i n t of XY (Figure 1). IGURI

This column is devoted to mathematics

I

similar); for the s a m e r e a s o n s Z Y 2 = ZP 9 ZQ, so Z X = ZY. It t u r n s out other examples circles parallel p a r a b o l a s . Here

that there are m a n y of s t a t e m e n t s a b o u t to s t a t e m e n t s a b o u t is one:

Let A B C b e a triangle. Let p o i n t B ' lie s o m e w h e r e on the line AC, p o i n t C' lie s o m e w h e r e on AB, and p o i n t A ' lie s o m e w h e r e on BC. Then the circles c i r c u m s c r i b e d a r o u n d triangles AB'C', A'BC', and A ' B ' C haCe a common intersection p o i n t (Figure 3).

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for fun. What better purpose is there for mathematics? To appear here, a theorem or problem or remark does not need to be profound (but it is allowed to be); it m a y not be directed only at specialists; it must attract and fascinate. We welcome, encourage, and

The solution u s e d s o m e c o m p u t a t i o n s in coordinates, and I s t a r t e d to think w h e t h e r a m o r e g e o m e t r i c a l one exists. Then I realized t h a t t h e r e was a similar p r o b l e m a b o u t circles:

frequently publish contributions f r o m readers--either new notes, or replies to past columns.

C o n s i d e r two circles intersecting in t w o points. Let l b e t h e i r c o m m o n chord, and m b e the c o m m o n tangent touching circles at p o i n t s X a n d Y. Then 1 i n t e r s e c t s m in the p o i n t Z that is the m i d p o i n t of XY (Figure 2). IGURE;

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The p a r a l l e l s t a t e m e n t a b o u t p a r a b o l a s r e a d s as follows: Let A B C b e a triangle. Let p o i n t B ' lie s o m e w h e r e on the line AC, p o i n t C' lie s o m e w h e r e on AB, a n d p o i n t A ' lie s o m e w h e r e on BC. Then the t h r e e p a r a b o l a s going t h r o u g h the p o i n t s A B ' C ' (the fLrst one), A ' B C ' (the s e c o n d one), and A ' B ' C (the third one) have a c o m m o n intersection p o i n t (Figure 4).

B

Please send all submissions to the

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Mathematical Entertainments Editor, A l e x a n d e r Shen, Institute for Problems of Information Transmission, Ermolovoi 19, K-51 Moscow GSP-4, 101447 Russia; e-mail:[email protected]

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The s e c o n d p r o b l e m h a s a simple solution: it is well k n o w n that Z X 2 : Z P 9 ZQ (triangles Z P X and ZXQ are

THE MATHEMATICAL INTELUGENCER 9 1999 SPRINGER-VERLAG NEW YORK

To give a n o t h e r example, w e have to e x t e n d o u r dictionary giving the corr e s p o n d e n c e b e t w e e n n o t i o n s related to circles and parabolas. F o r example, "concentric circles" s h o u l d b e transl a t e d as "parabolas with a c o m m o n axis o b t a i n e d one from the o t h e r by a shift along this axis". This translation is u s e d in the following statements:

(This s t a t e m e n t is s o m e t i m e s called N e w t o n ' s theorem.) The parallel statem e n t for p a r a b o l a s is: F o u r tangents to a p a r a b o l a inters e c t to form a quadrilateral. Then the line that goes through the midp o i n t s of the diagonals o f the quadrilateral is parallel to t h e axis o f the p a r a b o l a (Figure 8).

A line intersects c o n c e c t r i c circles at the four p o i n t s A1, B1, B2, A2. Then the s e g m e n t s AIB1 and B2A2 are equal (Figure 5).

@ A line intersects "concentric" p a r a b o las (in the sense e x p l a i n e d above) at the four points A1, B1, B2, A2. Then the s e g m e n t s AIB1 and B2A2 are equal (Figure 6).

Still a n o t h e r e x t e n s i o n o f o u r diction a r y w o u l d be the t r a n s l a t i o n o f an idi o m a t i c e x p r e s s i o n "line g o e s through the c e n t e r of a circle", w h i c h b e c o m e s "line is parallel to the axis of a parabola". This t r a n s l a t i o n is u s e d in t h e following statements: If a circle is i n s c r i b e d in a quadrilateral, then the m i d p o i n t s of the diagonals and the c e n t e r of the circle lie on a straight line (Figure 7).

I 9 I t

A similar a r g u m e n t can be a p p l i e d to o t h e r e x a m p l e s given above. However, s o m e additional tricks a r e needed. F o r example, c o n s i d e r t h e s t a t e m e n t a b o u t the four tangents (Figure 8). If w e try to use the s a m e method, w e c o m e to a picture that differs from Figure 7: see Figure 9. :IGURE

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T h e s e (and m a n y other) e x a m p l e s give us a strong feeling that t h e r e is s o m e general principle saying that every true s t a t e m e n t a b o u t circles (in a certain language) can be t r a n s l a t e d into a true s t a t e m e n t a b o u t p a r a b o l a s . Unfortunately, it is not clear h o w to f o r m u l a t e a p r e c i s e general principle of this type. Instead, let us see h o w the statem e n t a b o u t p a r a b o l a s can be d e r i v e d from the s t a t e m e n t a b o u t circles. Recall o u r first pair of statements; let us p r o v e t h a t the c o m m o n t a n g e n t to t w o p a r a b o l a s is divided into equal s e g m e n t s b y their c o m m o n c h o r d (Figure 1). A s s u m e that it is not the case, and the i n t e r s e c t i o n point is not the midpoint. A p a r a b o l a can be a p p r o x i m a t e d b y an ellipse that has one of its f o c u s e s far away. C o n s i d e r such ellipses for b o t h p a r a b o l a s . The c o m m o n c h o r d o f the ellipses will be close to the comm o n c h o r d of the parabolas, a n d the c o m m o n t a n g e n t for the ellipses will b e close to t h e c o m m o n t a n g e n t for the p a r a b o l a s . Therefore, if ellipses are close e n o u g h to the p a r a b o l a s , the c o m m o n c h o r d to t h e m will i n t e r s e c t the c o m m o n tangent n o t in the m i d d l e point. One m a y a s s u m e b o t h ellipses to have the s a m e ratio of axes. Then w e can a p p l y an affine t r a n s f o r m a t i o n to t r a n s f o r m the ellipses into t w o circles for w h i c h the c o m m o n c h o r d d o e s n o t i n t e r s e c t the c o m m o n t a n g e n t at the midpoint, w h i c h is impossible. Q.e.d.

However, if t h e s t a t e m e n t (saying that the c e n t e r o f the circle and t h e m i d p o i n t s of diagonals lie on a straight line) is true for Figure 7, it should b e also true for Fig. 9. The e x p l a n a t i o n goes as follows. C o n s i d e r p o l a r coordinates on the circle; let 4)1, 4)2, 4)3, 4)4 be the angle c o o r d i n a t e s o f the t a n g e n t points. Then t h e c o o r d i n a t e s of all other p o i n t s are rational functions o f sin 4)1, cos 4)1, sin r cos r etc. Using the substitution ti = tan(4)i/2), we s e e that the c o o r d i n a t e s of all points a r e rational functions o f tl, t2, t3, t4. The s t a t e m e n t in question (three points lie on a straight line) is an identity involving t h o s e coordinates. Therefore, if it is true in the n e i g h b o r h o o d of s o m e point (as Figure 7 shows), it should b e true for all values of ti, a n d therefore also for the Fig. 9 configuration. Acknowledgments: The author thanks M. Vyalyi and the p a r t i c i p a n t s of the " T o u r n a m e n t o f the Towns" (high s c h o o l m a t h competition) for interesting discussions. e-mail address: [email protected]

L e t t e r to t h e C o l u m n Editor In your article on 3-dimensional p r o o f s of p l a n a r t h e o r e m s , one o f m y favorite e x a m p l e s of t h a t kind w a s missing. Do you k n o w the following 3-dimensional p r o o f of B r i a n c h o n ' s T h e o r e m (saying that the m a i n diagonals o f a h e x a g o n

V O L U M E 21, N U M B E R

1, 1999

c i r c u m s c r i b e d a b o u t a circle meet at one point)? Proof: C o n s i d e r the circle as a plane section going t h r o u g h the center of a h y p e r b o l o i d o f one sheet. There are two families o f straight lines on the hyperboloid, s a y A a n d B, such that through every p o i n t on the h y p e r b o l o i d p a s s e s exactly one line of e a c h family, and every line in family A m e e t s every line in family B. Let 1, 2 , . . . , 6 be the six points w h e r e the sides of the hexagon are tangent to the circle. Consider the lines 1 1 , . . . , 16 that lie on the hyp e r b o l o i d a n d p a s s t h r o u g h 1 , . . . , 6, respectively. Lines 11,13,15 are in family A and 12,14,16 a r e in family B. Since the neighbor lines are in different families, they intersect to form a 3-dimensional hexagon; its t o p view is our original hexagon. (The vertices of this 3-dimensional h e x a g o n are d e n o t e d by 12, 23, 34, 45, 56, 61 in the sequel.) Consider n o w lines 11 and 14. They belong to different families, so they int e r s e c t each o t h e r and lie in s o m e plane P14. Planes P25 and P36 are defined in a similar way. Now let us look at the intersection line of p l a n e s P14 and P25. Points 12 a n d 45 lie on b o t h planes, so the i n t e r s e c t i o n of these t w o

p l a n e s is the diagonal 12-45. Since w e have three planes, t h e r e a r e three int e r s e c t i o n lines (12-45, 23-56, 34-61), a n d the p o i n t Q w h e r e the t h r e e p l a n e s m e e t is the point w h e r e t h e s e three lines meet. The top view of each of t h e s e lines is a main diagonal of o u r original plane hexagon, h e n c e the top view of Q is the point w h e r e the m a i n diagonals meet. There is a version of this p r o o f w h i c h w o r k s for all fields k of characteristic r 2 ("circle" m u s t b e r e p l a c e d by "conic"; in c h a r a c t e r i s t i c 2 the B r i a n c h o n t h e o r e m m a k e s no sense, since in that case all the t a n g e n t lines to a conic m e e t at one p o i n t and h e n c e the diagonals of a c i r c u m s c r i b e d hexagon are not defined). S u p p o s e that w e have a conic C in a p r o j e c t i v e plane p2 over k. We m a y a s s u m e that k is algebraically closed, that P~ lies in the 3d i m e n s i o n a l projective s p a c e p3 with h o m o g e n e o u s c o o r d i n a t e s (x:y:z:w), t h a t pe is given by the equation x + w = 0, and that the conic C is the int e r s e c t i o n of the "hyperboloid" H given b y the equation x w - y z = 0 with p2. Writing every point (x:y:z:w) E p3 as a m a t r i x with the rows ( x y ) and (zw), w e s e e that p o i n t s of p3 c o r r e s p o n d to

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56 non-zero 2 x 2 m a t r i c e s X, c o n s i d e r e d up to a s c a l a r factor. In t e r m s o f matrices the equation of H is d e t X = O; the equation of the plane p2 is Tr X = 0; a n d the equation of their intersection C is X 2 = 0. There are two families of lines on H, say A and B, a n d every t a n g e n t plane to H i n t e r s e c t s H in the union of an A-line a n d a B-line. F o r e v e r y p o i n t on C, r e p r e s e n t e d b y a (nilpotent) m a t r i x X0, the equation o f the p l a n e t a n g e n t to H at this p o i n t is Tr XX0 = 0. Since TrX0 = 0, s c a l a r matrices satisfy this equation. Thus all p l a n e s t a n g e n t to H at points of C p a s s t h r o u g h the p o i n t E E p3 r e p r e s e n t e d by s c a l a r matrices. N o w we can r e p e a t the a b o v e proof: given a h e x a g o n S in p2 w h o s e sides are tangent to C, construct a 3-dimensional h e x a g o n S', using in t u r n A- and B-lines, s u c h that S is the p r o j e c t i o n of S' from E o n t o p2. (In the real c a s e c o n s i d e r e d above, E was the p o i n t at infinity in the direction o f t h e axis of the hyperboloid. Note t h a t the a s s u m p t i o n c h a r k se 2 implies t h a t the point E is not on p2.) The m a i n diagonals of S' m e e t at one point, since the t h r e e planes P14, P25, and P36 through the o p p o s i t e sides o f S' m e e t at one point. It follows that the main diagonals of S also m e e t at one point. Vladimir V. Uspenskij e-mail address: [email protected]

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THE MATHEMATICAL INTELLIGENCER

c i r c u m s c r i b e d a b o u t a circle meet at one point)? Proof: C o n s i d e r the circle as a plane section going t h r o u g h the center of a h y p e r b o l o i d o f one sheet. There are two families o f straight lines on the hyperboloid, s a y A a n d B, such that through every p o i n t on the h y p e r b o l o i d p a s s e s exactly one line of e a c h family, and every line in family A m e e t s every line in family B. Let 1, 2 , . . . , 6 be the six points w h e r e the sides of the hexagon are tangent to the circle. Consider the lines 1 1 , . . . , 16 that lie on the hyp e r b o l o i d a n d p a s s t h r o u g h 1 , . . . , 6, respectively. Lines 11,13,15 are in family A and 12,14,16 a r e in family B. Since the neighbor lines are in different families, they intersect to form a 3-dimensional hexagon; its t o p view is our original hexagon. (The vertices of this 3-dimensional h e x a g o n are d e n o t e d by 12, 23, 34, 45, 56, 61 in the sequel.) Consider n o w lines 11 and 14. They belong to different families, so they int e r s e c t each o t h e r and lie in s o m e plane P14. Planes P25 and P36 are defined in a similar way. Now let us look at the intersection line of p l a n e s P14 and P25. Points 12 a n d 45 lie on b o t h planes, so the i n t e r s e c t i o n of these t w o

p l a n e s is the diagonal 12-45. Since w e have three planes, t h e r e a r e three int e r s e c t i o n lines (12-45, 23-56, 34-61), a n d the p o i n t Q w h e r e the t h r e e p l a n e s m e e t is the point w h e r e t h e s e three lines meet. The top view of each of t h e s e lines is a main diagonal of o u r original plane hexagon, h e n c e the top view of Q is the point w h e r e the m a i n diagonals meet. There is a version of this p r o o f w h i c h w o r k s for all fields k of characteristic r 2 ("circle" m u s t b e r e p l a c e d by "conic"; in c h a r a c t e r i s t i c 2 the B r i a n c h o n t h e o r e m m a k e s no sense, since in that case all the t a n g e n t lines to a conic m e e t at one p o i n t and h e n c e the diagonals of a c i r c u m s c r i b e d hexagon are not defined). S u p p o s e that w e have a conic C in a p r o j e c t i v e plane p2 over k. We m a y a s s u m e that k is algebraically closed, that P~ lies in the 3d i m e n s i o n a l projective s p a c e p3 with h o m o g e n e o u s c o o r d i n a t e s (x:y:z:w), t h a t pe is given by the equation x + w = 0, and that the conic C is the int e r s e c t i o n of the "hyperboloid" H given b y the equation x w - y z = 0 with p2. Writing every point (x:y:z:w) E p3 as a m a t r i x with the rows ( x y ) and (zw), w e s e e that p o i n t s of p3 c o r r e s p o n d to

23 12

61

56 non-zero 2 x 2 m a t r i c e s X, c o n s i d e r e d up to a s c a l a r factor. In t e r m s o f matrices the equation of H is d e t X = O; the equation of the plane p2 is Tr X = 0; a n d the equation of their intersection C is X 2 = 0. There are two families of lines on H, say A and B, a n d every t a n g e n t plane to H i n t e r s e c t s H in the union of an A-line a n d a B-line. F o r e v e r y p o i n t on C, r e p r e s e n t e d b y a (nilpotent) m a t r i x X0, the equation o f the p l a n e t a n g e n t to H at this p o i n t is Tr XX0 = 0. Since TrX0 = 0, s c a l a r matrices satisfy this equation. Thus all p l a n e s t a n g e n t to H at points of C p a s s t h r o u g h the p o i n t E E p3 r e p r e s e n t e d by s c a l a r matrices. N o w we can r e p e a t the a b o v e proof: given a h e x a g o n S in p2 w h o s e sides are tangent to C, construct a 3-dimensional h e x a g o n S', using in t u r n A- and B-lines, s u c h that S is the p r o j e c t i o n of S' from E o n t o p2. (In the real c a s e c o n s i d e r e d above, E was the p o i n t at infinity in the direction o f t h e axis of the hyperboloid. Note t h a t the a s s u m p t i o n c h a r k se 2 implies t h a t the point E is not on p2.) The m a i n diagonals of S' m e e t at one point, since the t h r e e planes P14, P25, and P36 through the o p p o s i t e sides o f S' m e e t at one point. It follows that the main diagonals of S also m e e t at one point. Vladimir V. Uspenskij e-mail address: [email protected]

30

THE MATHEMATICAL INTELLIGENCER

CHRISTIAN GOTTLIEB

Tho Simple and Straightforward Construction of tho Regular 257-gon

Regular Polygons Most easily c o n s t r u c t e d o f t h e m all, and m a y b e t h e m o s t beautiful also, is o f c o u r s e the regular hexagon. We all l e a r n e d at school that the side of the regular h e x a g o n equals the radius o f its c i r c u m s c r i b e d circle, and this is really all one has to know. Connecting every o t h e r v e r t e x of the hexagon, w e obtain an equilateral triangle (i.e., a regular 3-gon), a n d b y rep e a t e d l y bisecting angles, w e m a y obtain any r e g u l a r 2n'3 gon, n = 0, 1, 2 , . . . . Trisecting angles, however, is n o t p o s s i b l e in general using the ruler and c o m p a s s in the classical way, so, for example, the regular 9-gon is n o t constrnctible, but this w a s actually not definitely s e t t l e d until the nineteenth c e n t u r y by Wantzel [9]. The c o n s t r u c t i o n of the r e g u l a r p e n t a g o n is far less comm o n l y k n o w n to n o n m a t h e m a t i c i a n s than that o f the regular hexagon. It m a y be f o u n d in lots of t e x t b o o k s on elem e n t a r y g e o m e t r y or history of mathematics; a g o o d r e f e r e n c e is the b o o k b y A a b o e [1]. The c o n s t r u c t i o n o f the p e n t a g o n and its c o n n e c t i o n s with the "golden section" w e r e k n o w n a l r e a d y in a n c i e n t Greece. The h e x a g o n a n d p e n t a g o n lead directly to a constrnction of the regular 15-gon (this is simply b e c a u s e ~ - ~ ~). Analogously, the regular ab-gon can be found from the regular a- and b-gons w h e n e v e r a and b are relatively prime. 1

--

2

On the o t h e r hand, it m a y be s h o w n t h a t the regular p2-gon is n e v e r c o n s t r n c t i b l e for p odd. F o r t h e s e reasons, investigations on regular p-gons m a y b e confined to p-gons, w h e r e p is an o d d prime number. We have p = 3 a n d p = 5 so far. By the end of the eighteenth century, it was the c o m m o n o p i n i o n among mathematicians that no o t h e r p-gons w e r e constrnctible. Indeed, m o r e t h a n 2000 y e a r s had e l a p s e d since the 3- and 5-gons w e r e first c o n s t r u c t e d , and no o t h e r p - g o n h a d appeared. But on 30 March 1796, the young Carl-Friedrich Gauss w r o t e in his diary (see [3]) that he h a d found "the principles on w h i c h the partition of the circle is based, and in p a r t i c u l a r the geometrical divisibility o f the s a m e into seventeen parts" (this is a free translation from Latin). In o t h e r words, he had found that the r e g u l a r 17-gon is constrnctible. Naturally, he was very p l e a s e d with this discovery, a n d it is said that it influenced him strongly to devote his life to mathematics. Why a 17-gon? What do p = 3, 5, a n d 17 have in comm o n m o r e than being prime n u m b e r s ? First, note that w h e n we inscribe a p-gon in a circle, w e s t a r t with one first vertex, w h i c h w e m a y c h o o s e arbitrarily, and our t a s k is to fmd the p - 1 "unknown" vertices. Therefore, along with p, the n u m b e r p - 1 s e e m s r a t h e r r e l e v a n t to the situation. F o r p as above, w e get p - 1 = 2, 4, a n d 16, respectively,

9 1999 SPRINGER-VERLAG NEW YORK, VOLUME 21, NUMBER 1, 1999

31

and it can h a r d l y e s c a p e us now that 22 = 4 and 42 = 16. Continuing this series, we get 256, 6 5 5 3 6 , . . . (i.e., integers of the form 22k). In fact, as is easily proved, 2 n 4- 1 can only be a prime n u m b e r if n itself is a p o w e r of 2. Prime numb e r s of the f o r m 22k + 1 w e r e s t u d i e d b y F e r m a t and are n a m e d F e r m a t primes, after him. The first five are 3, 5, 17, 257, and 65,537 (for k = 0, l, 2, 3, a n d 4). F e r m a t conject u r e d that 22k 4- 1 is p r i m e for all k. This, however, w a s disp r o v e d by Euler, w h o found that 225 4- 1 is divisible b y 641. In part VII o f his D i s q u i s i t i o n e s A r i t h m e t i c a e [2], Gauss p r o v e d that the regular p-gon is c o n s t r u c t i b l e with ruler and c o m p a s s for all F e r m a t primes p. He further c o n t e n d e d that for the r e g u l a r p-gon to be constructible, it is necess a r y that p b e a F e r m a t prime, but, actually, he did not prove the necessity, although he c l a i m e d that it could be p r o v e d with all rigor. The p r o o f was given b y Wantzel [9], and later by P i e r p o n t [6], w h o is e a s i e r to follow. F o r a general survey of the t h e o r y of g e o m e t r i c constructions, b y classical as well as n o n c l a s s i c a l methods, I highly r e c o m m e n d t h e r e c e n t l y p u b l i s h e d b o o k by Martin [5]. Here, a m o n g o t h e r things, you m a y find a construction of the regular 17-gon. The 17-gon also a p p e a r s in Stewart's b o o k on Galois t h e o r y [8]. I refer to this b o o k for the a m o u n t of Galois t h e o r y I use below. The c o n s t r u c t i o n o f the regular 257-gon d o e s n o t s e e m to have b e e n p e r f o r m e d very often. A c o n s t r u c t i o n by Richelot was p u b l i s h e d in CreUe's J o u r n a l in 1832 [7]. What m a k e s one h e s i t a t e to tackle this p r o b l e m is, a m o n g o t h e r things, the h e a v y c o m p u t a t i o n a l work. With a m o d e r n computer, this is no l o n g e r a problem, a n d the 257-gon has bec o m e a natural challenge to anyone with a s h a r p pencil (a very sharp pencil).

A Sequence of Quadratic Field Extensions We shall inscribe t h e regular 257-gon in a given circle; let this circle b e t h e unit circle Izl = 1 in the c o m p l e x plane. Thus, I shall u s e e x p r e s s i o n s like "constructing a c o m p l e x number." Choose z = 1 as the first v e r t e x o f the 257-gon; our t a s k is to find the o t h e r 256 vertices. Put 0)= e 27r//257. Then, w generates a field e x t e n s i o n Q(0)) of the field Q o f rational numbers. The vertices 1, w, 0)2, . , . , 0)256 o f the 257-gon are p r e c i s e l y the zeros of the p o l y n o m i a l t 257 - 1, w h i c h i m m e d i a t e l y splits into two factors: t 257 -- 1 = (t - 1)(1 + t + t 2 + ... + t256). Here, the s e c o n d factor is irreducible over Q. N o w a d a y s , this fact is usually p r o v e d v i a the so-called Eisenstein criterion, as in [8], Chap. 17, b u t the r e a d e r also might like to r e a d Gauss's original p r o o f [2], w or the beautiful p r o o f of Kronecker [4], pp. 101-102. Anyhow, this m e a n s that 1 + t + t 2 4- ..- 4- t 256 is the minimal p o l y n o m i a l of 0), 0 ) 2 , . . . 0)256, and h e n c e the field extension Q(w) : Q is of degree 256, that is, Q(0)) is 256-dimensional as a v e c t o r space over Q. We shall find a series o f i n t e r m e d i a t e fields Q = Q0 c Q1 c Q2 c ... c Q7 c Qs = Q(0)), such t h a t e a c h extension Qi+t : Qi is of d e g r e e 2 and c o r r e s p o n d s to a series of simple constructions. Thus, drawing lines a n d circles and find-

32

THE MATHEMATICAL INTELLIGENCER

ing t h e i r points of intersections, we shall be able to climb u p w a r d in the chain o f fields a n d ultimately r e a c h Q(w). The fields Qi are the "fixed fields" of a sequence of subg r o u p s of the Galois group F = F(Q(0)) : Q) of the extension Q(0)) : Q. We recall that F is, by definition, the group o f all field a u t o m o r p h i s m s of Q(0)) that leave every rational n u m b e r fixed. To e a c h s u b g r o u p H of F is a s s o c i a t e d a f i x e d f i e l d H ~, defined b y H t = {x E Q(0)); ~-(x) = x for all 7 E H]. Thus, our Qi will a p p e a r as the fixed fields of a s e q u e n c e of s u b g r o u p s of F. It follows from Galois t h e o r y that F is a group of o r d e r 256. E a c h e l e m e n t of F is obviously c o m p l e t e l y d e t e r m i n e d b y its effect on 0), and it is not h a r d to see that 0) m u s t m a p to s o m e 0)i. Let us say F = {7~; i = 1, 2 , . . . , 256}, w h e r e 7i(0)) = 0)i. Then, the effect o f 7~ on any 0)k is taking 0)k to its i t h power. Moreover, w e have 7i o ~. = rij, w h e r e the multiplication ij is m o d u l o 257. Let Z~57 be the group o f units o f the field Z257. Then, it follows that w e have an i s o m o r p h i s m of groups F ~ Z*257 given b y 7i ~ 0)i. It is a w e l l - k n o w n fact in the t h e o r y o f finite fields that the group of units of a finite field is cyclic. This is v e r y easy to see in this special case. Indeed, every e l e m e n t o f Z~57 has a p e r i o d which is a p o w e r o f 2. If it s h o u l d h a p p e n that no e l e m e n t has m a x i m a l p e r i o d (i.e., p e r i o d 256), then a l r e a d y a 12s -- 1 for all a ~ Z~57, w h i c h is i m p o s s i b l e b e c a u s e the p o l y n o m i a l t t28 - 1 c a n have at m o s t 128 zeros in the field Z257. Letting 7 b e a g e n e r a t o r o f F, w e n o w face o u r first comp u t a t i o n a l problem: to find i such that 7 = 7~. More precisely, w e m u s t find an e l e m e n t i o f Z~57 having p e r i o d 256. This is a g o o d o p p o r t u n i t y to l e a r n the c o m p u t e r p r o g r a m M a p l e TM. So, the author of this article laid d o w n his p e n c i l for a while at this point a n d t r i e d to learn s o m e b a s i c Maple. I m u s t confess that it t o o k s o m e time before the c o m p u t e r a n d I c o u l d r e a c h an a g r e e m e n t a b o u t w h a t should b e done, b u t t h e n w e easily found t h a t 2 has p e r i o d 16, b u t a l r e a d y 3 has p e r i o d 256 (as have 5, 6, 7, 10, and a lot o f others, b u t w e shall n o t n e e d them). Thus, let us p u t 7 = 73, a n d w e have F = (7>. This i m m e d i a t e l y yields a chain o f subgroups

(732)D(764>D(712s>D1

(1)

and the corresponding chain of fLxed fields

Q = <7>t C (T2> t (Z (T4>t C "'" C (T128>t C (T256>t = Q(0));

(2)

h e r e (72) t = {x ~ Q(0)); 72(x) = x}, a n d so on. Recall that the effect o f 7 on w (and on the p o w e r s o f 0)) is to raise to the third power: 7(0)i) = 0)3i. Hence, 7 2 ( 0 ) ) : 0) 32, 73((,0) : (-033,a n d so on. 7 k raises ~o to its 3kth power. Likewise, 7k(w i) = (wi) 3k.

Our First Step Is to Reach (T2) t We shall w o r k a lot with polynomials m o d u l o p ( t ) = 1 + t + t 2 + ..- + t 256 with integer coefficients, that is, polynomials in the quotient ring Z[t]/p(t). Note that as p(t)It 257 - 1,

each e x p o n e n t occurring in such a polynomial m a y be reduced modulo 257. Also, note that each m ( t ) E Z[t]/p(t) has a well-defined value for w i for i = 1, 2 , . . . , 256. In fact, m ( w ~) does not depend on the choice of representative for m because p ( w i) = O. Thus, we are free to view m ( t ) as a function {w, w 2 , . . . , w256}--) C, where C is the field of complex numbers, and we shall let t denote any of w, o~2,..., ~o256. Consequently, I shall also write (for example) r(t) = t 3. Remembering that the period of 3 in Z~57 is 256, we ob255

256

tain ~ , t 3~ = ~, tJ = - 1 [modulo p ( t ) as usual]. Of course, i--O -1

j=l

is in the fLxed field of ~-, so it is no surprise that 255

~(Z t3')= i--0

255

255

i--0

i=0

~', 7(t3') = .~ [~'(t)]3~ 255 =~

t3i+1= i--0

256 255 Z t3i=Zt3i" i--1

i-- 0 255

The p o i n t is, of course, that T takes each term in ~ t3~ to its successor (and the last term to the first), i=o 127 Now, let us pick every other term and put ao(t) = ~ t 32i 127

and a t ( t ) = ~

i=0

t 32i+1, where it will be c o n v e n i e n t to con-

i-0

sider the indices 0 and 1 as elements of Z2. As the effect of ./.2 o n t is raising t to its n i n t h power, we obtain r2(ao(t)) = ao(t) a n d T2(al(t)) = a l ( t ) (for t = w, w 2 , . . . , w256, as usual). Also, note that ai+t(t) = ai(t 3) for i ~ Z2. Put ao = ao(w) and at = al(w). Then, ao, a l E (~-2)t; that is, ao and al both satisfy a quadratic equation over Q. It will be a pleasant surprise to us that they actually satisfy the s a m e equation. We already know that ao + at = - 1 . As for the p r o d u c t ao(t)al(t), it is a s u m of 128.128 m o n o m i a l s in t. A n a t u r a l guess is that these are uniformly distributed among t, t 2, . . . , t 256, and hence that ao(t)al(t) = (t + t 2 + ... + t256).128.128/256 = -64. It takes a few minutes for Maple to verify this, and then we may safely write a0

+ al = --1, aoat = - 6 4 .

Looking at this system of equations with algebraic eyes, we would say that ao a n d a l are the roots of the seconddegree equation t 2 + t - 64 = 0, and these roots are ( - 1 _+ 2V2~)/2. Note that the roots are real. But, which one is a0, a n d which one is a l ? To a n s w e r this, we let Maple estimate a0 a n d al. We may omit the imaginary parts in the defining 127

8i centered at -1/2, and let a0 and a l denote the intersections with the real axis. Then, it follows from elementary geonmtry (using similar triangles; a special case of the Intersecting Chords Theorem) that Laollal[ = 82 = 64, a n d h e n c e aoal = - 6 4 , as desired. We will repeatedly e n c o u n t e r the same problem: to find real n u m b e r s xt, x2 such that Xl + x2 = a a n d XlX2 = b. Therefore, let us make some general considerations before we proceed. If b < 0, we proceed as above and find Xl a n d x2, where the real axis intersects the circle through ~ i with c e n t e r at a/2. We were lucky this first time in that 4 is an integer. In general, we find H i , where the imaginary axis intersects the circle w h o s e diameter is the segm e n t of the real axis b e t w e e n - 1 a n d - b (this, of course, is j u s t the Intersecting Chords T h e o r e m again). See Figure 1. It will occasionally h a p p e n that b > 0. One way to deal with this case is to follow the idea of Descartes: First find a/2. Then draw the circle with radius la[/2 a n d center at X/bi. Suppose this circle intersects the real axis at _+c. Then, x = a/2 +_ c satisfies the system of equations, a n d hence xt a n d x 2 are easily constructed. See Figure 2. We shall make these simple c o n s t r u c t i o n s quite a few times, each time with n e w values of a a n d b. Let us p a u s e for a m o m e n t a n d see what we have achieved. We have found points a0 a n d a l which (separately) generate the quadratic field e x t e n s i o n ( r 2 ) r : Q. Our plan is to move upward in the chain (2) of fLxed fields, by successively constructing points bk ~ @4)*, Ck E (~S)t, a n d so forth. Our progress can be described in the following m u c h more e l e m e n t a r y way, which completely avoids the use of field theory. Take, as a starting point, the fact that the s u m of the 256 "unknown" vertices is equal to - 1 , b u t be careful to write the terms in the following order: w + oJ3 + w32 + w33 + . . . . . 1. Then, a0 is the s u m of every other term starting with ~o, and a l is the s u m of every other term starting with w3. We have thus, in this section, found the s u m of 128 vertices, which indeed is promising because our ultimate goal is to f'md the "sum" of one vertex alone.

FIGURE

127

sums, and hence obtain a0 = ~ , w3~i = ~ , cos(32i2~'/257) "~ i= 0 127

7.5 and a l = ~ i--0

i=O

127

w32i+1 --- Z cos(32i+12 Ir/257) ~ - 8 . 5 . i=0

F r o m a geometric p o i n t of view, the first equation, a0 + a t = - 1, shows that (the real n u m b e r s ) a0 a n d a l have the same distance to the p o i n t - 1 / 2 ; in other words, that a0 a n d a l are the points of intersection of the real axis and a certain circle with c e n t e r at - 1/2. We need to k n o w j u s t o n e point on this circle in order to draw it a n d h e n c e find a0 a n d al. Such a point is 8i. Indeed, draw a circle through

Xl

-1

V O L U M E 21, N U M B E R 1, 1999

r

a n d b2 satisfy the s a m e quadratic equation o v e r Q ( a o ) = (~-2)t ( a n o t h e r one of t h e s e nice surprises), and w e o b t a i n

:IGURE

bo + b2 = ao, bob2 = - 1 6 .

,X 1

.._

We a l r e a d y know t h a t bo a n d b2 are real. They are the r o o t s of the equation t 2 - aot - 16 -- 0. Again, Maple 63 W34i h e l p s us to distinguish b e t w e e n the r o o t s b0 = }~i=0 -Z63 o cos(34i2~r/257) = 9.2 a n d b2 = ~/63--0w3ai+2 = ~/63--0c o s (34i+22Ir/257) ~ - 1.7. D r a w a circle with c e n t e r at a0/2 through the p o i n t 4i. This circle intersects the real axis at b0 and b2. As for b] a n d b3, it is i m m e d i a t e that bl + b3 = ai. Moreover, w e get b l ( t ) b 3 ( t ) = b0(t3)b2(t3) = - 1 6 , and, hence, bl + b3 = al, bib3 = - 16.

In the n e x t section, w e shall split the set o f u n k n o w n vertices further, as a first step into 4 s u b s e t s o f 64 vertices e a c h with s u m s bo, bl, b2, and b3, respectively. It will app e a r that the bk t a k e n in pairs satisfy quadratic equations with coefficients rational in the ak. This is h o w w e shall proceed. W e { l e e , | t o t h e I'iekl (~.a)r We have so far d e t e r m i n e d the field (r2) *, and w e have found that (~.2), C R. In fact, even (T128)t C R, as w e prove next. We have T128(w) = wal2s. But (3128) 2 = 3256 =-- 1 ( m o d 257), and h e n c e 3128 = - 1 ( m o d 257), for the p e r i o d of 3 is 256 and n o t 128. Of course, 3128 = - 1 ( m o d 257) is also easily found b y a Maple computation. In fact, it will b e very convenient for future r e f e r e n c e to let Maple p r o d u c e a list o f all p o w e r s o f 3 m o d u l o 257, and the r e a d e r w h o w a n t s to c h e c k all c o m p u t a t i o n s is invited to do so. We n o w obtain ~'128(w) = 0 ; 1 = ~. Thus, the effect o f 7"128 is c o m p l e x conjugation; h e n c e (~.128)t C R, as w a s to b e proved. Now, w e m o v e on to (Ta) *. E l e m e n t s in this fixed field are easily f o u n d b y adding every fourth t e r m in the poly255

64

nomial ~ . t 3i, so p u t bk(t) = ~ . t 34i+k E Z [ t ] / p ( t ) for k = 0, i=O

i=O

1, 2, 3, w h e r e t h e i n d e x k is r e g a r d e d and w h e r e t h e k in the e x p o n e n t can sentative o f the index. The choice o f not affect bk(t) m o d u l o p ( t ) . Also, p u t Clearly, 63

bk = bk(w).

63

.ra(bk(t)) = ~" (t3t)3 " + k = ~ i=0

as an e l e m e n t of Za b e a n y fixed reprerepresentative does

t 3t(i+l)+k

i=0 64

63

= ~ . t 34i+k = ~ i=l

t 34i+k = bk(t)

A Maple c o m p u t a t i o n s h o w s that bl = 1.6 a n d b3 - 1 0 . 1 . We find bl and b3 w h e r e the real axis i n t e r s e c t s the circle t h r o u g h 4i with c e n t e r at a l / 2 . The n e x t s t e p is to go f r o m (ra) * to (T8) *. Define Ck(t) = 31

Z

t3si+k for k ~ Z8, where, as before, the k occurring in the

i=0

e x p o n e n t m a y b e any fixed r e p r e s e n t a t i v e of t h e i n d e x k. We have thus p i c k e d out every eighth term o f the s u m 255

~ , t3i. As usual, p u t

Ck =

Ck(W ). We have r8(ck(t)) = Ck(t)

i=0

for t = w, 0,2, . . . , w256, k E Z8, a n d w e shall also n e e d the f o r m u l a Ck+l(t) = Ck(t3), k ~ Z8. Clearly, co(t) + ca(t) = bo(t). T h e p r o d u c t co(t)ca(t) i s a s u m of 32.32 m o n o m i a l s in t, so r e m e m b e r i n g our experie n c e with a o ( t ) a l ( t ) a n d bo(t)b2(t), t h e natural guess this time is that Co(t)ca(t) should equal - 1.32.32/256 = - 4 . This, however, is far from the truth. In fact, a Maple c o m p u t a t i o n a n d a c o m p a r i s o n with the a i ( t ) a n d b i ( t ) yield co(t)c4(t) = - 5 - ao(t) - 2bo(t) ( m o d p ( t ) ) (I wish I could explain that). But this is good enough, b e c a u s e it s h o w s that Co a n d ca are r o o t s o f the s a m e quadratic equation over Q(ao, bo) = ('ca) r Satisfactory as this is, w e are, nevertheless, kept in s u s p e n s e c o n c e r n i n g the future. Can w e c o u n t on the s a m e g o o d luck as w e go on and introduce dk, ek, and so forth? One w o u l d wish here for a nice little t h e o r e m which settles this for good. However, I leave this discussion n o w and t a k e up the s u b j e c t again in the final s e c t i o n of the article. Using the formulas Ck + l ( t ) = Ck(t3), bk+ l ( t ) = bk(t3), a n d a k + l ( t ) = ak(t3), we get c l ( t ) c 5 ( t ) = co(t3)ca(t 3) = - - 5 a0(t 3) -- 2b0(t 3) = - 5 - a l ( t ) - 2bl(t) and, analogously, c2(t)c6(t) = - 5 - a0(t) - 2b2(t) and c3(t)c7(t) = - 5 - a l ( t ) 2b3(t). Thus, w e have the following four s y s t e m s o f equations:

i=O

C o -F C4 ---- b0,

for t = w, w2. . . . , w256. Also, note t h a t b k + l ( t ) = bk(t 3) ( m o d p ( t ) ) . To find the bk, in"st n o t e that b o ( t ) + b 2 ( t ) = ao(t). Next, c o m p u t e bo(t)b2(t), w h i c h is a s u m of 64.64 monomials. As a Maple c o m p u t a t i o n verifies, t h e s e are uniformly distributed a m o n g t, t 2, . . . , t 256, a n d h e n c e bo(t)b2(t) = - 1 6 . Thus, bo

34

THE MATHEMATICALINTELLIGENCER

el

CoCa = - 5 - ao - 2bo; c2 + c6 = b2, C2C6 = - - 5

--

ao - 2b2;

+ c5 = bl,

clc5 = - 5 - a l - 2bl; e3 + c7 = b3, c3c 7 = - 5 - a] - 2b3.

Here, Co ~ 11.9 and c4 ~ - 2 . 6 , so Co and c4 are readily c o n s t r u c t e d as the p o i n t s o f i n t e r s e c t i o n b e t w e e n the real

axis a n d the circle t h r o u g h ~ / 5 + ao + 2boi = y, say, with c e n t e r at b0/2. I recall h o w to fred y. First, find 5 + a0 + 2b0, which is on the positive real axis. Next, d r a w the circle w h o s e diameter goes from this point to - 1 . Then, y is where this circle intersects the imaginary axis. Thus, Co a n d c4 are constructed as i n Figure 1. We have c 2 = 2 . 3 a n d c 6 ~ - 4 . 0 , so c2 a n d c6 are c o n s t r u c t e d analogously. However, cl ~ 0.3 a n d c5-~ 1.3, so c~c5 > 0. Thus, we draw a circle with c e n t e r at X / - 5 - al - 2bli and radius bl/2. This circle intersects the real axis at + c, say, w h e r e c > 0; hence, cl = bl/2 - c and c5 = bl/2 § c are easily found. This is the situation pictured in Figure 2. Further, we have c3 = - 6 . 4 and c7 = - 3 . 7 (i.e., c367 > 0); hence, c3 a n d c7 are constructed in the same m a n n e r as el a n d c5. We go on and define dk, ek, fk, and gk as follows: 15

dk(t) = V / . t ~l~§ , i=0 3 V t 3~i§ , f k ( t ) = ~.. i=0

7

k E Z16 ,

ek(t) = ~ . t332i+k, k ~ Z32 , i=0 1

k E Z~,

gk(t) = ~ . t 3'2~+k,

k E Z12s.

could have been avoided had we c h o s e n as our primary goal not the vertex 0) of the regular 257-gon but some other vertex (appropriately chosen). For example, we could have chosen 0 27, because 0)27 + ~27 = g3 and g3g67,f3f35, andf27f59 are all negative, as is easy to verify. Thus, we may move from the e~ to 0)27 a s in Figure 1, and o n c e 0)27 is f o u n d , the other vertices of the 257-gon are readily constructed. However, we c a n n o t completely avoid the Figure 2 case, b e c a u s e it will t u r n out that we will need all the dk, and hence all the Ck.

The Missing Links Turn out to Be the Most Laborious We n o w go back again and r e m e m b e r that all the ak, bk, and Ck have b e e n constructed. The n e x t step is to construct the dk. We let Maple calculate do(t)ds(t) = ao(t) + Co(t) + c2(t) + 2c5(t). Applying the f o r m u l a dk+l = dk(t 3) and working as before, we obtain do + d8 = Co, dods = ao + Co + c2 + 2c5;

d l § d9 = Cl, dido = a l § cl + c3 § 2c6;

d2+dl0=C2, d 2 d l 0 = ao + c2 § c4 § 2C7;

d3 § dll -- c3, d 3 d u = al § c3 § c5 § 2c0;

d 4 + d12 =64,

d5 -I- d13 = c5, d5d13 = al § c5 § c7 § 2c2;

i:0

As before, the k in the e x p o n e n t can be a n y representative of the index k, b e c a u s e we always work m o d u l o p(t). It is n o w easy to check that :16(dk(t)) = dk(t), r32(ek(t)) = ek(t), ~ 6 4 ( f k ( t ) ) = f k ( t ) , a n d " r 1 2 8 ( g k ( t ) ) = gk(t) for t =0), 02 . . . . , 0256, a n d to verify the formulas d k + l ( t ) = d k ( t 3 ) , . . . , g k + l ( t ) = gk(t3). We can put dk = d k ( 0 ) ) , . . . , gk = gk(0)) for all k, b u t we shall n o t need all of them.

Working B a c k w a r d f o r a M o m e n t In fact, look at go = 0) + 0)3128 = 0) § ~ ---- 2 Re 0). Once go is d e t e r m i n e d (i.e., constructed), we are only a few steps from our goal 0). We take half of go and move parallel to the imaginary axis until we reach the unit circle Iz[ = 1. T h e n there is 0), the s e c o n d vertex of the regular 257-gon. The n u m b e r s go a n d g64 are the roots of a quadratic equation over <~64>t (we work b a c k w a r d for a m o m e n t ) . We have go(t) + g64(t) = fo(t), a n d a n easy c o m p u t a t i o n shows go(t)g64(t) = t 15 + t 17 + t 24~ + t 242 = f56(t). You do n o t n e e d Maple to c o m p u t e this ff you have already let Maple p r o d u c e a display of all p o w e r s of 3 modulo 257 [we have go(t) = t + t 256 and g64(t) = t 16 + t24]], as I r e c o m m e n d e d earlier. F r o m go = 0) -~ ~ a n d g64 = 0)16 § ~16, it is obvious that go > g64 > 0; hence, go a n d g64 are c o n s t r u c t e d as in Figure 2. Clearly, fo(t) + f32(t) = eo(t) and f24(t) + f56(t) = e24(t). A simple c o m p u t a t i o n shows fo(t)f32(t) = el(t) + e23(t), a n d therefore, f24(t)f56(t) = fo(t324)f32(t324) = el(t 324) + e23(t 324) = e25(t) + e47(t) = e25(t) + els(t). Here, we have fo = 0) + 0)16 § "~ § m16 a n d f 3 2 = 0)4 § 0)64 § ~ 4 § ~6a and, hence, fo >f32 > 0. Further, f24 = 0)6o + 0)6s + ~ + ~6s a n d f56 = 0)15 § 0)17 § ~15 § ~17, w h e n c e f56 >f24 > 0. Note that f24 is j u s t slightly greater than 0. This is b e c a u s e 0)6o + 0)68 = r0)64 for s o m e r e a l r > 0, and 0)64is j u s t slightly to the right of the imaginary axis! It is somewhat unfortunate that gog64,fof32, andf24f56 are all positive, for this brings us into the "Figure 2 case." This

d4d12 = ao + c4 + c6 § 2Cl;

d6 § d14 = C6, d6d14 = ao § c6 § Co + 2c3

d7§ dTd15 = al + c7 + cl § 2c4.

A list of approximations is: do ~-- 9.2, ds ~ 2.6, d l ~-" 4.9, d9 ~ - 4 . 6 , d2 ~ 2.4, dl0 ~ -0.11, d 3 ~ -2.96, d l l ~- - 3 . 4 ,

d4 -~ - 0 . 8 , d12 ~ - 1 . 8 , d5 -~ 3.3, d m - l . 9 , d 6 ~- -3.2, d14 - - 0 . 8 , d 7 ~ 2.7, d15 ~ -6.4. The missing link is n o w only to c o n s t r u c t the ek from the dk. This time, Maple yields eo(t)e16(t) = do(t) + dl(t) + d2(t) + ds(t). I skip the details a n d only give the equations a n d a p p r o x i m a t i o n s needed. e 0 § el6 = do,

e0e16 = do § dl + d2 + d5; el+el7=dl, elel7 = dl § d2 § d3 § d6; e7 + e23 = d7, e7e23 = d7 + d8 + d9 + d12;

eo ~ 5.9,

el6 ~ 3.4.

el ~ 4.6,

el7 ~ 0.3.

e7 ~ - 0 . 4 , e23 ~-- 3.1.

es + e24 = d8, ese24 = ds + do + dlo + d]3;

es ~ - 1 . 1 , e24 ~ 3.7.

e9 + e25 = (/9 e9e25 = d9 § dlo + dll § d14;

e9 ~ - 6 . 1 , e25 ~ 1.5.

e15 + e31 = d15, e15e31 = d15 + do + dl + d4;

el5 ~ -1.4, e31 ~ - 5 . 0 .

With eo, el, e15, e23, e24, a n d e25 n o w at hand, we cons t r u c t f o , f24,f32,f56, and, finally, go, g64, and 0), as already

VOLUME 21, NUMBER1, 1999

35

demonstrated. The actual p e r f o r m a n c e o f the c o n s t r u c t i o n is left to the reader. S o m e Final R e m a r k s Looking b a c k at o u r c o n s t r u c t i o n of the regular polygon o f two h u n d r e d a n d fifty-seven sides, I feel the n e e d to a d d a few remarks. First, have a l o o k at the two equations determining a0 and al. The ffist equation a0 + a l = - 1 is obvious, w h e r e a s the s e c o n d equation aoal = - 6 4 was found after a rather heavy computation, w h e r e Maple w a s of help. But, as I pointed out, the result aoal = - 6 4 m e r e l y confirmed our previous guess, so it w o u l d be rather a p t to a s k w h e t h e r we could prove this b~2~ome o t h e r me la~. And i n d e e d w e can! Consider a0 = ~

oJ32~ and a l = ~

i=O

so, in fact, a l l f ( s ) a r e equal. The s u m f ( 0 ) + f ( 1 ) + ... + f(1) + ... + f(255) is the total n u m b e r o f t e r m s in the exp a n s i o n of aoal, n a m e l y 1282, w h e n c e it follows t h a t f ( s ) = 255

1282/256 = 64 for all s. Thus, finally, aoal = ~ f ( s ) w 3~ = 255

s=0

64 ~ . w 3s = - 6 4 . s=O

Let us try to r e p e a t this argument with b0 a n d b2 in p l a c e 63

of ao and al. We have bob2 = ~ i=0

63

w3a~

w 34i+2, which

i--0

e x p a n d s to a s u m o f 642 t e r m s of the form w 34~+3~+2. It is e a s y to p r o v e t h a t this t e r m n e v e r equals 1 and, hence, that w u4i+34k+2 = w3~ for s o m e s E {0, 1 , . . . , 255}. L e t f ( s ) be the n u m b e r of t i m e s a certain s occurs, that is, the n u m b e r of solutions (i, k) ( c o u n t e d m o d u l o 64) to the equation 3 ~ -= 34i + 34k+2 ( m o d 257). Clearly, f ( 0 ) + f ( 1 ) + ..- + f ( 2 5 5 ) = 642. It follows from the i m p l i c a t i o n 3 ~ -= 34i + 34k+2 ~ 3 s+2 --- 34(k+l) § 34i+2 t h a t f ( s + 2) = f ( s ) for all s, so that we have f(O) = f ( 2 ) . . . . 255

obtain

and f(1) =f(3) .... 127

bob2 = ~ f ( s ) w 3s = f ( 0 ) ~

127

f(1) ~

s=0

w 32i +

i=0

w3ei+' = f ( O ) a 0 + f ( 1 ) a l .

i=0

This is r a t h e r interesting, b e c a u s e it s h o w s that bob2 Q(ao, a l ) and, hence, w e have found, w i t h o u t computing bob2, that bo a n d b2 a r e the two r o o t s o f a quadratic equation over Q(ao, a l ) . Morever, it is quite c l e a r (I o m i t the details) that a similar a r g u m e n t can be u s e d to prove that c0c4 = ~ 7

fiibi for a p p r o p r i a t e integers fii, that dod4 =

i=0

yici for a p p r o p r i a t e Yi, and so on, a n d this is the r e a s o n i=O

3~

2n-l--1

i--O

f ( s ) <--f(s + 1), a n d h e n c e f(0) -
we

T h e o r e m . F o r n >- 2, t h e r e a r e i n t e g e r s c~ s u c h t h a t

co32~+~. The p r o d u c t

is a sum of 1282 terms, each of the form w32i+3~+1. Let us prove first t h a t such a t e r m can never b e equal to 1; that is, that 32i + 32k+] ~ 0 ( m o d 257) for all i and k. Assuming the c o n t r a r y a n d using the fact that 312s ~- - 1 ( m o d 257), w e would o b t a i n 32k+1 ~- 32i+12s (mod 257), a n d h e n c e 2k + 1 ~- 2i + 128 ( m o d 256), which is clearly impossible. Thus, w32i+32k+1 = w3~ for s o m e s E {0, 1 , . . . , 255}. Let f ( s ) be the n u m b e r o f times a certain s o c c u r s here; that is, the n u m b e r o f solutions (i, k) (counted m o d u l o 128) to the equation 3~ ~- 3 2i § 3 2 k + l ( m o d 257). It follows from the implication 3s --~ 32i § 3 2 k + l ~ 3 s + l ~ 3 2 ( k + l ) § 3 2 i + 1 that

Thus,

w h y Co and c 4 are the r o o t s of t h e s a m e quadratic equation over Q(ao, b0), w h y do and ds a r e the r o o t s of the s a m e quadratic equation over Q(a0, b0, Co), and so on. But there is even m o r e to it t h a n this. We have m a d e no h e a v y u s e here o f the fact that p = 257. Indeed, let p = 2 r § 1 b e any F e r m a t prime, let oJ = e 2~/p, and let g b e an int e g e r w h i c h has p e r i o d 2 r m o d u l o p; that is, g is a generat o r o f Zp. F o r 1 < - n < - r - 1, O<--k<--2 n - 1, p u t an,k = ~. wgs, the s u m taken o v e r all s ~ {0, 1 , . . . , 2 r -- 1} s u c h t h a t s ~ k ( m o d 2n). F o r example, for p = 257 and g = 3, w e find that a3,4 is j u s t o u r old c4. Using the s a m e technique as above, the r e a d e r m a y p r o v e the following theorem.

THE MATHEMATICALINTELLIGENCER

an,oan,2n-1 ~

~

o~ian- l,i.

i=0

F o r n = 1, w e h a v e al,oa],l = - 2 r-2 = - ( P - 1)/4.

F o r example, f o r p = 65,537, w e have r = 16 and, hence, al,0 + a],l = - 1 and a1,0a1,1---214 , and from this w e know, at least, the first s t e p o f a c o n s t r u c t i o n of the 65,537gon. Most o f the calculations in this article w o u l d be ext r e m e l y l a b o r i o u s to do b y hand. The calculation w h i c h u s e d t h e m o s t c o m p u t e r time w a s the calculation of aoal. However, having a direct p r o o f n o w that aoal = - 6 4 , this calculation could be avoided, a n d a by-hand calculation s h o u l d s t a r t with the p r o d u c t bob2.

Even if we could imagine bob2 being c o m p u t e d b y hand, t h e r e is a n o t h e r o b s t a c l e that s e e m s m o r e difficult to overc o m e without a c o m p u t e r , namely, to find the a p p r o x i m a tions of the ai, bi, ci, di, a n d ei, which w e u s e d to distinguish b e t w e e n the r o o t s of the successive s e c o n d - d e g r e e equations. F o r this reason, have a c l o s e look at the equations determining the bi and imagine for a m o m e n t t h a t ao and a l have b e e n m i x e d up either b y mistake or ignorance. Then, the sets {bo, b2} and {bl, b3} a r e m i x e d up also, b u t it is quite clear from the equations that the set {bo, bl, b2, b3} will be correct. Thus, it is not n e c e s s a r y to distinguish b e t w e e n ao a n d a l in o r d e r to fred the set of all bi. Moreover, it is c l e a r from the nature of the s y s t e m s of equations determining the ci from the ai and b~ that w e do n o t n e e d to distinguish b e t w e e n the bi to arrive at the corr e c t set {Co,..., c7}. We m o v e n o w to the equations that d e t e r m i n e the di. These are of a different nature, the main difference being that t h r e e c~ o c c u r in the s a m e equation. It follows that a mixing up of the ci m a y l e a d us into a false c o n c l u s i o n a b o u t the set of di. Thus, after all, it is n e c e s s a r y for us to a s c e r t a i n w h o is w h o a m o n g the c~. However, I do n o t m e a n to s a y that we n e e d n u m e r i c a l a p p r o x i m a t i o n s of the ci to b e able to do this. I believe we

could successfully e m p l o y the m e t h o d a b o v e by which w e f o u n d t h a t go > g64,fo >f32, and f56 >f24. I wish t h e r e a d e r who w a n t s to p u r s u e the c o n s t r u c t i o n in all details g o o d luck. And p l e a s e r e m e m b e r to send m e a c o p y o f y o u r result. REFERENCES

1. A. Aaboe, Episodes from the Early History of Mathematics, Washington, D.C.: Mathematical Association of America (1964). 2. C.F. Gauss, Disquisitiones Arithmeticae, English translation by Arthur A. Clarke, New Haven, CT: Yale University Press, (1966). 3. Gauss Tagebuch 1796-1814, Math. Ann. 57 (1903), 1-34. 4. Kronecker, Werke, Vol. 1, Leipzig: Teubner Verlag (1895). 5. G. Martin, Geometric Constructions, New York: Springer-Verlag (1998). 6. J. Pierpont, On an undemonstrated theorem of the Disquisitiones Arithmeticae, Bull. Am. Math. Soc. 2 (1895-96), 77-83. 7. F.J. Richelot, De resolutione algebraica aequationis x 257 = 1, sive de divisione circuli per bisectionam anguli septies repetitam in partes 257 inter se aequales commentatio coronata, Crelle's Journal IX (1832), 1-26, 146-161,209-230, 337-356. I. Stewart, Galois Theory, 2nd ed., London: Chapman & Hall (1989). P.L. Wantzel, Recherches sur les moyens de reconnaftre si un Probleme de Geom6trie peut se r6soudre avec la regle et le compas, J. Math. 2 (1837), 366-372.

VOLUME 21, NUMBER 1, 1999

37

Ikd,[.-]mLhl.lVl|,[a,~.~[.-~.nlbE.,,nd~i

How Long Was the Coast of Atlantis? Wojciech S~'omczySski and Tomasz Zastawniak

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafd where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.

Please send all submissions to Mathematical Tourist Editor, Dirk Huylebreuck, Aartshertogstraat 42, 8400 Oostende, Belgium e-mail: [email protected]

Dirk H u y l e b r o u c k , E d i t o r I

But afterwards there occurred violent earthquakes and floods, and in a single day and night of misfortune all your warlike men in a body sank into the earth, and the island of Atlantis in like manner disappeared, and was sunk beneath the sea. Plato, Timaeus, 24d-25d ;asoned mathematical will be aware that a journey along a coastline, a river bank, or a mountain ridge can take an inordinate length of time if the wiggliness of the line is to be followed precisely. As was observed by Hugo Steinhaus in his celebrated book Mathematical Snapshots [Ste54], "When measuring the length of a river one comes across the problem of tiny meanders. Some countries have such winding rivers or mountain ridges as their boundaries. Using increasingly detailed maps and increasing the accuracy of measurement accordingly, it is possible to verify that the length will become as large as one wants it to be." In fact, Steinhaus wrote on this subject as early as in 1949 [Ste49a], [Ste49b]. Almost 20 years later, Benoit Mandelbrot posed the famous question, "How long is the coast of Britain?" [Man67]. As a mathematical model he proposed fractals reflecting the highly irregular and fragmented outline of a coast or river bank, and anaiysed their fractional dimension and self-similarity properties. In July 1996, when taking part in the Second World Congress of Nonlinear Analysts in Athens, we xdsited the National Archaeological Museum in an unsuccessful attempt to divert our thoughts from mathematics. Whoever enters the exhibition of wall-paintings

from Akrotiri, Thera will be deeply impressed by the masterpieces from an ancient land destroyed by a volcanic eruption about three and a haft thousand years ago. The ancient town of Akrotiri was discovered by the Greek archaeologist Spyridon Marinatos in 1967. The dig revealed a number of houses richly decorated with wail paintings. One of the best-preserved masterpieces is the 20foot-long Frieze with the T~eet fromroom 5, The West House. But it was a beautiful scene in a miniature frieze known as Shipwreck, which was found in the same room, that immediately attracted our attention (Fig. 1). Just above the naked bodies of drowned men is a striking blue shape, which brings to mind a wave crushing into a rocky shore, perhaps the cause of the disaster. Or it may be an outline of the rocks themselves. 1 In any case, it looks very much like a fractai. The resemblance to von Koch's curve, Fig. 2, is remarkable. We cannot help wondering if the ancient artist could sense the fractal structure of the rocky shore of his native land surrounded with water. Is it but a coIncidence that the curve has been proposed as a mathematical model of the seashore and it is referred to as the Koch coastline [Man82]? The island of Thera (also known as Santorini) lies 70 miles north of Crete (Fig. 3). According to the ancient tradition, it used to be called Strongyle, the Round One. As a result of a volcanic upheaval in about 1500 sc, part of the island was submerged into the Aegean sea. Gigantic tidal waves and volcanic ashes contributed to the destruction of the Minoan civilization on Crete and the neighbouring islands, which were taken over by the Myceneans [Mar39]. Three and a half thousand

lit is a pleasure to thank Peter Warren, Professor of Archaeology at Bristol University, for his help in identifying the blue shape in the Shipwreck scene as "rockwork" and for his enlightening comments on Minoan art. The interested reader can find more information on the miniature frieze from the West House in Peter Warren's paper [War79].

THE MATHEMATICALINTELLIGENCER9 1999 SPRINGER-VERLAGNEW YORK

Figure 1. Drowned defenders scene, miniature frieze Shipwreck from the West House at Akrotiri, There. The striking blue shape is reminiscent of von Koch's curve. (Original exhibited in National Archaeological Museum, Athens. 9 Archaeological Receipts Fund, 57 Panepistimiou Street, 105 G4 Athens, Greece.)

years later, the Thera volcano is still active. The disaster following the eruption in around 1500 Bc is often identified with the sinking of mythical Atlantis. This conjecture, put forward at the be-

Figure 2. The yon Koch curve.

ginning of this century by K. T. Frost [Fro09], was developed into a wellfounded theory by J. W. Luce [Luc69], following Marinatos's discovery of a Minoan town buried under a layer of pumice in Akrotiri, Thera.

The occupants of the town apparently had time to flee the disaster, taking many precious possessions with them. Among the artifacts they left behind was a tripod offering table featuring fractal-like marine motifs, probably aquatic plants, rocks, or corals. Although we have not come across any other examples of this kind from Thera, more can be found among Marine Style vases and other items from different parts of the Minoan world (Figure 4). According to Pierre Devambez, Curator of Greek and Roman Antiquities at the Louvre, "the decorations on their vases were composed in such a way as to lay particular emphasis on the form which they adorned: a flowing form, incompatible with any sharply angular or abstractly geometrical design" [Dev62]. The naturalistic Marine Style reached its peak during the period of greatest prosperity of the Minoan civilization. As was observed by Sir Arthur Evans [Eva28], the demise of the empire was followed by a stage of decadence in decorative art. The original fractal-like marine motifs evolved into simplified geometric ornaments, losing to a large extent their "strength, naturalism, humor and freshness" [War69]. In his beautiful book The Fractal Geometry of Nature, Mandelbrot pointed out several works of great artists who demonstrated an amazing insight into the intricate geometric structure of fractals long before mathematicians and other scientists became interested in the subject. Wellknown examples include the engraving The Great Wave at Kanagawa by the Japanese artist Katsushika Hokusai and Leonardo da Vinci's Deluge showing eddies and whirls in water. It appears that the oldest case of fractals in a work of art cited to date is a frontispiece in the thirteenth-century French Bible Moralisde, which depicts the Lord as Great Geometer creating the world made of circles, waves, and fractats [Man82], and serf-similar patterns on the Celtic Densborough mirror, made probably in the first century hl). [Bripea89] In our opinion the paintings of Akrotiri, Thera provide evidence that the ancient masters could perceive and tried to convey the fractal character of

VOLUME 21, NUMBER 1, 1999

39

Figure 3. Present-day outline of the island of Thera (Santorini).

Figure 5. Amphora with nautili from Pylos. Figure 4. Tripod offering table from the West House, Akrotiri. (Originals of Figs. 4 and 5 exhibited in the National Archaeological Museum, Athens. 9 Archaeological Receipts Fund, 57 Panepistimiou Street, 105 G4 Athens, Greece.)

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Nature. This finding m o v e s the point w h e n such shapes a p p e a r e d in art s o m e two thousand years prior to The Densborough Mirror. Since "the style is the first truly naturalistic style to be found in European, or indeed any art" [HigS1], it may well be that the Therean paintings contain the fLrst ever instances of man-made fractal-like shapes. As mathematicians, w e fmd it fascinating and astonishing that nearly four thous a n d y e a r s ago the a n o n y m o u s artists o f Akrotiri d e m o n s t r a t e d a degree of a w a r e n e s s of w h a t has only b e c o m e u n d e r s t o o d and a p p r e c i a t e d b y science in the last t w e n t y - o d d years. It s u p p o r t s the view that g r e a t art, nature, and m a t h e m a t i c s are inseparable. REFERENCES

[Bripea89] John Briggs and F. David Peat. Turbulent Mirror. Harper & Row, London, 1989. [Dev62] Pierre Devambez. Greek Painting. Weidenfeld & Nicholson, London, 1962. [Dou92] Christos Doumas. The Wall-Paintings of Thera. The Thera Foundation-Petros C. Nomikos, Athens, 1992. [Dou95] Christos Doumas. Santorini. A Guide

to the Island and Its Archaeological Treasures. Ekdotike Athenon S.A., Athens,

1995. [Eva28] Sir Arthur Evans. The Palace of Minos at Knossos. Macmillanand Co., London, 1928. [Fro09] K. T. Frost. The "Critias" and Minoan Crete. Journal of Hellenic Studies, 33:189206, 1909. [Hig81] R. Higgins. Minoan and Mycenaen Art. London, 1981. [Luc69] J. V. Luce. The End of Atlantis. New Light on an 9 Legend. Thames and Hudson, London, 1969. [Man67] Benoit B. Mandelbrot. How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science, 155:636-638, 1967. [Man82] Benoit B. Mandeibrot. The Fractal Geometry of Nature. W. H. Freeman and Co., San Francisco, 1977, 1982. [Mar39] Spyridon Marinatos. The volcanic destruction of Minoan Crete. Antiquity, 13:425439, 1939. [Ste49a] Hugo Steinhaus. O dl'ugo~ci krzywych empirycznych i ich pomiarze, zwl'aszcza w geografii (On the length of empirical curves and its measurement, especially in geography). Sprawozdania Wroofawskiecjo Towarzystwa Naukowego, 4:1-6, Dodatek 5, 1949.

[Ste49b] Hugo Steinhaus. On the length of empirical curves. Casopis pro p#stovani matematiky a fisiky, 74:303-304, 1949. [Ste54] Hugo Steinhaus. Kalejdoskop Matematyczny. Par~stwoweZaMady Wydawnictw Szkolnych, Warszawa, 1954. 2nd Polish edition; English translation: Mathematical Snapshots, Oxford University Press, New York, 1950. [War69] Peter Warren. Minoan Stone Vases. Cambridge University Press, Cambridge, 1969. [War79] Peter Warren. The miniature fresco from the West House at Akrotiri. Journal of Hellenic Studies, 99:115-29, 1979.

Institute of Mathematics Jagiellonian University Reymonta 4 30-059 Krak6w, Poland e-mail: [email protected]

Department of Mathematics University of Hull Cottingham Road Kingston upon Hull HU6 7RX United Kingdom e-mail: [email protected]

VOLUME 21, NUMBER 1, 1999

41

VICTOR VINNIKOV

We Shall Know: Hilbert'8 Apology This note is about a poem to the glory of Mathematics. On January 23, 1930, David Hilbert reached the mandatory retirement age of 68. Among the many honoum bestowed upon him, he was made an "honorary citizen" (Ehrenbi~rger) of his native town of KSnigsberg. The honorary citizenship was presented to Hilbert on September 8, 1930, at the meeting of the Society of German Scientists and Physicians which was held that year in KSnigsberg. Hilbert's acceptance address was entitled "Natural Philosophy and Logic" (Naturerkennen und Logik).l Though to the outside world his name was inextricably linked to the scientific glory of GSttingen, where he was called on Klein's initiative in 1895, Hilbert himself had always remained deeply attached to the town of his forefatheirs. It was in KSnigsberg that he had started his career, and formed a lifelong friendship, mathematical as well as personal, with Hurwitz and Minkowski. He was very pleased to be honoured by the town of KSnigsberg on his retirement year. It was a perfect place and occasion to express, towards the end of his mathematical life, his views on the meaning and the goals of mathematics, and of science in general. Hilbert was glad to address a general scientific audience rather than just professional mathematicians. Already in his talk [6] at the International Congress of Mathematicians at Paris in 1900 he had said, quoting an unnamed old French mathematician, "A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street." Reidemeister and Szeg5 made arrangements for Hilbert to repeat a precis of the last part of his acceptance address at the Society meeting over the local radio station. A 45-

rpm-record of this radio broadcast exists and was distributed with Hilbert's "Gedenkband" [9]. While the complete address at the Society meeting has been duly published [7] (though never translated to English except for selected passages in Constance Reid's book [12, Chapter 22]), the text of the radio broadcast has never been printed. Even though it is essentially extracted from the complete address, the text of the radio broadcast has a life and a meaning of its own, a short and concise statement on the goals and the place of mathematics quite different from the detailed discussion in [7]. I fLrst received a transcript of Hilbert's recording from my colleague Victor Katsnelson about two years ago, while on a sabbatical at MSRI in Berkeley. On my suggestion the text was translated to English by Joseph and Amelia Ball, and distributed among the happy few at MSRI and elsewhere. Everybody-including occasional non-mathematicians---was impressed by the force of Hilbert's message; maybe we felt we were hearing "from afar" (as Hermann Weyl wrote in his reminiscences) "the sweet flute of the Pied P i p e r . . . seducing so many rats to follow him into the deep river of mathematics" [18]. Soon another translation was made by Laurent Siebenmann. The translation below is the result of joint efforts of Joseph and Amelia Ball and Laurent Siebenmann, some very helpful remarks by Otto Siebenmann, and small adjustments by myself. I am very glad of this opportunity to present anew this powerfifl apology of our science. Das Instrument, welches die Vermittlung bewirkt zwischen Theorie und Praxis, zwischen Denken und Beobachten, ist die Mathematik; sie baut die verbin-

~The literal translation of the German Naturerkennen is knowledge (or understanding) of nature. But I think that the meaning is Natural Philosophy, in its older English sense.

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THE MATHEMATICALINTELLIGENCER9 1999 SPRINGER-VERLAGNEWYORK

dende Brticke und gestaltet sie immer tragf'fihiger. Daher kommt es, da~ unsere ganze gegenwfirtige Kultur, soweit sie auf der geistigen Durchdringung und Dienstbarmachung der Natur beruht, ihre Grundlagen in der Mathematik fmdet. Schon Galilei sagt: Die Natur kann nur der verstehen, der ihre Sprache und die Zeichen kennengelernt hat, in der sie zu uns redet; diese Sprache aber ist die Mathematik, und ihre Zeichen sind die mathematischen Figuren. Kant tat den Ausspruch: "Ich behaupte, da~ in jeder besonderen Naturwissenschaft nur soviel eigentliche Wissenschaft angetroffen werden kann, als darin Mathematik enthalten ist." In der Tat: Wir beherrschen nicht eher eine naturwissenschaftliche Theorie, als bis wir ihren mathematischen Kern herausgeschfilt und vtillig enthtillt haben. Ohne Mathematik ist die heutige Astronomie und Physik unmSglich; diese Wissenschaften l(isen sich in ihren theoretischen Teilen geradezu in Mathematik auf. Diese wie die zahlreichen weiteren Anwendungen sind es, den die Mathematik ihr Ansehen verdankt, soweit sie solches im weiteren Publikum genie6t. 2 The tool that serves as intermediary between theory and practice, between thought and observation, is mathematics; it is mathematics that builds the linking bridges, and gives them ever more reliable forms. From this it has come about that our entire contemporary culture, inasmuch as it is based on the intellectual penetration and the exploitation of nature, has its foundations in mathematics. Already Galileo said: one can understand nature only when one has learned the language and the signs with which it speaks to us; but this language is mathematics and these signs are mathematical figures. Kant made the pronouncement: "I assert that, in any particular natural science, one encounters genuine scientific substance only to the extent that mathematics is present." Indeed: we do not master a scientific theory until we have shelled and completely pried free its mathematical kernel. Without mathematics, the astronomy and physics of today would be impossible; these sciences, in their theoretical branches, virtually dissolve into mathematics. They, along with the m a n y other applications,

are responsible for whatever esteem mathematics m a y enjoy in the eyes of the general public. Trotzdem haben es alle Mathematiker abgelehnt, die Anwendungen als Wertmesser fiir die Mathematik gelten zu lassen. Gau~ spricht von dem zauberischen Reiz, den 3 die Zahlentheorie zur Lieblingswissenschaft der ersten Mathematiker gemacht habe, ihres unerschSpflichen Reichtums nicht zu gedenken, woran sie alle anderen Teile der Mathematik so weit tibertrifft. Kronecker vergleicht die Zahlentheoretiker mit den Lotophagen, die, wenn sie einmal von dieser Kost etwas zu sich genommen haben, nie mehr davon lassen kSnnen. Nevertheless, all mathematicians have refused to accept the applications as a valid measure of the worth of mathematics. Gauss speaks of the magical attraction that made n u m b e r theory the darling discipline of the earliest mathematicians, not to mention n u m b e r theory's inexhaustible wealth, so f a r surpassing that of any other branch of mathematics. Kronecker compares number-theorists with the lotus eaters, who, once they have savored this delight, can never give it up. Der grot~e Mathematiker Poincar~ wendet sich einmal in auffallender Schfirfe gegen Tolstoi, der erklfi,--t hatte, da~ die Forderung "die Wissenschaft der Wissenschaft wegen" tSricht sei. a Die Errungenschaften der Industrie zum Beispiel h~tten nie das Licht der Welt erblickt, wenn die Praktiker allein existiert h~tten und wenn diese Errungenschaften nicht von uninteressierten Toren gef~irdet worden w ~ e n . 5 The great mathematician Poincard replied with remarkable sharpness to Tolstoy who had declared the p u r s u i t of "Science f o r the sake of Science" to befoolish. The achievements of industry, f o r example, would never have seen the light of day, i f only practical m e n had existed, and i f these achievements had not been made possible by disinterested 'fools." Die Ehre des menschlichen Geistes, so sagt der berfihmte KSnigsberger Mathematiker Jacobi, ist der einzige Zweck aller Wissenschaft. 6 The glory of the h u m a n spirit, 7 so said the f a m o u s K6nigsberg mathematician Jacobi, is the one purpose of all science.

2In [7] this sentence is slightly different: "Diese und die zahlreichen weiteren Anwendungen sind es, denen die Mathematik ihr Ansehen verdankt, soweit sie soiches im weiteren Publikum genie6t." 3"der" in [7]; did Hilbert make a mistake in reading? 4In [7] this sentence starts simply with "Poincare ..."; there is a sentence preceding it, omitted in the radio broadcast, referring to Poincare as "der gl~inzendste Mathematiker seiner Generation". 5In [7]: "Man braucht nur die Augen zu 5ffnen, so SchlieBt Poincar~, um zu sehen, wie zum Beispiel die Errunganschaften der Industrie nie das Licht der Welt erblickt h&tten, wenn die Praktiker allein existiert h&tten und wenn diese Errungenschaffen nicht von uninteressierten Toren gef6rdet worden w&ren, die nie an die praktische Ausn0tzung gedacht haben." 6In [7] Hilbert refers to an argument between "unser grol3er K6nigsberger Mathematiker Jacobi" and "der ber0hmte Fourier"; Fourier was claiming that the purpose of mathematics lies in the explanation of natural phenomena. "Ein Philosoph, wie Fourier es doch sei, h&tte wissen sollen, so ruft Jacobi, dab die Ehre des menschlichen Geistes der einzige Zweck ailer Wissenschaff ist und dalB unter diesem Gesichtspunkt ein Problem der reinen Zahlentheorie ebensoviel wert ist als eines, das den Anwendungen dient." 7Die Ehre des menschlichen Geistes" is not easy to translate to English. "The honour of the human spirit" or "The glory of the human intellect" are also possible translations.

VOLUME21, NUMBER1, 1999 4 3

Wir dtirfen nicht denen glauben, die heute mit philosophischer Miene und fiberlegenem Tone den Knlturuntergang prophezeien und sich in dem Ignorabimus gefallen, s Ffir uns 9 gibt es kein Ignorabimus, und meiner Meinung nach auch f'tir die Naturwissenschaft tiberhaupt nicht. Statt des t6richten Ignorabimus heine im Gegenteil unsere Losung: Wir miissen w i s s e n - - w i r w e r d e n wissen. We ought n o t to believe those w h o today, adopting a philosophical a i r a n d a tone o f s u p e r i o r i t y , prophesy the decline o f culture and are content w i t h the "unknowable ''1~ i n a self-satisfied w a y . For u s there is no unknowable, a n d i n m y o p i n i o n there is also none w h a t s o e v e r f o r the n a t u r a l sciences. I n place o f this f o o l i s h "unknowable," let o u r w a t c h w o r d on the c o n t r a r y be: We m u s t k n o w - - w e s h a l l k n o w . Beyond the sheer poetic power, Hilbert has addressed the fundamental issues facing all of us mathematicians, or at least those among us who (to cite Hermann Weyl again) "are not indifferent to what their scientific endeavours mean in the context of man's whole caring and knowing, suffering and creative existence in the world" [19]. Is "We Shall Know!" still a valid motto for mathematics in the late 1990's? It is impossible to present Hilbert's apology of our science without saying something about one's opinion. To put the question in a proper perspective, let me recall that in the same fall 1930, on November 17, Kurt Gbdel's paper [4] containing the famous incompleteness theorems was submitted for publication. GSdel proved (the first incompleteness theorem) that given any consistent mathematical system ~ containing elementary number theory, we can (effectively) construct a statement a in ~ such that neither a nor the negation of a is provable in ~.11 Furthermore (the second incompleteness theorem), the consistency of ~ cannot be proved in ~.12 The discovery of paradoxes in set theory around the turn of the century had led to a "foundational crisis" in mathematics and a re-evaluation of standard mathematical arguments. To avoid the known paradoxes, Russell had introduced his theory of types; with perhaps less logical insight into the process of set formation, but in a way much more

suited to the usual mathematical practice, Zermelo, Fraenkel, von Neumann, and others had introduced a system of axioms for set theory which is now in common use under the name of ZFC (Zermelo-Fraenkel axiom system plus the axiom of choice). However, this gave no guarantee that no further paradoxes will appear. L. E. J. Brouwer proposed a much more radical reform with his intuitionism. As Weyl, Brouwer's follower at the time, wrote later, "Brouwer opened our eyes and made us see how far classical mathematics, nourished by a belief in the 'absolute' that transcends all human possibilities of realization, goes beyond such statements as can claim real meaning and truth founded on evidence" [19]. Brouwer's intuitionism accepts as valid only those mathematical arguments which provide an effective construction; in particular, intuitionism rejects the law of the excluded middle when applied to variables ranging over infmite domains. Only a limited part of classical mathematics can be reconstructed within intuitionism, and usually in a much more cumbersome way. m Hilbert, for whom mathematical knowledge came before the foundational or philosophical issues underlying it, however important they were, could not accept this "mutilation" of science. 14 But Hilbert also could not disregard Brouwer's criticism, or the paradoxes of set theory. Hilbert's response was first to replace mathematical theories (elementary number theory, mathematical analysis, and eventually set theory as axiomatized in ZFC) by corresponding formal systems, reducing mathematical arguments, via the formalized rifles of inference, to "formula games." Then it is required to prove by f i n i t a r y means (even more restricted than those of Brouwer) the consistency of these formal systems. This would clarify completely the nature of the foundations of mathematics and solve the foundational crisis once and for all without sac~i2]cing any meaningful mathematical results. There is little doubt that when Hilbert said in his K6nigsberg address that in mathematics there is no unknowable, he meant, among other things, to assert his faith in the ultimate success of this programme. Now, though Hilbert never specified precisely what finitary proofs were, they are defmitely more restricted than proofs in elementary number theory. Hence G6del's sec-

8In [7]: "Wer die Wahrheit der groBzQgigen Denkweise und Weltanschauung, die aus diesen Worten Jacobis hervorleuchtet, empfindet, der verfb.llt nicht r0ckschrittlicher und unfruchtbarer Zweifelschut; der wird nicht denen glauben . . . . 9In [7] Hilbert writes "FOr den Mathematiker" instead of "FOr uns". l~ in the original--a reference to "Ignoramus et ignorabimus" ("We are and we shall remain ignorant") of the physiologist and philosopher Emile DuBoisRaymond (1818-1896) and his followers. DuBois-Raymond was claiming that there are predetermined limits to human knowledge, hence there are certain problems that the human intellect cannot solve even in principle, among these the nature of matter and force, the origin of motion, the origin of sensation and consciousness, I~A mathematical system here means a formal system in the sense of mathematical logic, and a statement in ~ means a (well formed) formula in ~. It is assumed that has a finite number of symbols, rules of inference, and axiom schemes. Recall that (as one would expect) ~ is called (simply) consistent if there is no formula a in such that both a and the negation of a are provable in ~; it is intuitively obvious, and easy to prove rigorously, that if Tf is not consistent, then any formula in ~ is provable in 3. In his original paper Godel established the first incompleteness theorem under the additional assumption of the so-called ~-consistency of ~, but it was later shown by Rosser [14] that simple consistency of ~ is enough. 12A little more precisely, the consistency of ~ can be expressed by a formula in ~, but this formula cannot be proved in ~. ~3Though when available the intuitionistic proof often gives more information. ~4Hilbert's remark after Brouwer's talk to the Mathematics Club at G6ttingen in 1927, as reported by Reid [12, Chapter 21], is very apt: "With your methods, most of the results of modern mathematics would have to be abandoned, and to me the important thing is not to get fewer results but to get more results." This also expresses well the reasons for the eventual rejection of the intuitionism by the vast majority of the mathematical community.

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ond incompleteness theorem implies that Hilbert's programme, at least as originally conceived, is unfeasible. Furthermore, one may interpret the first incompleteness theorem as imposing certain apriori limits on mathematical knowledge. It may even be tempting to look down on Hilbert's belief in the power of the human intellect as somewhat naYve, akin to the widespread belief, in the beginning of the century, in the progress of civilization as the ultimate solution for all the problems of mankind. However, such an attitude is superficial and does not do justice to the real value of either G6del's incompleteness theorems or Hilbert's penetrating vision of the main driving force of the scientific enterprise---our quest for knowledge, and our certitude in the ultimate possibility of discovering it. The undecidable is not the unknowable, though Hilbert in his fondness for the axiomatic method and in his desire to resolve the foundational crisis may have somewhat confused the two. But in his talk [6] at Paris in 1900 he had said: Occasionally it happens that we seek the solution [of a mathematical problem] under insufficient hypotheses or in an incorrect sense, and for this reason do not succeed. The problem then arises: to show the impossibility of the solution under the given hypotheses, or in the sense contemplated. Such proofs of impossibility were effected by the ancients, for instance when they showed that the ratio of the hypotenuse to the side of an isosceles right triangle is irrational. In later mathematics, the question as to the impossibility of certain solutions plays a preeminent part, and we perceive in this way that old and difficult problems, such as the proof of the axiom of parallels, the squaring of the circle, or the solution of equations of the fifth degree by radicals have fmally found satisfactory and rigorous solutions, although in another sense than that originally intended. It is probably this important fact along with other philosophical reasons that gives rise to the conviction (which every mathematician shares, but which no one has as yet supported by a proof) that every definite mathematical problem must necessarily be susceptible of an exact settlement, either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution and therewith the necessary failure of all attempts. The "impossibility results" that Hilbert has mentioned are related to some of the most striking advances in mathematics: the discovery of irrational and transcendental numbers, noneuclidean geometries, and Galois theory. Similarly, G0del's incompleteness theorems have revolutionized mathematical logic and led to a subsequent deep and thorough study of (un)decidability, (un)solvability, and (un) computability. Rather than refuting Hilbert's "We must know--we shall

know!", G6del's theorems are a magnificent example of Hilbert's motto in action---one of the most outstanding achievements of mathematical knowledge in this century. I shall restrain myself from referring to other brilliant breakthroughs in the last 70 years, or even just the last decade. G6ders results and their subsequent development vindicate Hilbert on yet another point made in the address. While motivated originally only by our quintessential quest for knowledge, the theories of decidability, solvability, and computability became an essential component in modemday computer science. There is no better reminder to our public leaders, senators, and (alas) many university presidents 15 who accuse us, in the worst Tolstoyan way, of misspending public funds: our species would be still living in caves and/or jumping from tree to tree if not for this quest, pure and untainted. The fact that this quest eventually leads to the most utilitarian benefits in our lives may or may not be a fundamental law of nature, or a part of Divine Predestination--but it is a fact all the same. For Some Further Reading Constance Reid's biography [12] is the best source of details on Hilbert's personal and scientific life; this fascinating book should be read not only by any mathematician, but by anybody interested in the human creative process. There is a detailed account of the KOnigsberg address in Chapter 22. There is a very good general survey of Hilbert's work, including the work on axiomatics and the foundations of mathematics, in Hermann Weyl's obituary article [18]. I recommend also Weyl's paper [19] for a veteran's account of the battles of the foundational crisis. For a detailed account of G6del's theorems and their place in mathematical logic and the foundations of mathematics see, e.g., Kleene's books [10, 11]. A good short introduction may be found in [5, Chapter 39]. G6del's original paper [4] as well as an early survey by Rosser [15] still make a good reading; these two together with other foundational papers on decidability, solvability and computability have been conveniently translated and reprinted in Davis's anthology [2]. Even though Hilbert's programme cannot provide a complete and final answer for the problems of foundations of mathematics, it remains an extremely interesting research topic. See, e.g., [16, 17, 3]. REFERENCES [1] F. E. Browder (ed.), Mathematical developments arising from Hilbert problems, Proc. Symp. Pure Math., vol. 28, Amer. Math. Soc., Providence, 1976. [2] M. Davis (ed.), The undecidable, Raven Press, Hewlett, New York, 1965. [3] S. Feferman, Hilbert's program relativized: proof-theoretical and foundational reductions, J. Symb. Logic 5,3 (1988), 364-384.

151f only they would bear in mind, when turning on their desktops and laptops, that computer-inventor John von Neumann was, for all his appetite for applications, a mathematician with a strong abstract bent.

VOLUME21, NUMBER 1, 1999

[4] K. G6del, Uber formal unentscheidbare S#tze der Principia Mathematica und verwandter Systeme I, Monatsh. f. Math. u. Phys. 38 (1931), 173-198, translated as On formally undecidable propositions of Principia Mathematica and related systems I, [2, pp. 5-38]. [5] H. B. Griffiths and P. J. Hilton, A comprehensive textbook of classical mathematics, Van Nostrand Reinhold, London, 1970. [6] D. Hilbert, Mathematische Probleme, G6ttinger Nachrichten (1900), 253-297 (reprinted in Archiv f. Math. u. Phys. 3. Reihe 1 (1901), 44-63 and 213-237; and in [8, Vol. 3, pp. 290-329]), translated as Mathematical Problems, Bull. Amer. Math. Soc. 8 (1902), 437-479 (reprinted in [1, pp. 1-34]). Naturerkennen und Logik, Naturwissenschaften (1930), [7] - - , 959-963 (reprinted in [8, Vol. 3, pp. 378-387]). [8] - - , Gesammelten Abhandlungen, Springer Verlag, BerlinHeidelberg, 1935 (reprinted in 1970). [9]--, Gedenkband, Springer-Verlag, Berlin-Heidelberg, 1971, edited by K. Reidemeister. [10] S. C. Kleene, Introduction to metamathematics, Van Nostrand, New York 1952. [11] - - , Mathematical logic, Wiley, New York, 1967. [12] C. Reid, Hilbert, Springer-Verlag, New York, 1970 (new edition as part of [13]).

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[13] - - , Hilbert-Courant, Springer-Verlag, New York, 1986. [14] J. B. Rosser, Extensions of some theorems of Gddel and Church, J. Symb. Logic 1 (1936), 87-91 (reprinted in [2, pp. 231-235]). [15] - - , An informal exposition of proofs of G(3del's theorem and Church's theorem, J. Symb. Logic 4 (1939), 53-60 (reprinted in [2, pp. 223-230]). [16] W. Sieg, Hilbert's program sixty years later, J. Symb. Logic 53 (1988), 338-348. [17] S. Simpson, Partial realisations of Hilbert's programme, J. Symb. Logic 53 (1988), 349-363. [18] H. Weyl, David Hilbert and his mathematical work, Bull. Amer. Math. Soc. 50 (1944), 612-654 (reprinted in [20, Vol. 4, pp. 130-172]). [19] - - , Mathematics and Logic. A brief survey serving as a preface to a review of "The Philosophy of Bertrand Russell", Amer. Math. Monthly 53 (1946), 2-13 (reprinted in [20, Vol. 4, pp. 268-279]). Gesammelten Abhandlungen, Springer-Verlag, Berlin[20]--, Heidelberg, 1968. Department of Theoretical Mathematics Weizmann Institute of Science Rehovot 76100, Israel e-mail address: [email protected]

STEVEN GIVANT

Unifying Threads in Alfred Tarski's Work* Tarski is one of the great figures in the history of logic, along with Aristotle, 9b Frege, and Kurt Gddel. His work covers an astonishing range of subset theory, measure theory, topology, geometry, classical and universal alv

~, algebraic logic, various branches of formal logic and metamathematics,

philosophy, even economics [46]. How did a logician end up working in so many different areas? Were there interconnections in his work that led him from one field to another? What drew him to set theory in the first place, and what drew him away from it a few years later? What sparked his interest in algebra and geometry? How did he become involved in the problem of defining truth? Why did he work so intensively in algebraic logic--a field that does not attract many logicians---and what did this work have to do with his other research? Just why did he go into logic, anyway? The aim of this essay is to answer these questions, to fmd underlying unity in Tarski's work, to trace steps that may have led to some of his discoveries. Occasionally the remarks are speculative in nature, but I believe that there is substantial evidence to support them.

Beginnings Tarski was born Alfred Tajtelbaum on January 14, 1901, in Warsaw, Poland. He was an outstanding pupil in school and had very broad interests. At one point he was studying seven languages (classical Greek, Hebrew, Latin, French, German, Polish, and Russian). In Poland at that time it was obligatory for pupils to take a course in logic; it was the one course in which he did not get the equivalent of an A. 1 His favorite subject was biology.

From Biology to Mathematics Tarski entered the newly opened Warsaw University in 1918. Not unexpectedly, his declared field of study was biology. A year later he had switched to mathematics.l Why? As a beginning student, he took a seminar in set theory that was being taught by Stanistaw Lefiniewski. Le~niewski

*This essay is the outgrowth of a talk given at the International Symposium "Alfred Tarski and the Vienna Circle" in Vienna on June 12, 1997. The symposium was organized by the Institut Wiener Kreis under the direction of Eckehart K6hler, Ffiedrich Stadler, and Jan Woler~ski. I would like to thank them for making my participation possible. I am indebted to Solomon Feferman, Paul Halmos, Bjami J6nsson, Maria Moszyflska, Constance Reid, and Jan Woleriski for their help and suggestions regarding this article. Most of all, I am indebted to Tarski for sharing his memories and insights with me during our ten-year collaboration. A description of Tarski and his work that is more biographical and more personal in tone than this essay can be found in [11]. 1Personal communication of Tarski.

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mentioned to the class an open problem that Tarski managed to solve. It resulted in his first publication [40]. This initial flush of success seduced him away from biology into mathematics and logic.1 In many respects, Warsaw provided an exceptional enviroument for the study of logic and the foundations of mathematics. Two of the three professors of philosophy at the University, Legniewski and Jan Lukasiewicz, were active in mathematical logic and foundational research. The mathematician Wac]~aw Sierpiriski, w h o joined the faculty of the University in 1919, had a strong interest in the recently developed theory of sets. In fact, he had been one of the first people anywhere to give a systematic course of lectures on set theory (in Lw6w in 1909--see [32], p. 8), and he had published (in 1912) what seems to have been the first book on set theory [34]. Together with two younger mathematicians, Zygmunt Janiszewski and Stefan Mazurkiewicz, he founded in 1919 an international journal, Fundamenta Mathematicae, dedicated exclusively to articles on set theory (and its applications) and mathematical logic (see [32], p. 12, and [4]). Le~niewski and Lukasiewicz were the two other m e m b e r s of the editorial committee. (This was the first mathematical journal that specialized in a particular field of mathematics.) Thesis

In those early days there was quite a bit of interest in reducing to a minimum the primitive notions and axioms

Alfred Tarski in the

1930s.

THE MATHEMATICAL INTELLIGENCER

needed to develop various logical systems. It was k n o w n that logic can be based on a single primitive c o n n e c t i v e - for example, Sheffer stroke (logical "nor")--and universal quantification (logical "for all"), including universal quantification of propositional variables. Other connectives and quantifiers can then be introduced via definitions. Since definitions are usually formulated using the connective of logical equivalence ("if and only if") it is natural to ask whether logical equivalence can be used as the only primitive connective (together with quantification). In his thesis [41], Tarski showed (among other things) that this is indeed the case. The key was demonstrating that conjunction (logical "and") is definable (see footnote 21 in [25]). Tarski obtained his solution in an unusual way. He was in the countryside visiting his brother when his tooth became infected and had to be extracted. Of course, painkillers as we know them were not available. Sitting nervously in the country dentist's chair, he concentrated with all his might on his thesis problem. And he got the solution just as the dentist got his tooth.;

S e t Theory Most of Tarski's work immediately following his dissertation was in axiomatic set theory, which he had learned mainly from Sierpifiski and Kuratowski. In the first decades of this century set theory was at the center of a widespread debate caused by the discovery of various set-theoretical paradoxes. In 1904 Zermelo proved, with the help of a new principle

Stanisraw Le~niewski.

called the axiom of choice, that every set can be well ordered. His proof, and the validity of the axiom itself, generated heated exchanges in Europe. The majority of those who voiced an o p i n i o n - - a m o n g them Baire, Borel, Lebesgue, and R u s s e l l - - o p p o s e d the axiom; Baire, for instance, believed that everything in mathematics m u s t be reduced to the fmite [28]. Others, such as Hadamard, Hardy, and Hausdorff, defended its use. In the w a k e of this controversy, m a n y mathematicians felt it important to determine which t h e o r e m s in set theory really d e p e n d on the axiom of choice. In a 1924 p a p e r [42] Tarski p r o v e d that seven well-known p r o p o s i t i o n s in cardinal arithmetic whose proofs use the axiom of choice are actually equivalent to the axiom. In the s a m e y e a r he published the first systematic d e v e l o p m e n t of a t h e o r y o f finite sets [43], b a s e d on Zermelo's axioms, but with the negation of the a x i o m of infmity and no a x i o m of choice. Dedekind and Hausdorff had envisaged such a project, b u t Tarski was the first to realize it completely (see the introduction to [43]). The eminent French mathematician Denjoy later incorporated substantial portions of the p a p e r into one of his b o o k s [6] (pp. 615-665). Still in the s a m e year, Tarski published with Stefan B a n a c h a p a p e r [1] that quickly b e c a m e famous. Its m a i n t h e o r e m a s s e r t s that any t w o b o u n d e d sets with interior p o i n t s are equivalent by finite decomposition. F o r e x a m ple, a s p h e r e can be d e c o m p o s e d into a fmite n u m b e r of p i e c e s that can be r e a s s e m b l e d , using rigid m o t i o n s (translations, rotations, and reflections), into two spheres, e a c h of w h i c h is congruent to the original one. More dramatically, a s p h e r e the size o f a p e a can be d e c o m p o s e d into a finite n u m b e r of p i e c e s that c a n be r e a s s e m b l e d to m a k e a s p h e r e a s b i g as the sun. 2 The p r o o f m a k e s essential use o f the a x i o m of choice. The fact that the a x i o m h a s such p a r a d o x i c a l c o n s e q u e n c e s w a s s e e n by s o m e a s evidence that it should not be a c c e p t e d . B e t w e e n 1923 and 1926, Tarski d i s c o v e r e d that a numb e r o f implications in c a r d i n a l arithmetic that h a d traditionally b e e n p r o v e d using the a x i o m of choice c o u l d in fact be p r o v e d without it (at the p r i c e of using a m o r e comp l i c a t e d argument). He a n n o u n c e d these and m a n y o t h e r results in a 1926 p a p e r [21] that w a s jointly w r i t t e n with A d o l f Lindenbaum. There one also finds, for instance, the t h e o r e m that the generalized c o n t i n u u m h y p o t h e s i s implies the a x i o m of choice. A total of 146 t h e o r e m s are listed in the paper, all of t h e m w i t h o u t proof. We s e n s e that in Lindenbaum, Tarski h a d f o u n d a k i n d r e d spirit: the results c a m e so fast that they didn't have time to write t h e m up properly. Sierpifiski s p e n t s o m e of the difficult y e a r s during World War II, w h e n W a r s a w University w a s closed, w o r k i n g out the proofs o f the t h e o r e m s in this paper. 1 Tarski s e e m s to have h a d a p a s s i o n for set t h e o r y and, in particular, for cardinal a r i t h m e t i c during this period. A p r o o f of the c o n t i n u u m h y p o t h e s i s once c a m e to him in a dream, a n d the "proof" w a s so g o o d it t o o k him t w o w e e k s to find the mistake. 1 This i n c i d e n t is strong evidence t h a t

Adolf Lindenbaum.

Tarski w a n t e d to solve the c o n t i n u u m p r o b l e m and that he w o r k e d h a r d on it. In 1928 he gave a talk [45] on the historical d e v e l o p m e n t o f cardinal arithmetic, and r e p o r t e d on the r e c e n t p r o g r e s s in the field, citing the w o r k o f Sierpifiski, Kuratowski, Banach, Lindenbaum, and himself. However, he w a r n e d that one should n o t b e too optimistic. He d o u b t e d t h a t the m o s t f u n d a m e n t a l a n d difficult problems of the theory, like the c o n t i n u u m hypothesis, would ever be decided; it w a s much likelier, he said, that these p r o b l e m s w o u l d p r o v e to be i n d e p e n d e n t of the axioms that w e r e the b a s i s of present-day set theory. New Directions

By the late 1920's Tarski had b e c o m e m u c h m o r e interested in a l g e b r a a n d m e t a m a t h e m a t i c s , a n d - - s t a r t i n g at the end of the d e c a d e - - h e published a w h o l e series of fundamental p a p e r s in this direction. Here are s o m e of their titles: "On s o m e f u n d a m e n t a l c o n c e p t s of m e t a m a t h e m a t i c s " [48], " F u n d a m e n t a l c o n c e p t s of the m e t h o d o l o g y of deductive sciences" [49], "On definable sets of real numbers" [50], "The c o n c e p t o f truth in formalized languages" [54], " F o u n d a t i o n s o f the calculus of systems" [52], [53], "Some m e t h o d o l o g i c a l investigations into the defmability of concepts" [55], "On the c o n c e p t of logical consequence" [56],

2The Banach-Tarski theorem strengthened substantially an earlier theorem of Hausdorff concerning the possibility of such "paradoxical" decompositions; see [12], p. 469. A very nice description of both theorems can be found in [10].

VOLUME 21. NUMBER1. 1999 4 9

and "On the limitations of the means of expression of deductive theories" [22]. What caused this shift in interest? Three principal factors appear to have been at work. The first has already been mentioned: the most fundamental problems in set theory, involving p r o d u c t s and exponentiation, appeared to be unsolvable. Secondly, in 1926 there began at Warsaw University a seminar in mathematical logic, conducted by J:,ukasiewicz, in which Tarski played a very active role; other participants included such people as Lindenbaum, Sobocifiski, and Wajsberg (see the introduction to [25]). Thirdly, out of economic necessity, Tarski became a teacher of mathematics, first at a high school for girls, then at the Polish Pedagogical Institute (1922-1925), and finally at the Stefan Zeromski Ginmazjum (1925-1939). 1

Geometry At the high schools Tarski was responsible for teaching geometry. As early as 1924 we see references in his papers to treatises on geometry by Enriques, Amaldi, and Hilbert (e.g., in [44], p. 47). In a little book entitled "Grundlagen der Geometrie" [13] Hilbert had given an axiomatic development of Euclidean geometry that was close in spirit to Euclid, but that filled in the gaps that had become apparent in Euclid's axiomatization. The book had immediately become famous and had exerted an enormous influence on the development of the axiomatic method in mathematics (see [9], p. 86). Tarski was critical of Hilbert's axiom system from a logical perspective; since defmed notions were used in formulating these axioms, their true complexity was not evident. It was not even clear to him in what (logical) language Hilbert's completeness axiom (ensuring the uniqueness of the geometric model) was formulated. From Enriques, Tarski learned of the work of Mario Pieri, an Italian geometer who was strongly influenced by Peano. Tarski preferred Pieri's system [30], where the logical structure and the complexity of the axioms were more transparent. 1 With his typical thoroughness, he set out to create a modern, rigorous foundation of the subject himself. Between 1926 and 1927 he developed the essential ingredients of his system, and he lectured on it at Warsaw University during the 1926-27 academic year (see, for example, footnote 4 on p. 95 of [31], and footnote 34 in [67]). What was different about Tarski's approach to geometry? First of all, the axiom system was m u c h simpler than any of the axiom systems that existed up to that time. In fact the length of all of Tarski's axioms together is not much more than just one of Pieri's 24 axioms. It was the first system of Euclidean geometry that was simple enough for all axioms to be expressed in terms of the primitive notions only, without the help of defined notions. Of even greater importance, for the first time a clear distinction was made between full geometry and its elementary--that is, its firstorder--part. Instead of a second-order axiom of continuity (insuring that lines have no "gaps"), Tarski introduced

an axiom schema of continuity, a schema of first-order instances of the second-order axiom. 3 In a similar fashion Tarski developed a system of elementary algebra, what we would now call the elementary theory of real numbers. He then undertook a comparison of his systems of algebra and geometry with the full (categorical) systems. He wanted to k n o w which of the traditional algebraic and geometric notions were expressible in his systems and which of the traditional theorems were provable.

Quantifier Elimination The method that Tarski used to investigate these questions was "elimination of quantifiers"; it consisted in showing that (up to provable equivalence) all elementary formulas of the theory could be constructed from a small collection of basic formulas using only the logical connectives "and", "or", and "not"--no quantifiers. The technique had already been employed a few times by earlier authors like I~wenheim, Skolem, and Langford to analyze rather simple theories, for example the theory of monadic (unary) predicates. Tarski, characteristically, developed it into a general and systematic method (see footnote 1 on p. 97 of [31]). He saw that in cases when a theory did admit an elimination procedure, one could use it to obtain a deep structural analysis: a characterization of the defmable notions, the provable sentences, and the complete extensions of the theory (see [53], Section 5, [54], footnote 53, and [7], pp. 1-2). Between 1926 and 1928 he obtained the preliminary results that the geometry of the line (that is, one-dimensional elementary geometry) and the algebra of real addition are complete in the sense that every assertion is provable or refutable (see footnote 4 on p. 95 of [31], and footnote 4 in [60], second edition). He also obtained (using quantifier elimination) a classification of the complete theories of discrete linear orderings. Eventually, he seems to have b e c o m e so engrossed in trying to find an elimination procedure for elementary algebra and geometry that he didn't stop to investigate s o m e related problems that occurred to him. One example concerns Moj2esz Presburger. Tarski ran the exercise sessions that were associated with/Sukasiewicz's seminar. 4 It was perhaps through these sessions (or through the seminar) that he got to know Presburger, to w h o m he suggested the problem of finding an elimination procedure for the additive theory of the natural numbers. We don't know if Tarski actually "saw" a path to a solution, but he certainly sensed a positive solution. Presburger found it in May, 1928 (see footnote 1 on p. 91 of [31]). Tarski obtained his own elimination procedure around 1930 (see the bibliographic note in the English translation of [50] and footnote 6 in [67]). He once said that when working on an especially difficult problem, he sometimes "got sick" just before discovering the solution; 1 perhaps this was just such a case. At any rate, some consequences of the elimination procedure were mentioned in the paper "On defmable sets of real numbers" [50] that he published in 1931.

3Unpublished manuscript on Tarski's system of geometry, by Tarski and Givant. 4See the introduction to [8], footnote 1 on p. 93 of [31], the note to Theorem 21 in the English version of [25], and footnote 53 in [54].

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Definability The p u r p o s e of that p a p e r w a s to m a k e p r e c i s e the n o t i o n of definability. Tarski b e g a n as follows: Mathematicians, in general, do not like to deal with the n o t i o n o f definability; their attitude t o w a r d this n o t i o n is one of distrust a n d r e s e r v e . . . . I believe that I have f o u n d a general m e t h o d w h i c h allows us to c o n s t r u c t a rigorous m e t a m a t h e m a t i c a l definition of this notion. Moreover, by analyzing the definition thus o b t a i n e d it p r o v e s to be p o s s i b l e to r e p l a c e i t . . . b y a definition f o r m u l a t e d exclusively in m a t h e m a t i c a l terms. He p r o c e e d e d to give w h a t w e n o w regard as the s t a n d a r d m e t a m a t h e m a t i c a l definition of defmability. 5 A set, for example, is elementarily (first-order) definable in an a l g e b r a i c s t r u c t u r e if it consists o f all e l e m e n t s of the s t r u c t u r e that satisfy a first-order f o r m u l a with one free variable. As this e x a m p l e m a k e s clear, the k e y notion that Tarski n e e d e d w a s that of satisfaction. He wrote: The successful a c c o m p l i s h m e n t of this t a s k [defining satisfaction] raises difficulties which are g r e a t e r than w o u l d a p p e a r at first sight. H o w e v e r . . . the intuitive m e a n i n g . . , s e e m s c l e a r a n d unambiguous. After giving the m e t a m a t h e m a t i c a l definition of definability, he p r o c e e d e d to the m a t h e m a t i c a l d e f m i t i o n - - t h e definition he h o p e d w o u l d o v e r c o m e the "distrust a n d reserve" m a t h e m a t i c i a n s felt t o w a r d s the notion. First, he int r o d u c e d certain "basic" relations b e t w e e n real n u m b e r s , s u c h as the relation o f equality and the relation o f one n u m b e r being the s u m o f t w o others. Then he d e f i n e d sett h e o r e t i c o p e r a t i o n s on r e l a t i o n s that c o r r e s p o n d to the logical o p e r a t i o n s of negation, disjunction, conjunction, existential and universal quantification, and substitution. F o r e x a m p l e , the set-theoretic v e r s i o n of disjunction is union; existential quantification relative to the kth variable corres p o n d s to p r o j e c t i o n parallel to the kth axis in a suitable g e o m e t r i c space. He t h e n specified that a relation is definable ff and only if it b e l o n g s to every set that c o n t a i n s the b a s i c relations and t h a t is c l o s e d u n d e r the set-theoretic o p e r a t i o n s j u s t described.

Completeness and Decidability T o w a r d the end of the paper, Tarski c h a r a c t e r i z e d the definable r e l a t i o n s - - a n d in particular, the definable s e t s - - - o f

Maria and Alfred Tarski in Zakopane, 1930s.

real numbers. In T h e o r e m s 1 a n d 2 he r e s t r i c t e d himself to the case o f real addition, and he r e f e r r e d to his earlier elimination p r o c e d u r e . After the t h e o r e m s he r e m a r k e d that the m e t a m a t h e m a t i c a l analogue of T h e o r e m 1 p e r m i t t e d one to d e d u c e t h e c o m p l e t e n e s s of the e l e m e n t a r y t h e o r y of real addition. He then briefly d i s c u s s e d h o w to e x t e n d the results to the e l e m e n t a r y t h e o r y o f real addition and multiplication, t h a t is, to the e l e m e n t a r y t h e o r y o f algebra. Though it w a s not m e n t i o n e d explicitly, his p r o o f also yielded the c o m p l e t e n e s s of e l e m e n t a r y a l g e b r a and a decision m e t h o d for the t h e o r y - - a m e c h a n i c a l p r o c e d u r e for deciding the truth or falsity of every sentence. 6 In addition, it yielded the c o m p l e t e n e s s and d e c i d a b i l i t y of e l e m e n t a r y g e o m e t r y a n d "a constructive c o n s i s t e n c y p r o o f for the whole of e l e m e n t a r y geometry" (see [60], in p a r t i c u l a r footnote 18). 7

5According to Solomon Feferman, a metamathematical definition of the notion of definability had already been published by Weyl [70] in 1910, but this paper went unnoticed until it was mentioned in [69], p. 285. 6It is important to consult the original paper [50]. The later English translation [66] contains several remarks that were added to the text in the 1950's. In 1931 Tarski emphasized the consequences of his elimination procedures for the analysis of the definable notions, though he definitely knew that they also yielded an analysis of the complete extensions of the theories (see [67], in particular footnote 6). Starting in the mid 1930's--perhaps influenced by the growing interest in completeness questions that followed in the wake of G6del's completeness and incompleteness theorems--he began to stress completeness (see, for example, [53] and [67], in particular footnote 6). It wasn't until the 1940's, with the growing interest in decision questions, that he began to emphasize decidability (see [58], p. 88, and [60], but see also p. 10 in [67]). He himself (in a remark to the author) attributed this gradual change in emphasis to the changing fashions, to the evolving interests of logicians. 7Tarski began to publish these results (with a French publisher) in 1940 as a monograph entitled "The completeness of elementary algebra and geometry", and he even brought the corrected page proofs with him to America. However, publication was interrupted by the German invasion of France, and it was not until 1948eighteen years after its initial discovery--that the elimination procedure and its consequences were made available in printed form in [60]. The original monograph finally appeared in 1967 as [67], a photographic copy of the 1940 page proofs with some insertions and corrections.

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Hilbert had e x p r e s s e d his belief that every mathematical assertion is provable or refutable, and had emphasized the importance of proving the consistency of mathematical theories. He and others had raised the p r o b l e m of finding decision p r o c e d u r e s for various domains of mathematics. Tarski's elimination p r o c e d u r e resolved positively all of these questions for two of the oldest and historically most important branches of mathematics, algebra and geometry, at least as far as the elementary assertions are concerned. In particular, despite its simplicity, Tarski's axiom system was strong enough to prove every elementary geometrical assertion that is true (in the usual model of geometry).

Definability and Truth It s e e m s that Tarski w a s led to the p r o b l e m o f making precise the notion o f defmability by his desire to characterize the notions o f a l g e b r a and g e o m e t r y that w e r e definable in his systems, a n d m o r e generally by his n e e d to e x p r e s s precisely the results that he (and o t h e r p a r t i c i p a n t s in the Lukasiewicz s e m i n a r ) h a d o b t a i n e d for v a r i o u s theories using quantifier elimination (see the last p a g e of [53]). Def'ming definability led him to the p r o b l e m o f making precise the n o t i o n of satisfaction. It w a s this latter p r o b l e m that led him to the p r o b l e m of defining truth. In fact, his defmition o f truth w a s o b t a i n e d in the s a m e y e a r as his definition of defmability, 1929, and a b o u t a y e a r after he obt a i n e d his elimination p r o c e d u r e for the g e o m e t r y of the line a n d the a l g e b r a o f real addition (see, for example, the bibliographic n o t e to the English t r a n s l a t i o n of [54]). Tarski's teachers, Le~niewski and Kotarbifiski, h a d disc u s s e d in their l e c t u r e s the classical v i e w of truth, the various a t t e m p t s t h a t h a d b e e n m a d e to a d e q u a t e l y define it, and the p r o b l e m s a s s o c i a t e d with t h e s e a t t e m p t s (see [20], f o o t n o t e s 1 a n d 3 in [66], and f o o t n o t e 7 in [59]). Thus, Tarski was certainly a w a r e of the p r o b l e m of defining truth. The point is that he didn't j u s t decide to w o r k on the problem. He was l e d to it while working on p r o b l e m s related to his s y s t e m s o f a l g e b r a and geometry.

Sentential Logic Let's n o w turn to a n o t h e r a s p e c t of Tarski's work, Boolean algebra and a l g e b r a i c logic. In s o m e s e n s e this direction of his r e s e a r c h w a s t h e result of the third influence that I mentioned, the ongoing s e m i n a r on m a t h e m a t i c a l logic cond u c t e d by ~ u k a s i e w i c z . The p a r t i c i p a n t s v i e w e d the seminar as a kind of logico-mathematical l a b o r a t o r y w h e r e they could c o n d u c t e x p e r i m e n t s in assessing the e x p r e s s i v e and deductive p o w e r s o f various theories.1 Principal a m o n g the theories that t h e y investigated were the sentential calculus and its generalizations. They were i n t e r e s t e d in questions such as the following: What kind of a x i o m s y s t e m s do these theories have? H o w m a n y variables are n e e d e d in the axiom sets? H o w big o r small can i n d e p e n d e n t sets of a x i o m s be? H o w short can a single sentence that a x i o m a t i z e s such a t h e o r y be? Which single s e n t e n c e s axiomatizing such a t h e o r y are " i n d e c o m p o s a b l e " ? [25] Tarski eventually a s k e d himself m o r e general questions. What do we really m e a n by a deductive t h e o r y ? What does

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it m e a n for a theory to b e c o n s i s t e n t or c o m p l e t e ? H o w m a n y c o m p l e t e and c o n s i s t e n t e x t e n s i o n s can a t h e o r y have? What really is an a x i o m set, and w h a t d o e s it m e a n for it to be i n d e p e n d e n t ? H o w m a n y i n d e p e n d e n t a x i o m sets can a t h e o r y have? Characteristically, he p l a c e d the a n s w e r s that he found in a v e r y general algebraic framework. In a series of articles p u b l i s h e d b e t w e e n 1930 a n d 1935 he d e v e l o p e d a kind of calculus of deductive s y s t e m s (in a d d i t i o n to [25] see, for example, [48], [49], [52], a n d [53]) t h a t eventually led him to Boolean algebra.

Boolean Algebra In the 1920's Boolean a l g e b r a h a d n o t yet clearly e m e r g e d as an algebraic discipline distinct from the sentential calculus. The s e p a r a t i o n of the two, and the recognition o f the c o n n e c t i o n s b e t w e e n them, b e g a n with the w o r k of Huntington, Bernstein, Tarski, and Stone. One of Tarski's first results in this direction w a s a res p o n s e to p r o b l e m s of B a n a c h and Hausdorff c o n c e r n i n g the e x i s t e n c e of non-trivial m e a s u r e s on the c o l l e c t i o n of s u b s e t s o f an infinite set. In 1930 he a n n o u n c e d [47] that, on the collection of s u b s e t s of any infinite set X, t h e r e is a non-trivial, two-valued, finitely additive m e a s u r e w h i c h assigns t h e value 0 to every finite subset. Equivalently, t h e r e is a collection of s u b s e t s of X that forms a non-principal, m a x i m a l ideal in the B o o l e a n a l g e b r a of s u b s e t s o f X. An a b s t r a c t form of this result p l a y s a key role in the p r o o f of the r e p r e s e n t a t i o n t h e o r e m for Boolean algebras, t h e theo r e m w h i c h asserts that e v e r y a b s t r a c t Boolean a l g e b r a is i s o m o r p h i c to a c o n c r e t e a l g e b r a of sets. In a 1935 p a p e r [52] on the calculus of systems, Tarski p o i n t e d out that the f o r m u l a s o f the sentential calculus, w h e n f a c t o r e d by the r e l a t i o n o f p r o v a b l e equivalence, form a Boolean algebra u n d e r the logical o p e r a t i o n s of disjunction, conjunction, a n d n e g a t i o n - - w h a t t o d a y is called t h e B o o l e a n algebra of f o r m u l a s o r the Lindenbaum-Tarski algebra. This observation h e l p e d to clarify the e x a c t relat i o n s h i p b e t w e e n sentential logic a n d Boolean algebra. His p a p e r [51] on the f o u n d a t i o n s of Boolean a l g e b r a also a p p e a r e d in 1935. There he s t u d i e d several equivalent f o r m u l a t i o n s of the n o t i o n s o f a Boolean algebra, a comp l e t e Boolean algebra (a B o o l e a n algebra in which e a c h infinite set of elements also h a s a least u p p e r b o u n d a n d a g r e a t e s t l o w e r bound), a n d a complete, atomic B o o l e a n alg e b r a (a c o m p l e t e Boolean a l g e b r a in which each non-zero e l e m e n t is above a minimal non-zero element). At the e n d o f the p a p e r he p o i n t e d o u t that a complete, a t o m i c B o o l e a n algebra is i s o m o r p h i c to an algebra o f s e t s - n a m e l y the algebra of s u b s e t s of its set of atoms. In a 1936 note [57] he o b s e r v e d that the ideals of a B o o l e a n algebra are closely r e l a t e d to the deductive syst e m s in his calculus of systems. On the basis of this analogy he d e v e l o p e d an a b s t r a c t calculus of ideals of B o o l e a n a l g e b r a s o f sets. One s e n s e s that Tarski, in constructing this calculus o f ideals and in proving a r e p r e s e n t a t i o n t h e o r e m for complete, atomic Boolean algebras, was close to formulating a n d proving a general r e p r e s e n t a t i o n t h e o r e m for B o o l e a n

algebras. Stone p u b l i s h e d e x a c t l y such a t h e o r e m in 1936 a n d a stronger, t o p o l o g i c a l v e r s i o n in 1937. s It m u s t have c o m e as something of a s h o c k a n d d i s a p p o i n t m e n t to Tarski to learn that he, himself, h a d been so close.

Toward Algebraic Logic Yet d i s a p p o i n t m e n t s can act as stimuli. S o m e t i m e b e t w e e n 1937 and 1940 the s e e d s of an i d e a o c c u r r e d to Tarski: one could go b e y o n d Boolean algebra, one could d e v e l o p an algebraic version of all of m a t h e m a t i c s . Such a t h e o r y should have not only Boolean operations, but also o p e r a t i o n s corr e s p o n d i n g to existential a n d universal quantification, substitution, and equality. In his first a t t e m p t in this direction, Tarski turned to one of the oldest theories of logic, the calculus of relations, a (set-theoretic) arithmetic of b i n a r y relations that had b e e n d e v e l o p e d in the s e c o n d half o f the n i n e t e e n t h century by De Morgan [5], Peirce [29], and S c h r S d e r [33]. SchrSder's w o r k w a s well k n o w n in the earlier p a r t of this century. Certainly Tarski's thesis advisor, Le~niewsld, w a s a w a r e of it, a n d w e find r e f e r e n c e s to it in s o m e of Tarski's p r e - w a r p a p e r s (for example, footnote 69 in [54]). LSwenheim's f a m o u s 1915 p a p e r [23], "On possibilities in the calculus of relatives" (with which Tarski w a s acquainted), is written in the language of the c a l c u l u s of relations. Tarski m e t LSwenheim in Berlin s o m e t i m e in the p e r i o d 1938-39 and learned a b o u t the latter's u n p u b l i s h e d manuscripts. He urged LOwenheim to give him s o m e of t h e m so t h a t he could try to get t h e m published. Eventually L S w e n h e i m gave him three. 1 In all likelihood it w a s one of t h e s e m a n u s c r i p t s (delivered to the editors o f the J o u r n a l of Symbolic Logic on S e p t e m b e r 4, 1939--just a b o u t the time t h a t Tarski arrived in A m e r i c a - - a n d a p p e a r i n g as [24] in 1940) that p u s h e d Tarski in the direction of r e l a t i o n algebras. In this p a p e r L S w e n h e i m l a m e n t e d that p e o p l e h a d t u r n e d a w a y from the Peirce-SchrSder calculus, using ins t e a d the Peano-Russell symbolism. He e m p h a s i z e d the b e a u t y o f the f o r m e r s y s t e m in which precisely the mathe m a t i c a l l y meaningful a n d f ~ i t f u l was realized, a n d said that if he, himself, h a d w o r k e d with the Russell symbolism, he w o u l d not have m a d e m a n y of his discoveries. 9 It w a s generally believed that the S c h r S d e r calculus l a c k e d sufficient notions to form a f r a m e w o r k for m a t h e m a t i c s , he continued. He disagreed with this belief; in his opinion, all of m a t h e m a t i c s could b e "Schr(iderized", and in doing so

Leopold LSwenheim.

the k n o w n set-theoretical p a r a d o x e s c o u l d be avoided. The p u r p o s e o f his article was to give s o m e indication of h o w he thought this "Schr0derization" might be accomplished. He wrote: Every m a t h e m a t i c a l t h e o r e m is equivalent to an assertion in S c h r 6 d e r ' s calculus of relatives . . . . Every given m a t h e m a t i c a l p r o o f of such a t h e o r e m can be transf o r m e d into a p u r e l y logico-computational p r o o f [in the calculus of relatives] of the [corresponding] assertion. Even C a n t o r ' s set t h e o r y can be SchrSderized. 1~

Relation Algebras Around 1939 or 1940 Tarski b e g a n a p r o j e c t of formalizing m a t h e m a t i c s - - a n d in particular set t h e o r y - - i n the calculus of relations. His a p p r o a c h w a s radically different from LSwenheim's. A p p a r e n t l y he w a s dissatisfied with the form of the calculus o f relations as it o c c u r r e d in the w o r k s o f

8The representation theorem was first published in [38] and the stronger topological version in [39]. Apparently both theorems were obtained by Stone in 1932; they were announced in [35] (which was received by the editors on January 10, 1933) and discussed at greater length in [36]. The reason for the delay in publication is explained in [37]: just as he was completing a detailed exposition of the "independent theory of Boolean algebras," Stone discovered that Boolean algebras are definitionally equivalent to certain kinds of rings (now called Boolean rings), and the discovery compelled a radical revision of the theory. A closely related result--a representation theorem for distributive lattices--was published by Garrett Birkhoff in [2]. 9"lch bedaure es aus mehr als einem Grunde schwer, dab man yon dem eleganten Peirce-Schr6derschen Kalkul abgewichen ist und die Peano-Russellschen Zeichen benutzt, in denen . . . die sonst selbstverst~indliche Harmonie und SchOnheit der Mathematik preisgegeben wurde . . . . w~ihrend beim Peirce-Schr0derschen Kalkul auf Sch6nheit besonderer Weft gelegt und mit wunderbarer Instinktsicherheit gerade das mathematisch Bedeutsame und Fruchtbare getroffen wurde . . . . Mit Russells Zeichen h&tte ich manches nicht entdeckt, was ich gefunden habe...." lo"Jedem mathematischen Satz ist eine Aussage des Schr6derschen Relativkalkuls ~iquivalent .... Jeder vorliegende mathematische Beweis eines solchen Satzes I~t6t sich umwandeln in einen rein Iogisch-rechnerischen Beweis jener Aussage. Auch die Cantorsche Mengenlehre I~il3t sich verschr6dern."

VOLUME21, NUMBER 1, 1999 5 3

elegance. We obtain the calculus of relations in a very r o u n d a b o u t way, and in proving theorems of this calculus w e are forced to m a k e use of concepts and s t a t e m e n t s w h i c h are outside the calculus. To u n d e r s t a n d w h a t Tarski meant, c o n s i d e r the s i m p l e r exa m p l e that he cites: the calculus of classes. It consists of quantifier-free assertions ("laws") a b o u t sets that c a n be f o r m u l a t e d using variables that range over subsets of a universe o f discourse U and s y m b o l s that denote the operations of union, intersection, and complementation; individual variables ranging over e l e m e n t s o f U are not allowed. An e x a m p l e of such an a s s e r t i o n is XC

Y~-YC-X,

(1)

w h e r e - d e n o t e s the o p e r a t i o n of c o m p l e m e n t a t i o n . A s s e r t i o n (1) is equivalent to the following assertion in the e l e m e n t a r y (first-order) t h e o r y of classes (with unquantifled v a r i a b l e s that range o v e r sets and individual v a r i a b l e s that m a y b e quantified): Vx(x

Bjarni Jbnsson, circa 1946.

S c h r 6 d e r and L6wenheim, and this p r o m p t e d him to develop his o w n a p p r o a c h to the subject (just as he h a d done in the case of geometry). In [58] he wrote: a large p a r t o f the theory of relations can be p r e s e n t e d as a calculus w h i c h is formally much like the calculus of c l a s s e s . . , b u t which greatly e x c e e d s it in richness of expression and is therefore incomparably m o r e interesting from the deductive point of view . . . . The above w a y of constructing the elementary theory of relations [within the f r a m e w o r k of first-order logic, using individual variables that range over elements of the universe, but admitting also unquantified variables that range over binary relations on the universe] will p r o b a b l y s e e m quite natural to anyone w h o is familiar with m o d e r n mathematical logic9 If, however, we are interested n o t in the whole theory of relations but merely in the calculus of relations [that is, in t h o s e assertions that contain no quantifiers or individual variables], we must admit that this m e t h o d has certain defects from the point of view of simplicity and 9

THE MATHEMATICAL INTELLIGENCER

E X ----->x E Y ) ~

Vx(x

~ ~ Y ----) x E - X ) .

(2)

One a p p r o a c h to proving (1) is to p r o v e (2) using a s i m p l e set-theoretic argument a n d the defmition of c o m p l e m e n tation. However, such a p r o o f u s e s notions a n d s y m b o l s that a r e outside of the calculus of classes, for e x a m p l e the individual variable x. A n o t h e r a p p r o a c h is to specify a finite s e t o f a x i o m s for the c a l c u l u s of classes (consisting o f f o r m u l a s in w h i c h no individual variables occur) a n d t h e n to derive laws like (1) from t h o s e a x i o m s by m e a n s of app r o p r i a t e rules of inference (such as the rule of r e p l a c i n g equals b y equals). This is p r e c i s e l y w h a t is done in t h e theo r y o f Boolean algebras; t h e variables no longer d e n o t e sets, b u t r a t h e r a b s t r a c t o b j e c t s in s o m e universe, a n d the o p e r a t i o n s y m b o l s no l o n g e r d e n o t e union, intersection, a n d c o m p l e m e n t a t i o n , b u t r a t h e r a b s t r a c t o p e r a t i o n s on the universe that o b e y the l a w s f o r m u l a t e d in the axioms. F r o m this perspective, the calculus o f classes c o i n c i d e s with the (quantifier-free) t h e o r y of Boolean algebras. The situation for the calculus of relations is c o m p l e t e l y analogous, e x c e p t that v a r i a b l e s ranging over b i n a r y relations on the set U r e p l a c e the variables ranging o v e r subsets o f U, and the set of o p e r a t i o n s on these relations is a u g m e n t e d to include, for e x a m p l e , the o p e r a t i o n s of relational c o m p o s i t i o n and inverse. A large n u m b e r of the laws in the w o r k s of Peirce a n d S c h r 6 d e r were f o r m u l a t e d in the calculus of relations, t h a t is, w i t h o u t using individual variables. However, in deriving the laws those a u t h o r s u s e d set-theoretical arguments that involved individual v a r i a b l e s and the (set-theoretical) definitions of the operations. Tarski s a w his first t a s k to b e the creation of an a b s t r a c t t h e o r y of b i n a r y relations, j u s t as B o o l e a n algebra is an abs t r a c t t h e o r y of classes. In particular, he had to s e l e c t a finite s e t o f a x i o m s from w h i c h all laws in the calculus of r e l a t i o n s could be derived b y m e a n s o f a p p r o p r i a t e rules of inference without using set-theoretical a r g u m e n t s and w i t h o u t referring to individuals in the domain of discourse. In August, 1939, Tarski left P o l a n d to attend a c o n g r e s s

on the Unity of Science at Harvard. While he was en route, the Germans invaded Poland. He was stranded in America, yet despite tremendous financial and emotional difficulties he kept working. By late 1940 he had constructed an abstract algebraic theory of binary relations [58]. It had a finite set of axioms that could be written as equations. He seems to have gone through SchrOder's book and actually derived, on the basis of his axioms, all of the hundreds of relational laws that occur there. He asked whether every valid law (that is, every formula of the calculus of relations that is true in all algebras of binary relations) could be derived from his axioms; in other words, he asked whether his axiom set was complete. He also posed the analogue of the representation problem for Boolean algebras: is every abstract relation algebra--that is, every abstract model of his axioms--isomorphic to a concrete algebra of binary relations? In the same paper he announced that there is no decision procedure for the calculus of relations--no algorithm for deciding whether or not any given assertion is true. His proof (which he said would be given in another paper) simultaneously showed precisely how set theory--and hence mathematics--could be formalized in the calculus of relations. 11 By 1942 or 1943 he had succeeded in proving that set theory could even be formalized in his finitely axiomatized version of this calculus--what he later called the (equational) theory of relation algebras (see the introduction to [3], and also [68], in particular the footnote on p. 163). In describing his results, he later used phrases that are close to those of LOwenheim cited above (see [62] and [68], Section 2.4). During the early 1940's Tarski seems to have made little progress on the questions of completeness and representability that he had posed in his paper. He attacked them again around 1946, after teaching a course on relation algebras at the University of California at Berkeley (where he had fmally found a permanent position). From his perspective, the key point in Stone's representation theorem was that every Boolean algebra can be extended to a complete, atomic Boolean algebra with certain "topological" properties. Characteristically, Tarski gave these topological properties an elegant algebraic formulation.12 He then proved that every abstract relation algebra can be extended to a complete, atomic relation algebra with the same topological properties. This gave him a representation of abstract relation algebras as algebras of sets--sets of atoms---but not a representation as algebras of binary relations. He communicated his results to his former student, Bjarni J6nsson, who showed that the proofs go through in a much more general context, using only the additivity of the operations. 13 In this fashion the theory of Boolean algebras with operators was born (see [18] and

[19]). Today it plays a major role in algebraic logic, intensional logic, and theoretical computer science. The P r e s e r v a t i o n T h e o r e m In 1950 Roger Lyndon [26] proved that both of Tarski's questions had negative answers: there exist relation algebras that are not representable and there exist valid laws that are not derivable from Tarski's axioms. In fact, Lyndon claimed to prove more, namely that the class of representable algebras is not even axiomatizable by a set of quantifier-free formulas. To prove this, he constructed a relation algebra with the alleged property that it is not representable, although all of its finitely generated subalgebras are. He then raised the question whether the class of representable relation algebras is axiomatizable by a set of first-order formulas. In 1952 Tarski [61] announced a negative answer; his proof (which is outlined in the abstract) used Lyndon's "theorem" and a compactness argument. (Compactness in logic says that a set of first-order formulas has a model just in case every finite subset has a model.) Not long after publishing his abstract, Tarski realized that--in contradiction to Lyndon's theorem and his own announced coronary--the class of representable relation algebras is axiomatizable by a set of equations. 14 What led to this discovery? Already in 1949 or 1950 he had proved a (now famous) preservation theorem: if a class of algebras can be axiomatized by a set of first-order formulas, then the class of its subalgebras can be axiomatized by a set of quantifier-free formulas (see Theorems 1.6 and 1.13 in [64]). Even earlier, he and J6nsson had shown (see Theorems 4.29 and 4.31 in [19] and Result 3 in their 1948 abstract) that a relation algebra is representable if and only if it is a subalgebra of an atomic algebra with functional atoms (atoms satisfying an equation that characterizes binary relations which are functions). The class of atomic relation algebras with functional atoms is clearly first-order axiomatizable, so the class of its subalgebras--the class of all representable relation algebras--is axiomatizable by a set of quantifier-free formulas. A general property of relation algebras (proved in [58]) implies that each quantifierfree formula can be replaced by an equation. This surprising application of the preservation theorem underscored its importance and its potential for applications. The theorem (in various formulations) and its applications were published in one of Tarski's most famous series of papers, "Contributions to the theory of models" ([63], [64], and [65]) and stimulated extensive investigations into the possibility of giving logical characterizations of algebraic properties (for example, the property of a class of structures being closed under certain algebraic constructions such as homomorphisms) and algebraic char-

11Unpublished manuscript of Tarski. 12These topological properties of the extension algebra follow from Stone duality, the stronger representation theorem in [39]. Tarski formulated them algebraically as follows: (compactness) if the unit element is the sum of a set of elements of the original algebra, then it is the sum of a finite subset; (separability of atoms) for any two atoms there is an element of the original algebra that includes one and is disjoint from the other (see Definition 1.19 in [18]). ~3Oral communication of J6nsson; see also the beginning of Section 1.2 in [17]. ~4The mistake in Lyndon's proof was tracked down by Dana Scott; see the first paragraph in [27].

VOLUME21, NUMBER 1, 1999 5 5

acterizations of logical notions (for example, the notion of elementary equivalence--that is, of two structures having the same set of valid sentences). Algebraic logic--relation algebras--had led back to model theory. Cylindric Algebras Although the theory of relation algebras provides an algebraic framework for (formalizing) set theory, Tarski realized that it does not provide an adequate algebraic framework for fLrst-order logic. In the late 1940's, he began looking for a different approach. He returned to his mathematical description of the logical operations---that is, the set-theoretical and geometrical description that he had developed in 1929 in order to give a mathematical definition of definability. Abstracting the essential properties of these set-theoretical operations, he defmed an extension of the theory of Boolean algebras in which there are also abstract projection operations or "cylindrifications", as he called them--operations corresponding to existential quantifications. Together with Louise Chin and Frederick Thompson, he created the theory of cylindric algebras, a theory that he and his collaborators extensively developed (see [14], [15], and [16]). Summary Tarski's interest in quantifier elimination, and in particular in applying this method to study his systems of algebra and geometry, was influenced by his teaching of these subjects in high school. This study led him to the problem of making precise the notion of definability, which led him to the problem of defming satisfaction and ultimately to the problem of defining truth. The applications of the method of quantifier elimination by Tarski and others, and the deftnitions of definability, satisfaction, and truth, led eventually to the creation of model theory (see Section 5 and the appendix in [53], and footnote 1 in [63]). Tarski's work on Boolean algebras was an outgrowth of the investigations into sentential logic and its generalizations that he and other members of ~Lukasiewicz's logic seminar had carried out. And Boolean algebra led Tarski to algebraic logic. We have seen that algebraic logic owes a big debt to Tarski's work in model theory, in particular the theory of definitions. But it paid the debt by providing an important application of Tarski's preservation t h e o r e m - - a theorem that had a substantial influence on the further development of logic. REFERENCES [1] Stefan Banach and Alfred Tarski, "Sur la decomposition des ensembles de points en parties respectivement congruentes," Fundamenta Mathematicae 6, 1924, 244-277. [2] Garrett Birkhoff, "On the combination of subalgebras," Proceedings of the Cambridge Philosophical Society 29, 1933, 441-464. [3] Louise Chin and Alfred Tarski, "Distributive and modular laws in the arithmetic of relation algebras", University of California Publications in Mathematics, new series, vol. 1, 1951, 341-384.

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[4] Krzysztof Ciesielski and Zdzis~aw Pogoda, "The beginning of Polish topology," Mathematical Intelligencer 18, 1996, no. 3, 3239. [5] August De Morgan, "On the syllogism: IV, and on the logic of relations," Transactions of the Cambridge Philosophical Society 10, 1864, 331-358. [6] Arnaud Denjoy, L'enum~ration transfinie, Paris: Gauthier-Villars, 1954. [7] John Donor, Andrzej Mostowski, and Alfred Tarski, "The elementaw theory of well-ordering--A metamathematical study," in: Angus Maclntyre, Leszek Pacholski, and Jeff Paris (Eds.), Logic Colloquium 77, Amsterdam: North-Holland Publishing Company, 1978, pp. 1-54. [8] Editorial remark, Fundamenta Mathematicae 23, 1934, 161. [9] Howard Eves and Carroll Newsom, An introduction to the foundations and fundamental concepts of mathematics, New York: Holt, Rinehart, and Winston, 1958. [10] Robert French, "The Banach-Tarski paradox," Mathematical Intelligencer 10, 1988, no. 4, 21-28. [11] Steven Givant, "A portrait of Alfred Tarski," Mathematical Intelligencer 13, 1991, no. 3, 16-32. [12] Felix Hausdorff, Grundz(Jgeder Mengenlehre, Leipzig: Veit, 1914. (Reprinted: Bronx NY: Chelsea Publishing Company, 1949.) [13] David Hilbert, Grundlagen der Geometrie, Leipzig: Teubner, 1899. [14] Leon Henkin, J. Donald Monk, and Alfred Tarski, Cylindric algebras. Part I, Amsterdam: North-Holland Publishing Company, 1971.

[15] Leon Henkin, J. Donald Monk, Alfred Tarski, Hajnal Andreka, and Istvan Nemeti, Cylindric set algebras, Berlin: Springer Verlag, 1981. [16] Leon Henkin, J. Donald Monk, and Alfred Tarski, Cylindric algebras, Park II, Amsterdam: North-Holland Publishing Company, 1985. [17] Bjarni JOnsson, "A survey of Boolean algebras with operators", in: Ivo Rosenberg and Gert Sabidussi (Eds.), Algebras and orders, NATO ASI Series C: Mathematical and Physical Sciences, vol. 389, Dortrecht: Kluwer Academic Publishers, 1993, pp. 239-286 [18] Bjarni Jonsson and Alfred Tarski, "Boolean algebras with operators. Part I," American Journal of Mathematics 73, 1951,891-939. [19] Bjarni Jonsson and Alfred Tarski, "Boolean algebras with operators. Part I1,"American Journal of Mathematics 74, 1952, 127-162. (The abstract appeared as "Representation problems for relation algebras", Bulletin of the American Mathematical Society 54, 1948, 80.) [20] Tadeusz Kotarbir~ski, Elementy teerji posnania, Iogiki formalnej i metodologji nauk, Lw6w: Wydawnictwo Zak4"aduNarodowego im Ossolir~skich, 1929. [21] Adolf Lindenbaum and Alfred Tarski, "Communication sur les recherches de la theorie des ensembles," Sprawozdania z Pesiedzeh Towarzystwa Naukowego Warszawskiego, Wydzia# III Nauk Matematycznych i Pczyrodniczych (= Comptes Rendus des Seances de la Societ6 des Sciences et des Lettres de Varsovie, Classe 111)19, 1926, 299-330. [22] Adolf Lindenbaum and Alfred Tarski, "0ber Beschr&nktheit der Ausdrucksmittel deduktiver Theorien," Ergebnisse eines Mathematischen Kolloquiums 7, 1936, 15-22. (An English translation appeared as Article XIII in [66].) [23] Leopold L6wenheim, "0ber die M6glichkeiten im RelativkalkQl," Mathematische Annalen 76, 1915, 447-470. [24] Leopold L6wenheim, "Einkleidung der Mathematik in Schr6derschen Relativkalkul," Journal of Symbolic Logic 5, 1940, 1-15. [25] Jan Z_ukasiewicz and Alfred Tarski, "Untersuchungen Qber den Aussagenkalk~Jr', Sprawozdania z Posiedzeh Towarzystwa Naukowego Warszawskiego, Wydzia#"III Nauk Matematycznofizycznych (= Comptes Rendus des Seances de la Societe des Sciences et des Lettres de Varsovie, Classe III) 23, 1930, 30-50. (An English translation appeared as Article IV in [66].) [26] Roger Lyndon, "The representation of relational algebras," Annals of Mathematics 51, 1950, 707-729. [27] Roger Lyndon, "The representation of relation algebras, I1,"Annals of Mathematics 63, 1956, 294-307. [28] Gregory Moore, Zermelo's axiom of choice: its origins, development, and influence, Berlin: Springer Verlag, 1982. [29] Charles Saunders Peirce, "Note B: the logic of relatives," in: Charles Saunders Peirce (Ed.), Studies in logic by the members of the Johns Hopkins University, Boston: Little, Brown, and Company, 1883. [30] Marie Pieri, "La geometria elementare istituita sulle nozioni 'punto' 6 'sfera' ," Memorie di Matematica e di Fisica della Societa Italiana delle Scienze 15, 1908, 345-450. [31] Moj~'esz Presburger, "0ber die Vollst&ndigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt," Sprawozdania z Pierwszego Kongresu Matematyk6w Kraj6w Sfowianskich (= ComptesRendus du Premier Congres des Mathematiciens des Pays Slaves), Warsaw, 1930, pp. 92-101.

[32] Andrzej Schinzel, Waol'aw Sierpihski, Warsaw: Iskry, 1976. [33] Ernst Schr6der, Vorlesungen (~ber die Algebra der Logik, vol. 3, Algebra und Logik der Relative, Band I, Leipzig: Teubner, 1895. (Reprinted: Bronx NY: Chelsea Publishing Company, 1966.) [34] Waot'aw Sierpi6ski, Zarys teoryi mnogo~ci, Warsaw: Biblioteka Matematyczno-Fizyczna, 1912. [35] Marshall Stone, "On the structure of Boolean algebras," Bulletin of the American Mathematical Society 39, 1933, 200. [36] Marshall Stone, "Boolean algebras and their application to topology," Proceedings of the National Academy of Sciences 20, 1934, 197-202. [37] Marshall Stone, "Subsumption of the theory of Boolean algebras under the theory of Boolean rings," Proceedings of the National Academy of Sciences 21, 1935, 103-105. [38] Marshall Stone, "The theory of representations for Boolean algebras," Transactions of the American Mathematical Society 40, 1936, 37-111. [39] Marshall Stone, "Applications of the theory of Boolean rings to general topology," Transactions of the American Mathematical Society 41, 1937, 375-481. [40] Alfred Tarski, "Przyczynek do aksjomatyki zbioru dobrze uporz~dkowanego," Przegl~d Filozoficzny (= Revue Philosophique) 24, 1921, 85-94. [41] Alfred Tarski, "O wyrazie pierwontnym Iogistyki," Przegl~d Filozoficzny (= Revue Philosophique) 26, 1923, 68-89. (A French translation appeared in the two articles "Sur le terme primitif de la Iogistique," Fundamenta Mathematicae 4, 1923, 196-200, and "Sur les truth-functions au sense de MM. Russell et Whitehead," Fundamenta Mathematicae 5, 1924, 59-74. An English translation appeared as Article I in [66].) [42] Alfred Tarski, "Sur quelques theoremes qui equivalent & I'axiome du choix," Fundamenta Mathematicae 2, 1924, 47-60. [43] Alfred Tarski, "Sur les ensembles finis," Fundamenta Mathematicae 6, 1924, 45-95. [44] Alfred Tarski, "Q r6wnowa2no~ci wielok~t6w," Przegl~d Matematyczno-fizyczny 2, 1924, 47-60. [45] Alfred Tarski, "Geschichtliche Entwicklung und gegenw&rtiger Zustand der Gleichm&chtigkeitstheorie und der Kardinalzahlarithmetik," in: KsiC~la Pamiatkowa Pierwszego Polskiego Zjazdu Matematycznego, Krak6w, 1929, pp. 48-54. [46] Alfred Tarski, "Na marginesie 'Rozporz~dzenia Prezydenta Rzeczypospolitej o ubezpieczeniu procownikow umys4"owych z dnia 24 listopada 1927 r'," Ekonomista 29, 1929, 115-119. [47] Alfred Tarski, "Une contribution b, la theorie de la mesure," Fundamenta Mathematicae 15, 1930, 42-50. [48] Alfred Tarski, "0ber einige fundamentalen Begriffe der Metamathematik," Sprawozdania z Posiedzen Towaczystwa Naukowego Warszawskiego, Wydzia# III Nauk Matematycznofizyznych (= Comptes Rendus des Seances de la Societe des Sciences et des Lettres de Varsovie, Classe III) 23, 1930, 22-29. (An English translation appeared as Article Ill in [66].) [49] Alfred Tarski, "Fundamentale Begriffe der Methodologie der deduktiven Wissenschaften. I," Monatshefte fElr Mathematik und Physik 37, 1930, 361-404. (An English translation appeared as Article V in [66].) [50] Alfred Tarski, "Sur les ensembles definissables de nombres reels. I," Fundamenta Mathematicae 17, 1931, 210--239. (An English translation appeared as Article Vl in [66].)

VOLUME 21, NUMBER 1, 1999 5 7

[56]

[57]

[58] [59]

[60]

[61] [62] [63]

[64]

[51] Alfred Tarski, "Zur Grundlegung der Boole'schen Algebra. I," Fundamenta Mathematicae 24, 1935, 177-198. (An English translation appeared as Article Xl in [66].) [52] Alfred Tarski, "GrundzQge des SystemenkalkQIs. Erster Tell," Fundamenta Mathematicae 25, 1935, 503-526. (An English translation appeared as Article XII in [66].) [53] Alfred Tarski, "GrundzQge des SystemenkalkQIs. Zweiter Tell," Fundamenta Mathematicae 26, 1936, 283-302. (An English translation appeared as Article Xll in [66].) [54] Alfred Tarski, "Der Wahrheitsbegriff in den formalisierten Sprachen," Studia Philosophica 1, 1935, 261-405. (This is a German translation of "Pojgcie prawdy w jgzykach nauk dedukcyjnych, Prace Towarzystwa Naukewego Warszawskiego, Wydzia#" III Nauk Matematyczno-fizycznych (= Travaux de la Soci6t6 des Sciences et des Lettres de Varsovie, Classe III Sciences Math~matiques et Physiques), no. 34, Warsaw, 1933. An English translation appeared as Article VIII in [66].) [55] Alfred Tarski, "Einige methodologische Untersuchungen Qber die

58

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[65]

[66] [67] [68]

[69]

[70]

[71]

Definierbarkeit der Begriffe," Erkenntnis 5, 1935, 80-100. (This is a German translation of "Z badAn metodologicznych nad definjowalnosci~ termin6w," Przegl~d Filozoficzny (= Revue Philosophique) 37, 1934, 438-460. An English translation appeared as Article X in [66].) Alfred Tarski, "0ber den Begriff der Iogischen Folgerung," in: Actes du Congres Internationalde Philosophie Scientifique, vol. 7, 1936, pp. 1-11. (This is a German translation of "O pojgciu wynikania Iogicznego," Przegl~dFilozoficzny(= Revue Philosophique)39, 1936, 58-68. An English translation appeared as Article XVI in [66].) Alfred Tarski, "ldeale in den Mengenk6rpern," Rocznik Polskiego Towarzystwa Matematycznego (= Annales de la Soci~t6 Polonaise de Math#matique) 15, 1936, 186-189. Alfred Tarski, "On the calculus of relations," Journal of Symbolic Logic 6, 1941, 73-89. Alfred Tarski, "The semantic conception of truth and the foundations of semantics," Philosophy and Phenomenological Research 4, 1944, 341-376. Alfred Tarski, A decision method for elementary algebra and geom- . etry (prepared for publication by J. C. C. McKinsey), Santa Monica CA: RAND Corporation, 1948. (Revised edition: Berkeley: University of California Press, 1951 .) Alfred Tarski, "On representable relation algebras," Bulletin of the American Mathematical Society 58, 1952, 172. Alfred Tarski, "A formalization of set theory without variables," Journal of Symbolic Logic 18, 1953, 189. Alfred Tarski, "Contributions to the theory of models. I," Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, Series A, 57 (= Indagationes Mathematicae 16), 1954, 572-581. Alfred Tarski, "Contributions to the theory of models. I1,"Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, Series A, 57 (= Indagationes Mathematicae 16), 1954, 582-588. Alfred Tarski, "Contributions to the theory of models. II1,"Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, Series A, 58 (= Indagationes Mathematicae 17), 1955, 56-64. Alfred Tarski, Logic, semantics, metamathematics, Oxford: Clarendon Press, 1956. Alfred Tarski, The completeness of elementary algebra and geometry, Paris: Institut Blaise Pascal, 1967. Alfred Tarski and Steven Givant, A formalization of set theory without variables, Providence, RI: American Mathematical Society, 1987. Jean van Heijenoort, From Frege to Gddel. A source book in mathematical logic, 1879-1931, Cambridge, MA: Harvard University Press, 1967. Hermann Weyl, "0ber die Definitionen der mathematischen Grundbegriffe," Mathematisch-naturwissenschaftliche BlOtter 7, 1910, 93-95, 109-113. Ernst Zermelo, "Beweis, dab jede Menge wohlgeordnet werden kann," Mathematische Annalen 59, 1904, 514-516.

m,',.=-~.~--l~._~[.--

Jeremy

Gray,

The Classification of Algebraic Surfaces by Castelnuovo and Enriques

Column Editor's address: Faculty of Mathematics, The Open University, Milton Keynes, MK7 6AA, England

Editor

O

I

one o f the lasting c o n t r i b u t i o n s to m a t h e m a t i c s m a d e b y the Italian g e o m e t e r s of a century ago w a s the classification of algebraic surfaces. This article gives a brief a c c o u n t of the p a t h t a k e n by Enriques a n d Castelnuovo that culminated in the Enriques classification, as it is known. The classification w a s e x t e n d e d to all c o m p l e x 2-manifolds b y Kodaira in t h e early 1950s a n d has b e e n m u c h studied ever since. An a l g e b r a i c surface can be t h o u g h t of as being def'med in 3-space b y a polynomial in t h r e e variables x, y, a n d z. As such, t h e y w e r e heavily s t u d i e d t h r o u g h o u t t h e nineteenth century, and m a n y e x a m p l e s were k n o w n - - t h e cubic s u r f a c e with its 27 lines a n d K u m m e r ' s quartic surface with its 16 nodal p o i n t s a r e among those that remain of i n t e r e s t (see [18]). In the 1870s, Max N o e t h e r c r e a t e d the outline o f a general theory, w h i c h followed the outlines of the general t h e o r y o f algebraic curves (itself a r e c e n t creation). But it w a s c l e a r that m u c h m o r e rem a i n e d to b e done than h a d b e e n solidly a c c o m p l i s h e d . The a p p r o a c h that a p p e a l e d particularly to the Italian s c h o o l of g e o m e t e r s was initiated b y Veronese in his p a p e r [29]. He u s e d the m e t h o d o f p r o j e c t i o n and s e c t i o n to s h o w h o w s o m e singular curves in the plane a n d singular surfaces in 3-space could p r o f i t a b l y be thought of as nonsingular o b j e c t s in a high-dimensional space. To m a k e a s y s t e m a t i c a t t a c k on the s t u d y o f surfaces, Corrado Segre then s u g g e s t e d that the b e s t a p p r o a c h to s u r f a c e s w o u l d b e to study t h e m birationally a n d to l o o k for families o f curves sufficiently well b e h a v e d to yield an e m b e d d i n g of the surface in s o m e s u i t a b l e projective space. To see h o w this can be done, at least in principle, it is w o r t h looking at t h e s i m p l e s t kind o f birational transformation o f the plane, and its effect on a mildly singular curve ( d e s c r i b e d in Box 1).

A birational t r a n s f o r m a t i o n rep l a c e s one surface w i t h another, it suggests that singular s u r f a c e s might be b l o w n up in w a y s t h a t could simplify their singularities, a n d this w a s indeed to p r o v e to be the case. The birational t r a n s f o r m a t i o n in B o x 1 was found b y using a 3 - p a r a m e t e r family o f curves (in this case conics) with the p r o p e r t y that at almost every p o i n t of t h e plane, t h e r e was a t h r e e - d i m e n s i o n a l space of curves passing t h r o u g h the point. More general birational t r a n s f o r m a t i o n s are found by c h o o s i n g o t h e r families o f curves---it is a f a m o u s e x e r c i s e in the s u b j e c t to obtain t h e cubic surface with its 27 lines b y considering all the plane cubics t h r o u g h 6 points.

A Quick Trip Through Algebraic Curves After Riemann's death in 1866, leadership in the field o f algebraic geometry p a s s e d to Clebsch; a n d when he died in 1872, to his former colleagues Alexander B~_ll and Max Noether. They gave the entire t h e o r y a firm twist in the direction of polynomial algebra. They thought of an algebraic curve as a plane curve and defined b y a polynomial equation in two c o m p l e x variables F(z, w) = 0 (or, equivalently, three homogeneous variables). Its genus, p, was defined in terms of the degree, n, of the defining equation and the na~tre of the singular points, according to the formula 1 -z(n - 1)(n - 2) -

Z oli -i(i- - 1) i 2

- p -

1,

w h e r e the curve h a s a i p o i n t s o f o r d e r i. To distinguish it from o t h e r curves w h i c h e n t e r the B r i l l - N o e t h e r theory, this curve will b e called the ground curve. The e v e r y w h e r e - h o l o m o r p h i c integrands on the curve, as Clebsch had shown, following Abel, are of the form r dz/(aF/~w), w h e r e (b is of degree n - 3. Brill a n d N o e t h e r s h o w e d that a curve of o r d e r n - 3 that p a s s e s

9 1999 SPRINGER-VERLAG NEW YORK, VOLUME 21, NUMBER 1, 1999 5 9

(i - 1) t i m e s t h r o u g h e a c h p o i n t o n t h e g r o u n d c u r v e o f o r d e r i c u t s t h e original c u r v e in n(n

-

3) - ~ ,

aii(i

-

1) = 2p - 2

i

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THE MATHEMATICALINTELLIGENCER

f u r t h e r points, of w h i c h at l e a s t p - 1 c a n b e c h o s e n arbitrarily. S u c h a c u r v e is c a l l e d an adjoint curve; it c u t s o u t w h a t is c a l l e d a c a n o n i c a l p o i n t - g r o u p o n t h e original curve.

In p a r t i c u l a r , t h e v e r s i o n o f t h e R i e m a n n - R o c h T h e o r e m t h a t Bfill a n d N o e t h e r e s t a b l i s h e d in [2] says: Let an adjoint curve of order n - 3 be drawn through a set of Q points on the ground

curve, and s u p p o s e q of t h e s e p o i n t s d e t e r m i n e the rest (and so t h e adjoint curve). If q = Q - p + l+r (where 0 < r < p - 1), then this curve m e e t s the g r o u n d curve in 2 p - 2 - Q = R f u r t h e r p o i n t s that t h e m s e l v e s belong to a set in which r p o i n t s d e t e r m i n e t h e rest. This version is c o n n e c t e d to R i e m a n n ' s b y the simple o b s e r v a t i o n t h a t if ~b0and ~b~ are two adjoint curves (of a n y order), then their quotient ~b0/~b~ is a m e r o m o r p h i c function on t h e g r o u n d curve having p r e s c r i b e d zer o s w h e r e the curve q~0 = 0 m e e t s the g r o u n d curve and having p o l e s w h e r e the curve (be = 0 m e e t s t h e g r o u n d curve. Interestingly, this o b s e r v a t i o n w a s m a d e explicitly for the first time b y Klein in his lectures [22, p. 189n]. By omitting it, Brill and N o e t h e r m a d e t h e i r p r e f e r e n c e for a l g e b r a o v e r geometry known. The s t u d y of algebraic curves taught the Italians that a curve o t h e r than a rational or elliptic curve can be emb e d d e d in a projective space. The first to a p p r e c i a t e this had b e e n Kraus, w h o died at the age of 27 in 1881, a n d his a p p r o a c h w a s later t a k e n up b y Felix Klein. It rests on the insight that an algebraic curve of genus p -> 2 has a p d i m e n s i o n a l space of h o l o m o r p h i c tforms. Kraus [23] had the h a p p y i d e a that if a basis is c h o s e n for t h e s e iforms, s a y o~l,. 9 9 OJp,then the p - t u p l e ( w t ( z ) , . . . ,Ogp(Z))= ( f l ( Z ) d z , . . . . , fp(z)dz), p > 1, gives rise to a m a p from the curve to a projective s p a c e of d i m e n s i o n p - 1: z--> [ f i ( z ) , . . . ,fp(Z)]. The a r g u m e n t that this m a p is well defined ( i n d e p e n d e n t of the c o o r d i n a t e s y s t e m used) and is a m a p into proj e c t i v e s p a c e (the f i n e v e r simultaneously vanish) w a s d u c k e d b y him. It is a straighforward e x e r c i s e in the R i e m a n n - R o c h Theorem, as is the observation t h a t the degree o f t h e m a p is 2p - 2. Hyperelliptic curves also c a u s e p r o b l e m s : as Kraus saw, one i n s t e a d r e a l i s e s the curve as a b r a n c h e d double c o v e r of the R i e m a n n sphere. These c a s e s aside, questions a b o u t alg e b r a i c curves have b e e n r e d u c e d to questions in projective geometry. In particular, as Klein n o t e d [22, p. 117], the a b s o l u t e invariants o f t h e n o r m a l curve are the moduli o f the algebraic curve.

Castelnuovo and Enriques Guido Castelnuovo and his close friend Federigo Enriques met in Rome in the autumn of 1892, when Castelnuovo w a s 27 and Enriques just 21. Castehiuovo was a former student of Corrado Segre, and Segre e n c o u r a g e d them to t a k e up the s t u d y o f algebraic surfaces, in which he w a s deeply interested. They a p p r o a c h e d t h e s u b j e c t more obliquely than he had, a n d with a surer g r a s p o f w h a t the b e t t e r d e v e l o p e d t h e o r y o f algebraic c u r v e s h a d to say. In particular, t h e y p u r s u e d the idea that a R i e m a n n - R o c h T h e o r e m for s u r f a c e s would be a p o w e r f u l tool, a n d s o it turned out. F r o m Segre, they d r e w the lesson t h a t t h e ideas of Kraus a n d Veronese c o u l d be h a r n e s s e d to a suitable g e n e r a l i s a t i o n of the w o r k o f Brill and Noether. They a c c e p t e d that ideas from complex analysis should be p l a y e d down, but they w e r e m o r e g e o m e t r i c a n d less algebraic t h a n their G e r m a n p r e d e c e s sors. They l o o k e d for a w a y of des ing c a n o n i c a l curves, K, on a s u r f a c e birationally. Projectively, t h e s e are curves on t h e surface of degree n cut out b y a d j o i n t surfaces of degree n 4 t h a t p a s s often enough through the singular p o i n t s a n d singular curves o f the surface (this had been N o e t h e r ' s approach). The adjoint, A(C), o f a curve C s h o u l d be C + K. A suitable generalisation of the R i e m a n n - R o c h T h e o r e m s h o u l d a p p l y to the s u r f a c e and a curve C o r the m a x i m a l linear s y s t e m ]C] to w h i c h C belongs. However, as Enriques o b s e r v e d in his first m a j o r paper, his Ricerche [10], Noether's definition of an adjoint surface invokes the degree, so it is projective but not birational. Another def'mition of the canonical curve m u s t be sought. Moreover, as the example in Box 1 illustrates, whereas linear families of curves on the surface do yield m a p s to projective space, if there are points c o m m o n to all the curves, the image of the surface that they provide will have n e w singularities (the base points will "blow up" into curves). It will be n e c e s s a r y to control, and ideally to eliminate, t h e s e exceptional curves. This b e c a m e a long-running story; interestingly, t h e definitions offered by the Italian algebraic g e o m e t e r s h a d

nothing to do with h o l o m o r p h i c integrands. The topic o f integrals was deliberately and h a p p i l y left to the French, n o t a b l y P i c a r d b u t also H u m b e r t (see [20]). It is quite striking that Picard in F r a n c e and Castelnuovo and Enriques in Italy chose to develop their ideas in parallel, b u t n o t to merge their fields.

Castelnuovo and Enriques and the Riemann-Roch Theorem One m a j o r aim of t h e Ricerche [10] was to establish a R i e m a n n - R o c h T h e o r e m for surfaces. By a n a l o g y with algebraic curves, a R i e m a n n - R o c h T h e o r e m for surfaces should c o n c e r n an algebraic surface, a curve o n it cut o u t by ano t h e r surface, and a family of surfaces, one m e m b e r of w h i c h cuts the given curve. To prove s u c h a theorem, Enriques e n d e a v o u r e d to e x p r e s s the n u m b e r s involved in t e r m s o f quantities that apply to t h e given curve, and then to invoke t h e R i e m a n n - R o c h T h e o r e m for the curve. He found it nece s s a r y to e x c l u d e certain t y p e s of surface, which w a s n o t n e c e s s a r i l y a misfortune: it could m e a n that the e x c l u d e d surfaces w e r e in s o m e w a y significantly different from the rest. He set aside ruled s u r f a c e s b e c a u s e they o b s t r u c t e d his c h a r a c t e r i s a t i o n of the canonical series (the sets o f p o i n t s cut out on a generic curve of ICI b y an adj o i n t system) as d i d surfaces of geometric genus 0. Ruled s u r f a c e s had alr e a d y b e e n s h o w n b y Cayley in 1871 [7] to have different a r i t h m e t i c a n d geometric genera, so t h e y a l r e a d y s e e m e d to belong to a p a r t i c u l a r family of surfaces. Given a linear s y s t e m IC], Enriques defined ~4(C)I, the adjoint linear system to [C], as the c u r v e s w h i c h cut out a canonical group on any generic plane section o f the s u r f a c e and which have (i - 1)-fold p o i n t s at a n y / - f o l d point of the linear s y s t e m ]C]. The central difficulty lay in d e t e r m i n i n g the dimension of the adjoint s y s t e m o f a linear system of curves o f genus ~r, which, Enriques said, w a s o f the greatest imp o r t a n c e b e c a u s e o f the m a n y applications there w e r e for the i d e a of an adj o i n t system. He w a s able to d e t e r m i n e this dimension for regular surfaces (it is ~r § p - 1) and, therefore, in this, his

VOLUME 21, NUMBER 1, 1999

61

curves o f the adjoint s y s t e m - - t h i s is w m o r e t h a n you might expect. It w a s to prove an elusive concept. Writing to Castelnuovo in July 1894, Enriques said, "The s u p e r a b u n d a n c e to has ano t h e r significance which gives a r e a s o n for its name. If ICI is cut out b y m e a n s of adjoint surfaces, and the virtual dim e n s i o n p of {C} is calculated a c c o r d ing to the p o s t u l a t i o n f o r m u l a o f Noether, one has, b y the r e s i d u e theorem, r-p

= w-i."

(In t h e c a s e I'm describing, i = 0. The entire, fascinating collection o f letters from Enriques to Castelnuovo has b e e n p u b l i s h e d in [1].)

Plurigenera

first m a j o r p a p e r , Enriques r e s t r i c t e d his attention to r e g u l a r surfaces. Enriques s t a t e d a R i e m a n n - R o c h t h e o r e m for r e g u l a r algebraic surfaces of genus pg > 0 in t h e following form. Suppose that the generic curve of ICI has genus 7r, that, in general, a set of r p o i n t s on t h e surface specifies a unique curve of IC], a n d that any two m e m b e r s of ICI m e e t in s points. Suppose also t h a t the curves of ICI are not c o n t a i n e d in the canonical system (Enriques also dealt with the case w h e r e they are). Then,

62

THE MATHEMATICAL INTELLIGENCER

r=

pg + to-

qr + l + s ,

w h i c h can b e rewritten as r + rr-

l - s = p g + to,

w h e r e to is the so-called s u p e r a b u n dance. He d e d u c e d it b y an ingenious geometric argument from the R i e m a n n R o c h T h e o r e m on the curve C. The s u p e r a b u n d a n c e o f a linear syst e m IC[ is defined as the n u m b e r w -> 0 s u c h t h a t through a group of p o i n t s c o m m o n to two curves o f ICI, t h e r e p a s s 2 p g + to linearly i n d e p e n d e n t

C a s t e l n u o v o and Enriques s o o n bec a m e dissatisfied with the a r i t h m e t i c and g e o m e t r i c genera. Not only w e r e t h e r e two, w h i c h differed w h e n a surface w a s irregular, but t h e y did n o t c h a r a c t e r i s e surfaces. In particular, a regular s u r f a c e w h o s e genera v a n i s h e d was n o t n e c e s s a r i l y a rational surface. This w a s illustrated by a r e m a r k a b l e d i s c o v e r y o f Enriques in 1894. He cons i d e r e d a t e t r a h e d r o n in C P 3 a n d f o u n d a sextic s u r f a c e (one of degree 6) t h a t had the e d g e s of the t e t r a h e d r o n as its double curves. Any adjoint surface m u s t b e o f degree 4 less t h a n t h e surface, a n d so b e o f degree 2 and therefore a quadric surface, and it m u s t p a s s t h r o u g h t h e d o u b l e curves o f t h e surface. But, plainly, there is no quadric surface t h r o u g h the six e d g e s o f a tetrahedron. So the sextic s u r f a c e h a s no adjoints. However, there is a surface of d e g r e e 2(n - 4) = 2(6 - 4) = 4, w h i c h p a s s e s twice t h r o u g h the edges of t h e tetrahedron, n a m e l y the surface c o m p o s e d of the four p l a n e s that form the faces of the t e t r a h e d r o n . So the s u r f a c e of degree 6 h a s no adj o i n t s u r f a c e and its genus is 0, b u t it d o e s have a biadjoint surface, a n d its bigenus (which is one m o r e than the d i m e n s i o n of the s p a c e of s u c h surfaces, a n d is d e n o t e d P2) is 1, n o t 0. This i n s p i r e d Castelnuovo to an investigation, and he was able to s h o w in 1896 [3] that a surface with arithmetic a n d g e o m e t r i c genera equal to 0 and bigenus P2 -- 0 is indeed a rational

surface. This was the first birational c h a r a c t e r i s a t i o n of a surface. It also m a r k s the m o m e n t w h e n the so-called p l u r i g e n e r a e n t e r the analysis. They are r e l a t e d to the d i m e n s i o n s of the p l u r i c a n o n i c a l systems I/~1 b y the form u l a Pi = diml/~l + 1. The plurigenera Pi w e r e s t u d i e d b y Enriques a n d Castelnuovo in their 1901 p a p e r [5]. They s h o w e d t h a t considering II~l = l i A ( C ) - iC[ a n d applying the R i e m a n n - R o c h T h e o r e m to find the d i m e n s i o n of the linear s y s t e m ]iA(C')l , t h e y could be e x p e c t e d to g r o w quadratically with i, b u t u n d e r certain conditions, t h e y might only g r o w linearly. Particular c l a s s e s o f surf a c e s did less well; they m a y be only 0 a n d 1 or even always 0. Trivially, ifP,i > 0, t h e n P~i > 0 for all integers k. Enriques's Introduzione [ 11 ] m a r k s a c o n s i d e r a b l e a d v a n c e on the Richetvhe in its level o f generality. Irregular surfaces could n o w b e treated, b e c a u s e of r e c e n t d i s c o v e r i e s b y Castelnuovo in his 1896 p a p e r [3], one that drew strong praise from Severi in 1958 [28], toward the end o f his life. Enriques was also helped by progress m a d e b y French workers, notably Humbert, in the flmction-theoretic study of surfaces. To emphasise that he was working systematically with birational properties, Enriques spoke n o w not of surfaces but of doubly infmite algebraic entities (ente algebrico doppiamente i n f i n i t o ) - - a phrase that he t o o k from Segre's influential p a p e r on algebraic curves of 1894. It m e a n s a birational equivalence class of surfaces. Any representative of such a class he called an image ( i m m a g i n e ) of the class. 1 9 0 1 : Partial Classifications The last major innovation of Castelnuovo and Enriques in the 1890s c o n c e r n e d the t h e o r y of adjunction, which, as we have seen, w a s the m o s t i m p o r t a n t single technique available to them. It was published in Enriques's Intorno [12], which is a p r e l u d e to the m o r e thorough-going j o i n t paper, the Questioni [5]. In the Intorno, the n e w i d e a of the J a c o b i a n of a linear system l e a d s to a simple a c c o u n t of adjoints. The J a c o b i a n of a linear s y s t e m ]C] of d i m e n s i o n greater than 2 is o b t a i n e d as follows. Take a 2-dimensional subsys-

tern, ILl, of ICI; its double points are the J a c o b i a n of ILl. Enriques e s t a b l i s h e d that all the J a c o b i a n s of the 2-dimensional s u b s y s t e m s in IC] lie in the s a m e linear system, the c o m p l e t e J a c o b i a n linear system, ]J(C)I , a s s o c i a t e d to a linear s y s t e m ]C]. They then established the t h e o r e m that

IJ(C

+

K)I = IJ(C) + KI.

The concept of an adjoint system was r e c a p t u r e d in this f r a m e w o r k b y establishing t h a t the adjoint linear syst e m to IV], ~4(C)], is given b y

~4(C)l = ]J(C) -

2ci.

This also simplified the i m p o r t a n t concept o f the pluricanonical systems. When does adjunction stop? Castelnuovo and Enriques t a c k l e d this p r o b l e m in their Questioni [5] b y showing that if it s t o p s for o n e imm e r s i o n o f the surface in s o m e projective space, it always stops. Then, they s h o w e d that if all the p l u r i g e n e r a vanish, t h e n successive a d j u n c t i o n stops. Finally, they s h o w e d t h a t stopping implies that all the p l u r i g e n e r a vanish. This w a s much h a r d e r a n d required t h e m to distinguish the c a s e s p(1) _< 1 a n d pO) > 1. Their h a n d l e on the g r o w t h o f the p l u r i g e n e r a w a s o f c o u r s e p r o v i d e d b y the R i e m a n n - R o c h Theorem, in line with C a s t e l n u o v o ' s earlier results. Their p a p e r c o n c l u d e d with a classification o f surfaces: Surfaces for w h i c h p(1)_ 1 and for w h i c h all the plurigenera vanish a r e either rational or elliptic ruled surfaces (and p(1) = 1). Surfaces for w h i c h p(1) > 1 and for w h i c h s o m e plurigenera do n o t vanish. In this case, either p ( 1 ) > 1, w h i c h implies that for i sufficiently large no P i = 0, o r p (1) = 1, in w h i c h case nothing could p r e s e n t l y b e said. Surfaces for w h i c h p(1) < 1: all t h e s e s u r f a c e s w e r e birationally equivalent to r u l e d surfaces of genus p > 1, a n d p (1) = - 8 ( p - 1). In this p a p e r , Castelnuovo a n d Enriques also found, by making essential use of the R i e m a n n - R o c h T h e o r e m and a n o t h e r invariant (the Z e u t h e n Segre invariant, which I shall n o t de-

fine), that s u c c e s s i v e adjunction s t o p s if and only if the surface is ruled. They also p r o v e d that e x c e p t i o n a l curves can be eliminated on all b u t ruled surfaces. This i m p o r t a n t result, which t h e y had h o p e d for and believed in through the 1890s, greatly simplified the theory. 1 9 0 5 - 1 9 0 7 : M o r e Classifications After 1900, Castelnuovo a n d Enriques shifted their i n t e r e s t t o w a r d the classification o f algebraic surfaces. In 1905, Enriques w a s p l e a s e d to publish a p a p e r classifying certain kinds of irregular surface, t h o s e w h o s e geometric genus pg = 0 [13]. He s h o w e d that if the arithmetic g e n u s Pa < - 1 , t h e n the surface is birationally equivalent to a ruled surface, a n d if Pa = - 1 , t h e n the surface p o s s e s s e s an elliptic group o f a u t o m o r p h i s m s a n d is a pencil o f elliptic curves. The phirigenera told these classes apart. F o r a ruled surface, all the p l u r i g e n e r a vanish (pg = Pz = / ) 3 . . . . . 0), b u t for an elliptic surface not equivalent to a ruled one, either P4 -> 1 or P6 -> 1. It follows that the ruled s u r f a c e s are t h o s e for which pg = 0 = P4 = P6. In particular, this t h e o r e m gives n e c e s s a r y and sufficient conditions for a p o l y n o m i a l equation in t h r e e variables f ( x , y, z ) = 0, to b e written as a p o l y n o m i a l equation in t w o variables, r Y) = 0, b e c a u s e a ruled surface is birationally equivalent to a cylinder. As he p o i n t e d out, this result w a s n o t w h a t Castelnuovo h a d expected. A m o r e d e t a i l e d e x a m i n a t i o n of the elliptic case f o l l o w e d (see below), a n d then Enriques c o n c l u d e d the p a p e r with an e x a m i n a t i o n o f the ruled, nonrational surfaces, w h i c h could now b e c h a r a c t e r i s e d as t h o s e for which pg = P2 = 0 and P12 r 0 (the last condition is a c o n s e q u e n c e of either P4 r 0 o r P6 r 0). They t u r n e d out to be of four kinds, three of w h i c h w e r e elliptic pencils of elliptic curves. In [14], Enriques t o o k up the t h e m e of surfaces for w h i c h P2 = 1. His surface F6 of degree 6 h a d s h o w n a d e c a d e earlier that the equations Pa = 0 = pg w e r e not sufficient conditions for a surface to be rational. Since then, o t h e r e x a m p l e s of nonrational surfaces of genus 0 h a d b e e n found, he

VOLUME 21, NUMBER 1, 1999 ( } 3

said, but F6 differed from t h e m in being the only k n o w n regular surface of bigenus 1, a n d in being the only surface w h o s e biadjoint surface cut it precisely in its d o u b l e curve. This suggested w h a t Enriques could n o w prove: a regular surface is birationally equivalent to F6 if a n d only i f p a = 0 = /)3 and P2 = 1. This c h a r a c t e r i s e d a class of s u r f a c e s n o w a d a y s called Enriques surfaces. In proving this result, Enriques was led to two classes o f surfaces with different p l u r i g e n e r a The first corresponds to a surface with P l = P3 = P5 . . . . . O a n d P2 = P4 = P6 . . . . . 1 (the Enriques surface) and the other to surfaces with s o m e P i > 1, and, in fact, P6 > 1. The p a p e r c o n c l u d e d with further properties of the Enriques surface and examples of surfaces of the other kind. In particular, Enriques showed that the surfaces birationally equivalent to F6 have an infinite discontinuous group of birational automorphisms. In a s e c o n d p a p e r [15], Enriques a n a l y s e d s u r f a c e s for w h i c h p(1) = 1. He found that t h e r e w e r e three cases: 1. Pa = 1 = p g , in w h i c h c a s e P2 -> 1; 2. Pa =- 0 a n d p g = 1, in w h i c h case

/)2>1; = - - 1 a n d p g = 1, in which case P4 -> 1, a n d the surface will not have an effective c a n o n i c a l curve if and only if P4 = 1 (in w h i c h case all the p l u r i g e n e r a = 1).

3. Pa

Castelnuovo a n d Enriques next w r o t e the first o f several surveys of the field [6], the R d s u l t a t s N o u v e a u x , that a p p e a r s as an a p p e n d i x in the s e c o n d volume of P i c a r d a n d Simart [26]. They r e c a p i t u l a t e d the a p p r o a c h of 1901 as far as the i n t r o d u c t i o n of the pluricanonical linear systems, a n d devoted the s e c o n d p a r t of their R d s u l t a t s to the classification o f algebraic surfaces. They n o w said t h a t a n s w e r s e x p r e s s e d in t e r m s of n u m e r i c a l birational invariants b e l o n g e d to the quantitative theory, w h e r e a s a n s w e r s involving the indefinite c o n t i n u a t i o n or eventual termination of the s e q u e n c e of successive adjoints b e l o n g e d to w h a t they called the qualitative theory. This interestingly recalls Poincar~'s c o n t e m p o r a r y distinction b e t w e e n the quantitative

THE MATHEMATICAL INTELLIGENCER

a n d the qualitative in his o w n w o r k on differential equations. In b o t h settings, the i d e a is that the qualitative app r o a c h p r e c e d e s and guides the quantitative one. They t h e n s u r v e y e d the o l d e r results in the field, w h i c h gave criteria for a surface to b e rational, before listing their o w n results, divided into qualitative and quantitative. These i n c l u d e d the following: 1. A surface is rational if a n d only if Pa = P2 = 0 (which implies that pg = 0).

2. A m o n g irregular s u r f a c e s with pg = 0 a n d Pa ~ O, t h o s e with Pa ~ - 1 are ruled, and the rulings are curves o f genus - P a , w h e r e a s t h o s e for w h i c h Pa = - 1 form v a r i o u s sorts o f elliptic surface. 3. A surface is ruled if a n d only if P4 = P6=0. Joint w o r k with Severi led Enriques in 1908 to establish m o r e results about

surfaces with Pa = 0 = pa a n d P2 = 1. He s h o w e d that they are all double covers of the projective plane branched over a particular kind of curve of degree 8.

Various Kinds of "Elliptic" Surface Two o t h e r a p p r o a c h e s to s u r f a c e s were yielding results at the s a m e time. A class o f surfaces were f o u n d b y m e a n s o f a u t o m o r p h i c functions in two variables; they w e r e m u c h s t u d i e d by P i c a r d a n d Humbert, and c a m e to be called hyperelliptic surfaces. The p a r a m e t r i s a t i o n will b e r to 1, w h e r e r is the r a n k of the surface; K u m m e r ' s quartic s u r f a c e is of rank 2. A n o t h e r class o f s u r f a c e s w a s found b y putting t o g e t h e r p a i r s of curves. The C a r t e s i a n p r o d u c t of a curve X and a curve Y is a surface X • Y. If X is the p r o j e c t i v e line, the surface is a ruled surface. If it is an elliptic curve, the surface is elliptic; if Y is also elliptic, the s u r f a c e is s o m e t i m e s called bielliptic. B e c a u s e

elliptic curves have large automorphism groups, surfaces constructed in this way tend to have symmetries, quotients of them can be expected to be nonsingular surfaces. These surfaces should be expected to have a special place in any classification, akin to the elliptic curves in the classification of algebraic curves, and so they do. Indeed, surfaces other than these turned out to be either much simpler (ruled or even rational) or else much more complicated and inscrutable (the surfaces "of general type"). The hypereniptic surfaces were the subject of long and detailed studies by Enriques and Severi, for which they were awarded the French Prix Bordin in 1907, and their younger Italian rivals Bagnera and de Franchis, who were awarded the Prix Bordin in 1909. 1914: The Full Castelnuovo and Enriques Classification In 1914, Castelnuovo and Enriques pubfished two articles in the Encyklo-pddie

der Mathematischen Wissen-schaften. The second of these [17] culminated in a classification of algebraic surfaces (see Box 3). It is interesting to compare

it with a modern version, taken from Barth, et al. [0], p. 188 (see Box 4). Kodalra studied compact, complex 2-manifolds and found some that cannot be embedded in any projective space, and so escaped the EnriquesCastelnuovo classification. The integers which parametrise the families are also much better understood, but the qualitative behaviour of the plurigenera survives as the framework of the Kodaira classification. The Enriques-Castelnuovo classification and many of the results obtained on the way represent a remarkable achievement. Over the years, Segre himself had attempted to tackle the singularities directly. This proved to be too hard, although a resolution theorem was obtained by the German mathematician H.W.E. Jung in 1908 [21]. Segre's former student Castelnuovo and his colleague Enriques got further, and the focus of algebraic geometry in Italy shifted to them. BIBLIOGRAPHY

0. Barth, W., Peters, C. and Van de Ven, A., Compact Complex Surfaces, SpringerVerlag, New York & Heidelberg (1984).

1. Bottazzini, U., Conte, A., and Gario, P., Riposte Armonie, Lettere di Federigo Enriques a Guido Castelnuovo, Bollati Boringhieri, Torino (1996). 2. Brill, A., and Noether, M., Ueber die algebraischen Functionen und ihre Anwendung in der Geometrie, Math. Ann. 7 (1874), 269-310. 3. Castelnuovo, G., Alcuni risultati sui sistemi lineari di curve appartenenti ad una superficie algebrica, Mem. Soc. It. Sci. XL. (3) 10 (1896), 82-102, in Memorie (1937), 335-360. 4. Castelnuovo, G., Alcune propriet& fondamentali sui sistemi lineari di curve tracciati sopra ad una superficie algebrica, Ann. di Mat. Pura Appl. (2) 25 (1897), 235-318, in Memorie, 361-442. 5. Castelnuovo, G., and Enriques, F., Sopra alcune questioni fondamentali nella teoria delle superficie algebriche, Ann. di MaL Pura Appl. (3) 6 (1901), 165, in Enriques F. Memorie Scelte, 2, 85-144. 6. Castelnuovo, G., and Enriques, F., Sur quelques resultats nouveaux darts la theorie des surfaces algebriques, in Picard and Simart [26], 2, 485-522. 7. Cayley, A., On the deficiency of certain surfaces, Math. Ann. 3 (187), 526-529. 8. Clebsch, R.F.A., Ueber die Anwendung der Abeischen Functionen in der Geometrie, Jour. for reine und angew. Math. 63 (1863), 189-243. 9. Clebsch, R.F.A., and Gordan, P., Theorie der Abelschen Functionen, Teubner, Leipzig (1866). 10. Enriques, F., Ricerche di geometria sulle superficie algebriche, Mem. Acc. Torino (2) 44 (1893), 171-232, in Memorie Scelte, 1, 31-106. 11. Enriques, F., Introduzione alia geometria sopra le superficie algebriche, Mem. Soc. Ital. delle Scienze (3) 10 (1896), 1-81, in Memorie Scelte, 1, 211-312. 12. Enriques, F., Interne ai fondamenti della geometria sopra le superficie algebriche, Atti Acc. Torino 37 (1901), 19-40, in Memorie Scelte, 2, 65-83. 13. Enriques, F., Sulle superficie algebriche di genere geometrico zero, Rend. Circ. Mat. Palermo 20 (1905), 1, in Memorie Scelte, 2, 169-204. 14. Enriques, F., Sopra le superticie algebriche di bigenere uno, Mern. Sec. Ital. Scienze (3) 14 (1907), 327-352, in Memorie Scelte, 2, 241-272. 15. Enriques, F., Interne alle superficie algebriche di genere p(1) = 1, Rend. Acc. Bol.

VOLUME 21, NUMBER 1, 1999

65

16.

17.

18.

19.

66

11 (1907), 11-15, in Memorie Scelte, 2, 279-282. Enriques, F., Sulla classificazione delle superficie algebriche e particolarmente sulle superficie di genere lineare p(1) = 1. Note I e II, Rend. Accad. Lincei (5) 23 (1914), 206-214, 291-297, in Memorie Scelte, 3, 173-182, 183-190. Enriques, F., and Castelnuovo, G., Die algebraischen Flachen vom Gesichtspunkte der birationalen Transformationen aus, in Encyklep~die der Mathematischen Wissenschaften, B.G. Teubner, Leipzig (1914), 111.2.1C, pp. 674-768. Gray, J.J., Algebraic and projective geometry in the late 19th century, in Symposium on the History of Mathematics in the 19th Century, edited by J. McCleary and D. E. Rowe, Academic Press, New York (1989), pp. 361-388. Gray, J.J., The Riemann-Roch Theorem

THE MATHEMATICAL INTELLIGENCER

20.

21.

22.

23.

24.

and Geometry, 1854-1914, in Proceedings of the International Congress of Mathematicians, Documenta Mathematica, vol. 3, 811-822. Berlin (1998), 811-822. Houzel, C., Aux origines de la geom6trie algebrique: les travaux de Picard sur les surfaces (1884-1905), Cahiers HisL Phil. Sciences 34 (1991), 243-276. Jung, H.W.E., Darstellung der Funktionen eines algebraischen K6rpers zweier unabhAngigen Ver&nderlichen, Jour. fdr reine und angew. Math. 133 (1908), 289-314. Klein, F., Riemannsche Fl~chen, Vorlesungen, Teubner-Archiv, Leipzig (1892), Vol. 5. Kraus, L., Note 0ber aussergew6hnliche Specialgruppen auf algebraischen Curven, Math. Ann. 16 (1880), 245-259. Noether, M., Sulle curve multiple di superficie algebriche, Ann. di Mat. Pura Appl. (2) 5 (1871), 163-178.

25. Noether, M. Zur Theorie des eindeutigen Entsprechens algebraischer Gebilde, Zweiter Aufsatz, Math. Ann. 8 (1875), 495-533. 26. Picard, E., and Simart G., Th~orie des fonctions algebriques de deux variables independantes, Gauthier-Villars, Paris (1897,1906) (reprinted by Chelsea, New York, 1971 ). 27. Salmon, G., On the degree of a surface reciprocal to a given one, Cam. Dublin Math. d. (2) 2 (1847), 65-73. 28. Severi, F., II teorema di Riemann-Roch per curve, superficie e varieta; questioni co/legate, Springer-Verlag, Berlin (1938). 29. Veronese, G., Behandlung der projectivischen Verh<nisse der R&ume von verschiedenen Dimensionen, Math. Ann. 19 (1882), 161-234.

J.M. HENLE

Non-nonstandard Analysis: Real Infinitcsimals

~

n

athematics has had a troubled relationship with infinitesimals, a relationship that stretches back thousands of years. On the one ,hand, infinitesimals make intuitive sense. They're easy to deal with algebraically. They make calculus a lot of fun. On the other hand,

they s e e m impossible to nail down. They're h a r d to deal with intellectually. They c a n m a s k a fundamental l a c k of u n d e r s t a n d i n g of analysis. There was hope, w h e n A b r a h a m Robinson d e v e l o p e d n o n s t a n d a r d analysis [R], that intuition and rigor h a d at last j o i n e d hands. His w o r k i n d e e d gave infinitesimals a found a t i o n a s m e m b e r s o f the s e t o f h y p e r r e a i numbers. But it w a s an a w k w a r d foundation, d e p e n d e n t on the A x i o m o f Choice. Unlike s t a n d a r d n u m b e r systems, t h e r e is no c a n o n i c a l set of h y p e r r e a l numbers. There is a way, however, of constructing infinitesimals naturally. Ironically, t h e s e e d s c a n be found in a n y calculus b o o k o f sufficient age. At the turn o f the century, it w a s typical o f texts to define an infinitesimal as a "variable w h o s e limit is zero" [C]. That is the inspiration for the p r e s e n t a p p r o a c h to calculus. Its infmitesimals are seq u e n c e s tending to 0. I call the system "non-nonstandard analysis" to d r a w attention to its misfit nature. Having infinitesimals, it is n o t

"standard." Nor is it "nonstandard," however, as this term now has a well-def'med meaning. In w h a t follows, we manipulate s e q u e n c e s of real numbers. We t r e a t t h e m (mostly) as numbers. We a d d them, s u b t r a c t them, and p u t t h e m into functions. They a r e n ' t numbers, however. Trichotomy fails, for example. The central c o n s t r u c t i o n in this article is a rediscovery. Its first d i s c o v e r e r p r o b a b l y was D. Laugwitz. More on this later. N o t a t i o n . We d e n o t e a sequence {an}nE ~ b y a boldface a. We p e r m i t a s to be undefined for fmitely m a n y n. The key i d e a t h r o u g h o u t is that of "from some point on." An equation o r inequality involving s e q u e n c e s will be int e r p r e t e d as being true or false, d e p e n d i n g on w h e t h e r the a s s o c i a t e d equation or inequality involving the terms o f the s e q u e n c e s is true o r false from s o m e p o i n t on. For example, a>2

@ 1999 SPRINGERWERLAG NEW YORK, VOLUME 21, NUMBER 1, 1999

67

s i m p l y m e a n s "an > 2 f r o m s o m e p o i n t on," t h a t is, that for s o m e k, n -> k i m p l i e s that an > 2. We u s e o r d i n a r y letters for real n u m b e r s . Note t h a t a real n u m b e r r c a n b e v i e w e d as t h e s e q u e n c e a, w i t h an = r for all n. E q u a t i o n s a n d i n e q u a l i t i e s i n v o l v i n g s e q u e n c e s are interp r e t e d in the s a m e way; that is, a + b = c m e a n s an § b n = Cn f r o m s o m e p o i n t on, a n d s i n ( a ) = b m e a n s s i n ( a n ) = bn from s o m e p o i n t on. Note that f o r f ( a ) to b e defined, it is o n l y n e c e s s a r y t h a t f ( a n ) b e d e f i n e d f r o m s o m e p o i n t on. The s e q u e n c e b: 0, 0, 0, 1, 2, 3, 4 , . . . , for e x a m p l e , does h a v e a reciprocal, I/b, b e c a u s e , as n o t e d , finitely m a n y t e r m s of a s e q u e n c e m a y b e u n d e f i n e d . If a s t a t e m e n t P is true from s o m e p o i n t o n a n d statem e n t Q is also t r u e f r o m s o m e p o i n t on, t h e n "P a n d Q" is true from s o m e p o i n t on. This fact a l l o w s u s to do a l g e b r a on sequences. Suppose, for e x a m p l e , w e have a+4=3b

P r o p o s i t i o n 1. S u p p o s e a, b ~ 0. T h e n (1) a + b ~ 0. (2) a - b ~ 0. (3) I f e is f i n i t e , t h e n a c ~ 0. (4) I f Icl < lal, t h e n e ~ O. PROOF. Given a n y positive d, w e k n o w that lal < d/2 and Ibt < d/2. Then, la + b I -< lal + Ibl < d. T h i s p r o v e s P a r t (1). P a r t (2) is p r o v e d similarly. F o r P a r t (3), b e c a u s e Icl < r for s o m e real r, a n d lal < d/r for all positive d, w e h a v e lacl < d. F o r P a r t (4), o b s e r v e t h a t b e c a u s e lel < lal < d, Icl
and c <45. Then, an + 4

=

3bn, f r o m s o m e p o i n t on,

and

P r o p o s i t i o n 2. I f a ~- r a n d b ~- s, then (1) a+b~r+s. (2) a - b ~ r - s. (3) a b ~ rs. (4) If a -< s, t h e n r -< s. (5) a + b ~ r/s, i f s r O.

Cn < 45, f r o m s o m e p o i n t on. B e c a u s e t h e t w o are true t o g e t h e r f r o m s o m e p o i n t on, w e c a n a d d t h e m to get an + 4 + Cn < 3bn + 45, from s o m e p o i n t on. In o t h e r words, a+4+c<3b+45. W h a t w e are doing, w i t h this c o n s t r u c t i o n , is t a k i n g the e q u i v a l e n c e c l a s s e s o f s e q u e n c e s u n d e r t h e r e l a t i o n "equality from s o m e p o i n t on." To p r e s e n t it in t h a t way, however, w o u l d e n t a i l e x c e s s i v e formalism. O n e of m y purp o s e s is to d e m o n s t r a t e that t h e r e is a fairly s i m p l e a p p r o a c h to i n f m i t e s i m a l s , o n e that c a n r e a s o n a b l y b e pres e n t e d to o r d i n a r y c a l c u l u s s t u d e n t s .

Infinitely Small and Infinitely Close Definition. A for all p o s i t i v e write a ~ 0. If or bounded. A a~0.

s e q u e n c e a is i n f i n i t e l y s m a l l if lal < d real n u m b e r s d. F o r infinitely s m a l l a, we lal < r for s o m e real r, w e s a y a is f i n i t e s e q u e n c e a is i n f i n i t e s i m a l if a r 0 a n d

These d e f i n i t i o n s hide quantifiers. W h e n w e s a y that a r 0, w e are a c t u a l l y s a y i n g that t h e r e is a k s u c h t h a t an r 0 for all n --> k. Similarly, this d e f m i t i o n o f a ~ 0 is, in reality, the m o r e f a m i l i a r a n d c o m p l i c a t e d Ve > 0 3k, n ~ k

lanl

<

e.

(}8

THE MATHEMATICAL INTELLIGENCER

PROOF. P a r t s (1) a n d (2) f o l l o w easily from P r o p o s i t i o n 1. F o r P a r t (3), n o t e t h a t ( a - r ) s , (b-s)r, and (a - r ) ( b - s) are all ~ 0. Adding these gives us a b - rs ~ O. F o r P a r t (4), if r > s, t h e n w e w o u l d have b o t h a n -< s f r o m s o m e p o n t o n a n d l a n - r I < ( r - s)/2 f r o m s o m e p o i n t on, w h i c h is i m p o s s i b l e . In v i e w of Part (3), w e n e e d only s h o w lfo ~ 1/s to establish Part (5). F r o m s ~ b, w e get Is~21 ~ Ib/21 < Ibl, so b y P a r t (4), Is~21-< Ibl. T h e n I1/b - 1/s I = I(s - b ) / b s I -< (1/Is/21)(1/Isb]s- b I. By P r o p o s i t i o n 1, this is infinitesimal. 9 This is all we n e e d to get started.

Elementary Calculus D e f i n i t i o n 1. A f u n c t i o n f i s c o n t i n u o u s at x = r, r real, iff a~rimpliesf(a)~f(r). Equivalently, f is c o n t i n u o u s at r if A x ~ 0 i m p l i e s f ( r + A x ) - f ( r ) ~ O. We c a n also d i s c u s s c o n t i n u i t y at a s e q u e n c e a, as opp o s e d to a real r, b u t c o n t i n u i t y o n a set X c o r r e s p o n d s to c o n t i n u i t y at all real n u m b e r s in X. As in n o n s t a n d a r d analysis, c o n t i n u i t y at all p o i n t s (real n u m b e r s a n d s e q u e n c e s ) in a set X is e q u i v a l e n t to u n i f o r m continuity. I will p r o v e this later.

P r o p o s i t i o n 3. The s u m , difference, p r o d u c t , a n d quo-

D e f i n i t i o n 3. a is a s u b s e q u e n c e of b (written a C b) if

t i e n t ( w h e n the d i v i s o r is n o n z e r o ) of f u n c t i o n s c o n t i n u o u s at x = r are c o n t i n u o u s at x = r.

every term of a is a term of b; more precisely, if there is an increasing function, k : N ~ N such that a n = bk(n).

This follows easily from Proposition 2.

P r o p o s i t i o n 6. L e t a C b be given.

D e f i n i t i o n 2. For a f u n c t i o n f a n d a real r, we s a y f ' ( r ) =

(1) I f b is i n f i n i t e s i m a l , then so is a. (2) I f b ~ r, t h e n a ~ r. (3) I f b s a t i s f i e s a g i v e n e q u a t i o n o r i n e q u a l i t y , then so does a.

d iff for all infinitesimal Ax, f ( r + Ax) - f ( r )

~ d. The proof of this is routine.

Ax Here is what the c o m p u t a t i o n looks like of the s t a n d a r d first example: f ( x ) = x 2. (x + Ax) 2 - x 2

P r o p o s i t i o n 7. ( T h e Chain R u l e ) . I f y = f i x ) a n d z = g(y), t h e n d z / d x = ( d z / d y ) 9 ( d y / d x ) .

x 2 - 2xAx + Ax 2 - x 2 PROOF. For Ax infinitesimal, d y / d x ~ Ay/Ax, where Ay = f ( x + A x ) - f i x ) . By Proposition 4, Ay ~ 0 , so seemingly A z / A y --~ dz/dy, where Az = g ( y + Ay) -- g(y). But there is another way to write Az, since g(y + Ay) - g(y) = g ( f ( x ) + Ay) - g ( f ( x ) ) = g ( f ( x + Ax)) - g ( f ( x ) ) , so dz/dx ~ Az/h,x.

Ax 2xAx + AX 2 Ax

Ax

= 2x + Ax ~2X.

P r o p o s i t i o n 4. I f f is d i f f e r e n t i a b l e at r, it is c o n t i n u o u s at r.

Putting this together, Az Ax

dz dx

PROOF. Just multiply f(r + Ax) - f(r) ~ d

and

A x --~ O

Ax to get f ( r + Ax) - f ( r ) ~ 0.

9

The proofs of the differentiation rules are simple. The p r o d u c t rule is typical:

P r o p o s i t i o n 5. I f the f u n c t i o n h is the p r o d u c t o f differ-

Step functions are simply functions that are piece-wise constant. A s e q u e n c e of step functions, s, is a sequence where Sn is a step f u n c t i o n from some p o i n t on.

if(x) + Af)(g(x) + Ag) - f ( x ) g ( x )

Ax _ f(x)Ag + g(x)Af + AfAg Ax

D e f i n i t i o n 5. I f s is a step f u n c t i o n o n [a0, an] w i t h s ( x ) = ci on each s u b i n t e r v a l , (ai, ai+l), i = 0 , . . . , n - 1, then

(x) Af + A f A g = f(x) Ax + g

Ax

Ax

--~f ( x ) 9 g ' ( x ) + g ( x ) . f ' ( x ) + 0 9 g ' ( x ) =f(x) 9g'(x) + g(x) .f'(x).

n

f? 9

The proof of the Chain Rule, omitted or b a n i s h e d to the a p p e n d i c e s in virtually all texts today, is easy, b u t we n e e d to discuss subsequences.

For integration, we can use s e q u e n c e s of step functions.

terval, [a, b], i f i t is defined on the i n t e r v a l a n d c h a n g e s value o n l y a f i n i t e n u m b e r o f t i m e s o v e r the interval.

Thus,

Ax

9

D e f i n i t i o n 4. A f u n c t i o n s is a s t e p f u n c t i o n o n a n i n -

= h ( x + Ax) - h ( x ) = f ( x + Ax)g(x + Ax) - f ( x ) g ( x ) = (f(x) + Af)(g(x) + Ag) - f ( x ) g ( x ) .

Ah

dy dx"

d z = d z . d y. dx dy dx

+ g'f.

PROOF. Writing Af f o r f ( x + Ax) - f ( x ) and similarly for g a n d h, we have f ( x + Ax) = f ( x ) + Af, so

Ay ~ d z Ax dy

The only difficulty with this is that Ay, while infinitely close to 0, m a y n o t be infinitesimal b e c a u s e it may equal 0 infinitely often. In that case, we can't claim that d z / d y .~ Az/Ay, b e c a u s e the definition of derivative requires A y e 0. But then, let Ax* C Ax be the s u b s e q u e n c e such that the c o r r e s p o n d i n g Ay* is a sequence entirely composed of O's. Then, d y / d x ~ Ay*/Ax* = 0. And, b e c a u s e Ay* = 0, the corresponding Az* = g ( y + A y * ) - g ( y ) is also 0, so d z / d x ~ Az*/Ax* = 0. Thus, once again,

e n t i a b l e f u n c t i o n s f a n d g, t h e n h i s d i f f e r e n t i a b l e a n d h' =f'g

Az Ay

0

s(x)dx

1

~ . ci(ai+l - ai). i=0

D e f i n i t i o n 6. F o r a f u n c t i o n f a n d a n i n t e r v a l [p, q],

f; f c~ = r

VOLUME 21, NUMBER 1, 1999 6 9

i f f there are s e q u e n c e s o f step f u n c t i o n s [p, q] w i t h

s;

ddx

~ r ,~.

s;

d - < f - < u on

udx.

We continue in this way, finding d2 such that infinitely m a n y t e r m s begin " k . d l d 2 " a n d choosing c3 so t h a t it begins "k.dld2," and so on. W h e n w e are done, w e have a subs e q u e n c e c C a, and a real number, r = k.dld2dad4.. 9 and b y construction,

As usual, d -< f - < u simply m e a n s that dn -< f - < u,, on [p, q] from s o m e p o i n t on, and fpq d d x is the sequence {fq dn dx}. The integral, r, is unique, for if di, ui, a n d ri satisfy the conditions a b o v e for i = 1, 2, then rl ~

l;

dl d x -<

s;

u2 d x ~ r2 ~

s;

d2 d x ~

s;

u 1 d x ~ rl.

The basic t h e o r e m s on integrals are easily proved. The following is an example:

Proposition 8. I f f i s integrable over [a, b] a n d [b, c], then it i s integrable o v e r [a, c] a n d

;a

f dx=

f dx +

f;

f dx.

d-
fa

dab d x 4-

f; ;a

dbc d x --~

Uab d x q'-

f;

Ubc d X -~

U dx.

9

hnall$is The o u t s t a n d i n g p o w e r of Robinson's n o n s t a n d a r d analysis is evident in t h e n o n s t a n d a r d p r o o f o f t h e o r e m s , such as the I n t e r m e d i a t e Value Theorem, a n d the integrability of continuous functions. We can do that h e r e too, a n d the proofs are startlingly similar. Our tool will be the nonn o n s t a n d a r d equivalent of "every fmite n o n s t a n d a r d numb e r is infinitely close to a real." The following is effectively the B o l z a n o - W e i e r s t r a s s theorem.

Proposition 9. I f a i s f i n i t e , then f o r s o m e c C a a n d s o m e real r, c ~ r. PROOF. As a is finite, there is s o m e d s u c h t h a t la! < d. That m e a n s there are only a finite n u m b e r o f possibilities for the integer p a r t o f e a c h an. One o f these possibilities m u s t occur an infinite n u m b e r of times. Let k b e such t h a t for infinitely m a n y an, the integer p a r t of an is k. Let cl be the first term in a with integer p a r t k. Now of the infinitely m a n y {an} having integer p a r t k, there are only 10 possibilities (0, 1, 2 , . . . , 9) for the first digit after the d e c i m a l point. One of t h o s e possibilities m u s t o c c u r an infinite n u m b e r of times. Let dl b e such a digit. Let c2 be the first t e r m in a after ct, w h i c h begins "k.dl."

70

THE MATHEMATICALINTELLIGENCER

Thus, for any q > 0, Icn - r I < q from s o m e p o i n t on. That m e a n s Ic - r I ~ O, or c ~ r. 9

Proposition 10 (Intermediate Value Theorem). I f f i s c o n t i n u o u s on [p, q] a n d f ( p ) -< s -
PROOF. Let dab , Uab , dbc, Ubc b e the step-functions witnessing the integrability o f f over [a, b] a n d [b, c]. Define d and u on [a, c] b y gluing t o g e t h e r the r e s p e c t i v e l o w e r and upp e r functions. Then, w e certainly have

d dx =

Icl - r I < 1, Ic2 - r] < 0.1 Ic3 - rI < 0.01

p =X0 ~Xl

~ X 2 ~ "'" ~ X n

= q,

equally spaced, a d i s t a n c e o f (q - p ) / n apart. As f(x0) -< s <-f(Xn), there m u s t be t w o a d j a c e n t points, Xk and Xk+l s u c h that f(xk) -< s -
9

Proposition 11 (Extreme Value Theorem). I f f i s cont i n u o u s on [p, q], then f a t t a i n s a m a x i m u m

on [p, q].

PROOF: I will define a single s e q u e n c e a in [p, q]. F o r n, divide [p, q] as in the p r e v i o u s proof. Let an be the division p o i n t Xk for whichf(Xk) is greatest. Choose c C a and r so that c ~ r. I claim t h a t f r e a c h e s its m a x i m u m at x = r. To s e e this, t a k e any s in [p, q]. F o r m sequence b b y c h o o s i n g bn for e a c h n to be the division p o i n t Xk n e a r e s t to s. We have that b ~ s, as ]bn -- SI < (q -- p)/n, so Ibn - sl < w from s o m e p o i n t on for any positive w. We also have, b y t h e c o n s t r u c t i o n of a, t h a t f ( b ) -< f ( a ) . Now, let d be the s u b s e q u e n c e of b c o r r e s p o n d i n g to e. As before, f ( d ) -< f ( c ) a n d d ~ s. So, b y continuity, f ( s ) f ( d ) --
Proposition 12. I f f i s c o n t i n u o u s on [p, q], t h e n f i s integrable on [p, q]. PROOF. F o r any n, let In b e t h e step-function f o r m e d b y partitioning [p, q] into n equal subintervals and setting In(x)

:IGURE

on e a c h subinterval to b e the m i n i m u m value o f f (guara n t e e d to exist b y P r o p o s i t i o n 11). Similarly, define Un as t h e m a x i m u m value o f f . We have that I -<-f-< u. B e c a u s e f is b o u n d e d , t h e sequences L = fq 1 d x a n d U = fpq u d x are b o u n d e d , s o t h e r e is a s u b s e q u e n c e l* C 1 a n d real r such that the c o r r e s p o n d i n g L* = f~ l* d x ~ r. Let u* b e the s u b s e q u e n c e of u c o r r e p s o n d i n g to l*, and let, for each n, Ax~ b e t h e length of the c o r r e s p o n d i n g subinterval. I claim t h a t fq u* d x ~ r and so fq f d x = r. P r o o f o f claim. F o r any n, w e can find the g r e a t e s t difference, Rn, b e t w e e n In(x) a n d Un(X) on [p, q], a n d w e have that

s;*

Un d x --

s;"

In d x <- Rn(q - p).

The difference, Rn, can be r e p r e s e n t e d as If(Sn) - f(tn)l, for s o m e Sn and tn with ISn - tn] <-- AXn. Then,

s;

u* d x -

s;

l* d x - S ( q - p ) = ~f(s) - f ( t ) l ( q

- P).

u*dx-

less of our s e q u e n c e s than one a s k s o f numbers. They aren't totally o r d e r e d , for example. The s e q u e n c e s 0, 1, 0, t, . . . and 1, 0, 1, 0 , . . . are incomparable. T h e r e are also zero divisors. This d o e s n ' t c a u s e any p r o b l e m s . W h y d i d n ' t I n e e d t h e T r a n s f e r P r i n c i p l e ? The Transfer Principle is a powerful s c h e m a of n o n s t a n d a r d analysis that says that any s t a t e m e n t (in a particularly rich, well-defmed language) that is true a b o u t the real n u m b e r system is true a b o u t the hyperreal n u m b e r s y s t e m and vice versa. We have here a w e a k form of this, n a m e l y that all equations a n d inequalities true a b o u t reals are true a b o u t sequences. We also have c o n j u n c t i o n s o f these (but not disjunctions). In e l e m e n t a r y calculus a n d analysis, t h e Transfer Principle is u s e d chiefly to prove that the n o n s t a n d a r d defmitions are equivalent to the s t a n d a r d definitions. But, here, t h e s e equivalences are easy. H e r e ' s an example: Definition

7. f is u n i f o r m l y c o n t i n u o u s on C i f f

(Standard)

We w o u l d like to s a y that as is - tl -< A x ~ O, then, b y continuity, If(s) - f ( t ) l ~ O, so

P

H o w D i d W e Do It?. H o w d i d I a v o i d t h e A x i o m o f C h o i c e ? I simply a s k e d

s;

l*dx~O

and

s;

l*dx~r.

But continuity requires t h a t one o f s and t b e real. The p r o p e r t y s ~ t ~ f ( s ) ~ f ( t ) is actually equivalent to uniform continuity. We can easily w o r k around this, h o w e v e r , by finding a s u b s e q u e n c e s** C s and real r, with s** -~ r a n d using the c o r r e s p o n d i n g s u b s e q u e n c e s u**, 1"*, a n d t** ~ r ~ s** to finish the proof. 9

Ve > 0 38 > 0, Vx, y E C Ix - Yl < ~ ~ If(x) - f(Y)t < ~. ( N o n - n o n s t a n d a r d ) Va, b E C, a = b ~ f ( a ) ~ f ( b ) . 13. The s t a n d a r d a n d the n o n - n o n s t a n d a r d d e f i n i t i o n s o f u n i f o r m c o n t i n u i t y are equivalent.

Proposition

PROOF. S u p p o s e the n o n - n o n s t a n d a r d definition holds, and s u p p o s e w e a r e given an e for w h i c h t h e r e is no suitable & Then, for e a c h n a t u r a l n u m b e r n, c h o o s e an and bn such that la.n - bnI < 1/n, b u t If(an) - f(bn)I >- ~. Then, w e have a -~ b, b u t f ( a ) r f ( b ) , a contradiction.

:IGURE ;

VOLUME 21, NUMBER 1, 1999

71

Suppose, now, that the s t a n d a r d definition holds, and we are given a ~ b and d, a positive real. By the s t a n d a r d definition, t h e r e is a 6 such that Vx, y Ix - y] < 8 ~ If(x) f(y)] < d. Then, b e c a u s e ]a~ - b~] < 8 from s o m e p o i n t on, If(an) -f(b,~)] < d from s o m e point on, a n d so f ( a ) ~ f(b).

What happened to the Axiom of Completeness? I did use the c o m p l e t e n e s s of the real line, b u t in a m o s t inn o c u o u s a n d c o m p r e h e n s i b l e form: I s i m p l y a s s u m e d that every infinite d e c i m a l c o r r e s p o n d s to a real number. 1 W h a t h a p p e n e d t o a l l t h e q u a n t i f i e r s ? In the case of uniform continuity, for example, I w e n t from " r e 3 8 Vx, y . . . " (logicians call this a II3 s t a t e m e n t ) to "Va, b , . . . " (a II1 statement). S o m e of the quantifiers a r e buried. The s t a t e m e n t "a ~ b" is, in reality, s o m e t h i n g like "Vr > 0 lan - bnl < r from s o m e point on," or "Vr > 0 3k Vn > k . . . . " Essentially, I simplify definitions b y coding up the m o s t difficult part. One can also ask, W h a t is n e x t ? There are t h e o r e m s of analysis w h e r e n o n s t a n d a r d p r o o f s are a w k w a r d or do not exist. In the f o r m e r category is the t h e o r e m that the uniform limit of c o n t i n u o u s functions is continuous. This is p r o v e d in [HK] b y taking a n o n s t a n d a r d m o d e l of a nons t a n d a r d model. It can b e done in the p r e s e n t s y s t e m by taking s e q u e n c e s of sequences. Indeed, b y closing under the o p e r a t i o n of "taking sequences," a great deal m o r e analysis can b e handled. In [He], this is p u r s u e d to prove the Baire C a t e g o r y Theorem, for w h i c h no simple nons t a n d a r d p r o o f exists, and to develop m e a s u r e theory.

Pedagogy The n o n - n o n c o n s t r u c t i v e infmitesimals p r e s e n t e d here could i m p r o v e the teaching of calculus. In b o t h "standard" and "reform" calculus courses, rigor has b e e n a l m o s t entirely omitted. Consequently, s t u d e n t s are n o t a s k e d to prove t h e o r e m s until they have a fairly strong intuition for the subject a n d have m e t infinite sequences. This background m a k e s n o n - n o n s t a n d a r d analysis v e r y attractive for, say, an A d v a n c e d Calculus course, o r even Calculus III. F o r t h o s e wishing to r e m a i n as "standard" as possible, one can still use sequences. The chief a d j u s t m e n t is to rep l a c e a ~ 0 with a --~ 0. The definitions and p r o o f s in this article are easily converted. This a p p r o a c h is being used n o w in [CH].

Some History The G r e e k s e x p l o r e d the ideas of infmity a n d infinitesimals. On the whole, t h e y rejected them. To Aristotle, there was no a b s o l u t e infinity. C o m p l e t e d infinite sets such as the natural n u m b e r s {1, 2, 3 , . . . } , did n o t exist. There could only be p o t e n t i a l infinities; for example, for every number, there is a n o t h e r n u m b e r that is larger, and so on. The distinction is similar to that b e t w e e n an infinitesimal n u m b e r

a n d the p o w e r to find e v e r smaller numbers. In a w e a k sense, it is the difference b e t w e e n inf'mitesimals a n d limits. A r c h i m e d e s used infinitesimals for intuition [A1], t h e n verified his results by proving t h e m with (what w e w o u l d call t o d a y ) limits [A2]. Calculus, as formulated in the seventeenth century, was first e x p r e s s e d in t e r m s of infinitesimals. As infinitesimals t h e m s e l v e s w e r e not well understood, there w e r e critic i s m s and misunderstandings. On the whole, however, the d o u b t s were o v e r s h a d o w e d b y the outstanding s u c c e s s of the t h e o r y as d e v e l o p e d in the eighteenth century. In the nineteenth century, Cauchy, Weierstrass, a n d others m a d e infinitesimals unnecessary. Absolute infinities w e r e r e p l a c e d by limits, n o w rigorously defined. But linguistic habits didn't change. Mathematicians a n d physicists c o n t i n u e d to talk in t e r m s o f infinitesimals. Infinitesimals d i d n ' t d i s a p p e a r from calculus texts for over 100 years. Infinitesimals b e g a n to r e a p p e a r in the t w e n t i e t h century. The i d e a of s e q u e n c e s as infinitesimals a p p e a r s in a r e m a r k a b l e book, The L i m i t s o f Science, O u t l i n e o f Logic a n d Methodology of S c i e n c e b y Leon Chwistek, painter, philosopher, and m a t h e m a t i c i a n [Ch]. Published in 1935, t h e b o o k is not well k n o w n t o d a y (it is in Polish). Chwistek's definitions are similar to those p r e s e n t e d here, although there are differences and limitations. His w o r k f o r e s h a d o w s not only n o n - n o n s t a n d a r d analysis, b u t nons t a n d a r d analysis. In [L1] and [L2], Laugwitz f o r m u l a t e d the s y s t e m o f seq u e n c e s d e s c r i b e d in this article, b u t with different notation. Laugwitz used his "~-Zahlen" to investigate distributions and operators. An earlier p a p e r by S c h m i e d e n and Laugwitz [SL] used a m o r e primitive system with the i d e a o f justifying the infmitesimals of Leibniz. Laugwitz's w o r k w a s n o t carried further, p o s s i b l y b e c a u s e the d i s c o v e r y of n o n s t a n d a r d analysis m a d e real infinitesimals unglamorous.

In 1960, Abraham Robinson constructed n o n s t a n d a r d m o d e l s of the real n u m b e r system using mathematical logic [R]. Robinson credits the p a p e r s of Laugwitz and Schmieden with s o m e inspiration for his work. N o n s t a n d a r d analysis requires a substantial i n v e s t m e n t ( m a t h e m a t i c a l logic and the A x i o m of Choice) b u t p a y s great dividends. Nons t a n d a r d analysis has b e e n u s e d to discover n e w t h e o r e m s o f analysis. It has b e e n fruitfully applied to m e a s u r e theory, B r o w n i a n motion, a n d e c o n o m i c analysis, to n a m e j u s t a few areas. A t t e m p t s to r e f o r m calculus instruction along infinitesimal lines, however, did not have m u c h s u c c e s s [HK], [K]. Robinson's b o o k c o n t a i n s an excellent h i s t o r y of infinitesimals. There are other systems of standard infmitesimals, Conway's surreal numbers, for example [Co]. There are other systems for avoiding e's and fi's (see [Hi] for a recent example). There may also be other rediscoveries of this system. The intended contribution of the present article is to place the structure in an algebra suitable for students of calculus.

1This unremarkable statement is accepted easily by students. It is not equivalent to Completeness, but is stronger, and yields the Archimedean Principle as well.

72

THE MATHEMATICALINTELLIGENCER

REFERENCES

Acknowledgments I would like to thank Michael Henle, Ward Henson, Roman Kossak, Dan Velleman, and several anonymous reviewers for careful reading and helpful remarks.

[A1] Archimedes, The Method of Archimedes, recently discovered by Heiberg, edited by Sir Thomas L. Heath, Cambridge, Cambridge University Press (1912). [A2] Archimedes, "The quadrature of the parabola, in The Works of Archimedes, edited by Sir Thomas L. Heath, Cambridge: Cambridge University Press (1897). [C] Chandler, G. H., Elements of the Infinitesimal Calculus, 3rd ed., New York: John Wiley & Sons (1907). [Ch] Chwistek, L., The Limits of Science, Outline of Logic and Methodology of Science, Lwow-Warszawa: Ksiaznica-Atlas (1935). [CH] Cohen, D., and Henle, J. M., Conversational Calculus, Reading, MA: Addison-Wesley/Longman (in press). [Co] Conway, J. H., On Numbers and Games, New York: Academic Press (1976). [He] Henle, J. M., Higher non-nonstandard analysis: Category and measure, preprint. [HK] Henle, J. M., and Kleinberg, E. M., Infinitesimal Calculus, Cambridge, MA: M.I.T. Press, (1979). [Hi] Hijab, O., Introduction to Calculus and Classical Analysis, New York: Springer-Verlag, (1997). [K] Keis~er, H. J., Elementary Calculus, An infinitesimal Approach, 2rid ed., Boston: Prindle, Weber & Schmidt (1986). ILl] Laugwitz, D., Anwendungen unendlich kleiner Zahlen I, J. Reine Angew. Math. 207 (1960), 53-60. [L2] Laugwitz, D. Anwendungen unendlich kleiner Zahlen II, J. Reine Angew. Math. 208 (1961), 22-34. [R] Robinson, A., Nonstandard Analysis, Amsterdam: North-Holland, (1966). [SL] Schmieden, C., and Laugwitz, D., Eine Erweiterung der Infinitesimalrechnung, Math. Zeitschr. 69 (1958), 1-39.

m:,~,,t~,,,J-1

Jet

Wimp,

Editor

I

Erd6s on Graphs: His Legacy of Unsolved Problems by Fan Chung and Ron Graham WELLESLEY, MA: A. K. PETERS, 1998, 142pp US $30.00, ISBN: 1-56881-079-2 REVIEWED BY JOEL SPENCER

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Column Editor's address: Department of Mathematics, Drexel University, Philadelphia, PA 19104 USA.

74

N o w the legacy begins. Paul Erd6"s was a giant w h o s e mathematical career spanned a remarkable p r o p o r t i o n of our s o o n - t o - e n d century. F r o m csodagyerek (prodigy) to der Zauberer von Budapest (the magician f r o m Budapest), he r a p i d l y attained the very pinnacle of t h e m a t h e m a t i c a l world. There he s t a y e d for over half a century. Well past the n o r m a l retirem e n t age, he continued to follow his o w n famous a d m o n i t i o n - - P r o v e and Conjecture! At his d e a t h in 1996 he h a d j u s t c o m p l e t e d a Graph T h e o r y confere n c e in Warsaw and w a s p r e p a r i n g for a Probabilistic Number T h e o r y meeting in Vilnius that was to begin the following day. Another roof, a n o t h e r proof. A few w e e k s later m a t h e m a t i c i a n s c o n v e r g e d on B u d a p e s t for his funeral. Vera S6s, his longtime friend and collaborator, m a d e the a p p r o p r i a t e arrangements. Sixteen mathematicians s p o k e on the influence on their fields o f his work: V6rtesi on A p p r o x i m a t i o n Theory, F r e i m a n on S u b s e t Sums, Wirsing on Additive Functions, Nicolas on Partitions, Lovfisz on Set Systems and H y p e r g r a p h s - - t h e b r e a d t h of Erd6"s's contributions w a s astounding. Uncle Paul was the m o s t social of mathematicians. F o r him m a t h e m a t i c s w a s always a j o i n t endeavour. We were all s e a r c h e r s for The B o o k Proof. He w a s c o n s t a n t l y asking questions, and e a c h question, w h e n a n s w e r e d , w o u l d e x p a n d the b o u n d a r i e s o f o u r knowledge. S o m e t i m e s he a t t a c h e d a monet a r y r e w a r d for the solutions. The a m o u n t s were small, b u t the incentive to solve an Erd6"s p r o b l e m w a s enor-

THE MATHEMATfCALINTELLIGENCER9 1999 SPRINGERVERLAGNEWYORK

mous. Paul didn't state his c o n j e c t u r e s in a vacuum. When talking to a mathe m a t i c i a n - - e s p e c i a l l y , as this r e v i e w e r well r e m e m b e r s , a young m a t h e m a t i cian j u s t beginning to feel his o r h e r a b i l i t i e s - - h e w o u l d ask the questions that w e r e a p p r o p r i a t e for that person. E r d d s w o u l d c h o o s e the p r o b l e m carefully so t h a t the listener w o u l d have s o m e h o p e o f solving it. This s e d u c t i v e a p p r o a c h b r o u g h t m a n y into his circle. Once t h e y w e r e in they b e c a m e totally d e v o t e d to Paul. His p r o b l e m s b e c a m e their p r o b l e m s . "On a P r o b l e m o f Erd6"s" c o u l d well have b e e n the title for t h o u s a n d s of papers. We s e e n o w that the p r o b l e m s of Paul Erd6"s w e r e n o t r a n d o m l y chosen. Through their posing and their solutions entire a r e a s of m a t h e m a t i c s c a m e into being. N o w after his d e a t h it is time to t a k e s t o c k - - t o e x a m i n e his m a t h e m a t i c a l legacy. Erd6"s on Graphs is an e x c e l l e n t p l a c e to start. Graph t h e o r y (and, m o r e generally, Discrete Mathematics, which is well c o v e r e d in this v o l u m e ) fascin a t e d E r d 6 s from early days. The beginning might have b e e n the w i n t e r of 1931/2 in Budapest. E s t h e r Klein, fresh from a s u m m e r ' s stay in Gottingen, p r e s e n t e d an intriguing result: given any five p o i n t s in general p o s i t i o n in the plane, s o m e four form the vertices of a c o n v e x polygon. With h e r friends George Szekeres and the t e e n a g e r Paul E r d d s a generalization s o o n emerged: s h o w t h a t for all n there exists a g(n) so that a m o n g s t any g(n) v e r t i c e s in general p o s i t i o n in the p l a n e s o m e n form t h e vertices of a c o n v e x polygon. Both Szekeres and Erd6"s c a m e up with p r o o f s - - S z e k e r e s r e d i s c o v e r i n g R a m s e y ' s T h e o r e m and Erd6"s proving that any sequence of length n 2 + 1 has a monotone subsequence of length n § 1. Erd6"s r e f e r r e d to these results as the "Happy Ending Theorem" since George a n d E s t h e r later married. This r e v i e w e r h a d the j o y of s p e a k i n g on R a m s e y n u m b e r s in Sydney in March

1998 with the two of t h e m actively participating in the discussion. The b e s t value for g(n) remains an o p e n p r o b lem, with s o m e very r e c e n t results inc l u d e d in this volume. What is the m a x i m a l n u m b e r t(n, C2k) of edges that a graph on n vertices can have without containing a cycle of length 2k? F o r k = 2, 3, and 5, excellent b o u n d s are known. In general c k n l+l/k edges do force a cycle, and Erd6"s c o n j e c t u r e d in 1963 t h a t t h e r e w e r e g r a p h s with c'kn l+l/k e d g e s with no such cycle. To their credit Chung a n d G r a h a m give n o t only t h e conjecture. They place it in its historical c o n t e x t - - a central question in the d e v e l o p m e n t of E x t r e m a l G r a p h T h e o r y - - a n d give r e f e r e n c e s to partial results as r e c e n t as 1995. This b o o k is b e s t r e a d nonlinearly. H o w m a n y r-sets, a s k e d E r d d s and his long-time c o l l a b o r a t o r R i c h a r d Rado in 1960, n e e d there be so that s o m e t h r e e A, B, C of t h e m have A V1B = A A C = B A C - - w h a t is called a A-system? They s h o w e d 2rr! suffice, b u t d e s p i t e m u c h intervening w o r k their conjecture that c r sets suffice (c an a b s o l u t e c o n s t a n t ) remains open. Flipping s o m e m o r e pages: w h a t is t h e m a x i m u m n u m b e r o f vertices r(k) that a graph c a n have w h e n it c o n t a i n s n e i t h e r a clique n o r an i n d e p e n d e n t s e t o f size k? In 1947 Erd6"s s h o w e d s u c h graphs exist with roughly 2k v e r t i c e s b y [in m o d e r n language] showing t h a t the r a n d o m graph had this p r o p e r t y . The p r o b a b i l i s t i c m e t h o d w a s born. The u p p e r bound, using the p r o o f of R a m s e y ' s Theorem, is roughly 4 k. I m p r o v e m e n t s on u p p e r a n d l o w e r b o u n d s are carefully described. These u n f o r t u n a t e l y have only affected l o w e r - o r d e r terms. The value c = lim r(k) uk r e m a i n s o p e n a n d c o u l d be a n y w h e r e in [2,4]. Even the e x i s t e n c e of the limit is n o t known. The first p o p u l a r b o o k s on E r d d s ' s life are a l r e a d y at the b o o k s t o r e s . There will b e more to come. Paul's fierce d e d i c a t i o n to his subject, his rej e c t i o n of worldly goods, his inspirational qualities, his p e r i p a t e t i c life style, a n d his generosity all m a k e him

a suitable o b j e c t of such studies. It is for t h o s e o f us in the m a t h e m a t i c a l world to p r e s e r v e his t h e o r e m s a n d conjectures, to j u d g e his vast influence on o u r subject. Erdds on Graphs is a w o r t h y s t e p in that direction. May it have m a n y successors. Courant Institute of Mathematical Sciences New York University New York, NY 10012-1110 USA e-mail: [email protected]

Introduction to Combinatorics by Martin J. Erickson NEW YORK: JOHN WILEY AND SONS, 1996. xi + 195 PP. US $59.95 (paperback), ISBN 0-471-15408-3 REVIEWED BY JET WlMP

he r e g n a n t institution of higher learning in the o b s c u r e h a m l e t of Kirksville, MO has u n d e r g o n e m o r e transmutations than the legendary Vicar of Bray. Originally, it was North Missouri Normal School and Commercial College; then it b e c a m e Kirksville N o r m a l College, t h e n First District N o r m a l School, t h e n N o r t h e a s t Missouri State T e a c h e r s College, then N o r t h e a s t Missouri State College, then N o r t h e a s t Missouri State University, and now, w e h o p e fmally, T r u m a n State University. In its reincarnations, it has a c h i e v e d a sort o f flowering, b e c o m i n g the h o m e of two highly r e g a r d e d literary p e r i o d icals, 1 a n d it is n o w r a t e d fourth o f all institutions in the nation b y Money Magazine in t e r m s of b e s t e d u c a t i o n for the buck. It w a s j u s t a m a t t e r of time b e f o r e t e x t b o o k s , like this one, b e g a n to issue from T r u m a n State University. This excellent text s h o u l d prove a useful a c c o u t r e m e n t for any developing m a t h e m a t i c s program. The p r o b l e m s with writing a t e x t on c o m b i n a t o r i c s - - - o r any other s u b j e c t - are basically two: (1) keeping the b o o k both c l e a r a n d s h o r t and (2) m a k i n g the m a t e r i a l current. I have taught combinatorics several times; choosing a t e x t has always b e e n

T

a source of frustration. Texts, as they m e a n d e r through several editions, tend to gigantism, ending up engorged with material. What g o o d is 500 pages of information to a h a r r i e d instructor planning a course of one term or semester? I have noticed, too, that authors as they segue from one edition to another suecumb to pretentiousness, so that the statements of basic results b e c o m e insufferably abstract; s o m e of the available versions of the pigeonhole principle, for example, are ludicrously ethereal. The b o o k I like the best, W i l l s generatingfunctionology, doesn't provide the plenary selection of topics neeessary for a well-balanced course, although the b o o k is s o well written that I have sometimes m a d e it serve as a text by incorporating additional material. It a p p e a r s that with t h e p r e s e n t book, I can stop looking, at least for awhile. It's short, it's sweet, it's beautifully written; it c o n t a i n s material on two cutting-edge disciplines, Ramsey Theory and c o d i n g theory, that will give the s t u d e n t a hint of w h a t ' s really going on in c o m b i n a t o r i c s t h e s e days. I think it's the b e s t c o m b i n a t o r i c s t e x t available. It also h a s its o w n peculiar and satisfying fillips. The b o o k sets o u t with a nonstand a r d selection of preliminaries: sets and group theory, predictably, but then material on fields, n u m b e r theory, a n d linear a l g e b r a - - u n u s u a l . The author p r e p a r e s sedulously: information on quadratic residues, for example, is imp o r t a n t for R a m s e y theory. The b o o k is divided into t h r e e parts: Part I, Existence; P a r t II, Enumeration; Part III, Construction. A clever and original trichotomy. In Chapter 2, t h e a u t h o r introduces the pigeonhole principle in an elegant b u t not u n n e c e s s a r i l y a b s t r a c t formulation. There follow examples. This is one of the b o o k ' s m a n y strengths: each t h e o r e m is f o l l o w e d b y one, two, sometimes m o r e e x a m p l e s . Chapter 3 deals with sequences a n d partial orders, including the f a m o u s E r d d s - S z e k e r e s T h e o r e m - - a n y s e q u e n c e of n 2 + 1 real n u m b e r s contains a m o n o t o n i c subsequence o f n + 1 terms. (Actually, the

1Chariton Review and Paintbrush. Kirksville's other claims to fame are that it is the home town of the actress Geraldine Page and of Andrew Taylor Still, the founder of osteopathy (now indistinguishable from medicine). I know all these things because Kirksville is my own home town.

VOLUME21, NUMBER 1, 1999 75

a u t h o r p r o v e s a useful generalization.) The material in this c h a p t e r naturally segues into R a m s e y theory; as the aut h o r asserts, The Erd6"s-Szekeres T h e o r e m a n d Dilworth's l e m m a . . , guarantee the existence of p a r t i c u l a r substructures o f certain combinatorial configurations. In other words, they say that large random systems contain nonrandom subsystems. We continue this t h e m e by presenting t w o c o r n e r s t o n e s of R a m s e y theory . . . . [my italics] Of course, this is true. It had never occ u r r e d to m e to see R a m s e y t h e o r y o r the E r d 6 s - S z e k e r e s result in this light. A g o o d text will a l w a y s t e a c h us something new. The a u t h o r b e g i n s the discussion o f Ramsey's t h e o r e m with a problem, n o w famous, from a P u t n a m competition:

Six points are in a general position in space (no three in a line, no f o u r in a plane). The fifteen line segments j o i n i n g them in pairs are drawn and then painted, some segments red, some blue. Prove that some triangle has all its sides one color. The a u t h o r t h e n s t a t e s and p r o v e s Ramsey's t h e o r e m and a n u m b e r of related results, including Schur's l e m m a involving m o n o c h r o m a t i c solutions to equations involving integers. Part II o f the b o o k o p e n s with C h a p t e r 5, The F u n d a m e n t a l Counting Problem. As the a u t h o r has noticed, Enumeration is p r o b a b l y the trickiest branch of combinatorics. Sadly, it is c o m m o n to h e a r students say, "I just can't count," or "I don't count," or "I count, but I always get two different answers." There is confusion about w h a t is being counted and there are m a n y formulas to remember. Although the situation is fraught with difficulty---even d e s p e r a t i o n - there is a solution. Surprisingly, one

general principle and a few variations suffice for nearly all enumeration problems . . . . The a u t h o r d o e s n ' t claim that his epistemological a p p r o a c h is original, b u t he advances the view of representing the objects to b e c o u n t e d as set-to-set mappings. This is i n d e e d a novel antidote for the s t u d e n t ' s malady, but it will take an i n o r d i n a t e l y selfdefined student to i n c o r p o r a t e this p o i n t of view into a w o r k i n g conception of combinatorics. Chapter 6 introduces the meat and p o t a t o e s of classical combinatorics, and here the student can take a breather: the inclusion-exclusion principle, Stirling numbers, Bell numbers, recurrence relations, generating functions. Chapter 7 discusses Permutations and Tableaux, and Chapter 8, P61ya's Theory of Counting, which allows one to do more complicated counting, including nonisomorphic graphs. P a r t III, Construction, o p e n s with C h a p t e r 9, Codes, a n d t h e n Chapters 10 a n d 11, devoted to designs. The b o o k is stuffed with challenging exercises, a n d with historical asides. Throughout, the a u t h o r mentions appealing u n s o l v e d problems. Unfortunately, there a r e no solutions to the exercises, although hints are occasionally given. Department of Mathematics and Computer Science Drexel University Philadelphia, PA 19104 USA e-mail: [email protected]

A Primer of Real Functions by Ralph P. Boas, Jr. FOURTH EDITION: REVISEDAND UPDATED BY HAROLD P. BOAS THE CARUS MATHEMATICALMONOGRAPHS NUMBER 13 WASHINGTON, DC: THE MATHEMATICAL ASSOCIATION OF AMERICA, 1996. XIV + 305 PP. US $43.95 (hardcover), ISBN 0-88385-029-X REVIEWED BY JET WIMP

W

' h a t m a k e s a b o o k a classic? Why will T i t c h m a r s h ' s old-

fashioned Theory of Functions, R o y d e n ' s Introduction to Analysis, Rudin's Principles of Mathematical Analysis, and this book, be around long after dim legions of other b o o k s have vanished into the dust? Well, as one of the ecdysiasts in the Jule Styne musical Gypsy so convincingly avers, "You have to have a gimmick." The gimmick in this b o o k - - I ' m talking now about the first edition, b y the elder Boas, published in 1960---is that it so completely reflects the tastes of its author, and thus t a k e s on the captivating qualities of a personal narrative. "Experts," Boas cautions in the introduction, "are not s u p p o s e d to read this b o o k at all; since this statement will doubtless be taken as an invitation for t h e m to do so, I must explain what I have tried and not tried to do. I have set out to tell readers with no previous experience of the subject s o m e o f the results I find particularly interesting." Very surprisingly, Boas p~re says nothing a b o u t the integral a n d n o t m u c h a b o u t measure. After a treatm e n t o f p r o p e r t i e s of sets in m e t r i c spaces, he has the tools at h a n d to sail into Baire theory, to m y m i n d o n e of the m o s t beautiful intellectual achievem e n t s in mathematics. When I t e a c h reals, Baire t h e o r y c o m p r i s e s a m a j o r p a r t of the syllabus, and m o s t o f the c o n s t r u c t i o n s I give c o m e from this book. Let m e review the basics. A set in a m e t r i c s p a c e that can be repres e n t e d as the union of c o u n t a b l y m a n y n o w h e r e d e n s e sets is called a s e t o f first c a t e g o r y (or, sometimes, a m e a g e r set). Otherwise, it is a set of s e c o n d category. Sets of first c a t e g o r y are tiny, t h o s e of s e c o n d category are huge. Baire's t h e o r e m says that a complete metric space is of second category. So w h a t ? the r e a d e r m a y w a n t to know. Almost everyone knows that there are functions continuous everywhere which n o w h e r e possess a derivative. It is almost impossible to visualize such a function; any visualization will obscure the fimction's essential properties, but one can think of the graph of such a h m c t i o n as c o m p o s e d of a series of inf m i t e s i m a l crinkles. 2 That such functions exist, indeed, exist in profusion, is

2It is easy to construct such functions from Fourier series the frequencies of whose harmonics grow exponentially.

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one of the consequences of Baire theory. One can show that most continuous functions---not just a small n u m b e r - nowhere possess a derivative (for such functions are a set of s e c o n d category in the space of continuous functions with the sup metric). This profoundly anti-intuitive result requires a delicate and ingenious construction, which is itself of great intellectual interest. Another easy result from Baire theory: l e t f ( x ) be infinitely differentiable on [0, 1], and suppose that at each point some derivative of f is 0. Then, f is a polynomial. It was by encountering such romantic results in Boas's b o o k that I first b e c a m e h o o k e d on Baize theory. The s e c o n d part o f B o a s ' s b o o k is d e v o t e d to functions: c o n t i n u o u s functions, sequences o f functions, a p p r o x imations, linear functions, derivatives (the old-fashioned Dini d e r i v a t e s that are rapidly disappearing from books), m o n o t o n i c functions, c o n v e x functions. Boas's discussion of linear functions is spellbinding. A f u n c t i o n f i s line a r if f ( x + y) = f ( x ) + f ( y ) . Any c o n t i n u o u s linear function m u s t be of the f o r m f ( x ) = cx, c a constant. There are o t h e r linear functions, b u t a disc o n t i n u o u s linear function is necessarily wildly discontinuous: it is unb o u n d e d in every interval, indeed, its g r a p h is dense in R 2. In a l a t e r c h a p t e r in t h e current edition, H a r o l d Boas u s e s discontinuous linear functions to c o n s t r u c t n o n m e a s u r a b l e sets. There is a quality t h e e l d e r Boas's b o o k s h a r e s with Titchmarsh's: the style is discursive r a t h e r t h a n formal. A l m o s t nothing is s t a t e d as a theorem, p r o o f s are chatty, definitions are emb e d d e d in the text. It is a l m o s t as if B o a s is giving us a t o u r o f his private bestiary, pointing out with relish the strange colors and s e l c o u t h physical a t t r i b u t e s of the i n h a b i t a n t s - - t h o s e functions of a real v a r i a b l e - - - c a g e d and on display. This style will n o t be to e v e r y o n e ' s taste: it is b o t h leisurely and eccentric. To t h o s e a c c u s t o m e d to the gatling-gun epistemic style o f Royden a n d Lang, it m a y be disorienting. It is difficult to dip into the b o o k a n d retrieve a result, since t h e o r e m s and explications are woven together, but it is p r e c i s e l y this quality t h a t m a k e s the original b o o k such a fascinating read.

Harold P. Boas has substantially enlarged (from 196 to 305 p a g e s ) his father's book. As far as I could see, m o s t of the original material is present. He has a d d e d m a n y m o r e r e f e r e n c e s - - r e f e r e n c e s w h i c h augment the m a t e r i a l and w h i c h p r o v i d e additional e x a m ples; he h a s e x p a n d e d and fleshed out solutions to the exercises; and he h a s p r o v i d e d a substantial s e c t i o n on Lebesgue m e a s u r e and integration, which m a k e s the b o o k m o r e consonant with c u r r e n t real analysis texts. A m i n o r point: I w o u l d have liked to s e e here S h a p i r o ' s sly p r o o f that a n y locally integrable linear function is of t h e form f ( x ) = cx. It goes as follows. In the e q u a t i o n f ( x + y) = f ( x ) + f ( y ) , rep l a c e y b y t, integrate with r e s p e c t to t from 0 to y, then apply e l e m e n t a r y p r o p e r t i e s o f the integral to get

(

x+y

y f ( x ) = ~o

f ( t ) dt ] : f ( t ) dt - S: f(t)dt.

The right-hand side is s y m m e t r i c in x and y, so interchanging the t w o gives yf(x) xf(y) or =

f(x) x

_

tained in E a n d a n open set G containing E such t h a t the m e a s u r e s of G and F differ b y less t h a n s. The canonical w a y o f doing this nowa d a y s is to t a k e Carath6odory's approach, first introducing the c o n c e p t of the o u t e r m e a s u r e , m*, of a set, a n d t h e n stating as a def'mition A set E is said to be m e a s u r a b l e if for e a c h s e t A, w e have m * ( A ) = m * ( A n E) + m * ( A - E ) ; see Royden, Introduction to Analysis, p. 56. There are t h o s e w h o will w e l c o m e an updating of B o a s ' s classic, but there are others, r o m a n t i c i s t s to the core, w h o will c o n s i d e r it a despoliation. I, for one, w e l c o m e the w e a l t h of additional material; a n d I think that the changes are r e s p e c t f u l of the timeless c h a r a c t e r of the original. Department of Mathematics and Computer Science Drexel University Philadelphia, PA 19104 USA e-maih [email protected]

f(y) - const., y -

-

which is the d e s i r e d conclusion. Thus if a n o n i n t e g r a b l e solution of the equation exists, it m u s t define sets in the d o m a i n o f the function which are n o t measurable. I f o u n d it b o t h curious a n d t o u c h i n g that H a r o l d Boas in his s u p p l e m e n t a r y material h a s m a d e an a t t e m p t to imitate the discursive style of his father r a t h e r t h a n to modernize. To do so m u s t have c a u s e d him trepidation; t h e following p a s s a g e is not the w a y t h a t m o s t of us e x p e c t to see the c o n c e p t of m e a s u r e introduced, n o r could this have b e e n the m a n n e r of Harold B o a s ' s inauguration into real analysis during his o w n Ph.D. program, 40-some y e a r s after his father's: A set E is called (Lebesgue) m e a surable if it can be arbitrarily well a p p r o x i m a t e d b o t h from o u t s i d e b y o p e n sets and from inside b y c l o s e d sets. More formally, E is m e a s u r able if, for every p r e s c r i b e d positive ~, t h e r e exists a closed set F con-

Hyperbolic Geometry and Barbilian Spaces by W.G. B o s k o f f PALM HARBOR, FL: HADRONIC PRESS, 1996, 169 PP. US $60.00, ISBN 1o57485-007-5

REVIEWED BY VICTOR PAMBUCCIAN

an Barbilian (1895-1961) is, through two of his papers, the most influential Romanian mathematician of the first half of this century. The first, a summary of a talk at a congress in Prague in 1934, is two pages long and contains no proofs. The second is a long p a p e r in two parts on the axiomatics of projective ring geometry, published in 1940 in Jahresbericht der Deutschen MathematikerVereiningung. The latter has generated a continuous stream of papers on the subject of ring geometry, which Mathematical Reviews associates with his and Hjelmslev's name. The second p a p e r was undoubtedly much m o r e seminal than the first; it a l o n e w o u l d have suf-

D

VOLUME21, NUMBER1, 1999

"r~

riced to m a k e him a h o u s e h o l d n a m e in the f o u n d a t i o n s o f geometry. By his o w n admission, Barbilian o w e s his m a t h e m a t i c a l outlook to an "immersion into the w o r k of Gauss, Riemann and Klein [ . . . ] s t r e n g t h e n e d by r e p e a t e d stays in G6ttingen." It is there that t h e 26-year-old arrives in August 1921 to s t u d y n u m b e r t h e o r y u n d e r Landau, b u t d o e s not rind numb e r theory "as c o n c e i v e d by Landau (a c o m p e t i t i o n of f o r m u l a s of a s y m p t o t i c evaluation)" to his taste, so he continues the literary activity that he h a d started in 1917. (He is, to this day, the only m a t h e m a t i c i a n b e s i d e s 'Umar alKhayy-ami to have m a d e (under the p s e u d o n y m Ion Barbu) a name for himself in the w o r l d of p o e t r y as well.) Having r e t u r n e d from G6ttingen to Bucharest in 1924 without a doctorate, he a c c e p t s G. Tzitz6ica (1873-1939) as Doktorvater a n d d e f e n d s his thesis in 1929. The Prague c o n g r e s s summary, In-

corporation of Lobachevsky's metric in certain general metrics of Jordan domains (in German), got the attention of W. Blaschke, w h o wrote that s a m e y e a r a l e t t e r to Barbilian. The s p a c e s d e s c r i b e d therein were given 4 y e a r s later a one-page presentation, as "examples o f m e t r i c spaces," in L. M. Blumenthal's Distance Geometries. There they a r e r e f e r r e d to as "Barbilian spaces." Barbilian g e o m e t r y resurf a c e d in a n article b y P. J. Kelly in the American Mathematical Monthly in 1954; he also d e v o t e s an a p p e n d i x to it in his t e x t b o o k The Non-Euclidean, Hyperbolic Plane (Springer-Verlag, 1981). But, a p a r t from four p a p e r s Barbilian h i m s e l f d e v o t e s to the subj e c t in 1959-1960, containing notes he m a d e b e t w e e n 1934 and 1939, no res e a r c h paper, e x c e p t t h o s e r e c e n t l y published b y t h e a u t h o r o f this book, ever refers to them. This neglect is due to s o m e e x t e n t to the fact that t h o s e four p a p e r s w e r e in Romanian. An English p r e s e n t a t i o n of the t h e o r y o f Barbilian spaces, as generalized b y him in 1959, was long overdue, b u t I a m afraid that the b o o k u n d e r review will fall to be noticed, let alone to s p a r k ren e w e d interest in the subject. The b o o k h a s s e v e r a l serious shortcomings: it is w r i t t e n in a dry style,

78

THE MATHEMATICALINTELLIGENCER

w i t h o u t any m o t i v a t i o n for the theor e m s or definitions p r e s e n t e d , its halting English and its spelling m i s t a k e s a r e at b e s t annoying, a n d it is apparently c o m p l e t e l y u n e d i t e d and possibly unrefereed. The first 54 p a g e s are s p e n t on a lab o r i o u s d e v e l o p m e n t of p l a n e projective geometry, i n t e n d e d to m a k e the Cayley-Klein model of hyperbolic plane g e o m e t r y u n d e r s t a n d a b l e . Even if the novice, w h o is the i n t e n d e d a u d i e n c e o f the first c h a p t e r on p r o j e c t i v e geometry, w e r e to u n d e r s t a n d it, there is not m u c h of a rationale for including it as an introduction to Barbilian spaces, for t h e s e generalize the Poincar6 model, w h i c h is introduced as a special case o f a Barbilian space, w i t h o u t any n e e d for the projective g e o m e t r y prologue, on p. 55. The original version of a Barbilian space (which w a s called a Jdomain), i n t r o d u c e d in 1934, is defined a s the interior J of a J o r d a n curve K, e q u i p p e d with the m e t r i c d(A, B) := In M/m, w h e r e M = SUpp~K PA/PB, m = infpEKPA/PB, and PA a n d PB are E u c l i d e a n distances. Later v e r s i o n s rep l a c e K and J with a n y t w o sets and t h e Euclidean d i s t a n c e with a strictly positive function f defined on K x J s u b j e c t to the condition t h a t gAB(P) := f(P, A)/f(P, B) has a s u p r e m u m which it r e a c h e s as P varies in K. Subject to the additional condition that, for A #: B in J, the ratio gAB(P) is n o t c o n s t a n t a s P varies in K, (J, d) t u r n s out to be a metric space. In Chapter II, w h i c h r e p r e s e n t s a g o o d introduction to Barbilian spaces, t h e a u t h o r a t t e m p t s to s h o w that the t h r e e constant-curvature 2-dimensional s p a c e s are Barbilian spaces. F o r the negative-curvature case, this w a s d o n e b y Barbilian himself. F o r the E u c l i d e a n and elliptic cases, the aut h o r s h o w s only that t h e r e is, for all A =~B in a naturally c h o s e n J, a function gAB: K--~ (0,oo), w h o s e "logarithmic oscillation"

maxpEK gAB(P) d(A, B) = minp~g gAB(P) is the Euclidean (or the elliptic) distance. But the function gAB he p u t s forw a r d is not a ratio f(P, A)/f(P, B), so t h e s e have not b e e n s h o w n to be, according to definition, Barbilian s p a c e s

(the s a m e p r o b l e m r e s u r f a c e s in T h e o r e m 2.3.1). The m a t e r i a l pres e n t e d in C h a p t e r s II and IV is a selection, with generalizations to the n-dim e n s i o n a l case, of Barbilian's 19591960 results. No particular r e a s o n for the s e l e c t i o n is stated, and it is s o m e t i m e s h a r d to u n d e r s t a n d w h y a certain result w a s omitted. The c o n d i t i o n s und e r w h i c h an a b s t r a c t Barbilian s p a c e ( t h e s e s p a c e s w e r e i n t r o d u c e d in 1959 by a different path) p o s s e s s e s the property o f unique geodesic c o n n e c t i o n of each pair of its points are investigated and a t h e o r e m proved, yet there is no m e n t i o n of Barbilian's 1934 result that an original Barbilian s p a c e has this p r o p e r t y w h e n and only w h e n it coincides with the Poincar6 m o d e l o f plane hyperbolic geometry. Although this characterization of the circle is omitted, 9.5 pages are used in w to prove that the circle is the only simple closed curve with convex interior which has s y m m e t r y a x e s in all directions, a result w h o s e relevance to the main topic of the b o o k is at b e s t marginal. C h a p t e r III is a 14-page intermezzo d e v o t e d to the c h a r a c t e r i z a t i o n of the plane constant-curvature geometries, using a t h e o r e m of e l e m e n t a r y geometry d i s c o v e r e d by Tzitz6ica in 1908 (and r e d i s c o v e r e d by R.A. J o h n s o n in 1916.) In t h e E u c l i d e a n plane, its statem e n t is: "If t h r e e circles of equal r a d i u s r p a s s t h r o u g h a point O a n d have seco n d p o i n t s of intersection A, B, C, t h e n the r a d i u s rABC of the c i r c u m c i r c l e o f the triangle A B C is equal to r." The aut h o r s h o w s that plane Euclidean, hyperbolic, a n d elliptic g e o m e t r i e s can be c h a r a c t e r i z e d b y rABC = , <, and > r, respectively. Although this is a nice exercise in elementary non-Euclidean geometry, no special feature relates Tzitz6ica's t h e o r e m with the metric on the surfaces of constant curvature. In effect, any t h e o r e m of metric Euclidean geometry stating a congruence of segments as its conclusion, which is not a t h e o r e m of absolute geometry, c o u l d have se~ced as a test for the c u r v a t u r e of the s u r f a c e and t h e r e b y have provided s u c h a characterization. The simplest is m o s t likely this: "IfABC is a triangle a n d M, N, and P are the m i d p o i n t s of the sides AB, AC, and BC, then M N is =, <, or > BP if and only

if the curvature is =, < , o r >0, respectively." Chapters V and VI are d e v o t e d to differential-geometric a p p l i c a t i o n s of Barbilian spaces. Certain Barbilian s p a c e s are s h o w n to b e generalized Lagrangean s p a c e s that are not Lagrangean, although the r e a d e r is not p r e s e n t e d with a definition of a Lagrangean (or generalized Lagrangean) structure. These were i n t r o d u c e d b y R. Miron in 1976 as m o d e l s for the t h e o r y of relativity. They are generalizations of Lagrangean spaces, w h i c h are generalizations of Finsler s p a c e s , which

are generalizations of R i e m a n n i a n spaces. Let's r e t u r n to the question o f w h y Barbilian s p a c e s w e r e a l m o s t forgotten. Traditionally, interest in the found a t i o n s o f g e o m e t r y a r o s e from t w o s o u r c e s : t h e a x i o m a t i c and t h e "realworld" r e p r e s e n t a t i o n aspect. The latt e r w o u l d a s k in the c a s e of h y p e r bolic g e o m e t r y w h e t h e r t h e r e is a s u r f a c e in s o m e E u c l i d e a n s p a c e w h o s e n a t u r a l l y inherited g e o m e t r y is the h y p e r b o l i c one. Unlike h y p e r b o l i c g e o m e t r y , Barbilian s p a c e s a r e n o t m o d e l s o f a c e r t a i n a x i o m s y s t e m for-

m u l a t e d in p u r e l y g e o m e t r i c t e r m s (say, in t h e T a r s k i a n t e r m s o f equidist a n c e and b e t w e e n n e s s ) , s o they d i d n o t lend t h e m s e l v e s to a x i o m a t i c investigation. This w o u l d c h a n g e if at least a subclass of them would be s h o w n to b e the c l a s s of m o d e l s of a certain s y n t h e t i c a l l y f o r m u l a t e d axiom system. It is a l s o not k n o w n w h a t s u b c l a s s o f t h e m a d m i t s globally a real-world r e p r e s e n t a t i o n . Department of Integrative Studies Arizona State University West Phoenix, AZ 85069-7100 USA

STATEMENT OF OWNERSHIP, MANAGEMENT, AND CIRCULATION (Required by 39 U.S.C. 3685). (1) Publication title: Mathematical Intelligencer (2) Publication No. 001-656. (3) Filing Date: 10/98. (4) Issue Frequency: quarterly. (5) No. of Issues Published Annually: 4. (6) Annual Subscription Price: $54.00. (7) Complete Mailing Address of Known Office of Publication: 175 Fifth Avenue, New York, NY 10010-7858. Contact Person: Joe Kozak, Telephone: 212-460-1500 EXT 303. (8) Complete Mailing Address of Headquarters or General Business Office of Publisher: 175 Fifth Avenue, New York, NY 10010-7858. (9) Full Names and Complete Mailing Addresses of Publisher, Editor, and Managing Editor: Publisher: Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010-7858. Editor: Chandler Davis, Department of Mathematics, University of Toronto, Toronto, Canada M5S 1A1. Managing Editor: Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010-7858. (10) Owner: Springer-Verlag Export GmbH, Tiergartenstrasse 17, D-69121 Heidelberg, Germany, and Springer-Verlag GmbH & Co. KG, Heidelberger Platz 3, D-14197 Berlin, Germany. (11) Known Bondholders, Mortgagees, and Other Security Holders Owning or Holding 1 Percent or More of Total Amount of Bonds, Mortgages, or Other Securities: Dr. Konrad Spnnger, Heidelberger Platz 3, D-14197 Berlin, Germany. (12) The purpose, function, and nonprofit status of this organization and the exempt status for federal income tax purposes: Has Not Changed During Preceding 12 Months. (13) Publication Name: Mathematical Intelligencer. (14) Issue Date for Circulation Data Below: Fall 1998. (15) Extent and Nature of Circulation: (a.) Total No. Copies (Net Press Run): Average No. Copies Each Issue During Preceding 12 Months, 7700; Actual No. Copies of Single Issue Published Nearest to Filing Date, 6900. (b.) Paid and/or Requested Circulation: (1) Sales Through Dealers and Carriers, Street Vendors, and Counter Sales: Average No. Copies Each Issue During Preceding 12 Months, 1295; Actual No. Copies of Single Issue Published Nearest to Filing Date, 665. (2) Paid or Requested Mall Subscriptions: Average No. Copies Each Issue During Preceding 12 Months, 4869; Actual No. Copies of Single Issue Published Nearest to Filing Date, 4869. (c.) Total Paid and/or Requested Circulation: Average No. Copies Each Issue During Preceding 12 Months, 6164; Actual No. Copies of Single Issue Published Nearest to Filing Date, 5534. (d.) Free Distribution by Mail: Average No. Copies Each Issue During Preceding 12 Months, 137; Actual No. Copies of Single Issue Published Nearest to Filing Date, 137. (e.) Free Distribution Outside the Mall: Average No. Copies Each Issue During Preceding 12 Months, 175; Actual No. Copies of Single Issue Published Nearest to Filing Date, 175. (f.) Total Free Distribution: Average No. Copies Each Issue During Preceding 12 Months, 312; Actual No. Copies of Single Issue Published Nearest to Filing Date, 312. (g.) Total Distribution: Average No. Copies Each Issue During Preceding 12 Months, 6476; Actual No. Copies of Single Issue Published Nearest to Filing Date, 5846. (h.) Copies Not Distributed: (1) Office Use, Leftovers, Spoiled: Average No. Copies Each Issue During Preceding 12 Months, 1067; Actual No. Copies of Single Issue Published Nearest to Filing Date, 1054. (2) Return from News Agents: Average No. Copies Each Issue During Preceding 12 Months, 157; Actual No. Copies of Single Issue Published Nearest to Filing Date, 0. (i.) Total (sum of 15g, 15h(1), and 15h(2)):Average No. Copies Each Issue During Preceding 12 Months, 7700; Actual No. Copies of Single Issue Published Nearest to Filing Date, 6900. Percent Paid and/or Requested Circulation: Average No. Copies Each Issue During Preceding 12 Months, 95.1895; Actual No. Copies of Single Issue Published Nearest to Filing Date, 94.66%. (16) This Statement of Ownership will be printed in the Winter 1998 issue of this publication. I certify that all information furnished on this form is true and complete.

o s oone Senior Vice-President and Chief Financial Officer

VOLUME 21, NUMBER 1, 1999

79

~"]i~.ana,~m.*'J[,]a*[=-]i

Robin

Wilson

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The

Ir g' _y p. z. t_a nL . '

Mathematics

main achievements of early Egyptian mathematicians involved the practical skill of measurement. The magnificent p y r a m i d s o f Gizeh (or Giza)date from about 2600 BC and attest to this extremely accurate measuring ability. In particular, the Great Pyramid of Cheops, constructed from over two million blocks averaging over 2 tons in weight, has a square base of side 755 feet, where the sides agree in length to within 0.01% and the right angles are equally accurate. The oldest pyramid is King D j o s e r ' s p y r a m i d in Saqqara, which dates from about 2700 Be; it was supposedly designed by the celebrated court physician and architect I m h o t e p . Our knowledge of later Egyptian mathematics derives mainly from just

step

King Djoser's step pyramid

Pyramids of Gizeh

two primary sources, the 'Moscow papyrus' (c.1850 BC) and the "Rhind papyrus" (c. 1650 BC). These papyri contain tables representing certain numbers as "unit fractions" (reciprocals) and include a variety of problems in arithmetic and geometry, probably designed for the teaching of scribes. These problems range from the sharing of loaves in specified proportions to finding the volume of a cylindrical granary; the latter gives rise to a value for ~r of 256/81 (-~ 3.16). Among other Egyptians interested in mathematics was A m e n h o t e p (c. 1400 BC), a high official during the reign of Amenhotep III. During the Ptolemaic era 1000 years later his name came to be associated with the ibis-headed Thoth, god of reckoning.

Imhotep

Egyptian scribes/accountants

Please send all submissions to the Stamp Corner Editor,

An Egyptianpapyrus

Amenhotep

Robin Wilson,

Faculty of Mathematics and Computing, The Open University, Milton Keynes, MK7 6AA, England

80

THEMATHEMATICALINTELLGENCER9 1999SPRINGER-VERLAGNEWYORK

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