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Letter to the Editors

Response to Spencer and Graham Article The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to either of the editors-in-chief, Chandler Davis or Marjorie Senechal. n a recent issue, J. Spencer and R. Graham wrote an article (‘‘The Elementary Proof of the Prime Number Theorem’’, Mathematical Intelligencer vol. 31 (2009), no. 1, 18–23) which presented a posthumous note by E. G. Straus. In the article, they also include excerpts from a lengthy interview we had with Atle Selberg; parts of this appeared in N. A. Baas and C. F. Skau, ‘‘The Lord of the Numbers, Atle Selberg. On his Life and Mathematics’’, Bull. Amer. Math. Soc. (2008), 617–649. In order to understand Selberg’s point of view, we think it is important to read his complete account in the interview. This is available at www.math.ntnu.no/ Selberg-interview/PNT/. There, one finds Selberg’s complete statement—some of it was left out in the Bulletin article, including a letter and report from Hermann Weyl to which Selberg refers. We think that this material, especially the Weyl report, is of great historical interest, and we recommend it to the reader. Hermann Weyl’s role in all this becomes clearer and more balanced than in Straus’s somewhat biased view. The reader is also referred to the interesting article by D. Goldfeld, ‘‘The Elementary Proof of the Prime Number Theorem: An Historical Perspective’’ (in Number Theory: New York Seminar 2003, Eds. D. Chudnovsky, G. Chudnovsky and M. Nathanson, New York: Springer, 179–192) (www.math.columbia.edu/*goldfeld/ErdosSelbergDispute.pdf).

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Here, Selberg refutes in his correspondence with Goldfeld several of the claims that E. G. Straus makes, in particular, the claim that he, Selberg, did not appropriately refer to Erd} os in his published paper. In the interview, Selberg also openly explains how he tried to get Erd} os off the track. In the excerpt (see the webpage above) from our interview at the bottom of page 9, Selberg says: ‘‘I kind of tried to scare him away from the prime number theorem itself. It was, one may say, a little dishonest that I did not tell him that my counterexample was based on a nonmonotonic function.’’ Furthermore, in the interview Selberg states that his goal was to prove the PNT by using his fundamental formula; see the excerpt page 5, lower part. Goldfeld writes in his article on page 8 (lower part), quoting a letter from Selberg: ‘‘This attempt to throw Erd} os off the track (clearly not succeeding!) is somewhat understandable given my mood at the time.’’ It is our impression that Selberg wanted to work towards the prime number theorem at his own pace using his fundamental formula, and in his attempt to lead Erd} os away from it, he apparently gave Erd} os the impression that he thought this would not lead to a proof. This seems to have caused much of the controversy. We think that this additional information provides a better picture of the circumstances around the elementary proof of the prime number theorem and should be of great interest for the mathematical community.

Nils A. Baas Department of Mathematical Sciences Norwegian University of Science and Technology NO-7491 Trondheim Norway e-mail: [email protected] Christian F. Skau Department of Mathematical Sciences Norwegian University of Science and Technology NO-7491 Trondheim Norway e-mail: [email protected]

Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

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Letter to the Editors

About the Spencer and Graham Article VICTOR PAMBUCCIAN n ‘‘The Elementary Proof of the Prime Number Theorem’’ by Joel Spencer and Ronald Graham (The Mathematical Intelligencer Vol. 31 (2009), No. 3, pp. 18–23), Ernst Straus is quoted as follows: ‘‘The elementary proof has so far not produced the exciting innovations in number theory that many of us expected to follow.’’ But it was the essential step in reassigning this theorem to the realm of pure arithmetic from that of real or complex analysis, in line with Hilbert’s concern for the purity of the method: ‘‘In der modernen Mathematik [wird] solche Kritik sehr ha¨ufig geu¨bt, woher das Bestreben ist, die Reinheit der Methode zu wahren, d. h. beim Beweise eines Satzes womo¨glich nur solche Hilfsmittel zu benutzen, die durch den Inhalt des Satzes nahe gelegt sind (1898–1899).’’ (In modern mathematics one often applies such a critique, the

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THE MATHEMATICAL INTELLIGENCER Ó 2010 Springer Science+Business Media, LLC

objective being to preserve the purity of the method, i.e., to use in the proof of a theorem only those means that are suggested by its content). Specifically, the logarithm function showing up in the Prime Number Theorem can be replaced by a function definable entirely inside number theory, and the resulting theorem can be proved in a fragment of first-order Peano Arithmetic (ID0+ exp, to be precise), as shown in Cornaros and Dimitracopoulos (1994), ‘‘The prime number theorem and fragments of PA.’’

REFERENCES

Cornaros C. and C. Dimitracopoulos. The prime number theorem and fragments of PA. Arch. Math. Logic 33(4), 1994, 265–281. Spencer J. and Ronald Graham, The elementary proof of the Prime Number Theorem, Math. Intelligencer 31(3), 2009, 18–23, DOI: 10.1007/s00283-009-9063-9. Division of Mathematical and Natural Sciences Arizona State University-West Campus Phoenix, AZ 85069-7100 USA e-mail: [email protected]

Letter to the Editors

Contradict or Construct? The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to either of the editors-in-chief, Chandler Davis or Marjorie Senechal.

urrah for Michael Hardy and Catherine Woodgold! [‘‘Prime Simplicity,’’ Intelligencer, Fall 2009, 44–52.] They have provided another brilliant example of what the mathematician Lacroix once described as ‘‘the ease with which errors pass from book to book.’’ (Lacroix was writing in 1797 about the way in which so many of his predecessors had misunderstood and disparaged the foundations of Leibniz’s calculus.)

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However, I think they overlook an essential point when they fail to deny that Euclid proves ‘‘the existence of infinitely many prime numbers.’’ Euclid does not deal in infinites. He proves that, given any finite set of primes, there is another prime not in the set, and his proof is constructive: Form the product, add one, and factor the result into primes. There is of course at least one prime factor, and none of the prime factors are in the set that was given. Whether a proof by contradiction is appropriate depends on what is being proved. Euclid does pﬃﬃﬃ prove Proposition 9 of Book X (of which the irrationality of 2 is a consequence) by contradiction. This is appropriate because the statement to be proved says something is impossible, whereas the statement about primes (as Euclid formulated it) says something is possible. In Essay 5.2 of my book ‘‘Essays in Constructive Mathematics’’ I argue that a proof by contradiction deserves to be called constructive if it proves a construction is impossible by deducing pﬃﬃﬃ a contradiction from it, and I prove both the irrationality of 2 and Sylow’s theorem in group theory by this method.

Harold Edwards New York University New York USA e-mail: [email protected]

Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

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Erratum

Erratum to: The Double Twist The online version of the original article can be found under doi:10.1007/s00283-009-9065-7.

Erratum to: Math Intelligencer 2009 31(3): 57–61 DOI 10.1007/s00283-009-9065-7

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n The Mathematical Intelligencer, vol. 31, no. 3 (2009), Page 60, the Book Review: The Double Twist ‘‘From Ethnography to Morphodynamics’’ and ‘‘The Artist and the Mathematician: The Story of Nicolas Bourbaki, The Genius Mathematician Who Never Existed,’’ reviewed by Osmo Pekonen, the reference to James M. Cargal’s work, should read: Cargal, James M. (2001). ‘‘The problem with algebraic models of marriage and kinship structure’’, The UMAP Journal 22 (4), pp. 345–353.

Osmo Pekonen Agora Centre University of Jyva¨skyla¨ P.O. Box 35, Jyva¨skyla¨ FI-40014, Finland e-mail: [email protected]

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THE MATHEMATICAL INTELLIGENCER Ó 2010 Springer Science+Business Media, LLC

Erratum

Erratum to: Kronecker’s Algorithmic Mathematics The online version of the original article can be found under doi:10.1007/s00283-009-9028-z.

Erratum to: Math Intelligencer 2009 31(2): 11–14 DOI 10.1007/s00283-009-9028-z

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n The Mathematical Intelligencer, vol. 31, no. 2, the Viewpoint column: ‘‘Kronecker’s Algorithmic Mathematics’’ by Harold M. Edwards, p.13, lines 10 and 11 of the first column should read: See Page 39 of the critical edition (1962) of Galois’s works. Galois’s Mathematics,… Harold M. Edwards Courant Institute of Mathematical Sciences New York University New York, NY 10012 USA e-mail: [email protected]

2010 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

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Note

A Visual Proof for the Sum of the First n Triangular Numbers HASAN UNAL

N

elson (1993) gave a visual proof that

Tk ¼ 1 þ 2 þ . . . þ k )

n X

Tk ¼

k¼1

nðn þ 1Þðn þ 2Þ : 6

Goldoni (2002) has given a visual proof of the formula for the sum of the first n squared numbers and for the sum of the first n factorials of order two. In a similar fashion, we can find a formula for the sum of the first n triangular numbers, Tk.

Figure 1. Combination of two sums of triangular numbers.

Figure 2. Separation of two sums. 6

THE MATHEMATICAL INTELLIGENCER Ó 2009 Springer Science+Business Media, LLC

After stacking sets of unit cubes representing the triangular numbers, first combine two sums of triangular numbers visually (Figure 1). Then separate/decompose this sum of two sums into two parts (Figure 2). This decomposition creates two shapes: the first one might be called a unit-stair and the second one a pyramid-shaped stairway. Figure 3 shows the combination of three equal pyramidshaped stairs. Three unit-stair shapes have been left over. When we combine the first unit-stair with the shape above, it forms a square prism of dimensions n by n + 1 by n + 1 (Figure 4). Finally, combining this square prism with the two leftover unit-stair shapes results in a rectangular box n by n + 1 by n + 2 (Figure 5). The outcome is that six sums of triangular numbers have been combined into a rectangular box n by n + 1 by n + 2. Thus, 6

n X k¼1

as desired.

Tk ¼ nðn þ 1Þðn þ 2Þ

Figure 3. Combination of three pyramids.

Figure 4. Combining one unit-stair with the preceding shape.

Figure 5. Combining two unit-stairs with the preceding shape.

REFERENCES

Nelsen, B.R. (1993). Sum of Triangular Numbers II, in Proofs Without Words: Exercises in Visual Thinking, Washington, D.C.: Mathematical Association of America, p.95. Goldoni, G. (2002). ‘‘A Visual Proof for the Sum of the First n Squares and for the Sum of the First n Factorials of Order Two,’’ Mathe-

Mathematics Department Yildiz Technical University Davutpasa Campus Istanbul 34210 Turkey e-mail: [email protected]

matical Intelligencer 24(4), 67–69.

Ó 2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

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Note

The Broken Stick Reconsidered Again HANS SCHUPP n his article in The Mathematical Intelligencer, Vol. 30 (2008), No. 3, pp. 43–49, Gerald S. Goodman analyzes the genesis of the concept ‘‘uniform distribution’’—or more generally ‘‘at random’’—within the framework of geometric probability. For that he uses the example of the ‘‘broken stick.’’ I concur with Goodman’s analysis, but I would like to point out that the history of this famous problem is a bit more complex than sketched. Already in 1866 William Clifford (1845–1879) solved a generalization: A line is broken up into pieces at random. Prove that the chance that they cannot be made into a polygon of n sides is n 21–n. He presented several proofs (see Smith 1959), of which one is especially interesting (see Figure 1 for n = 4). Let x, y, z be three of the four stick parts so that x + y + z \ 2m (stick length). Now each subdivision (x; y; z; 2m – (x + y + z)) may be represented by a point P(x; y; z). Then these possible points fill the tetrahedron ODEF. Favorable points (in the sense of the statement) are those with x + y + z \ m, because then the fourth part exceeds half the stick length; and likewise those with one

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z F

2m

m

m o

E

y

2m

m 2m

D x

Figure 1. Clifford’s Solution. 8

THE MATHEMATICAL INTELLIGENCER 2009 Springer Science+Business Media, LLC

of x, y, z at least m. Each of these four conditions holds in a tetrahedron at a vertex of the big tetrahedron, having half its edge length and therefore 1/8 its volume. Thus the probability looked for is 1/2 . If there are n parts we follow an analogous reasoning with (n – 1)-dimensional simplices. The favorable points belong to subsimplices having half the edge length and therefore 1/2n–1 the volume. There are n such subsimplices, one at each vertex of the original simplex. h Here, as in the second proof of the Cambridge text, we already have a solution by comparing the area of the favorable points with that of the possible points fitting the geometric context of the problem. Lemoine’s choice in 1872–1873 (not 1875) to treat a discrete, combinatorial version of the problem may have been made in ignorance of earlier continuous approaches. Clifford’s work did not appear in a scientific journal but in the Educational Times (which often offered its readers mathematical problems–even those of leading mathematicians). Georges-Henri Halphen (1844–1889) may have known of Clifford’s study. In the same journal and the same volume as Lemoine, he solved Clifford’s problem (but regarding as favorable the event ‘‘polygon possible’’). His approach was continuous, but his tools were arithmetic. Each unfavorable case can be characterized by an n-tuple (x1; x2; …; xn–1; 2m – (x1 + x2 + … + xn–1)) with 0 B x1 + x2 + … + xn–1 B m and 2m – (x1 + x2 + … + xn–1) [ m. Such a tuple comes about if each of the n – 1 intermediate marks lies in the left half of a stick of length 2m. This happens with probability 1/2n–1. When we consider that each of the n parts can be the one leading to the unfavorable case, we have again the total probability n 1/2n–1 for the complementary event ‘‘construction impossible’’ (3/4 for n = 3). Halphen used in his proof the fact that each side of an n-gon of perimeter 2m has length \ m, which follows readily from the triangle inequality. After a solution in 1879 of Lemoine’s original problem by Le´on Lalanne (1811–1892) corresponding to the second Cambridge solution—indicating that even at that date it wasn’t known in France—we find in a publication of Ernesto Cesa`ro (1882) the nice version using the theorem of Viviani (and thus comparing an equilateral triangle with its midtriangle). Only later did Poincare´ take it up. It may be of interest that Lemoine in 1883 in a second publication looked into some variants of his problem. Questions he dealt with: What is the probability that the perpendiculars from a point to the sides of a general triangle can form a second triangle? What is the probability that this triangle is acute angled? What result do we get when instead of the perpendiculars we take the line segments to the three vertices of the given triangle? To all of these, his solutions again use suitable area comparisons.

The first book about geometric probability appears in 1884 (1902 in French translation), written by Emanuel Czuber (1851–1925). It has a lot of relevant problems, among them the broken stick and many variants of it. Moreover, Czuber sketches a general way to obtain such probabilities by means of multiple integrals obtained by obvious limit processes (as Clifford and Halphen had done already for the broken stick), and uses this approach to solve the nontrivial ones among his problems. Thus, he anticipated some arguments in Goodman’s article. But he concedes (p.7; translation mine): It happened several times that problems about geometrical probabilities led to different solutions. The reason is found in different views of the concept ‘‘at random,’’ whose meaning indeed is not always sufficiently obvious to exclude disagreements. He himself avoided such problems in his book. But at least with the Calcul des probabilite´s of Joseph Bertrand (1822–1900) in 1889 containing the famous paradox (several well-founded but different solutions of the same problem) the discussion about a meaningful and useful definition of geometric probability arose again. The contribution of Poincare´ to this discussion is modest. In his Calcul de probabilite´s of 1896 he devotes to the Probabilite´s du continu only 13 of 333 pages. After a description of the limit processes up to multiple integrals (shorter than given by Czuber), he offers the well-known (but he thinks now justified) solutions to some key problems: Bertrand’s paradox, the baˆton brise´ (since then the French name for the broken stick), Buffon’s needles. Final remark: In the computer age it is possible to simulate the randomly-directed breaking of the stick, to

check the inequalities classifying the case as favorable or unfavorable, to repeat this as often as needed, and to calculate the relative frequency of the favorable cases as approximation of the probability. But the computer cannot help us in defining ‘‘at random.’’ REFERENCES

Cesa`ro, E.: Une question de probabilite´s. Mathesis 2 (1882), 177–180. Czuber, E.: Geometrische Wahrscheinlichkeiten und Mittelwerte. Teubner, Leipzig, 1884. Goodman, G.S.: The problem of the broken stick reconsidered. Math. Intelligencer 30 (2008), no.3, 43–49, DOI: 10.1007/BF02985378. Halphen, G.H.: Sur un proble`me de probabilite´s. Bull. Soc. Math. de France 1 (1872/1873), 221–224. Lalanne, L.: Emploi de la ge´ome´trie pour re´soudre certains questions de moyennes & de probabilite´s. J. math. Liouville 5 (1879), 107–115. Lemoine, E.: Sur une question de probabilite´s. Bull. Soc. Math. de France 1 (1872/1873), 39–40. Lemoine, E.: Quelques questions de probabilite´s re´solues ge´ome´triquement. Bull. Soc. Math. de France 11 (1882/1883), 13–25. Poincare´, H.: Calcul de probabilite´s. Gauthier-Villars, Paris, 1896. Smith, D.E. (Ed.): A Source Book in Mathematics. Dover, London, 1959.

Universita¨t des Saarlandes Fakulta¨t 6: Mathematik und Informatik Postfach 151150, D-66041 Saarbru¨cken Germany e-mail: [email protected]

2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

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Baserunner’s Optimal Path DAVIDE CAROZZA, STEWART JOHNSON

hen you hit that final long ball in the World Series of Baseball and know you need the home run, what is your optimal path around the bases? If you run straight for first, you either have to slow to a near stop or go sailing far beyond into the outfield. The standard recommended ‘‘banana’’ path follows the baseline maybe halfway and then veers a bit to the right to come at first base from a better angle to continue toward second. That cannot be ideal. It would have been better to start at an angle to the right to head directly to an outer point on the banana path. So what is the optimal path? Using a very simple model, we obtain the path of Figure 1. You start out heading about 25 right of the base line and run with acceleration of constant maximum magnitude r, as illustrated by the vectors decorating the path. You slow down a bit coming into first, hit a local maximum speed as you cross second, and start the final acceleration home a bit before crossing third base (see Fig. 2). The total time around the bases is about 52.7/Hr, about 16.7 seconds for r = 10 ft/sec2, about 25% faster than following the baseline for 22.2 seconds (coming to a full stop at first, second, and third base), and about 6% faster than following a circular path for 17.8 seconds. The record time according to Guiness [G] is 13.3 seconds, set by Evar Swanson in Columbus, Ohio, in 1932. His average speed around the bases was about 18.5 mph or 27 ft/sec. Is it legal to run so far outside the base path? The relevant official rule of Baseball says: 7.08 Any runner is out when—(a) (1) He runs more than three feet away from his baseline to avoid being tagged unless his action is to avoid interference with a fielder fielding a batted ball. A runner’s baseline is established when the tag attempt occurs and is a straight line from the runner to the base he is attempting to reach safely. The rule just says that after a tag attempt the runner cannot deviate more than three feet from a straight line from that point. The rule doesn’t apply until the slugger is almost home, when our fastest path is nearly straight. So our path is legal.

AND

FRANK MORGAN

W

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THE MATHEMATICAL INTELLIGENCER 2009 Springer Science+Business Media, LLC

Figure 1. Second picture shows the fastest path around the bases given a bound r on the magnitude of the acceleration vector, shown at each point. First picture from http://www. bsideblog.com/images/2008/03/baseball-diamond.jpg.

Our model simply assumes a bound r on the magnitude of the baserunner’s acceleration (which includes deceleration and curvature). The locus of the fastest path around the bases is independent of r because you can scale velocity by k, acceleration by k2, and time by 1/k. So slow runners should follow the same route as fast ones. At first you might think that a very slow, awkward runner should just walk directly from base to base, except that he’d likely fall down trying to make the sharp turn at first. To find the fastest path around the bases, we consider the simpler problem of finding the fastest path between two points, given the initial and final velocities, which has a unique solution. Intriguingly enough, for this problem, total time is not continuous in the prescribed conditions. Even on the line, consider starting at the origin with initial velocity 1 and going at maximal acceleration for a second, ending with velocity 2; now if, instead, the prescribed final velocity were increased a bit, you would have to start out by decelerating to velocity 0, go backward to well left of the origin, and then accelerate right to the terminus. (See Remark after Lemma 1. Fortunately time is lower-semicontinuous, which is what we need to prove the existence of fastest paths.) For a critical path between bases, the acceleration vector a has constant magnitude r and remarkably is given by At + B normalized, for some constant vectors A, B. In velocity space, such paths are portions of catenaries (the famous leastenergy shape of hanging cables as for suspension bridges), which in general can be absolute minima, local minima, or unstable critical points (see Remarks after Lemma 2). It is easy to see that a fastest path for bounded |a| also minimizes max |a| for given time, since if you could reduce max |a|, then by increasing speed along an appropriate portion of the locus in space, you could reduce time. There are, however, more solutions to the second problem. In the example at the end of the Remarks after Lemma 2, all three paths minimize max |a| for given times T1 \ T2 \ T3. Given the fastest path between bases for prescribed velocities, we find the shortest path around the bases by minimizing over all choices of velocity at the bases, specifying velocity 0 at the start. We think that the solution is unique, but we know no proof.

Figure 2. Speed as a function of time. For r = 10 ft/sec2, each unit of time represents 3 seconds and each unit of velocity represents 30 ft/sec. The times for each segment are about 5.1, 4.1, 4.4, and 3.1 seconds, for a total of about 16.7 seconds.

Our model is, of course, an oversimplified one, since it assumes that maximum deceleration equals maximum acceleration and that maximum acceleration remains possible at high speeds; taking r = 10 ft/sec2, it leads to a final speed coming into home of about 42 ft/sec, faster than the highest recorded human speed as of August 2009 of 40.5 ft/ sec by Usain Bolt, even though his initial acceleration exceeded 18 ft/sec2 [S].

Fastest Paths Lemmas 1 and 2 provide existence and structure for the shortest path between two bases, given initial and final velocities. Proposition 1 considers the full baserunner problem with all four bases. We conclude by explaining our numerical solution of Figure 1.

LEMMA 1 There exists a fastest path from one point to another in the plane, given initial velocity, final velocity,

AUTHORS

......................................................................................................................................................... graduated in 2009 from Williams College, where he presented his senior colloquium (a requirement for all senior math majors) on some preliminary analysis of optimal base-running. In his present job he teaches Algebra I.

DAVIDE CAROZZA

Department of Mathematics and Statistics Williams College Williamstown, MA 01267 USA e-mail: [email protected]

STEWART JOHNSON is a professor working in

dynamics, optimal control, mathematical biology, game theory, differential equations, and statistics. He is also the Quantitative Skills Coordinator for Williams College, and as such he gives tutorials and courses for developing quantitative skills. Department of Mathematics and Statistics Williams College Williamstown, MA 01267 USA e-mail: [email protected] 2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

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and a bound r [ 0 on the acceleration. The minimum time is a lower-semicontinuous function of the initial and final positions and velocities.

REMARK The minimum time is not continuous in the prescribed conditions. For example, for r = 1, the fastest path from (0, 0) to (v0 + 1/2, 0) with initial velocity v0 [ 0 and final velocity v0 + 1 + e is for e = 0 simply forward motion for 1 second at unit acceleration, but for small e [ 0 one must decelerate for v0 seconds to velocity 0 at (.5 v20, 0), move backward, accelerating and decelerating for another H2 v0 seconds to come to rest just left of (0, 0), and then move forward for a bit more than a second at unit acceleration, for total time a bit more than 1 + v0(1 + H2) seconds, a huge discontinuity if v0 is large. See Figure 3 for the case v0 = 1. In summary, increasing the final velocity of a linear path with maximum acceleration involves backing up and a discontinuous increase in total time.

this is obvious. Otherwise just follow the given initial direction with maximum negative acceleration until obtaining velocity 0, and similarly backwards from the terminal point, to reduce to the obvious case. This path bounds the minimum time and hence the positions and velocities. Except for the trivial case when the initial and final position and velocities coincide, there is also a lower bound on the total time. To prove simultaneously existence and lower-semicontinuity in the prescribed conditions, consider a sequence of paths with conditions converging to the prescriptions and times Ti converging to the infimum T. We may assume that the velocities are bounded functions from [0, Ti ] into R2 with Lipschitz constant at most r and that Ti B 2T. Rescale time to change the domain to [0, T]. Now each velocity has Lipschitz constant at most rTi /T B 2r and the conditions still converge to the prescriptions. By the compactness of uniformly bounded Lipschitz functions, we may assume that the velocities and hence the paths converge; the limit has time T as desired.

LEMMA 2 A fastest C1,1 path from one point to another in the plane, given initial velocity, final velocity, and a bound r [ 0 on the magnitude of the acceleration a, At þ B a¼r jAt þ Bj for some constant vectors A, B, is unique.

Figure 3. As the prescribed final velocity increases past that obtained by constant maximum acceleration, the fastest path has to back up, with a discontinuous increase in total time.

PROOF OF LEMMA 1 First we note that there exists some path satisfying the conditions. If the given velocities are 0,

......................................................................... is a specialist in minimal surfaces. One of his books is ‘‘The Math Chat Book’’, based on the call-in TV show he used to run and his column at MathChat.org. He is founder of Williams’s ‘‘SMALL’’ Undergraduate Research Project. As Vice-President of the American Mathematical Society, he launched the blog http://mathgradblog.williams.edu/ by and for mathematics graduate students.

FRANK MORGAN

Department of Mathematics and Statistics Williams College Williamstown, MA 01267 USA e-mail: [email protected] 12

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REMARKS By a translation in time, we may assume that B A = 0 and that A is a unit vector. The path is real-analytic in time unless B = 0 and t = 0, when a flips direction. In addition, the path in space can have a singularity where the velocity vanishes, as in Figure 4B. Up to rotation and translation in the plane and scaling in time and space, we may assume that ð1; tÞ a ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ; 1 þ t2

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ v ¼ ðarcsinh t; 1 þ t 2 Þ þ v0 ; pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x ¼ ðt arcsinh t 1 þ t 2 ; :5 t 1 þ t 2 þ :5 arcsinh tÞ þ v0 t; pictured for v0 = 0, v0 = -(0,1), v0 = –(0,10), and v0 = -(1,0) in Figure 4; or in the degenerate case a ¼ ðsign t ¼ 1; 0Þ; v ¼ ð1 t; 0Þ þ v0 ; x ¼ ðt :5t 2 ; 0Þ þ v0 t; pictured for v0 = -(0,1) in Figure 5. Some such critical paths are not minimizing. Indeed, the translation in velocity space of a minimizer need not be minimizing. For example, for r = 1, the following path P is minimizing, but its translation P 0 by v0 = (1, 0) is not. The path P starts at (0, 0), accelerates left for 1 second to (-1/2, 0), decelerates for 1 second to (-1, 0), and then accelerates

Figure 4. (A, B, C, D) Some critical paths with acceleration At + B normalized.

Figure 5. A symmetric critical path with acceleration ± (0,1), which is the special case A = (0,1), B = 0. 2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

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to the right for 2 seconds, ending up at (1, 0) with velocity (2, 0). Its translation P 0 starts at (0, 0) with velocity (1, 0), decelerates for 1 second to (1/2, 0) and velocity (0, 0), and then accelerates for 3 seconds ending at (5, 0) with velocity (3, 0), for a total time of 4 seconds. A minimizer P} accelerates for H10 - 1 seconds and then decelerates for H10 - 3 seconds for a total time of 2H10 – 4 & 2.32 seconds. In summary, the translation P 0 of a backtracking minimizer P may decelerate unnecessarily and fail to be minimizing. Note that up to translation, rotation, and scaling, the path in velocity space is the famous catenary v = cosh u or in the degenerate case a line. It is well known that such paths minimize energy $v dt = Dy for given length r Dt. There are relative minima that are not absolute minima. Consider given velocities (-1, 1), (1, 1) and change in position (0, Dy) vertical. Possible paths in velocity space are catenaries (or horizontal lines), a 1-parameter family. A horizontal line yields minimum time, but a catenary v = a cosh (u/a) yields local minimum Dy. Rotating such a catenary about the u-axis generates the famous minimal catenoid surface, with area 2p times the potential energy $v dt = Dy of the catenary. It is well known that for two relatively close congruent vertical circles about the same horizontal axis there are two catenoids, a slightly bowed area minimum and a deeply bowed unstable one [TF, chap. I, §3]. Bowing upward from the catenary generator of the stable catenoid, Dy increases; time decreases to the horizontal line, then increases. Downward, time increases; Dy decreases to the generator of the stable catenoid, then increases to the generator of the unstable catenoid, then decreases, eventually going very negative. So Dy values between the generators of the two catenoids are obtained three times, with times T1 \ T2 \ T3. All have the same r. The first is the global minimum. The third is a local minimum, since by the energy-minimizing property of the catenary, decreasing time requires increasing Dy. All three paths minimize max |a| for given time, because if you could reduce max |a| for given time, you could rescale to reduce time and Dy in the same proportion instead, but for reduced time, the minimum Dy is the catenary in velocity space, for which Dy (the potential energy of the catenary) is reduced less than proportionately, because the average value of velocity increases.

0¼

oH d oH d 2 oH þ ¼00€ k; ox dt ox_ dt 2 o€ x

so that k(t) = At + B for constant vectors A, B, and second that 0 B qH/qa. Since a is constrained to lie in the disc of radius r, this second condition just says that a ¼ rk=jkj¼ rðAt þ BÞ=jAt þ Bj: Suppose that there were two fastest paths x1(t), x2(t). Then their average x3(t) would also be a fastest path. Since the acceleration a3(t) must, like a1 and a2, have constant length r, a1 = a2 and x1 = x2.

PROPOSITION 1. Given r [ 0 and points x1, x2, . . ., xn in R2 and optionally velocities v1, vn, there is a fastest path from x1 to xn passing in order through x2, . . ., xn-1 with initial velocity v0, final velocity vn, and acceleration bounded by r. The acceleration is continuous of magnitude r, with at most one possible exception from xk to xk+1: it may flip direction between xk and xk+1 or it may change discontinuously at xk or xk+1; the former can occur only if a is otherwise constant along the segment (as it is on the last segment), the latter only if a is constant along both incident segments. (If no velocities are prescribed, we must assume that the points do not lie in order along a line, the one case in which arbitrarily small time is possible.)

PROOF. Since the set of all possible velocities at the points xi is compact, existence follows from Lemma 1. Lemma 2 implies the asserted regularity except at the points x2, …, xn-1. Free velocity at xi adds a boundary term k dvxxii þ to the first variation, so that the Lagrange multiplier k is continuous at xi. Therefore the acceleration a = k/|k| is continuous at xi, unless k(xi) = 0, in which case a is constant on both incident segments. At xn, k = 0, so on the last segment k = B(t - tn) (see Remarks after Lemma 2), and a is constant on the last segment, except possibly for a flip.

PROOF OF LEMMA 2 For variable position x(t) in C1,1 and variable acceleration a(t) in L?, we want to minimize ZT dt 0

subject to the constraints x€ ¼ a (a.e.) and |a| B r. Since T is smooth in a, for some Lagrange multiplier k(t), a minimizer is a critical point for $H dt where € H ¼ 1 þ k ða xÞ: The Euler conditions of vanishing first variation (see, e.g., [M, 29.2]) say first that weakly 14

THE MATHEMATICAL INTELLIGENCER

Figure 6. The fastest path to second base.

The fastest path (see Fig. 1). Computing the fastest path proceeds in two steps. First, for prescribed velocities at two sequential bases, we use Lemma 2, a finite difference boundary-value method [F, §14.2], and multidimensional Newton’s method [F, §7.1], to find a solution with velocities that match the prescriptions. This problem can be highly nonlinear, and requires close guesses for Newton’s method to converge, which we achieved by deforming an easily computed symmetric path. Second we minimize total time over varied choices of prescribed velocities at the bases as in the proof of Proposition 1, which we achieve with a gradient descent method [F, §7.2]. Since there is no general uniqueness result for relative minima, we cannot be sure that our solution reflects a global minimum. Our MATLAB code is given as an Appendix to the web version of this article, available at the blog entry at blogs.williams.edu/ Morgan. Figure 6 shows the fastest path to second base for a double, taking 10.4 seconds for r = 10 ft/sec2, as compared to 12 seconds along the baseline, coming to a full stop at first and second base. The runner slows down a bit before rounding first base.

who was in the audience, discovered the remarkably simple critical condition and computed the fastest path of Figure 1. Morgan acknowledges NSF support.

REFERENCES

[F] Laurene V. Fausett, Applied Numerical Analysis using MATLAB, 2nd ed, Prentice-Hall, 2008. [G] Guiness World Records, http://www.baseball-almanac.com/ recbooks/rb_guin.shtml. [M] Frank Morgan, Real Analysis and Applications, Amer. Math. Soc., 2008. [N] Johannes C. C. Nitsche, Lectures on Minimal Surfaces, Cambridge Univ. Press, New York, 1989. [S] http://speedendurance.com/2009/08/19/usain-bolt-10-metersplits-fastest-top-speed-2008-vs-2009/ [TF] Dao Trong Thi and A.T. Fomenko, Minimal Surfaces, Stratified Multivarifolds, and the Plateau Problem, American Mathematical Society, 1991.

ACKNOWLEDGMENTS

This work stemmed from a Williams College undergraduate colloquium talk by Carozza advised by Morgan. Johnson,

2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

15

Mathematically Bent

Colin Adams, Editor

Job Solicitation COLIN ADAMS

The proof is in the pudding.

Dear Recent Math Ph.D., We are sending you this letter because we believe you may be interested in a position in the Mathematics Department at Berbunnion University. Berbunnion, It’s Not Your Typical University. es, we here at Berbunnion are proud to be able to say we are not your typical university. And that means we are not looking to hire your typical math professor. But you have received this letter precisely because you are not typical. A typical math professor teaches two or three courses a term and then does research on the side. We are not interested in that! On the contrary, we are looking for those extraordinary individuals who strive to do more. How much more? How does five courses a semester sound? ‘‘Wait a second,’’ you might say, ‘‘you think I could possibly teach five courses a semester?’’ Yes, we do. Or we wouldn’t have sent you this letter. You are in a small pool of individuals who we believe to be capable of handling a load like that. How would you do it?

Y Opening a copy of The Mathematical Intelligencer you may ask yourself uneasily, ‘‘What is this anyway—a mathematical journal, or what?’’ Or you may ask, ‘‘Where am I?’’ Or even ‘‘Who am I?’’ This sense of disorientation is at its most acute when you open to Colin Adams’s column. Relax. Breathe regularly. It’s mathematical, it’s a humor column, and it may even be harmless.

Step 1: Give up TV. Step 2: Move out of your home and onto a cot in your office. Step 3: Eliminate family ties. Step 4: Stop wasting time on your personal hygiene. Step 5: Make your entire focus your students and your job.

â Column editor’s address: Colin Adams, Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 01267, USA e-mail: [email protected] 16

THE MATHEMATICAL INTELLIGENCER Ó 2009 Springer Science+Business Media, LLC

That sounds like a lot to ask of a faculty member, but remember, you are special. What makes you so special? We know how hard you worked in grad school. You were trying to finish the research for your thesis while teaching three sections of calculus a semester. In addition, you had your social obligations, financial difficulties, your drinking problem, and that nasty incident with the undergraduate who later turned out to be the department chair’s son. And yet, in spite of everything that was going on, you managed to complete your Ph.D. That makes you the kind of faculty member we want here at Berbunnion. What else makes you special? If you come, you will be one of only two members of the faculty with an active fungal infection. Yes, you are truly unique and as such, you contribute to the diverse environment we foster at Berbunnion.

Berbunnion, Loving Learning How toLove Learning.

Berbunnion, the Only College Education You Will Ever Need.

The enclosed piece of paper with the handwritten table of values of sine and cosine at important angles is our free gift to you. You need not return it or send us payment of any kind. It is meant to demonstrate to you the high esteem with which we hold you. You are special and you deserve gifts like this. If you come to work for us, you can expect gifts like this on a regular basis. Oftentimes, it will be this very same gift again! That is how we do things at Berbunnion. We don’t have a rigid salary structure with health benefits and a mortgage plan. No, we work under a different model, a merit-based model. A model that encourages success. Every once in a while, when you are doing a good job, you receive gifts from the university. They vary widely. Sometimes, it might be something as large as a lawn tractor, or as useful as an old blackboard. Other times it might be athletic socks, or a tube of fungicide. But rest assured, if you do well, you will regularly receive gifts like these, often when you least expect them. It’s our way of saying thank you for a job well done.

Perhaps you don’t know a lot about us, and that makes you nervous. I know I become nervous when I receive letters of solicitation from nonfamily members. So let me tell you a little about us. Much like other venerable institutions of higher learning, Berbunnion University has a rich history chock full of funny traditions and heartwarming anecdotes. However, many of ours are copyrighted, so I cannot divulge them. But I can tell you that our school was founded by Janet Babbler Berbunnion over 30 years ago, after she discovered a linear algebra textbook buried deep in one of her closets, which was very mysterious, as no one in her family had ever previously shown any interest in mathematics. She interpreted this discovery to be an omen directing her to establish a university in her garage. That university has grown well beyond that single garage to now encompass close to 1,575 garages in the greater Dayton metropolitan area. In the process, we have become one of the most educationally oriented garage-based universities in the country. In fact, we consider education our primary mission.

Berbunnion Puts the You Back in Youniversity. Our newest hire in Transfinite Gender Studies was thrilled to return home to his office to find a brand new lounge chair and a pair of Berbunnion logo pajamas. In his words, ‘‘Berbunnion isn’t like a home away from home. It is my actual home.’’ A junior member of the Quantum Teleportation Department was recently overheard talking to herself in the bathroom, as she washed up in the sink: ‘‘At least I have a job.’’ That is a sentiment we encourage. Because jobs build self-esteem. And even if there is no salary associated with them, it is still great to have a job. Berbunnion. It’s Not About the Books. It’s About What’s in Them. What about research expectations? At Berbunnion, we are proresearch. Does that mean you must publish some fixed number of papers in order to receive tenure? It would be a sad state of affairs reflecting very poorly on the university if we believed that research output could be measured by the number of papers produced. No! We have no bar that you must hurdle. Because we don’t have tenure. That might make you nervous, but it shouldn’t. Because if you do well at Berbunnion, you have a job for life. We firmly believe that high-performing faculty should not be fired. They become part of the family that is Berbunnion University. And even if they become incapacitated in some way, perhaps due to an altercation with a student, or through an encounter with an automatic garage door, we can often find a place for them within our community, maybe washing dishes, or helping to clean the office/living quarters. Do we care about the different fields within mathematics? Yes, we do. We aren’t interested in someone who divides by zero or takes square roots of negative numbers. No, we seek to hire a low-dimensional topologist specializing in the Floer homology of pseudo-Anosov maps. What’s that? That’s your specialty? We said you were special, didn’t we? A match made in heaven.

Berbunnion, Part School, Part Family. And what about advancement? Will you be trapped teaching remedial math courses to hordes of students packed into a variety of garages for the rest of your career? Certainly not! If you do well, we will move you up into administration. Yes, you will be the one writing these letters, soliciting new faculty members. You will become one of the hundreds of Deans of Faculty we have here at Berbunnion. Why so many? Because we believe that too many cooks don’t spoil the broth. No, they help to solicit even more cooks, who help to make even more broth until pretty soon, almost everybody is teaching or cooking or eating soup or cleaning up after meals. And if you do well as a Dean of Faculty, we move you up to the silver circle, the inner circle of the most powerful administrators at the university, including the Head of Dining Services and the Vice President for Deans of Faculty. Above the silver circle, there is only one level, the highest level attainable at Berbunnion, which we call the Platinum Sphere. This level is reserved for the best of the best, those select few who ascend to become one of the presidents of Berbunnion. Who knows? One day, you may be invited to become a president and step inside the platinum sphere to partake of the luxuries hidden therein. I hope by now I have given you enough information to convince you that Berbunnion is the place for you. How to apply? Must you find three individuals who will testify positively to your success in research or teaching, a daunting task for any applicant? Not at all! Fill out the postage paid postcard that serves as your application file. The minute we receive it, we will send out a contract with all of the details and obligations. So don’t wait! Return it today! You will be very glad you did! Berbunnion, an Equal Opportunity/Affirmative Action employer. We seek to attract a diverse faculty of the highest caliber. That would be you!

Ó 2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

17

Mathematical Communities

Analytic Number Theory in China CHUANMING ZONG

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

introduce their mathematical work and tell some of their legendary stories. Before starting the main story, let us review some basic facts about China. Throughout its history, China had made great contributions to world civilization as well as to ancient mathematics. For example, at the beginning of the second century Lun Cai invented a technique for producing paper; in the third century Hui Liu obtained p = 3.14… by approximating a circle by polygons; in the fourth century Chong-Zhi Zu knew the first seven digits of p and how to calculate the volume of a ball; in the middle of the tenth century Sheng Bi invented moveable type. However, modern science, including modern mathematics, came to China rather late. The first university in China, Peking University, was founded in 1898. In the next decades several other universities were founded in Beijing, Shanghai, Tianjin, and other cities. The first mathematics department in China was formally opened at Peking University in 1913 with two professors. At the beginning there were only a few students studying mathematics and the courses were limited to calculus, linear algebra, and linear geometry. Within the next two decades, the number of mathematics students grew rapidly and abstract algebra, differential equations, set theory, differential geometry, and function theory gradually became university courses. By 1930, there were dozens of mathematics professors working in Chinese universities, most of them educated in Europe and America. The Chinese Mathematical Society was founded in 1935 and its first journal, Acta Mathematica Sinica, started publishing papers a year later.

Waring’s Problem rom 1930 to 1980, China suffered several national disasters, such as the Japanese invasion, the civil war, and the cultural revolution. These catastrophes destroyed not only people’s lives, but also culture and science. However, during that extremely difficult time China produced several famous mathematicians in analytic number theory: Loo-Keng Hua, Jingrun Chen, Yuan Wang, and Chengdong Pan. Their work and their legendary lives made them national heroes in China. Hua’s and Chen’s eventful lives have been documented in popular biographies and TV series. In this report I briefly

F

â

Please send all submissions to Marjorie Senechal, Department of Mathematics, Smith College, Northampton, MA 01063 USA e-mail: [email protected]

18

THE MATHEMATICAL INTELLIGENCER 2009 Springer Science+Business Media, LLC

In 1770, the British mathematician E. Waring asserted without proof in his Meditationes Algebraicae that every natural number is a sum of at most four integral squares, a sum of at most nine positive integral cubes, also a sum of at most nineteen biquadrates, and so on. It is usually assumed that, in modern notation, Waring meant: For each integer k C 2 there exists an integer s = s(k) depending only on k such that every positive integer n can be expressed in the form n ¼ z1k þ z2k þ þ zsk ;

ð1Þ

where the zi are non-negative integers. Proving this statement and its analogues is known as Waring’s problem. For convenience, let g(k) denote the smallest s(k) with this property and let G(k) denote the smallest s(k) such that the statement holds for all sufficiently large n. The k = 2 case had been studied by Fermat

and Euler even before Waring made his assertion. It was conjectured by Bachet in 1621 that g(2) = 4; both Fermat and Euler tried but failed to prove it. In 1770, Bachet’s conjecture was proved by Lagrange. Afterwards, new proofs for this beautiful theorem were discovered by Euler, Cauchy, Jacobi, Davenport, etc. In 1909, Hilbert solved Waring’s problem. His proof is based on algebraic identities by which one can only deduce a very poor bound for g(k). It seems that G(k) is harder than g(k). At least, so far we understand g(k) much better than G(k) (see [8]). In 1918 and 1920, Hardy, Littlewood, and Ramanujan devised a very powerful method (the circle method) to treat some additive problems in number theory, including Waring’s problem and Goldbach’s conjecture. We define f ðaÞ ¼

q X

k

e 2piaj ;

Taking j = k in Hua’s lemma, one can easily deduce that (3) holds for s C 2k + 1. Thus, the condition for (4) can be improved to s C 2k + 1 and (5) can be improved to GðkÞ 2k þ 1:

Hua’s lemma is fundamental in analytic number theory. Vaughan’s famous book [7] discusses several examples of its applications. Clearly, estimating values of the exponential sums is a key problem in analytic number theory, in particular in the circle method. Let q be a positive integer and let f ðxÞ ¼ ak x k þ ak1 x k1 þ þ a1 x be a polynomial over Z with (a1, …, ak, q) = 1. Then we define

j¼0

Sðq; f ðxÞÞ ¼

1=k

where q ¼ bn c; and let rk,s(n) denote the number of representations of n in the form (1). Then Z 1 f ðaÞs e 2pina da ð2Þ rk;s ðnÞ ¼ 0

and to prove Waring’s statement it is enough to show that rk,s(n) [ 0 for all sufficiently large positive integers n and some positive integer s depending only on k. Take v = 1/100, N ¼ bn1=k c and P = Nv. For 1 B a \ q B P with (a, q) = 1, we define Mða; qÞ ¼ fa : ja a=qj N vk g: Let M denote the union of all such M(a, q) and define m ¼ ðN vk ; 1 þ N vk n M: For historical reasons, M and m are called the major arcs and the minor arcs, respectively. Then the integral in (2) can be divided into Z Z f ðaÞs e 2pina da þ f ðaÞs e 2pina da: M

x¼1

If f(x) = ax , the sum is the well-known Gauss sum. Gauss proved that 8 pﬃﬃﬃ < q if q 1 ðmod 2Þ jSðq; ax 2 Þj ¼ p0ﬃﬃﬃﬃﬃﬃ if q 2 ðmod 4Þ : 2q if q 0 ðmod 4Þ: For general polynomials, it was first shown by L.J. Mordell that jSðq; f ðxÞÞj\k q 11=k when q is a prime. In this case, A. Weil improved Mordell’s upper bound to pﬃﬃﬃ jSðq; f ðxÞÞj\k q : In 1940, Hua studied the general case and proved, Hua’s inequality.

ð3Þ

when s C (k - 2)2k-1 + 5. In other words, we have ð4Þ

provided s C (k - 2)2k-1 + 5 and thus

In 1938 Hua proved, Hua’s Lemma. Suppose that 1 B j B k. Then Z 1 j j jf ðaÞj2 da N 2 jþ : 0

holds for any positive integer q. This inequality is a fundamental result in number theory. Combining with a result of I.M. Vinogradov it produces GðkÞ 2kðlog kÞð1 þ oð1ÞÞ

m

GðkÞ ðk 2Þ2k1 þ 5:

e 2pif ðxÞ=q :

jSðq; f ðxÞÞj q 11=kþ

Through the work of Hardy, Littlewood, and Weyl, we know that Z Cð1 þ 1=kÞ s=k1 f ðaÞs e 2pina da F ðnÞ ; n Cðs=kÞ M

Cð1 þ 1=kÞ s=k1 n rk;s F ðnÞ Cðs=kÞ

q X

2

m

where F ðnÞ 1 is a certain singular series, and Z f ðaÞs e 2pina da ¼ oðns=k1 Þ;

ð6Þ

as k ! 1 (see [3]). As for the values of g(k), among many partial results, it was shown by K. Mahler that ð7Þ gðkÞ ¼ 2k þ bð3=2Þk c 2 holds when k is sufficiently large. It was conjectured by Euler that (7) holds for all k C 2. The exact values of g(k) for small k are listed in the following table.

ð5Þ k

g(k)

Authors

2

4

J. L. Lagrange

3

9

A. Wieferich

4

19

R. Balasubramanian, F. Dress, J.M. Deshouiller

5

37

J.R. Chen

6

73

S.S. Pillai

2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

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For a comprehensive survey on Waring’s problem we refer to Vaughan and Wooley [8]. Loo-Keng Hua was born on November 12, 1910, in Jintan, Jiangsu Province, China. His father managed a small store. After junior middle school, his family was too poor to support him for high-school. Instead, his father sent him to a free professional school in Shanghai for about two years. Afterwards, he worked in his father’s store and studied mathematics during his free time. In 1926, Jia-Ju Su published a paper claiming a general solution for algebraic equations of degree five. In 1930, Hua published a paper pointing out the mistake in Su’s paper. It caught the attention of Professor Qing-Lai Xiong, the chairman of the mathematics department of Tsinghua University at that time, and changed Hua’s life. In 1931, invited by Prof. Xiong, Hua joined the mathematics department of Tsinghua University, first as a clerk in the library, then as an assistant. Three years later he was promoted to the rank of lecturer. At that time, S.S. Chern was there as a graduate student. In 1934 and 1935, Hua published fifteen papers, one of them in Mathematische Annalen and ten of them in Toˆhoku Mathematical Journal. This distinguished him from the other mathematicians in China at that time. During 1935–1936, J.S. Hadamard and N. Wiener visited Tsinghua University. Hua was one of the very few, if not the only one, who could discuss mathematics with them at the research level. Impressed by Hua’s talent and hard work, Wiener recommended him to G.H. Hardy at Cambridge. In 1936, invited by Hardy and supported by a scholarship of the Culture and Education Foundation of China, Hua arrived at Cambridge University for a two year stay (Figure 1). Hardy was visiting the United States when Hua arrived at Cambridge, but he soon made friends with H. Davenport, T. Estermann, H. Heilbronn, R.A. Rankin, E.C. Titchmarsh, and E.M. Wright. In Cambridge, Hua mainly worked on exponential sums and Waring’s problem, and published a

dozen papers in the journals of the London Mathematical Society. In particular, the lemma discussed previously was proved during that time. However, Hua did not try to get a Ph.D. degree: he could not afford the registration fee, and he preferred to study several important topics during his limited time instead of concentrating on a thesis. Several of Hua’s results were introduced in Hardy and Wright’s famous An Introduction to the Theory of Numbers, published in 1938. Perhaps these are the earliest modern results achieved by a Chinese researcher appearing in a standard mathematical book. In 1937, the Japanese invaded China from the north. Peking University, Tsinghua University, and Nankai University had to move to the deep southwest city Kunming as one temporary Associated University. Hua returned to China as a full professor at that university in 1938, and his family joined him there. More or less at the same time, Chern returned from France and joined the mathematical faculty as a full professor. Needless to say, conditions were extremely difficult, the food supply was limited, scientific connection with the west was almost suspended, and they were in danger of bombardment every day. However, Hua produced some of his best work during that time, for example, his inequality, as mentioned previously. In 1945, Japan was defeated. In 1946, invited by I.M. Vinogradov, Hua visited the Soviet Union for three months. There he was treated as royalty, and was received by ministers, academicians, national artists, and others. In particular, he met almost all the well-known Soviet mathematicians: P.S. Alexandrov, B.H. Delone, A.N. Kolmogorov, Y.V. Linnik, A. Markov, I.G. Petrowski, L.S. Pontryagin, I.M. Vinogradrov, etc. Hua was so impressed by this visit that he wrote and published an enthusiastic diary. From 1947 to 1948, Hua was a member of the Institute for Advanced Study in Princeton. From 1948 to 1950, he was a professor at the University of Illinois at UrbanaChampaign. In 1949, the civil war in China ended and the People’s Republic of China was born. In 1950, Hua returned to China as a professor at Tsinghua University. On the way from the United States to China, he published a very moving letter in a newspaper appealing to the Chinese overseas to serve the motherland (Figure 2).

Figure 1. Loo-Keng Hua at work.

Figure 2. Hua and Mao.

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THE MATHEMATICAL INTELLIGENCER

them are exceptional since they occur only in dimensions 16 and 27. The other four are the so-called ‘‘classical domains’’. In a certain sense, classical domains may be regarded as the higher dimensional analogues of the unit disc in the complex plane. They are crucial in complex analysis of several variables. In 1953, using group representation theory, Hua obtained the orthonormal system for each of the four classical domains and gave the Bergman kernel, the Cauchy kernel, and the Poisson kernel for each of them. According to S.T. Yau (see Gong [2]), Hua’s work in this area was at least 10 years in advance of his contemporaries elsewhere. In algebra textbooks, we often meet Hua’s theorem, the Cartan-Brauer-Hua theorem, Hua’s identity, etc. That Hua is Loo-Keng Hua. Figure 3. Front, from left to right : Pan, Lu, Hua, Chen and Yue; second line: Wang on the right and Wan the second on the left.

Hua’s Theorem. Every semi-automorphism of a skew field is either an automorphism or an anti-automorphism. Cartan-Brauer-Hua Theorem. Every normal subfield of a skew field is contained in its center.

Though modern science came to China at the beginning of the 20th century, the Japanese invasion and the civil war destroyed almost everything. The new republic was faced with the difficult job of rebuilding the country. Hua was, by far, the best mathematician and one of the best scientists in China. Of course, he took a leadership role in the mathematical community. He was elected president of the Chinese Mathematical Society in 1951, appointed founding director of the Mathematics Institute of the Academia Sinica in 1952, and was elected to the Academia Sinica in 1955. From 1949 to 1956, China flourished economically, culturally, and scientifically. Although conditions were very poor, Hua organized different research groups and seminars in the new institute. Many of his students at that time became leading figures in the Chinese mathematical community afterwards, for example, Chen, Wang and Pan in number theory, Zhexian Wan in algebra, Qikeng Lu and Sheng Gong in complex analysis (Figure 3). Unfortunately, the situation changed in 1956. During the next two decades, China experienced much political turmoil, in particular the cultural revolution. It is hard to explain to foreign colleagues what the Chinese culture revolution was, since there is no analogue abroad. The universities and research institutes were run by the revolutionaries with crazy ideas; professors and learned people were forced to do physical labor in factories, in farms or in labor camps, or even put in jail; talented students were replaced by ideologically favorable youths regardless their ability. Even under the special protection of En-Lai Zhou, the Chinese prime minister at that time, Hua had a very difficult time. Of course, his mathematical research work was fatally affected. Nevertheless he did a lot to popularize mathematics in China. Although Hua only had a formal education up to junior middle school, he was a universal mathematician, as Hilbert and Poincare´. He did fundamental work in number theory, algebra, the geometry of matrices, complex analysis, numerical integration, optimization, etc. In 1935, E. Cartan proved that there are six types of irreducible homogeneous bounded symmetric domains. Two of

On June 12, 1985, Hua presented a lecture at Tokyo University, Japan. At the end of the lecture he collapsed from a heart attack. He died a few hours later (Figure 4). For his distinguished contribution to mathematics, Hua was awarded honorary doctorates by the University of Nancy (1979), the Chinese University of Hong Kong (1982), and the University of Illinois at Urbana (1984). In 1956, he received a National Science Prize of first rank in China; it was the first time the People’s Republic of China honored a scientist at the national level. He was elected to the Academia Sinica and the Third World Academy, and a foreign member of the National Academy of Sciences, USA, the Bavarian Academy of Sciences, and the Deutsche Akademie der Naturforscher Leopoldina. Within China, Hua is a symbol for mathematics. His stories appear in the textbooks and on TV, his statues appear on the campuses of universities and high-schools, and his pictures appear in textbooks and on stamps. He will be remembered forever as the first modern mathematician in China (Figure 5).

Figure 4. Hua at his last lecture. 2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

21

If A is the set of integers in the interval [1, z2] and P the set of all primes, then (8) is Eratosthenes’ sieve. Let Ad denote the subset of elements of A divisible by d, let |X| denote the number of the elements of X, and let l(n) be the Mo¨bius function, defined for n ¼ pa11 par r by 8 if n ¼ 1; < 1 lðnÞ ¼ 0 ai 2 for some i; : ð1Þr otherwise: Then SðA; P; zÞ ¼

X

lðdÞ jAd j:

ð9Þ

djPðzÞ

Figure 5. Hua on a Chinese stamp.

Let P denote the set of primes that are not in P: We choose a suitable X [ 1 and a multiplicative function x(d) defined on the integers d satisfying both lðdÞ 6¼ 0 and (d, p) = 1 for all p 2 P; and take rd ¼ jAd j

Goldbach’s Conjecture Goldbach made his famous conjecture in a letter to Euler in 1742:

(a) Every even integer n C 6 is the sum of two odd primes. (b) Every odd integer n C 9 can be represented as the sum of three odd primes. Clearly, (a) implies (b). In 1923, under the assumption of the Generalized Riemann Hypothesis (explained below), Hardy and Littlewood were able to prove that (b) is true for sufficiently large n. In 1937, based on the circle method and his ingenious estimation of trigonometric sums with prime variables, I.M. Vinogradov was able to remove the assumption of the GRH. In other words, he proved. Goldbach-Vinogradov theorem. Every sufficiently large odd integer can be represented as the sum of three odd primes. However, the circle method does not solve case (a). For that, sieve methods turn out to be more powerful. Mark any natural number that is divisible by the first prime 2, and repeat the same with all other primes less than a given z [ 2. Then any natural number less than z2 that remains unmarked is either 1 or a prime in the interval [z, z2). This is the well-known sieve method of Eratosthenes of Alexandria. The idea of Eratosthenes’ sieve method is clear and important. However, it cannot handle more sophisticated problems. In 1919, Viggo Brun improved Eratosthenes’ sieve to a quantitatively effective device, and became the founder of the modern theory of the sieve method. Let A be a sequence of integers and let P be an infinite set of primes. Assume that z C 2 is a real number and define P(z) to be the product of all primes p satisfying both p \ z and p 2 P: Then we define the sieve function X 1: ð8Þ SðA; P; zÞ ¼ a2A ða;PðzÞÞ¼1

22

THE MATHEMATICAL INTELLIGENCER

It follows from (9) that X

X xðdÞ lðdÞ rd Xþ d djPðzÞ djPðzÞ X X xðpÞ þh ¼X 1 jrd j; p p\z djPðzÞ

SðA; P; zÞ ¼

Goldbach’s conjecture.

xðdÞ X: d

lðdÞ

p2P

where h is a number satisfying |h| B 1. Clearly, the keys to success with sieve methods are a skillful choice of x(d) and sharp estimates of the last two sums in this formula. For convenience, we use (a, b) as shorthand for the following proposition: There are positive integers a, b such that, for all sufficiently large integers n, 2n ¼ Pa þ Pb where Pk is a product of at most k primes. Clearly, the case (a) of Goldbach’s conjecture is basically equivalent to (1, 1). Let N be a sufficiently large even integer and let k be a fixed number not less than 2. We let A ¼ fjðN jÞ :

1 j N g;

P be the set of all primes, and z = N1/k. Then, it can be shown that SðA; P; zÞ [ 0 implies (a, a), where k1 a¼ bkc

ð10Þ

if k is an integer; otherwise:

In 1920, Brun was able to prove (9,9) along this line, thereby shedding light on Goldbach’s conjecture. Afterward, many mathematicians made improvements to sieve methods and to (a, b) type propositions. In 1953, when the Mathematics Institute of the Academia Sinica was founded, Hua organized two seminars on number theory: an introduction to number theory and Goldbach’s conjecture. These seminars produced several well-known number-theorists such as Chen, Wang, and Pan.

Wang was the first Chinese to contribute to Goldbach’s conjecture. In 1956 and 1957, he was able to prove (3, 4) and (2, 3), respectively, which started a race in China as well. On the other hand, if we take A ¼ fN p :

p N g;

then (10) implies (1, a). Let p(x, q, l) denote the number of primes p B x in the arithmetic progression l, l + q, l + 2q, …. In 1948, A. Re´nyi proved the following result, from which it follows that (1, c) holds for some positive integer c: There are constants d [ 0 and a C 6 such that Z x x X 1 dt max pðx; q; lÞ ¼O a ; uðqÞ 2 ln t ln x ðl;qÞ¼1 q xd

ð11Þ

where u(q) is the Euler function. However, he could not determine the values of d and c. It was Pan who first gave an exact number c, in 1961, proving (11) for d = 1/3 - e and therefore (1, 5). Sieve methods have been developed and improved by many authors such as M.B. Barban, E. Bombieri, Brun, A.A. Buchstab, Chen, Estermann, P.X. Gallagher, H. Halberstam, H. Iwaniec, P. Kuhn, Linnik, Pan, H. Rademacher, Re´nyi, H.E. Richert, A. Selberg, A. Vinogradov, Wang, and many others, and (a, b) type results have been improved by many of them. In 1966 Chen proved (1, 2). Chen’s theorem. We take B = {p: p B N, N - p = P2}, then ! Y p1 Y 1 N jBj [ 0:67 1 2 ðp 1Þ 2\pjN p 2 log2 N p[2 holds for all sufficiently large even integers N. In particular, every sufficiently large even integer N can be represented in the form N = p + P2. Jingrun Chen (Figure 6) was born on 22 May, 1933, in Fujian Province, China. His father was a clerk in a small post office. Compared to Hua, he was much luckier, since he had a formal university education, entering the mathematics department of Xiamen University in 1950. In 1953, Chen became an high-school teacher in Beijing. However, he was fired in 1954 by the school because he was unsuited for teaching. In 1955, he obtained a job in the library of the mathematics department of Xiamen University. There he carefully studied Hua’s book Additive Prime Number Theory and wrote a paper entitled On Tarry’s problem. Impressed by his work, Hua invited him to join the Institute of Mathematics, Academia Sinica. That was the starting point of Chen’s academic career. Chen announced his (1, 2) in 1966 and published it in full detail in 1973. The delay was mainly caused by the cultural revolution, during which Chen had a difficult time. One day, a politician of very high rank visited him in his small room on the third floor of a dormitory and asked him to sign a letter claiming that Hua stole his scientific work. If he signed such a letter, Hua would be in very serious political trouble. This visit put him in a very difficult

Figure 6. Chen (1933–1996).

situation: Certainly, he could not tell a lie against his beloved and respected teacher, the best known mathematician in the whole country. However, if he did not do as the politician requested, he would have endless troubles of his own. After a hard struggle, he chose to commit suicide by jumping from the window of his room. Fortunately, there was a balcony at the first floor. Chen was injured but not fatally. By this extreme action, among other reasons, he managed to avoid political trouble. When the nightmare of the culture revolution ended in 1976, people realized that Chen was one of the very few persons who had continued his work in secret during the hard time and had made remarkable contributions to mathematics (among others, Wentsun Wu, who has made an important contribution to algebraic topology, and Lo Yang and Guanghou Zhang, who have done important work in valuedistribution theory; see [1]). In 1978, Chen’s heroic story was published by a national magazine and was soon reprinted by newspapers and the popular press. Overnight, Chen became a national hero. Since it was hard to explain Chen’s mathematical contribution to the layman, the writer simply claimed that Chen had proved ‘‘1 + 2’’. Although thousands of people were moved by his story, many of them wondered why one could not simply show 1 + 2 = 3. Besides Goldbach’s conjecture and Waring’s problem, Chen made contributions to several other topics in analytic 2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

23

Figure 7. From left to right: Hua, Chen, Yang and Zhang.

number theory, including the lattice-point problem. Let r(n) denote the number of integer solutions of n = z21 + z22 and define X rðnÞ: AðxÞ ¼ 0nx

Clearly, A(x) is the pﬃﬃﬃ number of the lattice points (z1, z2) in a circle of radius x and centered at the origin. Assume that AðxÞ ¼ px þ O x hþ holds for some constant h and any given positive e. That h = 1/4 is one of the best known conjectures in number theory. Gauss proved h = 1/2, W. Sierpinski improved it to 1/3, J.G. van der Corput to 37/112, Hua to 13/40, Chen to 12/37, and Iwaniec and J. Mozzochi to 7/22. In 1978, Chen was promoted to a position as research professor at the Institute of Mathematics, the Chinese Academy of Sciences. In 1980, he was elected to the Chinese Academy of Sciences. He was honored by a National Science Prize of first rank and a Hua prize of the Chinese Mathematical Society. Chen suffered from Parkinson’s disease from 1984 to his death in 1996. During that time, he was mainly in the hospital. Yuan Wang (Figure 8) was born on 29 April in 1930 in Zhejiang Province, China. His father was the governor of Lanxi county at that time; in the 1940s, he was a high-ranking secretary of the Academia Sinica. From 1949 to 1952, Yuan Wang studied at Zhejiang University, where both Jian-Gong Chen and Bu-Qing Su were teaching. In 1952, recommended by Chen and Su, Wang was hired by the Institute of Mathematics of the Chinese Academy of Sciences. Soon he joined the number theory group under Hua’s supervision. In 1978, he was promoted to full professor at the Institute of Mathematics of the Chinese Academy of Sciences. Besides Goldbach’s conjecture, Prof. Wang has made important contributions to uniform-distribution theory, to Diophantine approximation, and even to statistics. Let Ean(C) denote the set of the n-variable functions X f ðxÞ ¼ cðm1 ; . . .; mn Þe 2piðm1 x1 þþmn xn Þ satisfying

where a [ 1 and C [ 0 are two constants and kmk ¼ m01 m0n ; m0i ¼ maxf1; mi g: In 1960 Hua and Wang proved the following basic result: Let F0, F1, F2, . . . be the Fibonacci sequence defined by F0 = F1 = 1 and Fi+2 = Fi + Fi+1. Then Z Z Fn 1 1 1 X k Fn1 k sup f ðxÞdx f ; Fn k¼1 Fn Fn 0 f 2E2a ðCÞ 0 C ln 3Fn : ¼O Fna Afterward, in a series of papers, Hua and Wang generalized this result from two to n dimensions. For his distinguished contribution to mathematics, Prof. Wang was elected as a member of the Chinese Academy of Sciences in 1980. He has been honored by a National Science Prize of first rank (together with Chen and Pan), by a Hua prize of the Chinese Mathematical Society, and by several other prizes. From 1984 to 1987, he was the director of the Institute of Mathematics of the Chinese Academy of Sciences. From 1988 to 1992 he was the president of the Chinese Mathematical Society.

The Least Primes in Arithmetic Progressions Let l and q be positive integers satisfying both l \ q and (l, q) = 1. Dirichlet proved in 1837 that there are infinitely many primes in the arithmetic progression l; l þ q; l þ 2q; . . .: In fact, similar to the prime number theorem, we have pðx; q; lÞ ¼

x ð1 þ oð1ÞÞ: uðqÞ log x

Let p(l, q) denote the least prime in this progression. S. Chowla conjectured in 1934 that pðl; qÞ q log2 q:

jcðm1 ; . . .; mn Þj

24

Figure 8. Prof. Wang in a TV interview.

THE MATHEMATICAL INTELLIGENCER

C ; kmka

Let v(n) denote the characteristic function for a given modulus q and let L(s, v) denote the Dirichlet L function

defined by Lðs; vÞ ¼

1 X

vðnÞns ;

s ¼ r þ si:

n¼1

The Generalized Riemann Hypothesis claims that not only f(s) but all the functions L(s, v) have their nontrivial zeros in the critical strip on the line r = 1/2. In 1944, Linnik proved that pðl; qÞ q k holds for some positive number k, but he was not able to determine its value. In 1957, by studying the zeros of L functions, Pan obtained the first effective upper bound, pðl; qÞ\q 10

4

and soon after reduced it to pðl; qÞ\q 5448 : This breakthrough led to a long list of improvements by Chen, M. Jutila, S. Graham, W. Wang, D.R. Heath-Brown, and others. So far the best known record is pðl; qÞ q 5:5 ; which is still far from the conjectured one. Chengdong Pan (Figure 9) was born on 26 May, 1934, in Suzhou, Jiangsu Province, China. In 1952, he entered Peking University to study mathematics. At that time, several professors in the faculty had obtained their Ph.D degrees from Cambridge, Princeton, and other institutions. Attracted by the beauty of number theory, Pan chose to study analytic number theory under Professor Sihe Min’s

supervision. Since Peking University is near the Institute of Mathematics of the Chinese Academy of Sciences, Pan was a regular member of Hua’s seminars. In 1961, Pan joined the mathematical faculty of Shandong University. In 1978, he was promoted to professor at that university, and became its president in 1986. He died in 1997. Besides Goldbach’s conjecture and the least primes in arithmetic progressions, Pan made a contribution to the estimate of the exponential sums with prime variables, prime numbers in small intervals, and in other areas as well. He supervised many successful students and, with Chengbiao Pan, wrote many books in Chinese on number theory. Pan was elected to the Chinese Academy of Sciences in 1991. He was once a vice-president of the Chinese Mathematical Society, and was once the chairman of the Shandong Science and Technology Association. Together with Chen and Wang, he was honored by a National Science Prize of first rank in China. Many other Chinese mathematicians have made contributions to analytic number theory, in particular the active young generation, for example, Chaohua Jia, Jianya Liu, Tianze Wang, and Tao Zhan. But in this article I have discussed only the period from 1930 to 1980. ACKNOWLEDGMENTS

I am very grateful to Prof. Marjorie Senechal for her kind help in improving the English of this paper, to Prof. Chaohua Jia, and to the referees for their helpful comments and suggestions. REFERENCES

[1] A. Fitzgerald and S. MacLane (eds), Pure and Applied Mathematics in the People’s Republic of China, National Academy of Sciences, Washington, D.C., 1977. [2] S. Gong, The life and work of famous Chinese mathematician Loo-Keng Hua, Adv. Appl. Clifford Algebras, 11 (2001), 1–7. [3] H. Halberstam, Loo-Keng Hua: Obituary, Acta Arithmetica, 51 (1988), 99–107. [4] H. Halberstam and H.E. Richert, Sieve Methods, Academic Press, London, 1974. [5] Y. Motohashi, An overview of the sieve method and its history, Sugaku Expositions, 21 (2008), 1–32. [6] C. D. Pan and Y. Wang, Chen Jingrun: A brief outline of his life and works, Acta Math. Sinica, 12 (1996), 225–233. [7] R.C. Vaughan The Hardy-Littlewood method, Cambridge University Press, Cambridge, 1981. [8] R.C. Vaughan and T.D. Wooley, Waring’s problem: A survey, Number Theory for the Millennium III (eds, M.A. Bennett, etc.), A.K. Peters, Massachusetts, 2002, 301–340. [9] Y. Wang, Hua Loo-Keng (in Chinese), Beijing, 1994; English version, Springer, Singapore, 1999. [10] Y. Wang, Pan Chengdong: A brief outline of his life and works, Acta Math. Sinica (Chinese version), 3 (1998), 449–454.

Figure 9. Chengdong Pan (1934–1997).

School of Mathematical Sciences Peking University Beijing 100871 China e-mail: [email protected]

2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

25

Chromogeometry N. J. WILDBERGER

hree-fold symmetry is at the heart of a lot of interesting mathematics and physics. In this article I show that three-fold symmetry also plays an unexpected role in planar geometry, in that the familiar Euclidean geometry is only one of a trio of interlocking metric geometries. I refer to Euclidean geometry here as blue geometry; the other two geometries, called red and green, are relativistic in nature and are associated with the names of Lorentz, Einstein, and Minkowski. The three geometries support each other and interact in a rich way. This transcends Klein’s Erlangen program, in that there are now three groups acting on a space. Algebraic identities lie at the heart of the explanations. The results described here are just the tip of an iceberg, leading to many generalizations of results of Euclidean geometry, with much waiting to be explored; see for example [4] for applications to conics and [3] for connections with one-dimensional metric geometry. In basic structure, all three geometries are similar—they are ruled by the laws of rational trigonometry as developed recently in [1], which hold over a general field not of characteristic 2. Although over the rational numbers (or the ‘real numbers’) there are significant differences between the Euclidean (blue) version and the other two (red and green), it is the interaction of all three that yields the biggest surprises. To start the ball rolling, I first introduce the phenomenon in the context of the classical Euler line and nine-point circle of a triangle. Then I will recall the main laws of rational trigonometry, introduce the basic facts about the three geometries, and state some explicit formulas, and then show how chromogeometry allows us to enlarge our understanding of the geometry of a triangle. In particular I associate to each triangle A1 A2 A3 in the Cartesian plane a second interesting triangle, which I call the X-triangle of A1 A2 A3 : The results are verified by routine but sometimes lengthy computation, which inevitably reduce to algebraic identities, some of which are lovely. The development takes place in the framework of universal geometry, so that we are interested primarily in what happens over arbitrary fields. The article [2] shows that universal geometry also extends to arbitrary quadratic forms, and

T

26

THE MATHEMATICAL INTELLIGENCER 2009 Springer Science+Business Media, LLC

embraces both spherical and hyperbolic geometries in a projective version.

Euler Lines and Nine-point Circles in Relativistic Settings Recall that for a triangle A1 A2 A3 the intersection of the medians is the centroid G, the intersection of the altitudes is the orthocenter O, and the intersection of the perpendicular bisectors of the sides is the circumcenter C, which is the center of the circumcircle of the triangle. Remarkably, it was left to Euler to discover that these three points are collinear, and that G divides OC in the (affine) proportion 2:1. Furthermore, the center N of the circumcircle of the triangle M1 M2 M3 of midpoints of the sides of A1 A2 A3 (called the nine-point circle of A1 A2 A3 ) also lies on the Euler line, and is the midpoint of OC:

4.5 4 3.5 3 2.5 2 1.5 1 0.5

1 -0.5 -1 -1.5 -2 -2.5 -3

2

3

4

5

6

This is shown here for the triangle A1 A2 A3 with points A1 ½0; 0 A2 ½6; 1

A3 ½2; 3:

4.5 4 3.5

The triangle A1 A2 A3 is in black, whereas the circumcircle and nine-point circle are in blue (the latter more boldly), as are the Euler line and the points O, C, and N, which are given the subscript b for blue, and are henceforth referred to as the blue Euler line, the blue orthocenter, etc. Planar Euclidean geometry rests on the blue quadratic form x2 + y2 (or, if you prefer, on the corresponding symmetric bilinear form, or dot product). But now let us change the setting and consider the red quadratic form x2 - y2 that figures prominently in two-dimensional special relativity. With respect to this form, two lines are redperpendicular precisely when one can be obtained from the other by ordinary Euclidean reflection in a red null line, which is red-perpendicular to itself, and has usual slope ±1. It turns out that for any triangle A1 A2 A3 the three red altitudes also intersect, now in a point called the red orthocenter and denoted Or, and the three perpendicular bisectors also intersect in a point called the red circumcenter and denoted Cr. This latter point is the center of the unique red circle through the three points of the triangle, but attention! a red circle is given by an equation of the form ðx x0 Þ2 ðy y0 Þ2 ¼ K : This is what we would usually call a rectangular hyperbola, with axes in the red null directions. This diagram shows the same triangle A1 A2 A3 as before, along with its red circumcircle, its red nine-point circle, and its red orthocenter, circumcenter, nine-point center, and

AUTHOR

......................................................................... NORMAN J. WILDBERGER studied at the

University of Toronto and Yale and taught at Stanford and Toronto before settling at the University of New South Wales. He has posted several dozen YouTube videos about his pet invention Rational Trigonometry. He also enjoys music, playing Go, and bushwalking. As for modern mathematics, he thinks what it needs most is fixing. School of Mathematics and Statistics UNSW, Sydney 2052 Australia e-mail: [email protected]

3 2.5 2 1.5 1 0.5

1

2

3

4

5

6

-0.5 -1 -1.5 -2 -2.5 -3

centroid G. Note that the centroid is independent of color (because the medians are). These points all lie on a line—the red Euler line, and the affine relationships between these points are exactly the same as for the blue Euler line, so that for example Nr is the midpoint of Or Cr : In the classical framework, there are some difficulties in setting up this relativistic geometry, as ‘distance’ and ‘angle’ are problematic. In universal geometry one regards the quadratic form as primary, not its square root. This approach was introduced in [1], see also [5], and works over a general field (with characteristic 2 excluded for technical reasons), as shown in [2]. Distance and angle are avoided, their place being taken by rational functions of points called quadrance and spread. Euclidean geometry can be built up so as to allow generalization to the relativistic geometries. The possibility of relativistic geometries over other fields seems particularly attractive. Now consider the third geometry—that associated to the green quadratic form 2xy. Two lines are green-perpendicular when one is the ordinary Euclidean reflection of the other in a line parallel to either axis, the latter being a green null line. Since x2 - y2 and 2xy are conjugate by a simple change of variable, it should be no surprise that the corresponding relations among the green orthocenter Og, the green circumcenter Cg, the green nine-point center Ng, and the centroid G hold as well. Here is the relevant diagram for our triangle A1 A2 A3 : I have found, so far, that most theorems of planar Euclidean geometry, when formulated algebraically in the 2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

27

4.5 4 3.5 3 2.5 2 1.5 1 0.5

1

2

3

4

5

6

-0.5 -1

We obtain remarkable collinearities, for example between Ob, Cr, and Og, and between Cb, Nr, and Cg; furthermore, Cr turns out to be the midpoint of Ob Og ; and Nr the midpoint of Cb Cg : Also we observe that, for example, Or and Og lie on the blue circumcircle of A1 A2 A3 ; whereas Cr and Cg lie on the blue nine-point circle of A1 A2 A3 : This latter fact is a consequence of the well-known theorem of Euclidean geometry that the center of any rectangular hyperbola through the vertices of a triangle lies on the ninepoint circle. But now we see that all these facts hold even if we permute colors, so there is a three-fold symmetry here. However sometimes this symmetry is broken. The blue geometry, as we shall see, behaves somewhat differently from the red and the green in certain contexts, and when we come to explicit formulas we will see that the green geometry is often simpler. The red geometry seems less inclined to distinguish itself.

-1.5

Rational trigonometry

-2 -2.5 -3

context of universal geometry, extend to the red and green situations. However, there are exceptions. For example, over the ‘real numbers’ there are no equilateral triangles in the red or green geometries, so Napoleon’s theorem and Morley’s theorem will not have direct analogs. Much could be said further to support this generalization, but this support is not what I wish to pursue here. Instead, let’s consider a completely new phenomenon. Observe what happens when the three diagrams are put together!

Let’s now proceed more formally, beginning with the main definitions and laws of rational trigonometry. I work over a fixed field, not of characteristic 2, whose elements will be called numbers. The plane will consist of the standard vector space of dimension 2 over this field. A point, or vector, is an ordered pair A : [x, y] of numbers. The origin is denoted O : [0, 0] . A line is a proportion l ha : b : ci where a and b are not both 0. The point A : [x,y] lies on the line l ha : b : ci; or equivalently the line l passes through the point A, precisely when ax þ by þ c ¼ 0: This is not the only possible convention, and the reader should be aware that it is prejudiced towards the usual Euclidean (blue) geometry. For distinct points A1 : [x1, y1] and A2 : [x2, y2] there is a unique line l : A1A2 that passes through them both. Specifically, we have A1 A2 ¼ hy1 y2 : x2 x1 : x1 y2 x2 y1 i:

4.5 4

Three points [x1, y1], [x2, y2] and [x3, y3] are collinear precisely when they lie on the same line, which amounts to the condition

3.5 3

x1 y2 x1 y3 þ x2 y3 x3 y2 þ x3 y1 x2 y1 ¼ 0:

2.5

ð1Þ

Three lines ha1 : b1 : c1 i; ha2 : b2 : c2 i and ha3 : b3 : c3 i are concurrent precisely when they pass through the same point, which amounts to the condition

2 1.5 1

a1 b2 c3 a1 b3 c2 þ a2 b3 c1 a3 b2 c1 þ a3 b1 c2 a2 b1 c3 ¼ 0:

0.5

1

2

3

-0.5 -1

5

6

Fix a symmetric bilinear form, denoted by the dot product A1 A2. In practice we will take this bilinear form to be nondegenerate. The line A1A2 is perpendicular to the line B1B2 precisely when ðA2 A1 Þ ðB2 B1 Þ ¼ 0:

-1.5

A point A is a null point or null vector precisely when A A = 0. The origin O is always a null point, but there may be others. A line A1A2 is a null line precisely when the vector A2 - A1 is a null vector.

-2 -2.5 -3

28

4

THE MATHEMATICAL INTELLIGENCER

A set {A1, A2, A3} of three distinct noncollinear points is a triangle and is denoted A1 A2 A3 : The lines of the triangle are l3 : A1A2, l2 : A1A3, and l1 : A2A3. A triangle is non-null precisely when each of its lines is non-null. A side of the triangle is a subset of {A1, A2, A3} with two elements, and is denoted A1 A2 ; etc. A vertex of the triangle is a subset of {l1, l2, l3} with two elements, and is denoted l1 l2 ; etc. The quadrance between the points A1 and A2 is the number QðA1 ; A2 Þ ðA2 A1 Þ ðA2 A1 Þ: The line A1A2 is a null line precisely when Q(A1, A2) = 0. The spread between the non-null lines A1A2 and B1B2 is the number sðA1 A2 ; B1 B2 Þ 1

ððA2 A1 Þ ðB2 B1 ÞÞ2 : QðA1 ; A2 ÞQðB1 ; B2 Þ

This is independent of the choice of points lying on the two lines. Two non-null lines are perpendicular precisely when the spread between them is 1. The theorems that follow are the five main laws of planar rational trigonometry in this general setting, replacing the usual Sine law, Cosine law, etc. Proofs can be found in [2]. I venture to recommend these laws not only to fellow geometers, but also for teaching high-school mathematics, because the laws are simpler, and they allow faster and more accurate calculations in practical problems. But the advantage for us here is that they hold for general quadratic forms, and in particular for each of the blue, red, and green geometries. Suppose we have three distinct points A1, A2, and A3 with non-zero quadrances Q1 QðA2 ; A3 Þ; Q2 QðA1 ; A3 Þ; and Q3 : Q(A1, A2), and spreads s1 sðA1 A2 ; A1 A3 Þ; s2 sðA2 A1 ; A2 A3 Þ; and s3 : s(A3A1, A3A2).

T HEOREM 1 (Triple quad formula) The points A1, A2, and A3 are collinear precisely when ðQ1 þ Q2 þ Q3 Þ2 ¼ 2 Q21 þ Q22 þ Q23 : T HEOREM 2 (Pythagoras’s theorem). The lines A1A3 and A2A3 are perpendicular precisely when Q1 þ Q2 ¼ Q3 :

A useful observation deduced from the Triple spread formula is that if s3 = 1 then s1 þ s2 ¼ 1:

Three-fold symmetry The vectors A1 : [x1, y1] and A2 : [x2, y2] are parallel precisely when x1 y2 x2 y1 ¼ 0: Attention in this article is on three main examples of symmetric bilinear forms. Define the blue dot product ½x1 ; y1 b ½x2 ; y2 x1 x2 þ y1 y2 ; the red dot product ½x1 ; y1 r ½x2 ; y2 x1 x2 y1 y2 ; and the green dot product ½x1 ; y1 g ½x2 ; y2 ¼ x1 y2 þ x2 y1 : Note that between them these four expressions yield all possible bilinear expressions in the two vectors, up to sign, that involve only coefficients ±1. Two lines l1 and l2 are blue, red, and green perpendicular, respectively, precisely when they are perpendicular with respect to the blue, red, and green forms. For lines l1 ha1 : b1 : c1 i and l2 ha2 : b2 : c2 i these conditions amount to the respective conditions a1 a2 þ b1 b2 ¼ 0 a1 a2 b1 b2 ¼ 0 and a1 b2 þ a2 b1 ¼ 0 [green]: In terms of coordinates, the formulas for the blue, red, and green quadrances between points A1 : [x1, y1] and A2 : [x2, y2] are Qb ðA1 ; A2 Þ ¼ ðx2 x1 Þ2 þðy2 y1 Þ2 Qr ðA1 ; A2 Þ ¼ ðx2 x1 Þ2 ðy2 y1 Þ2 Qg ðA1 ; A2 Þ ¼ 2ðx2 x1 Þðy2 y1 Þ:

T HEOREM 6 (Colored quadrances) For any points A1 and A2, let Qb , Qr , and Qg be the blue, red, and green quadrances between A1 and A2, respectively. Then

THEOREM 3 (Spread law) s1 s2 s3 ¼ ¼ : Q1 Q2 Q3

T HEOREM 4 (Cross law) ðQ1 þ Q2 Q3 Þ2 ¼ 4Q1 Q2 ð1 s3 Þ: Note that the Cross law includes as special cases both the Triple quad formula and Pythagoras’s theorem. The next result is the algebraic analog of the formula for the sum of the angles in a triangle.

T HEOREM 5 (Triple spread formula) ðs1 þ s2 þ s3 Þ2 ¼ 2 s12 þ s22 þ s32 þ 4s1 s2 s3 :

[blue] [red]

Q2b ¼ Q2r þ Q2g :

P ROOF . This is a consequence of the identity 2 2 r 2 þ s2 ¼ r 2 s2 þð2rsÞ2 : The formulas for the blue, red, and green spreads between lines l1 ha1 : b1 : c1 i and l2 ha2 : b2 : c2 i are ða1 a2 þ b1 b2 Þ2 ða1 b2 a2 b1 Þ2 ¼ 2 2 2 2 2 ða1 þ b1 Þða2 þ b2 Þ ða2 þ b22 Þða21 þ b21 Þ ða1 a2 b1 b2 Þ2 ða1 b2 a2 b1 Þ2 ¼ 2 sr ðl1 ; l2 Þ ¼ 1 2 2 2 2 ða1 b1 Þða2 b2 Þ ða1 b21 Þða22 b22 Þ 2 ða1 b2 þ a2 b1 Þ ða1 b2 a2 b1 Þ2 sg ðl1 ; l2 Þ ¼ 1 ¼ : 4a1 a2 b1 b2 4a1 a2 b1 b2

sb ðl1 ; l2 Þ ¼ 1

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29

Note carefully the minus signs that precede the final expressions in the red and green cases.

18 16 14

T HEOREM 7 (Colored spreads) For any lines l1 and l2, let

12

sb, sr, and sg be the blue, red, and green spreads between l1 and l2, respectively. Then

10 8

1 1 1 þ þ ¼ 2: sb sr sg

6 4

PROOF. This is a consequence of the identity a21

þ

b21

a22

þ

b22

a21 2

b21

a22

b22

4a1 a2 b1 b2

2 -5

5

10

15

20

25

-2

¼ 2ða1 b2 a2 b1 Þ :

-4

Quadreas The most important single quantity associated to a triangle A1 A2 A3 with quadrances Q1, Q2 and Q3 is the quadrea A defined by A ðQ1 þ Q2 þ Q3 Þ2 2 Q21 þ Q22 þ Q23 : By the Triple quad formula, this is a measure of the noncollinearity of the points A1, A2 and A3. Denote by Ab ; Ar ; and Ag the respective blue, red, and green quadreas of a triangle A1 A2 A3 :

T HEOREM 8 (Quadrea) For three points A1 : [x1, y1], A2 : [x2, y2], and A3 : [x3, y3], the three quadreas Ab ; Ar , and Ag satisfy Ab ¼ Ar ¼ Ag ¼ 4ðx1 y2 x1 y3 þ x2 y3 x3 y2 þ x3 y1 x2 y1 Þ2 :

P ROOF . A calculation. So each quadrea of a triangle is ±16 times the square of its signed area, and that area is defined purely in an affine setting, without any need for metrical choices. I now adopt the convention that if no proof is given, ‘a calculation’ is to be assumed.

Altitudes T HEOREM 9 (Altitudes to a line) For any point A and any line l, there exist unique lines nb, nr, and ng through A, which are respectively blue, red, and green perpendicular to l. If A : [x0, y0] and l : ha : b : ci then nb ¼ hb : a : bx0 þ ay0 i nr ¼ hb : a : bx0 ay0 i ng ¼ ha : b : ax0 þ by0 i: The lines nb, nr, ng are respectively the blue, red, and green altitudes from A to l, and if l is non-null they intersect it at the feet Fb, Fr and Fg, respectively.

T HEOREM 10 (Perpendicularity of altitudes) For any point A and any line l, let nb, nr, ng be the blue, red, and green altitudes from A to l respectively. Then nb and nr are green perpendicular, nr and ng are blue perpendicular, and ng and nb are red perpendicular.

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THE MATHEMATICAL INTELLIGENCER

The figure shows an example of the three color altitudes from a point A to a line l, and their feet Fb, Fr, and Fg.

T HEOREM 11 (Pythagorean means) Let l : ha : b : ci be a line that is non-null in each of the three geometries. If A is a point and Fb, Fr, and Fg are the respective feet of the altitudes nb, nr, and ng from A to l, then we have the affine relation 2 2 a b2 2ab 2 F þ Fg : Fb ¼ r a2 þ b2 a2 þ b2

P ROOF . Suppose that A : [x0, y0] and l : ha : b : ci. Elimination yields 2 b x0 aby0 ac abx0 þ a2 y0 bc ; Fb ¼ a2 þ b2 a2 þ b2 2 b x0 aby0 ca abx0 þ a2 y0 þ bc Fr ¼ ; a2 b2 a2 b2 ax0 by0 c ax0 þ by0 c ; Fg ¼ 2a 2b from which we deduce the result. Note again the connection with Pythagorean triples.

Anti-symmetric polynomials Let me adopt the notation for antisymmetric polynomials used in [page 28, 1]. If m is a monomial in the variables x1, x2, x3, y1, y2, y3, z1, z2, z3,… with all indices in the range 1, 2, and 3, then define [m]- to be the antisymmetric polynomial consisting of the sum of all monomials obtained from m by performing all six permutations of the indices and multiplying each term by the sign of the corresponding permutation. I often write such polynomials in the order described by the successive transpositions ð23Þ; ð12Þ; ð23Þ; ð12Þ; ð23Þ: For example,

½x1 y2 x1 y2 x1 y3 þ x2 y3 x3 y2 þ x3 y1 x2 y1 x12 x2 y2 x12 x2 y2 x12 x3 y3 þ x22 x3 y3 x32 x2 y2

x13 y1

þ x32 x1 y1 x22 x1 y1 x13 y1 x13 y1 þ x23 y2 x33 y3 þ x33 y3 x23 y2 :

The polynomial [x1y2]- is of particular importance, since it occurs in (1), is twice the signed area of the triangle A1 A2 A3 ; appears in the Quadrea theorem, and is often a denominator in formulas in the subject.

Orthocenters

2 x12 y2 þ y12 y2 x1 y2 þ x1 x22 ; Cb ¼ 2½x1 y2 2½x1 y2 2 2 2 x1 y2 y1 y2 x1 y2 x1 x22 Cr ¼ ; 2½x1 y2 2½x1 y2 ½x1 x2 y2 ½x1 y1 y2 Cg ¼ ; : ½x1 y2 ½x1 y2

Given a triangle A1 A2 A3 ; for each point Am (m = 1, 2, 3) we may construct the blue, red, and green altitudes abm, arm and agm respectively to the opposite side.

T HEOREM 15 (Circumcenters as midpoints) For any

T HEOREM 12 (Orthocenter formulas) The three blue

triangle, a colored circumcenter is the midpoint of the two orthocenters of the other two colors.

altitudes ab1, ab2, and ab3 meet in a point Ob called the blue orthocenter. The three red altitudes ar1, ar2, and ar3 meet in a point Or called the red orthocenter. The three green altitudes a1g, a2g, and a3g meet in a point Og called the green orthocenter. For A1 : [x1, y1], A2 : [x2, y2] and A3 : [x3, y3] these points are given by ½x1 x2 y2 þ y1 y22 ½x1 y1 y2 þ x12 x2 Ob ¼ ; ½x1 y2 ½x1 y2 ½x1 x2 y2 y1 y22 ½x1 y1 y2 x12 x2 ; Or ¼ ½x1 y2 ½x1 y2 2 x1 y2 þ½x1 x2 y1 x1 y22 ½x1 y1 y2 Og ¼ ; : ½x1 y2 ½x1 y2

Circumcenters When A1 and A2 are distinct points with l = A1A2, and M is the midpoint of A1 and A2, then the blue, red, and green altitudes from M to l are respectively called the blue, red, and green perpendicular bisectors of the side A1 A2 :

T HEOREM 13 (Perpendicular bisectors) If A1 : [x1, y1] and A2 : [x2, y2] are distinct points, then the blue, red, and green perpendicular bisectors of A1 A2 have respective equations x12 x22 þ y12 y22 2 x12 x22 y12 þ y22 ðx1 x2 Þx ðy1 y2 Þy ¼ 2 ðy2 y1 Þx þ ðx2 x1 Þy ¼ y2 x2 x1 y1 : ðx1 x2 Þx þ ðy1 y2 Þy ¼

Given a triangle A1 A2 A3 ; we may construct the blue, red, and green perpendicular bisectors of the three sides, denoted by bbm, brm and bgm respectively for m = 1, 2 and 3, where bb1 for example is the blue perpendicular bisector of the side A2 A3 ; and so on.

T HEOREM 14 (Circumcenter formulas) The three blue perpendicular bisectors bb1, bb2 and bb3 meet in a point Cb called the blue circumcenter. The three red perpendicular bisectors br1, br2 and br3 meet in a point Cr called the red circumcenter. The three green perpendicular bisectors bg1, bg2 and bg3 meet in a point Cg called the green circumcenter. For A1 : [x1, y1] , A2 : [x2, y2] and A3 : [x3, y3], these points are given by

P ROOF . This follows from the Orthocenter formulas and Circumcenter formulas.

Nine-point centers Suppose that the respective midpoints of a triangle A1 A2 A3 are Mm for m = 1, 2 and 3, where M1 is the midpoint of the side A2 A3 and so on. We let Nb, Nr and Ng be the blue, red, and green circumcenters respectively of the triangle M1 M2 M3 ; and call these the blue, red and green nine-point centers of the original triangle A1 A2 A3 :

T HEOREM 16 (Nine-point center formulas) For A1 : [x1, y1] , A2 : [x2, y2] and A3 : [x3, y3], the blue, red and green nine-point centers of A1 A2 A3 are 2 2 x1 y2 y1 y2 þ2½x1 x2 y2 Nb ¼ ; 4½x1 y2 2 x1 y2 x1 x22 þ2½x1 y1 y2 4½x1 y2 2 2 x1 y2 þ y1 y2 þ2½x1 x2 y2 ; Nr ¼ 4½x1 y2 2 x1 y2 þ x1 x22 þ2½x1 y1 y2 4½x1 y2 2 2 x1 y2 x1 y2 : ; Ng ¼ 2½x1 y2 2½x1 y2

T HEOREM 17 (Nine-point centers as midpoints) In any triangle, a colored nine-point center is the midpoint of the two circumcenters of the other two colors. P ROOF . This follows from the Circumcenter formulas and Nine-point center formulas.

The X-triangle and the Euler lines The X-triangle of a triangle A1 A2 A3 is the triangle X X A1 A2 A3 Ob Or Og of orthocenters of A1 A2 A3 : From the theorems of the last two sections, the corresponding midpoints of the sides of X are Cb, Cr and Cg, with Cb the midpoint of Or and Og, etc., and the midpoints of the triangle Cb Cr Cg are Nb, Nr and Ng, with Nb the midpoint of Cr and Cg, etc. We also know that the centroid of X is the same as the centroid G of the original triangle A1 A2 A3 : 2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

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T HEOREM 18 (Blue Euler line) The points Ob, Nb, G and Cb lie on a line called the blue Euler line. Furthermore Nb is the midpoint of Ob and Cb, and we have the affine relations

M2, M3 of the triangle A1 A2 A3 will be called respectively the blue, red and green nine-point circles of the triangle A1 A2 A3 :

1 2 1 2 G ¼ Ob þ Cb ¼ Cb þ Nb : 3 3 3 3

T HEOREM 22 (Orthocenters on circumcircles) Any col-

P ROOF . This follows from the Orthocenter, Circumcenter,

ored orthocenter of a triangle A1 A2 A3 lies on the circumcircles of the other two colors.

and Nine-point center formulas.

T HEOREM 23 (Nine-point circles) Any colored nineT HEOREM 19 (Red Euler line) The points Or, Nr, G and Cr lie on a line called the red Euler line. Furthermore Nr is the midpoint of Or and Cr, and 1 2 1 2 G ¼ Or þ Cr ¼ Cr þ Nr : 3 3 3 3

P ROOF . Likewise. THEOREM 20 (Green Euler line) The points Og, Ng, G and Cg lie on a line called the green Euler line. Furthermore Ng is the midpoint of Og and Cg, and 1 2 1 2 G ¼ Og þ C g ¼ C g þ N g : 3 3 3 3

point circle of a triangle A1 A2 A3 passes through the feet of the altitudes of that color, as well as the midpoints of the segments from the same colored orthocenter to the points A1, A2, and A3. In addition, it passes through the circumcenters of A1 A2 A3 of the other two colors. The following figure shows some of the points on the nine-point circles of different colors. Others are off the page. 26 24 22 20

P ROOF . Likewise.

18

The geometry of the X-triangle clarifies the various ratios occurring along points on the Euler lines, since these are just the medians of X. The lines joining the circumcenters are the lines of the medial triangle of X, and so are parallel to the lines of X.

16 14 12 10 8 6

Circles

4

A blue, red, or green circle is an equation c in x and y of the form

2

2

5

10

15

20

25

30

35

2

ðx x0 Þ þð y y0 Þ ¼ K ðx x0 Þ2 ð y y0 Þ2 ¼ K 2ðx x0 Þð y y0 Þ ¼ K respectively, where the point [x0, y0] is then unique and called the center of c, and K is the quadrance of c. A blue circle is an ordinary Euclidean circle. Red and green circles are more usually described as rectangular hyperbolas. A red circle has asymptotes parallel to the lines with equations y = ±x, and a green circle has asymptotes parallel to the coordinate axes.

I hope that this taste of chromogeometry will encourage others to explore this rich new realm. See [3] and [4] for more in this direction.

REFERENCES

[1] N. J. Wildberger, Divine Proportions: Rational Trigonometry to Universal Geometry, Wild Egg Books (http://wildegg.com), Sydney, 2005. [2] N. J. Wildberger, ‘Affine and Projective Universal Geometry’, to

T HEOREM 21 (Circumcircles) If A1, A2, and A3 are three distinct noncollinear points, then there are unique blue, red, and green circles passing through A1, A2 and A3.

appear in Journal of Geometry. [3] N. J. Wildberger, ‘One dimensional metrical geometry’, Geometriae Dedicata 128, no.1, 145–166, 2007. [4] N. J. Wildberger, ‘Chromogeometry and relativistic conics’

The circles in Theorem 21 will be called respectively the blue, red, and green circumcircles of the triangle A1 A2 A3 ; whereas the circumcircles of the triangle of midpoints M1,

32

THE MATHEMATICAL INTELLIGENCER

arXiv:0806.2789 [5] N. J. Wildberger, ‘A Rational Approach to Trigonometry’, Math Horizons, Nov. 2007, 16–20, 2007.

Mathematical Entertainments

Michael Kleber and Ravi Vakil, Editors

Twenty-Two Moves Suffice for Rubik’s CubeÒ TOMAS ROKICKI

This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, surprising, or appealing that one has an urge to pass them on. Contributions are most welcome.

â

1. We extend Kociemba’s near-optimal solving algorithm to consider six transformations of a particular position simultaneously, so it finds near-optimal positions more quickly; 2. We convert his solving algorithm into a set solver that solves billions of positions at a time at a rate of more than 200 million positions a second; 3. We show how to eliminate a large number of the sets from consideration, because the positions in them only occur in conjunction with positions from other sets; 4. We combine the three contributions above with some simple greedy algorithms to pick sets to solve, and, with a huge amount of computer power donated by Sony Pictures Imageworks, we actually run the sets, combine the results, and prove that every position in the cube can be solved in 22 moves or fewer.

Ò

he Rubik’s Cube is a simple, inexpensive puzzle with only a handful of moving parts, yet some of its simplest properties remain unknown more than 30 years after its introduction. One of the most fundamental questions remains unsolved: How many moves are required to solve it in the worst case? We consider a single move to be a turn of any face, 90 degrees or 180 degrees in any direction (the ‘face turn metric’). In this metric, there are more than 36,000 distinct positions known that require at least 20 moves to solve [9]. No positions are yet known that require 21 moves. Yet the best theoretical approaches and computer searches to date have only been able to prove there are no positions that require more than 26 moves [4]; this gap is surprisingly large. In this paper, we prove that all positions can be solved in 22 or fewer moves. We prove this new result by separating

T

the cube space into two billion sets, each with 20 billion elements. We then divide our attention between finding an upper bound on the distance of positions in specific sets, and by combining those results to calculate an upper bound on the full cube space. The new contributions of this paper are:

Please send all submissions to the Mathematical Entertainments Editor, Ravi Vakil, Stanford University, Department of Mathematics, Bldg. 380, Stanford, CA 94305-2125, USA e-mail: [email protected]

Author’s photo of the cube Ó Rubik’s/Seven Towns. All Rights Reserved.

Colors, Moves and the Size of Cube Space The Rubik’s cube appears as a stack of 27 smaller cubes (cubies), with each visible face of the cubies colored one of six colors. Of these 27 cubies, seven form a fixed frame around which the other 20 move. The seven that form the fixed frame are the center cubies on each face and the central cubie. Each move on the cube consists of grabbing the nine cubies that form a full face of the larger cube and rotating them as a group 90 or 180 degrees around a central axis Ó 2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

33

shared by the main cube and the nine cubies. Each move maintains the set of fully visible cubie faces. The eight corner cubies each always have the same set of three faces visible, and the twelve edge cubies each always have the same set of two faces visible. We will frequently use the term ‘corner’ to mean ‘corner cubie’, and ‘edge’ to mean ‘edge cubie’. In the solved position, each face of the main cube has a single color. By convention, we associate these colors with their orientation on the solved cube: U(p), F(ront), R(ight), D(own), B(ack) and L(eft). Each move that uses a 90 degree clockwise twist is denoted by writing the face with no suffix; each move that uses a 90 degree counterclockwise twist is specified with the face followed by a prime symbol (0 ), and each move that uses a 180 degree twist is specified with the face followed by the digit 2. So, a clockwise quarter turn of the right face is represented by R, and the move sequence R2L2U2D2F2B2 generates a pretty pattern known as Pons Asinorum. We write the set of all moves, containing the 18 combinations of faces and twists, as S. The way the moves in S can combine to generate different positions of the cube is not obvious, but is well known [1]; we state the relevant results here. The corner cubies may be permuted arbitrarily, or the edge cubies arbitrarily, but not both at the same time; the parity of the two permutations must match. This contributes a factor of 12!8!/2 toward the total number of reachable positions. Every corner cubie has exactly one face with either the U- or D-colored. We define the default orientation for the corner cubies to be that where the U- or D-colored face is on the whole-cube up or down face; the corner cubies may also be twisted 120 degrees clockwise or counterclockwise with respect to this default orientation (looking toward the center of the cube). Note that these orientations for each cubie are preserved by the moves U, D, R2, L2, F2, B2, but not by the moves R, L, F or B. This corner cubie orientation is fully arbitrary, except that the sum of all the twists for all the corner cubies must be a multiple of 360 degrees. These corner orientations contribute an additional 38/3 factor toward the total number of reachable positions. We define the default edge orientation to be that orientation in the solved state of the cube that is preserved by

AUTHOR

......................................................................... designed and built his first computer at age 15 in 1978, received his Ph.D. in computer science from Stanford University in 1993, and founded Instantis with a few friends in 1999. He presently lives in Palo Alto, California with his wife Sue and dog Andy, where he is in perpetual training for the Big Sur International Marathon.

TOMAS ROKICKI

Palo Alto CA USA e-mail: [email protected] 34

THE MATHEMATICAL INTELLIGENCER

the moves U, D, R, L, F2, B2 (but changed by F and B). Each edge is either flipped from this orientation or not; the count of flipped edges must be even. These edge orientations contribute an additional 212/2 factor toward the total number of reachable positions. The total number of reachable positions, and thus the size of the cube group, is the product of these factors, which is about 4.33 9 1019. We call the set of reachable positions G. For each of these positions, an infinite number of move sequences obtain that position. We define d(p), the distance of a position p, to be the shortest length of any move sequence that obtains that position from the identify. We define the distance of a set of positions to be the maximum of the distances of all the positions in that set. As a convention, we will denote the successive application of two move sequences by concatenation. We will also denote the application of a sequence to a position, or set of positions, by concatenation of the position and the sequence. A sequence s will have its length denoted |s|; the set of all sequences of length n of moves from set S will be denoted by Sn, and the set of all sequences from S will be denoted by S*.

Symmetry The Rubik’s cube is highly symmetrical. There is no distinction among the faces except for the color; if we were to toss the cube in the air and catch it, the cube itself remains the same. Only the color corresponding to the up face, the right face, and so on changes. Indeed, by tossing the cube, catching it, and noting the colors on the various faces in the new orientation, we can enumerate a total of 24 different ways we can orient the cube, each with a distinct mapping of colors to faces. Specifically, there are six different colors the up face can have, and for each of those six colors, there are four colors possible for the front face. These two face colors fully define the orientation of the normal physical cube. If we peer in a mirror while performing this experiment, we notice that our alter ego holds a cube with mirror-reversed orientations; these mirror-reversed orientations present an additional 24 possible mappings from colors to oriented faces. We further notice that whenever we do a clockwise move, our alter ego does a counterclockwise move. If we choose a canonical color representation, then each of these 48 orientations is a permutation of the cube colors. We call this set of color permutations M. If a particular cube position p is obtained by a move sequence s, we can obtain fully corresponding positions by applying one of the 48 color permutations (say, m), performing the sequence s, and then applying the inverse permutation of m. The resulting position shares many properties with the original one (especially, for us, distance). If we repeat this operation for all 48 permutations in M, we will obtain 48 positions. These positions are not always unique, but for the vast majority of cube positions they will be. Using this form of symmetry, we can reduce many explorations of the cube space by a factor of 48. Each cube position has a single specific inverse position. If a position is reached by a move sequence s, then the

inverse position is reached by inverse move sequence s0 . To invert a move sequence, you reverse it and invert each move; the face remains the same, but clockwise becomes counterclockwise and vice versa. The set of symmetrical positions of the inverse of position p is the same as the inverses of the symmetrical positions of p. Some properties of a position are shared by its inverse position (specifically, distance). We can partition the cube space into symmetry-plusinverse reduced sets by combining each position with its symmetrical positions and their inverses; there are only 4.51 9 1017 such sets.

Calculating the Diameter We are primarily interested in finding the maximum of the distance for all positions; this is known as the diameter of the group. For context, we review previous techniques for solving the cube using a computer, since our technique is derived from these. Speedsolvers, cube aficionados who compete in how fast they can solve the cube and other related puzzles, have a wide variety of manual algorithms, from very simple beginners’ methods to highly sophisticated methods that require the memorization of dozens of move sequences. Any of these algorithms are straightforward to implement on the computer, but the best of these tend to require many more moves than the actual distance of the position, so these techniques are, in general, useless in calculating the diameter. Simple approaches to optimally solving a single position fail because the size of cube space is so large. A simple breadth-first search exhausts both memory and CPU time. Iterative deepening, which uses depth-first search limited by a given maximum depth that increases from zero until the first solution is found, solves the memory exhaustion problem but still requires an impractical amount of CPU time. A more practical algorithm is to compute all positions that are within some small distance of solved (say, seven moves, totaling 109,043,123 positions [10]), and store these positions and their distances in memory. Then, iterative deepening can be used from the position to be solved, at each node examining the hash table to obtain a lower bound on the remaining distance and terminating that search branch if the sum of that bound and the current depth is greater than the current maximum depth. Various refinements are possible, such as only including one representative of the set of symmetrically equivalent positions in the hash table, or using a distance table of a subgroup of the cube rather than just the close positions, or only storing the distance mod 3 rather than the full distance. The first such program was written in 1997 [5] and required several days per position on average, but a recent version by Herbert Kociemba using eight threads on an i7 920 processor can find about 300 optimal solutions an hour. If we were to use such a program to solve the reduced set of 4.51 9 1017 positions, one at a time, with today’s hardware, we would require more than one million computers for more than one hundred thousand years. No better algorithm to optimally solve a single position is known.

It is not strictly necessary to optimally solve every position to compute the diameter. We know that some positions require at least 20 moves. The first such position found is called superflip; it has every cubie in the correct place, all corners correctly oriented, and all edges flipped [8]. Because we have a lower bound on the diameter, we need not optimally solve each position; once we find a solution of length 20 or less, we can move on to the next position. Kociemba devised an algorithm to quickly find reasonably short but not necessarily optimal solutions to arbitrary positions. That program (slightly improved as we shall describe) can find move sequences of length 20 or less at a rate of about 240 positions per second (subject to the condition that there is such a sequence; no exceptions have been found yet). Even with this kind of speed, proving all 4.51 9 1017 positions would require more than seven thousand computers for more than seven thousand years. Rather than using a tremendous amount of CPU time, we can instead use a large amount of memory. If we have enough memory or disk space to store two bits for each of the 4.51 9 1017 positions, we can perform a breadth-first search; some clever bit twiddling and some nice fast multicore processors should allow us to extend this table at a rate of billions of positions a second. Unfortunately, this approach would require over one hundred petabytes of memory. All hope is not lost. Technology marches onward; when we get to the point we can solve a billion positions a second, we will need only four computers for four years to finish the proof. In the meantime, we can come up with better techniques to refine the upper bound and improve our techniques.

Kociemba’s Algorithm Several techniques have been used to find an upper bound on the diameter of the cube group. Thistlethwaite gave a four-stage algorithm that requires a maximum of 52 moves. Kociemba improved this to an algorithm that requires a maximum of 29 moves (as shown by Michael Reid [7]). Our work is based on Kociemba’s algorithm, so we will describe it a bit further here. Kociemba himself has a much more detailed explanation on his web site [3]. In 2006, Silviu Radu reduced the upper bound to 27 [6], and in 2007 Kunkle and Cooperman reduced it to 26 [4]. Kociemba’s algorithm identifies a subset of 20 billion positions, called H. Reid showed that every position in this subset is solvable in at most 18 moves, and further that every cube position is at most 12 moves from this subset. Phase one finds a move sequence that takes an arbitrary cube position to some position in the subset H, and phase two finds a move sequence that takes this new position to the fully solved state. To describe this subset, we will introduce some new terminology. A cubie belongs in a particular place, if it is in that place in the solved cube. Thus, all cubies that have some face colored U belong in one of the top nine cubies. The middle layer consists of the nine cubies between the top and bottom layers; only four of these cubies (edges all) move. Ó 2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

35

The subset H is composed of all positions that have the following characteristics: 1. All corners and edges are in their default orientation (as defined earlier). 2. The edge cubies that belong in the middle layer are located in the middle layer. The number of positions for which these conditions hold are the permissible permutations of the corners, the top and bottom edges, and the middle edges, with the condition that the parity between the edge permutation and the corner permutation must match. This is 8!8!4!/2, or 19.5 billion positions. These characteristics are preserved by the moves U, U2, U0 , D, D2, D0 , R2, L2, F2, B2, which we call the set A. Further, these moves suffice to transform every position in H to the solved state. (This is a nontrivial result, but it can easily be shown by brute force enumeration.) For more than 95% of the positions in H, the shortest move sequence consisting only of moves from A is the same length as the shortest move sequence consisting only of moves from S, as shown in Table 1. Further, the worst case is 18 in both cases. Fitting a distance table for all 20 billion positions of H into memory may seem challenging, but there are a few tricks we can use. Because the defining characteristics of this set treat the up and down faces differently than the other B faces, all 48 symmetries of the cube cannot be used; however, 16 can be used; we need only store one entry per equivalence class. Further, instead of storing the distance, which is an integer between 0 and 18 and would require more than four bits each entry, we can store only the dis-

Table 1. The number of positions in H at a given distance using moves from S and moves from A; the numbers are strikingly similar d

moves in S

moves in A

0

1

1

1

10

10

2

67

67

3

456

456

4

3,079

3,079

5

20,076

19,948

6

125,218

123,074

7

756,092

736,850

8

4,331,124

4,185,118

9

23,639,531

22,630,733

10

122,749,840

116,767,872

11

582,017,108

552,538,680

12

2,278,215,506

2,176,344,160

13

5,790,841,966

5,627,785,188

14

7,240,785,011

7,172,925,794

15

3,319,565,322

3,608,731,814

16

145,107,245

224,058,996

17

271,112

1,575,608

18

36

1,352

19,508,428,800

19,508,428,800

36

THE MATHEMATICAL INTELLIGENCER

tance mod 3, requiring only two bits each entry. This can be achieved by only performing lookups for positions that are adjacent to a position at a known depth. By maintaining a current position and a current distance, and updating the distance as we perform each move, the distance mod 3 of the new position gives us enough information to know whether that position has a distance less than, equal to, or greater than that of the previous position. The remaining problem is how we can transform an arbitrary cube position into a position in H in 12 or fewer moves. To illustrate how this can be done, we describe a way to relabel the cube so that all positions in H have the same appearance, and all positions not in H have a different appearance. Consider an arbitrary position p. To be in H, the permutations of the corners are irrelevant; only the orientation matters. To represent this, we remove all colored stickers from the corners, replacing the stickers colored U or D with U and leaving the other faces, say, the underlying black plastic. (To make it easy to follow, we also replace the D sticker in the center of the down face with U.) All corner cubies are now interchangeable, but we have sufficient information to note the orientation of the corners. The permutation of the middle edges does not matter either, but they must lie in the middle layer and be oriented correctly. We thus remove the colored stickers from four edge cubies that belong in the middle layer, replacing the F and B colors with F and leaving the L and R colors as black. (We also replace the B center sticker with F for convenience.) The permutations of the top and bottom edges also does not matter; for these we do the same color change we did for the corners (U and D get turned into U, and the other four colors get removed). With this transformation, all positions in H get turned into the same solved cube: Eight corners, each with a U sticker on either the up or down face; four middle edges, each with an F sticker on either the front or back face; eight top/bottom edges, each with a U sticker on the up or down face. Every position not in H has a different appearance. This relabeled puzzle has a much smaller state space than the full cube space. Specifically, the space consists of by 212/2 edge orienta38/3 corner orientations multiplied 12 ways to distribute four middle tions multiplied by 4 edges among 12 edge positions, for a total of 2.22 9 109 positions. We call this set of positions R. With 16 ways to reduce this by symmetry and using only two bits per state, a full distance table is easy to fit in memory, and the full state space can be explored easily. We shall call this relabeling process r; it takes a position in G and transforms it into a position in R. Kociemba’s algorithm, then, is to take the original position, call it a, compute r(a), the relabeling; solve the relabeled puzzle with some sequence b 2 S ; apply those moves to an original cube yielding ab which lies in H, and then finish the solution with another sequence c 2 A such that abc is the solved cube. The final solution sequence is bc.

Algorithm 1. Kociemba’s Algorithm 1: d / 0 2: l / ? 3: while d \ l do 4:

for b [ Sd, r(ab) = e do

5:

if d + d2(ab) \ l then

6: 7:

Solve phase two; report new better solution l / d + d2(ab)

8:

end if

9:

end for

10:

one solution. This is motivated by the fact that we had already explored that prefix earlier (since we consider phase-one solutions by increasing length). 4. The last move at the end of phase-one is always a quarter turn of F, B, R or L; the inverse move is also a solution of phase-one, so candidate solutions are always found in pairs at the leaves of the phase-one search tree. 5. There are a number of optimizations that can be performed for the phase-one search when the distance to H is small, such as storing specifically which moves decrease the distance from that point.

d/ d + 1

11: end while

Kociemba’s algorithm splits the problem into two roughly equal subproblems, each of which is easy to exhaustively explore using a lookup table that fits in memory, yielding a fairly good solution to the much larger problem. This algorithm can find a solution of distance 29 or less almost instantaneously (in well under a millisecond). This defines an upper bound on the worst-case position distance. Kociemba extended this algorithm for another purpose: To quickly find near-optimal solutions for a given position. He proposed finding many phase-one solutions, starting with the shortest and increasing in length, and then, for each, finding the shortest phase-two solution. By considering dozens, thousands, or even millions of such sequences, he has found that, in practice, nearly optimal solutions are found very quickly. Given an input which is the initial cube position denoted by a, his algorithm is given as Algorithm 1. The algorithm can either run to completion, or it can be terminated by the user or when a solution of a desired length is attained. In Kociemba’s algorithm, d2 is a table lookup that takes a position in H and returns the distance to the identity element (e) using moves in A. (Kociemba actually uses a smaller, faster table that gives a bound on this value; see [3] for details.) The for loop is implemented by a depth-first recursive routine that maintains ab incrementally and has a number of further refinements, such as not permitting b to end in a move in A. The phase-two solution process is omitted both because it is straightforward and because it takes much less time than enumerating phase-one solutions. This algorithm is extremely effective. Some reasons are: 1. Phase-one solutions are found very fast and mostly access the portions of the phase-one lookup table near the solved position; this locality enhances the utility of caches significantly. 2. When searching for a phase-two solution, almost always the very first lookup shows that the distance to the solved position would make the total solution longer than the best found so far; thus, almost all phase-one solutions are rejected with a single lookup in the phasetwo table. 3. Kociemba has found that, in practice, the algorithm runs considerably faster if he does not consider phase-one solutions that contain a strict prefix that is also a phase-

Kociemba’s algorithm can be run as described above, or it can be run in triple-axis mode. Note how the algorithm treats the up and down faces differently than the other four. Instead of just exploring a single given position a, in tripleaxis mode we explore three rotated positions, one with the cube rotated such that the right and left faces correspond to upper and down, one such that the back and front faces correspond to upper and down, and the original unrotated position. We try each rotation for a given phase-one depth before moving on to the next phase-one depth. Our tests show that this finds smaller positions much faster than the standard single-axis mode; when trying to find solutions of length 20 or less, this works approximately six times faster on average than a single-axis search. We have taken this idea one step further; we also consider the inverse position in three orientations for a new six-axis mode. We find this gives, on average, a further doubling of speed when trying to find positions of 20 moves or fewer.

Our Set Solver Reid showed a bound of 30 by proving it takes no more than 12 moves to bring an arbitrary cube position to the H set (by solving the restickered cube), and then showing that every cube position in H can be solved in 18 moves. (He then reduced that to 29 with a clever insight we omit for brevity [7].) Our proof of 22 is similar, but instead of using just the H set, we use a union of over a million sets all related to H. Consider how Kociemba’s solver solves an arbitrary position to find a near-optimal solution. It first brings the position (a) into H, by solving the restickered puzzle using some sequence of moves (b). It applies that sequence of moves to the original cube, then looks up how far that position is from solved by using a sequence c containing only moves from A (those moves that stay within H), and determines if the total sequence is better than the best known one. It then finds another way to bring the position into H, and checks how close it is to solved at that point. It does this dozens, or hundreds, or thousands, millions or even billions of times, each time checking for a shorter solution. We turn this technique inside out. Each sequence b that solves the restickered position r(a) is a solution to some full cube position that has the same restickering as the given input position; so is each sequence bc where c 2 A : For every full-cube position in Ha, each of which has the same restickering, there is some c A* such that bc solves that Ó 2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

37

position. Rather than throwing most of these solutions away, we keep track of what full cube position each bc sequence solves, marking them off in a table, until we’ve found some solution for every position that has the same restickering as the original position. Where Kociemba’s algorithm searches for b and c such that abc = e, we instead search for b and c such that r(abc) = r(e). This way we find optimal solutions to 20 billion positions at a time. We are careful to do this in order of increasing length of bc, so that every time we find a bc that leads to a position we haven’t seen before, we know we have a optimal solution to that position. Since abc 2 H ; we can implement this by simply replacing the lookup table d2 on H with a bitmap on H indicating whether the position abc has already been seen. When every bit in the table has been set, we know we have found an optimal solution to every position in Ha. For our purposes, we do not need an optimal solution to every position; all we need is a bound on the distance of the entire set. Just as in Kociemba’s solver, the deeper the phase-one search is allowed to go, the longer the program takes; yet, a shallow phase-one search will still find a solution to every position in the set. We use a tunable parameter m that limits the depth of the phase-one search to trade off execution time against the optimality of the solutions found. The main input to our set solver is a sequence a 2 S ; which takes the solved cube into some position; the set that will be solved is Ha. Another input is the maximum depth m to run the phase-one search; we have found the value m = 16 is usually sufficient to prove an upper bound for the distance of the set to be 20. To find the exact distance, m should be set to ?. Our algorithm is given as Algorithm 2. At the end of each iteration of the main loop, f contains all positions abc such that |bc| \ d. The prepass (line 4), corresponding to Kociemba’s phase-two, extends the set f

by sequences ending with a move from A; the search (lines 9–11), corresponding to Kociemba’s phase-one, extends the set f with move sequences not ending in a move from A. Unlike Kociemba’s algorithm, we do permit our phaseone search to enter and then leave the H group; we do this in order to compute the precise set bound. We have not yet explored the performance impact of this on our running time. The set f is represented by a bitmap, one bit per position. For the prepass (line 4), we need to have both a source and destination set, so we need to have two of these bitmaps in memory at once. Our memory requirements are completely dominated by these bitmaps. The indexing of f is done by splitting the cube position into independent coordinates, representing the permutation of the corners, the permutation of the up/down edges, and finally the permutation of the middle edges. The time spent in the code is split between the prepass and the search phases. The prepass is a simple scan over the entire f set, multiplying by the 10 moves in A; this can be done efficiently by handling the coordinates from most significant to least significant in a recursive algorithm so that the inner loop only need deal with the permutation of the middle edges, and the more expensive corner coordinate computation is performed early in the recursion and thus substantially fewer times. In the innermost loop, we perform the move and bitmap update on all possible middle edge permutations using a lookup table and some bit-parallel logic operations. The time in the search phase (lines 9–11) is very small for low d, because there are few sequences s that satisfy the conditions, but as d grows, so does the time for the search phase, exponentially. Typically a search at level d + 1 will require 10 times as much time as a search at level d. By limiting m to 16 in the typical case, we limit the total time in the search phase, and the whole program runs fast. For values of m of 17 or higher, the search phase will dominate the total runtime.

The Set Graph Algorithm 2. Set Solver 1: f / [ 2: d / 0 3: loop 4:

f / f [ fA {prepass}

5:

if f = H then

6:

return d

7:

end if

8:

if d B m then

9: 10: 11:

for b [ Sd, r(ab) = e do {search} f / f [ ab end for

12:

end if

13:

if f = H then

14:

return d

15:

end if

16:

d/ d + 1

17: end loop

38

THE MATHEMATICAL INTELLIGENCER

The set R of relabeled positions of G has about two billion elements. Consider a position a 2 R; we can define the parent set of a to be all elements g 2 G such that r(g) = a. Let us pick a single one of the elements i in the parent set of a; the entire parent set can be represented by Hi. Each such set has precisely the same number of elements, about 20 billion; every pair of sets is either identical or disjoint; and the union of all of the sets is G, the full cube space. (This can be shown with elementary group theory because H is a subgroup of G and each set Hi is a coset.) These sets are all related by the full set of cube moves (S). Consider a cube position a and its set Ha. The set Hab for b 2 S is adjacent to the set Ha. We can consider R as a graph, where the vertices are the sets represented by the positions of R, and the edges are moves in S. Clearly, for any given position, |d(ab) - d(a)| B 1, and therefore the same is true for sets as a whole: |d(Hab) - d(Ha)| B 1. If we have shown that d(Ha) B c for some value of c, we have also shown that d(Has) B c + |s|, where s is a sequence of moves of length |s|. This allows us to find an

upper bound for one set and use it to infer constraints on upper bounds of neighboring sets in the graph of R. The relabeled puzzle shows 16-way symmetry, so there are only about 139 million relabeled positions when reduced by this symmetry. This reduced graph easily fits into memory, and operations on this graph can be performed reasonably quickly. For each vertex, we maintain a value which is the least upper bound we have proved to date. These values are initialized to 30, since we know every position and, thus, every set has a distance of no more than that. As we solve new sets, we update the value for the vertex associated with that set, and update adjacent vertices recursively with the new upper bound implied by this value.

Improving the Bound Some sets we solve have relatively few positions in the furthest distance. Since for lower values of m our set solver only gives us an upper bound on the set distance, in many cases the true distance of all these positions is less than the calculated upper bound. By solving these explicitly using a single-position cube solver, and proving they do not require as many moves as our set solver found, we can frequently reduce our bound on the distance for the set by 1. To facilitate this, if the count of unsolved positions in one of the sets falls below 65,536 at the top of the loop, we print each of these positions to a log file. To solve these positions, we first use our six-axis implementation of Kociemba’s solution algorithm. Since the solution distance we seek is almost always 19 or 20, this algorithm finds solutions very quickly, usually in a fraction of a second. For those positions that resist Kociemba’s solver, we solve them using our optimal solver.

Reducing Memory Use During the prepass, we compute f / f [ fA, where both the original and the new f is represented by a bitmap with one bit per position. Since the set size is almost 20 billion, this would normally require 2.4 GB per set for a total of about 4.8 GB. This is more memory than can be allocated on 32bit operating systems, and is more memory than can be added to many modern computers. We can reduce the memory requirements substantially by keeping only a portion of the source and destination bitmaps in memory at any given time. We do this by splitting the bitmap index into two parts, one calculated from the corner permutation and the other calculated by the edge permutation. We then split each bitmap into pieces, one piece per corner permutation; there are 8! such pieces. For every source bitmap part, corresponding to some source corner permutation, and every single move from A, there is only a single destination bitmap part, and this is found by performing the move from A on the corner permutation corresponding to the source bitmap part. As we proceed through the prepass, we consider each corner permutation in turn, allocating destination bitmap parts only as we need them, and freeing source bitmap parts as soon as we are finished with them. With a small program that performs a randomized search guided by some ad hoc heuristics, we have found a good ordering of the corner

permuations such that the maxiumum amount of memory required at any one time during the prepass is only 3.2 GB, which enables the set solver to be run on machines with only 4 GB of physical memory.

Choosing Sets to Solve This work grew out of a search for distance 21 positions [9] that involved solving a number of these sets exactly. We thus started this work with a few thousand sets already solved; we used those as our base set. At every point during this exploration we maintained the symmetry-reduced graph R on disk annotated with the best upper bound we had proven for each corresponding set. To select a new set to solve, we used a simple greedy strategy. We chose a vertex that, when we pushed its bound down to 20 and propagated its implications over the graph R, would reduce the maximum number of vertices from above 22 to 22 or fewer; we call this value the ‘impact’ of the vertex. We evaluated the impact of a few hundred vertices, and chose the one with the greatest impact to solve. Once we had selected a vertex, we added it to the list of sets to solve, updated the relevant vertices on the in-memory copy of the graph (not the persistent one on disk), and repeated this process to select another vertex. We typically generated lists of a few thousand sets to solve in this manner. Since some of the sets actually were found to have a bound of 19 or even 18, and this changed the graph in different ways than our above algorithm assumed, we generated a brand new list of vertices to solve every few days based on the updated R graph.

Results Approximately 6,000 sets, sufficient to prove an upper bound of 25, were all computed on home machines between October 2007 and March 2008. When those results were announced, we were contacted by John Welborn of Sony Pictures Imageworks, offering some idle computer time on a large render farm to push the computation further. Using these machines, we were quickly able to solve sets to prove bounds of 24 (26,380 sets requiring approximately one core year) and 23 (180,090 sets requiring approximately seven core years). With some additional time, we managed to finally prove a bound of 22 (1,265,326 sets requiring 50 core years). The sets were run on a heterogeneous collection of machines, some multicore, some single-core, some older and slower and some more modern. Since these sets were run, the set solver has seen significant performance improvement and processor technology has advanced; on a single Intel i7 920 processor, we believe we can reproduce all these results in only 16 core years (four CPU years on this processor). All of these sets were shown to have a distance of 20 or less, using searches through depth d = 16 or depth d = 15. Approximately 4.2% were shown to have a distance of 19. We continue to execute sets, and we are making progress toward proving a bound of 21. Once this is done, we believe that with only a few more core centuries, we can show a new bound of 20 on the diameter of the cube group. Ó 2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

39

The Quarter-Turn Metric These general ideas apply nearly equally well to the quarter-turn metric, where each 180-degree twist requires two (quarter) moves. The fundamental algorithms remain the same, except each 180-degree move (half move) has weight two. Implementing this in our set solver did introduce one complication: The prepass operation f / f [ fA does not properly handle the half moves. This problem can be solved by considering permutation parity. Every permutation is either of odd or even parity; it is of odd parity if an odd number of element swaps is needed to restore the permutation to the identity, and even if an even number of swaps is needed. Every quarter move performs a permutation of odd parity on the corners and also on the edges; every half move performs a permutation of even parity. Thus, the parity of the corner permutation always matches the parity of the edge permutation, and this is always equal to the parity of the number of quarter turns performed from the solved state. The positions in the set H are evenly divided between those of odd parity (H1) and those of even parity (H0). Similarly, we can consider our intermediate set of positions f to be split into odd (f1) and even (f0) parity, and the moves in A to be split into quarter moves (A1) and half moves (A0). At step d in the quarter-turn metric, we can only find positions whose parity is the same as the parity of d. Thus, before the prepass, the half of f that has the opposite parity to d represents positions at distance d - 1 or less, but the half that has the same parity as d represents positions at d - 2 or less. To reflect newly reachable positions at distance d, we can apply the half moves (A0) to the half with the same parity as d, and apply the quarter moves (A1) to the half with the other parity. Line four in Algorithm 2 must be replaced by the code shown in Algorithm 3. The distances in the phase-one pruning table (d2) are of course different in the quarter-turn metric, and in general Kociemba’s algorithm is somewhat less effective; the solutions found quickly tend to be somewhat further from optimal than with the half-turn metric. Similarly, the quarterturn metric version of our set solver requires searching deeper in phase one. Specifically, for almost all sets, searching through d = 19, taking about five minutes on our i7 920, proves almost all positions in that set can be solved in 25 or fewer moves. Typically, only one or two positions are left, and these are very quickly solved by Kociemba’s algorithm in 24 moves, leaving a bound of 25 for the whole set. Algorithm 3. Prepass for the Quarter-Turn Metric if

odd(d) then f1 / f 1 [ f 1 A 0 [ f 0 A 1

else f0 / f 0 [ f0 A 0 [ f1 A 1 end if

40

THE MATHEMATICAL INTELLIGENCER

In the quarter-turn metric, there is only one position known that has a distance of 26; this position was found by Michael Reid. We solved 24,759 sets in the quarter-turn metric to a depth of 19; each of these was found to have a bound of 25 or less, except for the single set which included Reid’s position. These sets sufficed to show that there is no cube position that requires 30 or more moves, lowering the upper bound in the quarter-turn metric from 34 [6] to 29. ACKNOWLEDGMENTS

This work was greatly helped by discussions with Silviu Radu; it was he who directed us to the subgroup (called H here) used by Kociemba. We are also grateful to Herbert Kociemba for both his original 1992 algorithm (and its implementation in Cube Explorer) and for ongoing e-mail discussions that have led to significant simplifications and performance improvements in the set solver. Many thanks also to John Welborn and Sony Pictures Imageworks, who donated massive computer time toward this project. The list of cosets and our calculated distance bounds are available at http://johnwelborn.com/rubik22/.

REFERENCES

[1] Joyner, David. Adventures in Group Theory: Rubik’s Cube, Merlin’s Magic and Other Mathematical Toys. Baltimore: The John Hopkins University Press, 2008. [2] Kociemba, Herbert. ‘‘Close to God’s Algorithm.’’ Cubism for Fun 28 (April), 1992, pp. 10–13. [3] Kociemba, Herbert. Cube Explorer (Windows Program). http:// kociemba.org/cube.htm [4] Kunkle, D., Cooperman, G. ‘‘Twenty-Six Moves Suffice for Rubik’s Cube.’’ Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC ’07), 2007. ACM Press, Waterloo. [5] Korf, Richard E. ‘‘Finding Optimal Solutions to Rubik’s Cube Using Pattern Databases.’’ Proceedings of the Workshop on Computer Games (W31) at IJCAI-97, 1997, Nagoya. [6] Radu, Silviu. ‘‘New Upper Bounds on Rubik’s Cube,’’ 2006. http://www.risc.uni-linz.ac.at/publications/download/risc_3122/ uppernew3.ps [7] Reid, Michael. ‘‘New Upper Bounds.’’ Cube Lovers, 7 (January), 1995. http://www.math.ucf.edu/^reid/Rubik/Cubelovers/ [8] Reid, Michael. ‘‘Superflip Requires 20 Face Turns.’’ Cube Lovers, 18 (January), 1995. http://www.math.ucf.edu/^reid/Rubik/Cubel overs/ [9] Rokicki, Tomas. ‘‘In Search Of: 21f*s and 20f*s; a Four Month Odyssey,’’ May 7, 2006. http://www.cubezzz.homelinux.org/ drupal/?q=node/view/56 [10] Sloane, N. J. A. ‘‘Sequence A080601 (Sum of the First Seven Terms).’’ Online Encyclopedia of Integer Sequences.

Years Ago

David E. Rowe, Editor

Debating Grassmann’s Mathematics: Schlegel Versus Klein

produced. Friedrich Engel, author of the definitive Grassmann biography (Engel 1911) published in Volume 3 of the collected works, went out of his way to praise Schlegel’s biography as well as his numerous efforts to promote interest in Grassmann and his work. Yet Engel also distanced himself from what he viewed as Schlegel’s one-sided hero worship, so typical among Grassmann’s closest followers. As we shall see, in taking this critical stance Engel was by no means alone.

Grassmann’s Ausdehnungslehre

DAVID E. ROWE

athematical fame can be a fickle thing, little more enduring than its mundane counterparts, success and recognition. Sometimes it sticks, but for odd or obscure reasons. Take the case of a largely forgotten figure named Victor Schlegel (1843–1905): Googling for ‘‘Schlegel diagrams’’ immediately brings up scads of colored graphics depicting plane projections of four-dimensional polytopes. It seems these figures are aptly named (Schlegel 1883, 1886), but how and when they came to be called Schlegel diagrams remains a mystery. In fact, clicking through Wikipedia, MacTutor and their progeny for Victor Schlegel turns up nothing; nor does he appear in standard compendia, like the Lexikon bedeutender Mathematiker. Nevertheless, during his lifetime Victor Schlegel was a well-known mathematician, though not primarily for his contributions to the study of figures in 4-space. To his contemporaries, Schlegel was a leading proponent of Hermann Gu¨nther Grassmann’s ideas and life’s work.1 Indeed, Schlegel was in an excellent position to write about this subject, having taught alongside Grassmann at the Stettin Gymnasium from 1866–1868. Afterward, he went on to publish over 25 works dealing with Grassmannian ideas, perhaps the most valuable for the historian being his biographical essay (Schlegel 1878) and his retrospective article (Schlegel 1896). Schlegel’s biography covers all facets of Grassmann’s far-ranging scholarly life, from theology and philology to mathematics and politics, presenting one of the most vivid personal portraits of this struggling genius ever

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It has often been observed that Grassmann’s mathematics was not widely appreciated during his lifetime. Although awareness of his achievements had begun to spread by the early 1870s, few in Germany appear to have been well acquainted with either the original 1844 edition of his Ausdehnungslehre or the mathematically more accessible revision of 1862.2 Among Grassmannians, the first edition was the true Ausdehnungslehre, a work of audacious and daring vision. Victor Schlegel described it at length in his Grassmann biography, claiming that it occupied a singular place in the history of mathematics: Such heights of mathematical abstraction as those reached in the Ausdehnungslehre had never before been attained. Like Pallas from the head of Zeus, it sprang suddenly to life, full and ready, leaping over a generation in the course of mathematical developments, it stood as a new science there, and today, 33 years later, it remains new and, unfortunately, for many just as incomprehensible as before (Schlegel 1878, pp. 19–20). No one would claim this book was an easy read, but sheer bad luck also had something to do with its weak reception. As Grassmann himself noted in the introduction to the second edition, his ideas might have become better known had not two distinguished voices passed on before having a chance to be heard (Grassmann 1894–1911, I.1, 18– 19). These were Hermann Hankel (1839–1873) and Alfred Clebsch (1833–1872), both of whom drew attention to the importance of Grassmann’s ideas before they abruptly died in the early 1870s (Tobies 1996). Hankel presented Grassmann’s theory in Hankel (1867), whereas Clebsch paid tribute to the same in his obituary for Julius Plu¨cker (Clebsch 1872). Afterward, no one of comparable stature arose to champion Grassmann’s cause, whereas some of those who did so tended to be seen as fanatics with a narrow, sectarian

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Send submissions to David E. Rowe, Fachbereich 08, Institut fu¨r Mathematik, Johannes Gutenberg University, D-55099 Mainz, Germany. e-mail: [email protected]

1

On the reception of Grassmann’s work in Germany in the 1870s, see (Rowe 1996), which serves as the basis for much of what follows. 2

For an account of Grassmann’s work and its influence, see Crowe (1967, pp. 54–95).

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agenda. It also seems that Grassmann’s own efforts to highlight the significance of his Ausdehnungslehre during the last years of his life mainly served to reinforce this tendency. These circumstances contributed to the tensions between a zealous band of Grassmannians and an equally committed group who promoted Hamilton’s calculus of quaternions (Crowe 1967). Their dispute raged into the 1890s until, with the emergence of the new vector analysis which drew on both systems, it largely subsided. Leaving this international conflict aside, I will focus on earlier events within Germany, in particular, Victor Schlegel’s role in promoting Grassmann’s work. In the early 1870s, Schlegel’s efforts were resisted by a leading member of Clebsch’s school, young Felix Klein (1849–1925), then a fastrising star in German mathematics. In the fall of 1872, Klein left Go¨ttingen to begin teaching as a full professor in Erlangen at just twenty-three years of age, a circumstance that compelled him to present a Programmschrift for his future activity. Thus, Klein’s (Klein 1872) was by no means the only ‘‘Erlangen Program,’’ it was merely the most famous—in fact, after 1900 it became so well known that even writers outside the field of mathematics, philosophers like Ernst Cassirer and Oswald Spengler, were discussing its ideas. The ‘‘Erlangen Program’’ sets forth a unified view of geometrical research by focusing on structures left invariant under various transformation groups. Working closely with Sophus Lie (1844–1899), Klein was especially intent on promoting this new approach, which had already shed much light on a number of topics and fields (Rowe 1989). By the time his Programmschrift was printed, however, Clebsch had succumbed to diphtheria, leaving a major gap in leadership among those in his circle. Klein quickly filled that vacuum, emerging as the self-appointed leader among ‘‘southern German’’ mathematicians (meaning those outside the Prussian universities, dominated by Berlin). Keenly aware of the fault lines of power within Germany, he was eager to adapt Grassmann’s concepts to his own new vision while asserting his own authority as to their relevance for geometrical research. Before turning to these matters directly, however, a few general observations should be made regarding Grassmann’s career and the mathematical world of his day (Figure 1).

Mathematics at University and Gymnasium The belated recognition of Hermann Grassmann’s importance and stature led some of his closer followers to view him as a martyr, a man forced to toil away his life as a school teacher in Stettin and whose brilliant genius was only appreciated after his death in 1877. Of course, no one today would dispute that Grassmann was a man of extraordinary gifts and impressive accomplishments. Nevertheless, his situation was hardly unique; nor was his the most striking example of a creative genius whose work failed to win swift acclaim. For a more balanced assessment, one must bear in mind the times and culture in which he lived, an era when professional research in pure mathematics was still in its infancy (Klein 1926, pp. 181–182). In 1852, Grassmann succeeded his father, Justus Gu¨nther Grassmann (1779–1852), as Oberlehrer at the Stettin 42

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Figure 1. Hermann Gu¨nther Grassmann was appreciated by leading geometers, like A. F. Mo¨bius, but his contributions to higher-dimensional affine spaces were long neglected.

Gymnasium. He apparently took no great pleasure in his duties there and longed instead to become a university professor. This circumstance has often been seen as the crux of Grassmann’s dilemma, for he never gained such a post, leading many of his latter-day followers to conclude that the German mathematical establishment failed to appreciate the merits of his new ideas and methods. Had they recognized his genius, so ran the argument, surely he would then have taken his place among Germany’s mathematical elite with the opportunity to spread his ideas through a close band of intellectual disciples. Maybe, but the market for research mathematicians in Germany circa 1850 was close to infinitesimally small. Moreover, however justified Grassmann’s desire for a university professorship may seem today, the fact that it remained unfulfilled is no qualification for martyrdom. In the 1850s, Prussian Oberlehrer were treated with considerable respect and deference, if not by their pupils then at least by their peers. What is more, they often consorted with members from the upper echelons of local society. Indeed, Grassmann himself was the product of a Prussian culture that not only honored, even revered, its teachers, but also attached extraordinarily high value to scholarly productivity. As such, he was in many ways a representative figure in an era when the pursuit of higher learning was almost taken for granted; it lay at the very heart of Germany’s neohumanist tradition (Steven Turner 1971). In Prussian secondary schools, this research ethos was

cultivated by and transmitted through numerous scholars with considerable e´lan. During Grassmann’s day, the gulf separating those who taught at the universities and their counterparts in the Gymnasien was not nearly as wide as it would later become at the end of the century. Thus, both academically and socially, an Oberlehrer was only one step from a university professor, and those who held the position were referred to by their proper title: Herr Professor. One should also note that gymnasium professors like Grassmann played a major part in Germany’s swift rise in the world of mathematics, which until around 1830 had been totally dominated by France. Ernst Eduard Kummer (1810–1893) and Karl Weierstrass (1815–1897), the two most influential mathematics teachers in Germany during the 1860s and 1870s, had both begun their careers teaching at secondary schools. Other distinguished mathematicians like Hermann Schubert (1848–1911), inventor of the Schubert calculus, spent their entire professional careers teaching young charges elementary mathematics. Some highly significant, nonmathematical findings took place in these settings. Thus, Leopold Kronecker (1823–1891) was discovered by Kummer when he taught at the Gymnasium in Liegnitz, whereas Schubert befriended young Adolf Hurwitz (1859–1919), his most gifted pupil, in Hildesheim. These particularly striking examples attest to a pattern of truly impressive quality. Indeed, it is safe to say that had it not been for the high standards and intense dedication demonstrated by scores of now forgotten Oberlehrer, Germany’s sudden ascent in mathematics during the second half of the nineteenth century would have been unthinkable. The scholarly ideals that animated professors at the Prussian Gymnasien did not differ markedly from the research ethos at institutions like Berlin University, where Grassmann studied theology, philosophy and philology in the late 1820s. Today, this would not seem an ideal preparation for one aspiring to a career in mathematics at the university level, but it was by no means atypical for this time. Both Gauss and Jacobi studied classical philology before ultimately turning to mathematics, and like Grassmann, they were both self-taught. Only very few students in Germany seriously contemplated pursuing a career as a research mathematician, for two obvious reasons: There were too few positions available, and one faced a long, arduous struggle that posed considerable financial hardships. Little wonder that even during the late 1800s some of the most promising talents, like David Hilbert (1862– 1943), opted to take the Staatsexamen, which qualified a person to teach in the secondary schools, rather than risk having nothing to fall back on later. Most of those who did choose to go on, by habilitating at a German university, came from families with fairly substantial means. Habilitation gave one the right to teach as a Privatdozent and to collect fees from the students, but nothing more. This unsalaried state of limbo discouraged many from even pursuing a professorship. Thus the fact that Grassmann never attained this more exalted position—although he was given serious consideration for a chair at Greifswald University in 1847—can hardly be considered unusual or surprising (Figure 2).

Figure 2. Grassmann taught at the Marienstifts-Gymnasium in Stettin from 1852 until his death in 1877.

Professionalisation and Patterns of Reception With regard to the slow diffusion of Grassmann’s ideas within wider mathematical circles, it should also be noted that this type of transmission pattern was close to the norm throughout most of the nineteenth century. Even when the ideas issued from highly esteemed individuals, like Gauss or Riemann, it could take decades before important new breakthroughs were absorbed and understood. As a general rule, the more novel or ‘‘revolutionary’’ the ideas (e.g., Galois’s theory of algebraic equations), the longer it took to assimilate them. Yet within Germany’s decentralized institutional structures, even relatively routine transmission of information was not easy, as there were few publication outlets and even fewer possibilities for meeting face to face. Before the founding of the Mathematische Annalen in 1869, Crelle’s Journal fu¨r die reine und angewandte Mathematik was the only specialized mathematics journal in Germany, and it was not until 1890 that German mathematicians managed to establish a national organization, the Deutsche Mathematiker-Vereinigung, which provided a forum for meetings and other professional activities. Thus, right until the end of the nineteenth century, most university mathematicians in Germany conducted their research in rather closed and isolated settings. Among the leading geometers from the mid-1800s, Julius Plu¨cker (1810–1868) worked in quiet solitude in Bonn, August Ferdinand Mo¨bius (1790–1868) was employed as an astronomer in Leipzig, and Karl von Staudt (1798–1867) taught for many years in the solitude of Erlangen. Much of Mo¨bius’s mathematical work became more widely accessible only in the 1880s, and it took several decades before Staudt’s fundamental contributions to the foundations of projective geometry became widely known. Thus, it was not until the mid-1880s, when he co-edited Mo¨bius’s collected works (see Klein 1921–1923, Vol. 1, p. 497), that Klein became aware that the long since deceased Leipzig geometer had anticipated many key ideas in his Erlangen Program. No doubt, Grassmann faced even graver obstacles than his contemporaries, Plu¨cker, Mo¨bius and Staudt. But more Ó 2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

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than his professional situation, it was his writing style that hindered his success. To read him, one had to be fairly steeped in idealistic philosophy, as his early work was pervaded by esoteric philosophical notions closely akin to Schliermacher’s thought (Lewis 1977). Even those who were motivated enough to penetrate into the misty realm of ideas set forth in Grassmann’s 1844 edition of the Ausdehnungslehre felt they had entered into a strange new world. One of the few foreigners who undertook the journey was William Rowan Hamilton (1805–1865), whose own work on quaternions had been partly inspired by German idealism, in particular, Kant’s notion that our conception of number derives from an intuition of pure time (Hankins 1980).3 Hamilton, who began reading Grassmann’s book in 1852, undoubtedly possessed the proper philosophical temperament (and linguistic abilities) needed to tackle this work. Though his principal concern appears to have been to reassure himself that he alone had discovered the quaternions, he expressed considerable admiration for Grassmann’s originality, an opinion shared by Luigi Cremona (1830–1903) and Giusto Bellavitis (1803–1880) in Italy. In Germany, it seems fair to say, Grassmann’s work failed to attract serious and sustained interest among leading contemporary mathematicians, not for lack of sympathy with his goals but rather because of the way he approached them. The elderly Carl Friedrich Gauss wrote Grassmann that he had no time to study his book, but in glancing through it he was reminded of his own longstanding interest in complex numbers and the metaphysical views he had expressed in his 1831 note on this topic (Rowe 1988). But he also admitted that he had published nothing like Grassmann’s calculus for spatial quantities (Schlegel 1878, pp. 22–23). Many expressed bewilderment when faced with the convoluted formulations in the Ausdehnungslehre. Particularly telling was Mo¨bius’s reaction, since his Bary€ of 1827 revealed strong affinities with zentrische Calcul Grassmann’s fundamental conceptions. Grassmann even visited Mo¨bius after completing work on his Ausdehnungslehre, but when he asked the latter to write a review of the book Mo¨bius replied: I was sincerely pleased to have come to meet in you a kindred spirit, but our kinship relates only to mathematics, not to philosophy. As I remember telling you in person, I am a stranger to the area of philosophic speculation. The philosophic element in your excellent work, which lies at the basis of the mathematical element, I am not prepared to appreciate in the correct manner or even to understand properly. Of this I have become sufficiently aware in the course of numerous attempts to study your work without interruption; in each case, however, I have been stopped by the great philosophic generality (A. F. Mo¨bius to H. G. Grassmann, 2 February 1845 (Schlegel 1878, p. 23)). Mo¨bius proved more helpful, however, in suggesting to Grassmann that he compete for a prize announced the previous year by the Jablanowski Society in Leipzig for a 3

Kant’s ideas in this respect exerted a similarly strong influence on Brouwer, for which see Freydberg (2009).

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work which would develop the ‘‘geometrical calculus’’ Leibniz had sketched in 1679 in a letter to Huygens, first published in 1833 (Schlegel 1878, pp. 28–29). Grassmann submitted his ‘‘Geometrische Analyse, geknu¨pft an die von Leibniz erfundene geometrische Charakteristik’’ (Grassmann 1894–1911, I.1, pp. 321–398) and won the prize (as the only entry submitted) in 1846. It was published the following year along with a commentary by Mo¨bius (Grassmann 1894–1911, I.1, pp. 613–633.). Like the Ausdehnungslehre, however, it failed to elicit sustained interest, despite favorable comments from some leading European mathematicians. The 14 articles Grassmann published in Crelle’s journal between 1842 and 1856 were somewhat better known. Most of them concerned methods for generating algebraic curves and surfaces. Although these results were both elegant and easy to comprehend, few geometers appear to have followed up on them. Nevertheless, many of these papers were read and appreciated (Tobies 1996, pp. 123–124). Felix Klein later emphasized that Grassmann’s methods—unlike those of the celebrated geometer Jacob Steiner—enable one to construct all possible algebraic curves synthetically, a truly important breakthrough (Klein 1926, pp. 180–181). After 1856, Grassmann’s mathematical productivity slackened, but certainly not his zeal for propagating what he had achieved. He presented his Ausdehnungslehre again in 1862, this time in strictly Euclidean fashion and without reference to its philosophical roots. Yet Grassmann’s unfamiliar terminology and the algorithms he employed posed major obstacles for his readers, and it seems most who tried to read the book eventually lost heart. Thus, like the 1844 edition, this account, too, failed to awaken substantial interest. Soon after its appearance Grassmann, discouraged by the indifference of professional mathematicians, turned to other endeavors, most notably philological studies.

Promoting Grassmannian Mathematics at the Gymnasium In the meantime, however, Grassmann and his brother, Robert, had turned to a different audience, hoping for a more enthusiastic response from mathematics teachers at the Gymnasien. Thus, in 1860 they published the first of a projected three volumes on elementary mathematics which were to serve as modern textbooks for the secondary schools (Grassmann 1860). Yet the response of the mathematics teachers was as discouraging as that of the research mathematicians. Whatever they might have thought about the mathematical merits of this new, fully rigorous approach to arithmetic, they found it altogether inappropriate for teaching (Engel 1911, pp. 225–228). Undaunted, Grassmann was sure his ideas could be and one day would be incorporated into the standard mathematical curriculum at German secondary schools, and others came to share that conviction. The most important new convert was Victor Schlegel, Grassmann’s colleague at the Stettin Gymnasium from 1866 to 1868. Afterward, Schlegel accepted a position as Oberlehrer at the Gymnasium in Waren, a small town in Mecklenburg. It was there that he began an intensive study of Grassmann’s Ausdehnungslehre, an experience that prompted him to write a

textbook entitled System der Raumlehre, which he dedicated to the master. Part I (Schlegel 1872) dealt with elementary plane geometry, whereas Part II (Schlegel 1875) presented a Grassmannian version of those portions of algebraic geometry closely connected with invariant theory. By virtue of this two-part study, but especially his biography of Grassmann published in 1878, one year after the latter’s death, Schlegel emerged as the leading spokesman for the Grassmannian cause. Like his hero, he hoped to show that the Ausdehnungslehre offered more than just another useful tool for the research mathematician. Indeed, he saw Grassmann’s methods as applicable to mathematics at nearly every level, and consequently as holding the key to a much needed reform of the mathematics curriculum in the secondary schools. Following the master’s lead, Schlegel’s System der Raumlehre had two principal objectives: To show the importance of Grassmannian ideas for pedagogical purposes, and to indicate the superiority of vector methods over techniques based on coordinate systems in presenting the main results of geometry, particularly projective geometry. He pursued the first of these goals in Part I, which dealt with the basic theorems of school geometry; the second objective was relegated to Part II, which appeared three years later. In the introduction to Part I, however, Schlegel not only made the two-fold purpose of his study more than clear, he went on to attack both research mathematicians and pedagogues for ignoring Grassmann’s work. Adopting a strident tone, he suggested that Grassmann’s Ausdehnungslehre of 1844 had simply been too radical an advance forward for a ‘‘generation that viewed imaginary quantities as impossible and that shook its head in dismay at non-Euclidean geometry’’ (Schlegel 1872, p. vi.). He also intimated that the playing field for mathematicians in Germany was anything but fair, and that Grassmann’s ideas had largely been neglected because he had not held a university chair, a circumstance that had prevented him from establishing a school for the promulgation of his ideas. Whereas younger mathematicians, as Schlegel observed, ‘‘had been preoccupied with the elaboration of the theories set forth by Jacobi, Dirichlet, Steiner, Mo¨bius, Plu¨cker, Hesse and others, all of whom gathered a circle of active pupils around them, fate failed to grant the author of the Ausdehnungslehre a similar influence’’ (Schlegel 1872, p. vi.). Schlegel seemed to further imply that Grassmann could claim priority for various methods usually ascribed to others. For, having ignored Grassmann’s work, other mathematicians (he referred to Otto Hesse explicitly) had in the meantime developed alternative techniques very close to Grassmann’s, but without fully realizing the advantages of the latter’s coordinatefree, n-dimensional mode of representation. Still, priority claims were the least of Schlegel’s concerns. Far more pressing, from his point of view, was the need to clarify the foundations for the treatment of extensive magnitudes in mathematics.4 He decried the state of arithmetic and algebra, calling it a ‘‘conglomeration of loosely 4

In this connection, Schlegel mentioned Martin Ohm’s Versuch eines vollkommen consequenzen Systems der Mathematik as about the only noteworthy recent effort to clarify the foundations of mathematics. On Ohm, see Bekemeier (1987) and Schubring (1981).

connected rules for calculation’’ supplemented by ‘‘a collection of arbitrary foundational principles, more or less vague explanations, and geometric artifices’’ (Schlegel 1872, p. xi.). Rather than developing a sound basis for their analytic methods, contemporary mathematicians tended to invent new symbolisms on an ad hoc basis, creating a Babel-like cacophony of unintelligible languages. This type of criticism, in fact, had long been hurled at analytic geometers, leading many to conclude that the only legitimate approach to geometry was the purely synthetic one. Schlegel went on to argue that this weakness had exerted an adverse effect not only on higher mathematics, but also on school mathematics. Indeed, he contended that such chaos inevitably reinforced the widespread view of mathematics as a kind of ‘‘black magic’’ accessible only to those with a predisposition for its abstruse formalisms. Schlegel further asserted that the neglect of foundations had led to the widely acknowledged lack of interest in mathematics in the schools. Echoing the views expressed in the preface of Hermann and Robert Grassmann’s textbook, he proclaimed that the standard mathematics curriculum contained nothing whatsoever that could serve as a basis for a truly ‘‘scientific method.’’ His goal in Part I of System der Raumlehre was to show how Grassmann’s ideas could provide just such a foundation. While Schlegel’s claims were, no doubt, exaggerated and his remedy far-fetched, his book did offer some fresh perspectives that might well have borne fruit. Certainly it appeared at a propitious time, as by now the sorry state of mathematics instruction in the Gymnasien had become a matter of considerable concern. In fact, Schlegel could even call attention to an editorial statement in the newly founded Zeitschrift fu¨r mathematischen und naturwissenschaftlichen Unterricht in favor of a rational reform of the teaching methods in geometry (Schlegel 1872, p. xii). But, as he realized, any significant reform would face strong resistance from those with a vested interest in the status quo. Not mincing words, he confronted the conservative opposition head-on, stating that he was well aware that his book would not please ‘‘those who regard the present state of elementary mathematics as satisfactory’’ (Schlegel 1872, p. xiii). Drawing battle lines in advance, he appealed not to these readers, but to all those capable ‘‘of‘ an unbiased consideration of new viewpoints’’ (Schlegel 1872, p. xiii).

Klein’s Critique Schlegel’s book did not receive the response its author had hoped for, nor did it garner accolades from the German mathematical community. Friedrich Engel later wrote that ‘‘Schlegel was not the man to put the old Grassmannian wine into new vessels’’ (Engel 1911, p. 324). Three years after its publication, however, the book was reviewed at considerable length by Felix Klein in Jahrbuch u¨ber die Fortschritte der Mathematik (Klein 1875). Ironically, but perhaps not by chance alone, Klein’s unusually lengthy review appeared in the Jahrbuch alongside a favorable review of his ‘‘Erlangen Program’’ written by his friend, the Austrian mathematician Otto Stolz. At first sight, this would seem like an unlikely pairing, since the two works were Ó 2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

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clearly written for different audiences and with quite different goals. Nevertheless, both dealt with geometry in a programmatic manner, and a comparison of their approaches sheds light on the dynamics surrounding the reception of Grassmannian ideas in Germany during the early 1870s. It also reveals the clashing agendas of research mathematicians at the universities and Grassmann’s followers, most of whom were Gymnasium teachers. Klein had been exposed to Grassmann’s Ausdehnungslehre in 1869 as a member of Clebsch’s school. He had studied both the 1844 and 1862 editions and clearly held Grassmann’s ideas in high regard, referring to both works prominently in his ‘‘Erlangen Program’’ (Klein 1872; pp. 478, 480, 483, 489, 492.) Like Schlegel, he also favored a reform of the mathematics curriculum, and indeed the whole philosophy behind mathematics instruction at the Gymnasien (Rowe 1985). His reaction to Schlegel’s book, however, was decidedly negative, not so much due to its substance as its tone and tendency. Thus, while he found the proofs of familiar results, like the Pythagorean Theorem, by means of Grassmannian methods interesting, he also noted that these proofs often required extensive calculations. This, Klein wryly observed, stood in flat contradiction to the book’s stated objective. He could have added that such excessive formalism was precisely what Schlegel had criticized in the work of contemporary analytic geometers, a criticism that Klein and other members of the Clebsch school had heard often enough. Beyond this, however, Klein largely ignored the pedagogical issues that Schlegel’s book sought to address. He was far less interested in the book under review than in the opportunity to expound on the merits and shortcomings of Grassmann’s mathematics as a whole from the perspective of his own ‘‘Erlanger Program.’’ Thus, he pointed out the limitations of Grassmannian conceptions for projective geometry, noting that neither Grassmann nor Schlegel introduced imaginary elements in their treatments of projective constructions. He also called attention to their general neglect of key aspects of the theory, such as cross ratios, polar curves, etc. After remarking on these drawbacks, Klein went on to say: Perhaps instead of presenting Grassmann’s conceptions as such, and only in elementary form, it would have been more important to show how these connect with and compare to similar directions that research has taken afterward, independent of Grassmann. So far as the present book is concerned, the principal accomplishments of Grassmann are essentially three. First, it is due to him that formal algebra gained an unsuspected depth, as he showed how to grasp the essence of the operations of addition and multiplication in a much more general way than had been done before him. In this regard, Grassmann stands alongside the English investigators, such as Hamilton. Furthermore, he was the first to develop the theory of higher-dimensional manifolds, which contains as special cases the theory of space and, especially, the theory of linear (projective) manifolds. Finally, he opened a new, wide-reaching field of research through his methods for generating all algebraic structures by means of linear mechanisms (Klein 1875, pp. 233–234). 46

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This praise for Grassmann’s achievements was followed by two critical observations. First, with regard to invariant theory, Klein found that Grassmann’s methods—with the exception of the mixed product, which had not been fully incorporated into algebra—offered nothing new. ‘‘The two forms of representation [Grassmann’s and those employed by invariant-theorists] are, at bottom, barely distinguishable and even the formulas that express them are often identical’’ (Klein 1875, p. 235).5 Second, with respect to Grassmann’s notion of n-dimensional manifolds, Klein noted that, from the present-day standpoint, this marked only a beginning. Grassmannian manifolds were ‘‘only the direct generalization to higher dimensions of ordinary space with its positional and metrical properties, whereas Riemann’s investigations opened a much more general line of inquiry that has since been extended significantly in various directions’’ (Klein 1875, p. 235). In his ‘‘Erlangen Program’’ Klein had sought to integrate a wide spectrum of geometrical research by placing the concepts of manifold, transformation group, and the associated group invariants at the center stage. This offered a broad new conceptual framework, but its design was clearly dominated by the particular field of research Klein knew best, namely projective geometry and the related theory of algebraic invariants of the projective group. Remarkably enough, the theory of affine spaces was not even mentioned in the ‘‘Erlangen Program,’’ a clear indication that Grassmannian ideas played a peripheral role in Klein’s overall conception of geometry at that time. Klein later openly admitted that this omission stemmed from the predominance of projective geometry in his background (Klein 1921–1923, pp. 320–321). This limitation along with Klein’s strong predilection for projective methods must, therefore, be taken into account when reading these reflections on the significance of Grassmann’s mathematics for geometrical research in the year 1872. Klein summed up his opinion of Schlegel’s book in a single sentence: ‘‘If, like the author, one presents Grassmann’s ideas divorced from such comparisons, it is this reviewer’s opinion that the reader will tend to be repelled rather than attracted by them; he will be forced to accept that Grassmann’s methods are absolutely superior, and such a claim always contains something improbable about it’’ (Klein 1875, p. 235).

Schlegel’s Response Schlegel surely must have been dismayed that one of Clebsch’s closest associates could have written such an unsympathetic, self-serving review of Part I of his book. Not only had Klein largely ignored the core issues at stake, namely the efficacy of Grassmannian methods for the presentation of elementary geometry, but the whole thrust of his final remarks seemed to say: Why didn’t the author write the kind of book I would have written instead of this one? Clearly, Klein had a very different mathematical agenda from the one Schlegel wished to pursue. The whole 5 This opinion was repeated at considerable length by Eduard Study in his commentary on Grassmann’s last mathematical works; see Grassmann (1894– 1911, II.1, pp. 431–433).

aim and tendency of his ‘‘Erlangen Program’’ was to unite disparate strands of geometrical research within a broad conceptual framework. Several passages in the ‘‘Erlangen Program’’ criticized tendencies toward disciplinary fragmentation and methodological purism, trends Klein spent his whole life fighting (Klein 1872, pp. 461, 490–491). Felix Klein was not one to back down from a fight, but he rarely felt inclined to take issue with Grassmann’s followers. His true enemies sat in Berlin, where elitism and purism reigned supreme, whereas the Grassmannians were a powerless faction without prominent standing in the German mathematical community. Shortly after Klein’s review appeared, Schlegel responded in the introduction to Part II of his System der Raumlehre. Klein had claimed that Schlegel failed to draw comparisons between Grassmann’s results and those obtained by others; after noting where this had been done in Part II, Schlegel curtly pointed out that this criticism was vacuous with respect to Part I, ‘‘since this part only comprises theories of elementary geometry’’ (Schlegel 1875, p. viii). As for the assertion that his book forced the reader to accept Grassmann’s methods as absolutely superior, Schlegel replied straightforwardly that, in his opinion, Grassmann’s methods offered ‘‘the shortest and easiest approach to the results of ancient and modern geometry and algebra’’ (Schlegel 1875, p. viii). But he also emphasized that the reader stood under no compulsion to accept this view; all he asked of the mathematical public was that they study the material presented in the book before judging it. As to whether Grassmann’s Ausdehnungslehre was merely of historical interest, whether it was ‘‘also at present worthy of further development and capable of being usefully employed in scientific advancements, that is a question that cannot be answered in three lines; nor can it be answered only on the basis of my merely introductory writings, but rather alone by a thorough study of Grassmann’s original works’’ (Schlegel 1875, p. viii). Klein apparently did not bother to reply to these comments, at least not in print; nor was Part II of Schlegel’s System der Raumlehre ever reviewed in Jahrbuch u¨ber die Fortschritte der Mathematik, though its mathematical content went far beyond that in Part I. Needless to say, neither party in this dispute succeeded in persuading the other to alter his views, and, in any case, it seems unlikely that any sort of compromise could have been reached. Klein later played a major role in engineering the appointment of Friedrich Engel as editor of Grassmann’s Gesammelte Mathematische und Physikalische Werke, a disappointing turn of events for Schlegel. Grassmann’s leading disciple made his displeasure known by noting that his own participation in the work on the Grassmann edition had been restricted to the preparation of a bibliography (Schlegel 1896, p. 3). It seems, in fact, that Engel later adopted a position with respect to ‘‘ fanatical Grassmannians’’ very similar to Klein’s own (Engel 1910, pp. 12–13). We do not know how Grassmann himself responded to Klein’s review of 1872. But if his last mathematical works serve as any indication, he felt very strongly that his methods and ideas, as presented by Schlegel, had, once again, failed to receive a fair hearing (Grassmann 1874, 1877). Grassmann tried to recoup some losses in these final works, but his

efforts probably only resulted in a hardening of opinion. In 1904, Engel wrote that they reveal ‘‘a striking imbalance between what is actually accomplished and the claims Grassmann made in them. This, however, has not restrained Grassmann’s unconditional admirers from praising even these works far beyond their merits’’ (Grassmann 1894–1911, II.1, p. vi). Eduard Study sounded a similar critical note in his commentary in (Grassmann 1894–1911, II.1, pp. 431–435). Grassmann chose to publish these last papers in Mathematische Annalen, the journal associated with Alfred Clebsch and his school. Clearly he considered Clebsch a strong and sympathetic supporter and his premature death in November 1872 a serious blow to the cause. Grassmann must have felt stung by the critical remarks published by Clebsch’s leading disciple three years later. These events of the mid 1870s surely only widened the gulf that already separated the Gymnasium teachers in Grassmann’s camp from those who taught at the German universities, marking a turning point in the tense relations between Grassmann’s growing contingent of followers and influential members within Germany’s still fragmented and often divisive mathematical community. Thus, by the time of the master’s death in 1877, the myth of his martyrdom was already firmly in place, shaped and cultivated by true believers who felt increasingly marginalized by professional mathematicians at German universities, many of whom adopted Grassmannian ideas in their own research. REFERENCES

Bernd Bekemeier. 1987. Martin Ohm (1792–1872): Universita¨tsmathematik und Schulmathematik in der neuhumanistischen Bildungsreform, Studien zur Wissenschafts-, Sozial- und Bildungsgeschichte der Mathematik, Bd. 4. Go¨ttingen: Vandenhoeck & Ruprecht. Alfred Clebsch. 1872. ‘‘Zum Geda¨chtnis an Julius Plu¨cker,’’ Abhandlungen der Ko¨niglichen Gesellschaft der Wissenschaften zu Go¨ttingen, 16,1–40. Michael J. Crowe. 1967. A History of Vector Analysis, Notre Dame: University of Notre Dame Press. Friedrich Engel. 1910. ‘‘Hermann Grassmann,’’ Jahresbericht der Deutschen Mathematiker-Vereinigung, 19:1–13. Friedrich Engel. 1911. ‘‘Grassmanns Leben,’’ in ed. Friedrich Engel, Hermann Grassmanns Gesammelte Mathematische und Physikalische Werke, 3 vols. in 6 pts., Leipzig: Teubner. Bernard Freydberg. 2009 ‘‘Brouwer’s Intuitionism vis a` vis Kant’s Intuition and Imagination,’’ Mathematical Intelligencer, 31(4). Hermann Grassmann. 1860. Lehrbuch der Arithmetik fu¨r ho¨here Lehranstalten, Berlin: Enslin. Hermann Grassmann. 1872. ‘‘U¨ber zusammengeho¨rige Pole und ihre Darstellung durch Produkte,’’ Go¨ttinger Nachrichten, 28:562– 576; reprinted in (Grassmann 1894–1911, II.1, 250–255). Hermann Grassmann. 1874. ‘‘Die neuere Algebra und die Ausdehnungslehre,’’ Mathematische Annalen, 7:538–548; reprinted in (Grassmann 1894–1911, II.1, 256–267). Hermann Grassmann. 1877. ‘‘Der Ort der Hamilton’schen Quaternionen in der Ausdehnungslehre,’’ Mathematische Annalen, 12:375– 386; reprinted in (Grassmann 1894–1911, II.1, 268–282). Hermann Grassmann. 1894–1911. Hermann Grassmanns Gesammelte Mathematische und Physikalische Werke, ed. Friedrich Engel, 3 vols. in 6 pts., Leipzig: Teubner.

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Hermann Hankel. 1867. Theorie der complexen Zahlensysteme,

Victor Schlegel. 1875. System der Raumlehre, nach den Prinzipien der

Leipzig: Teubner. Thomas Hankins. 1980. Sir William Rowan Hamilton, Baltimore: Johns

Grassmannschen Ausdehnungslehre und als Einleitung in dieselbe dargestellt, Zweiter Teil: Die Elemente der modernen

Hopkins University Press. Felix Klein. 1872. Vergleichende Betrachtungen u¨ber neuere geometrische Forschungen, Erlangen: Deichert; reprinted in (Klein 1921– 23, I, 460–497).

Geometrie und Algebra, Leipzig: Teubner. Victor Schlegel. 1878. Hermann Grassmann: Sein Leben und seine Werke, Leipzig: Brockhaus. Victor Schlegel. 1883. ‘‘Theorie der homogen zusammengesetzten

Felix Klein. 1875. ‘‘Review of Victor Schlegel, System der Raumlehre,

Raumgebilde,’’ Nova Acta Leopoldina Carolinium. (Verhandlun-

Erster Teil,’’ Jahrbuch u¨ber die Fortschritte der Mathematik,

gen der Kaiserlichen Leopoldinisch-Carolinischen Deutschen

Jahrgang 1872, Berlin: Georg Reimer, 231–235. Felix Klein. 1921–1923. Gesammelte Mathematische Abhandlungen, 3

Akademie der Naturforscher), Band XLIV, Nr. 4. Victor Schlegel. 1886. U¨ber Projectionsmodelle der regelma¨ssigen vier-dimensionalen Ko¨rper. Waren.

vols., Berlin: Springer. Felix Klein. 1926. Vorlesungen u¨ber die Entwicklung der Mathematik im 19. Jahrhundert, vol. 1, Berlin: Springer. Albert C. Lewis. 1977. ‘‘H. Grassmann’s 1844 Ausdehnungslehre and Schleiermacher’s Dialektik,’’ Annals of Science, 34:103–162.

Victor Schlegel. 1896. ‘‘Die Grassmannsche Ausdehnungslehre. Ein Beitrag zur Geschichte der Mathematik in den letzten fu¨nfzig Jahren,’’ Zeitschrift fu¨r Mathematik und Physik, 41:1–21, 41–59.

David E. Rowe. 1985. ‘‘Felix Klein’s ‘Erlanger Antrittsrede’: A

Gert Schubring. 1981. ‘‘The Conception of Pure Mathematics as an

Transcription with English Translation and Commentary,’’ Historia Mathematica, 12:123–141.

Instrument in the Professionalization of Mathematics,’’ in eds. H. Mehrtens, H. Bos, I. Schneider, Social History of Mathematics, Basel: Birkha¨user, 111–134.

David E. Rowe. 1988. ‘‘Gauss, Dirichlet, and the Law of Biquadratic Reciprocity,’’ Mathematical Intelligencer, 10(2):13–25. David E. Rowe. 1989. ‘‘The Early Geometrical Works of Sophus Lie and Felix Klein,’’ in The History of Modern Mathematics, vol. 1, ed. D. E.

Gert Schubring, ed. 1996. Hermann Gu¨nther Grassmann (1809– 1877): Visionary Mathematician, Scientist, and Neohumanist Scholar, Dordrecht: Kluwer.

Rowe and J. McCleary, Boston: Academic Press, 1989, 209–274.

Renate Tobies. 1996. ‘‘The Reception of Grassmann’s Mathematical

David E. Rowe. 1996. ‘‘The Reception of Grassmann’s Work in

Achievements by A. Clebsch and his School,’’ in ed. G.

Germany in the 1870s,’’ in ed. G. Schubring, Hermann Gu¨nther Grassmann (1809–1877): Visionary Mathematician, Scientist, and

Schubring, Hermann Gu¨nther Grassmann (1809–1877): Visionary Mathematician, Scientist, and Neohumanist Scholar, Dordrecht:

Neohumanist Scholar, Dordrecht: Kluwer, 131–146.

Kluwer, 117–130.

Victor Schlegel. 1872. System der Raumlehre, nach den Prinzipien der

R. Steven Turner. 1971. ‘‘The Growth of Professional Research in

Grassmannschen Ausdehnungslehre und als Einleitung in die-

Prussia, 1818–1848—Causes and Contexts,’’ Historical Studies

selbe dargestellt, Erster Teil: Geometrie, Leipzig: Teubner.

in the Physical Sciences, 3:137–182.

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The Mathematical Tourist

Dirk Huylebrouck, Editor

A Commemorative Plate for Wilhelm Killing and Karl Weierstraß U. REHMANN

AND

A. SZCZEPAN´SKI1

Does your hometown have any mathematical tourists attractions such as statues, plaques, graves, the cafe´ 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 an essay to this column. Be sure to included a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.

ä

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

search for ‘‘The Greatest Mathematical Paper of All Time’’ on Google, MathSciNet, or ‘‘Zentralblatt fu¨r Mathematik’’ unvaryingly leads to the work of Wilhelm Killing (1847–1923). His classification of simple Lie algebras over the complex numbers is a favorite keyword as well. Killing discovered them while a professor at the Lyceum Hosianum in Braniewo, Poland. In the 19th century, Braunsberg or Braniewo belonged to East Prussia, in Germany, but it became a Polish city after the Second World War, and thus its name changed.

A

The ‘‘Greatest Mathematical Author’’ and His Mentor Killing published four consecutive papers in the Mathematische Annalen, in a time span of two years, from 1888 to 1890. He entitled them ‘‘Die Zusammensetzung der stetigen endlichen Transformationsgruppen’’. The papers were the reason for A. J. Coleman’s biographic article [1] in The Mathematical Intelligencer, written on the occasion of the centennial anniversary of their publication. Coleman’s admiration for Killing’s work was supported by others, such as Jean Dieudonne´, in a review of Coleman’s article in Mathematical Reviews [3]. Dieudonne´ even added ‘‘Killing’s result became a most important milestone in modern mathematics’’. The history of mathematics of the last century shows Killing’s classification result has been revisited, revised, simplified, and extended into broader and different areas by eminent mathematicians, such as E. Cartan (1894), H. Weyl (1925), B. L. van der Waerden (1933), H. S. M. Coxeter (1934), E. Witt (1941), E. Stiefel (1942), E. D. Dynkin (1947), C. Chevalley (1955, 1961 ff.), J. Tits (1966 ff.), V. G. Kac and R. V. Moody (1968), F. Bruhat (1972 ff.), just to mention but a few. A detailed description of the history of Killing’s classification, including bibliographic remarks, is provided in [9]. Killing’s influence is still present in current common mathematical expressions, such as ‘‘characteristic equation of a matrix’’ (‘‘charakteristische Gleichung’’ in German, cf. [6] II, p. 2), or ‘‘semi-simple group’’ (‘‘halbeinfach’’ in German, [6], III, p. 74). He introduced the latter as follows: Solange ein besserer Name fehlt, mo¨ge es gestattet sein, eine solche (Gruppe) als eine halbeinfache zu bezeichnen. That is: As long as a better name is lacking, it might be permissible to denote such a group as semi-simple. Apparently, a better name never came up.

1

U. Rehmann was supported by the DFG grant CRC 701: Spectral Structures and Topological Methods in Mathematics; A. Szczepan´ski was supported by the Polish grant -0524/H03/2006/31

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Map of Poland.

Killing was a student of Karl Weierstraß, in Berlin, where he got his PhD in 1872. He worked as a high-school professor at various places for several years, simultaneously writing several publications on geometric subjects. Finally, Weierstraß’s recommendation led to an appointment for a chair of mathematics at the ‘‘Lyceum Hosianum’’, where his advisor Weierstraß had held a teaching position himself from 1848 until 1856. Killing would teach there for ten years, from 1882 to 1892. A detailed biographical description of both Killing’s and Weierstraß’s life is provided in [8] and [4]; see also [5] and [7].

The Memorial Plate In November 1996, the first author gave a lecture on ‘‘Linear algebraic groups and related structures’’ at the University of Bielefeld, and he mentioned Coleman’s paper [1]. The

second author of the current article, who teaches at the Gdan´sk University, was surprised to learn that Wilhelm Killing had been a professor at Braunsberg, located 110 kilometers east of Gdan´sk. The Institute of Mathematics of the University of Gdan´sk then organized a workshop at Braniewo entitled ‘‘The Second Days of Hyperbolic Geometry, in memoriam of Wilhelm Killing’’ from August 31 to September 2, 1998. Coleman wrote an address entitled ‘‘Killing in Braniewo’’; see [2]. At that time, the authors realized Weierstraß had been a school teacher too in Braniewo, from 1848 to 1856. During this workshop, the idea of a memorial plate in honor of Killing and Weierstraß was born. Its realization took 10 years, because of lack of financial support and difficulties to obtain all kinds of permissions. The ceremony of unveiling the commemorating plate took place on July 24–25, 2008. It was organized by the Institute of Mathematics of the University of Gdan´sk and Braniewo’s local

Images of the ceremony (right: J. Elstrodt during his ceremonial address). 50

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The Memorial Plate.

government. The plate shows a simple text, in German and in Polish, with the information explaining that Killing and Weierstraß were teachers in Braniewo. It is signed by the Polish Mathematical Society and the Deutsche Mathematiker-Vereinigung [10]. The rectors of the University of Gdan´sk and of the University of Warmia and Mazury in Olsztyn belonged to the honorary Committee. The ceremony began with a Mass in memory of Killing and Weierstraß, celebrated by the bishop Jacek Jezierski. Then the group of about 60 participants went to the former Lyceum Hosianum to unveil the plate. Prof. Falko Lorenz

(Department of Mathematics, University of Mu¨nster) informed the authors that this actually is the front of the former Catholic Gymnasium as there is nothing left of the buildings of the former Lyceum Hosianum. After the mandatory official speeches, L. Dzia˛g and A. Szczepan´ski unveiled the plate and the bishop gave it his benediction. A memorial colloquium was held, including lectures about the life of Killing, by F. Lorenz (University of Mu¨nster), and about the life of Weierstraß, by J. Elstrodt (University of Mu¨nster). The involvement of Mu¨nster is not surprising, because Killing held a position at the University of Mu¨nster Ó 2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

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and also because the cities of Braniewo and Mu¨nster are related by a European city partnership. This historical part was followed by more mathematically inclined lectures, by F. Knopf (University of Erlangen), T. Januszkiewicz (University of Ohio and University of Wrocław), F. E. A. Johnson (University College of London), and W. Soergel (University of Freiburg). Finally, in the evening of the first day, a concert and a party sponsored by the mayor of the city Braniewo festively concluded the event. The authors want to thank the University of Gdan´sk, the University of Mu¨nster, the University of Warmia and Mazury in Olsztyn, the GWO publishing house of Gdan´sk, the city of Braniewo, and the Deutsche MathematikerVereinigung for their support. Photos and full texts of the lectures by Elstrodt and Lorenz are available on the web (see [11]).

[5] F. Engel: ‘‘Wilhelm Killing’’, Jahresbericht der Deutschen Mathematiker-Vereinigung 39, (1930), 140–154. [6] W. Killing: ‘‘Die Zusammensetzung der stetigen endlichen Transformationsgruppen’’, I Math. Ann. 31, (1888), 252–290; II ibid 33, (1889), 1–48; III ibid 34, (1889), 57–122; IV ibid. 36, (1890), 161–189. [7] K. Lampe: ‘‘Karl Weierstraß’’, Jahresbericht der Deutschen Mathematiker-Vereinigung 6, (1899), 27–44. [8] F. Lorenz: ‘‘Wilhelm Killing (1847–1923)’’, http://math.univ. gda.pl/pdf/W.Killing-komplett.pdf, 2008. [9] U. Rehmann: ‘‘On Reflection Groups and semi-simple Lie algebras, Remarks on the Article by E. Witt; Spiegelungsgruppen und Aufzahlung halbeinfacher Liescher Ringe’’, in: Ernst Witt, Collected papers, Gesammelte Abhandlungen. Ed. by Ina Kersten. Springer-Verlag, Berlin, (1998), 247–255. [10] Institute of Mathematics web page’’, http://math.univ.gda. pl/img/braniewo/slides/P1010161.html, 2008. [11] Institute of Mathematics web page’’, http://math.univ.gda.pl/ braniewo, 2008.

REFERENCES

[1] A. J. Coleman: ‘‘The Greatest Mathematical Paper of All Time’’, The Mathematical Intelligencer, 11 (1989) no. 3, 39–38. [2] A. J. Coleman: ‘‘Killing in Braniewo’’, (in Polish), Wiadomos´ci Matematyczne XXXV (1999), pp. 141–144.

Institute of Mathematics University of Gdan´sk ul. Wita Stwosza 57, 80-952 Gdan´sk Poland e-mail: [email protected]

[3] J. Dieudonne´: Review of Coleman’s paper ‘‘The Greatest Mathematical Paper of All Time’’: MR1007036 (90f:01047), Math. Reviews (1990). [4] J. Elstrodt: ‘‘Karl Weierstrass (1815–1897)’’, http://math.univ. gda.pl/pdf/text-elstrodt.pdf, 2008.

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Fakulta¨t fu¨r Mathematik Universita¨t Bielefeld Postfach 100131, 33501 Bielefeld Germany

Implicit Differentiation with Microscopes JACQUES BAIR

AND

VALE´RIE HENRY

mplicit functions arise throughout mathematical analysis. They are especially prominent in applications to economics. From the beginning of mathematical economics one must work with implicit functions, as in analysis of a consumer and his indifference curves, or of a firm and its isoquants. Consequently, in teaching calculus to business students we come early to the challenge of explaining differentiation of implicit functions. The aim of this note is to give a systematic approach to computing the successive derivatives of an implicit function of one real variable, an approach that we think brings students naturally to the main results. The idea is that smooth curves looked at closely enough are straight, so that analysis problems are locally problems of linear algebra. Our method invokes that intuition: we look at curves through (virtual) microscopes—but we pay for this by having to deal with hyperreal numbers. Although Leibniz and Newton, for instance, already worked with ‘‘infinitesimals’’, the rigorous treatment called ‘‘nonstandard analysis’’ was introduced only in 1961 by A. Robinson [4]. An especially simple presentation for didactical purposes was given by Keisler [3]; we shall mostly adopt his definitions and notations. We recall that the hyperreal numbers extend the real ones with the same algebraic rules; technically, the set *R of the hyperreal numbers is a non-archimedean ordered field in which the real line R is embedded. Moreover, *R contains at least one infinitesimal (and then it must contain infinitely many). An infinitesimal is a number e such that its absolute value is less than every real number but which is unequal to 0; its reciprocal 1e is infinite, that is, is a number whose absolute value is greater than every real number. Clearly, non-zero infinitesimals and infinite numbers are not real. A hyperreal

I

number x, which is not infinite, is of course said to be finite; for any such x there exists one and only one real number r, which is infinitely close to x, that is, such that the difference x - r is an infinitesimal: this r is called the standard part of x and is denoted by r = st(x). Formally, st is a ring homomorphism from the set of finite hyperreal numbers to R; and its kernel is the set of infinitesimals. Moreover, a function of one or several real variables may have a natural extension to the hyperreal numbers, with the same definition and the same properties as the original one. Indeed, if a real-valued function is defined by a system of formulas, its extension can be obtained by applying the same formulas to the hyperreal system. In this article, we adopt the same notation for a real function and for its natural extension. The concept of (virtual) microscope is well known (see, for example, [1, 2, 5]). For a point P(a, b) in the hyperreal plane *R2 and a positive infinite hyperreal number H, a microscope pointed on P and with H as power magnifies the distances from P by a factor H; more explicitly, it is a map, denoted by MPH ; defined on *R2 as follows: MPH : ðx; yÞ 7! ð X; Y Þ with X ¼ H ðx aÞ and Y ¼ H ð y bÞ: Then we also have x ¼aþ

X H

and

y ¼bþ

Y : H

For a real function f of two real variables x and y, we first consider, in the classical euclidean plane R2 , the curve C defined by f ðx; yÞ ¼ 0:

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We consider a point P(r, s) belonging to C and assume that f is of class C p, for p sufficiently large, in a neighbourhood of P. For simplicity, we denote by d1, d2, d11, d12, …, the corresponding partial derivatives of f at P: d1 = fx(r, s), d2 = fy(r, s), d11 = fxx(r, s), d12 = fxy(r, s), …. Moreover, we assume that d2 6¼ 0. When we look at C through a microscope MPH , where H is an arbitrary positive infinite hyperreal number, we see a curve with the equation X Y f rþ ; sþ ¼ 0: H H Taylor’s Formula for f at P leads to X Y 1 þ d2 þ o 2 ¼ 0; f ðr; sÞ þ d1 H H H where the notation o H1k , for any integer k, means something whose product with Hk-1 is infinitesimal. Because f(r, s) = 0, we also have 1 ¼ 0: ð1Þ d1 X þ d2 Y þ o H But we are really interested only in hyperreal numbers X and Y that are finite (otherwise they could not be really observed); so let us now suppress all the infinitesimal details by taking the standard parts of the two members in (1); then we get d1 stð X Þ þ d2 stðY Þ ¼ 0; d1 stðY Þ ¼ stð X Þ: d2

ð2Þ

Of course the coefficient m ¼ dd12 is the slope of the tangent line T to C at P; T has the equation y s ¼ m ðx r Þ:

This non-vertical line T approximating C suggests intuitively that C is, close to P, the graph of a real function g of the variable x; the implicit function theorem ([3], p. 708) ensures the existence of a function g such that g(r) = s, the domain of g is an open interval I containing r, and the graph of g is a subset of C: Although g is not found by this reasoning, it is possible to compute its derivatives at r. For that, we first look at the graph of g through the microscope MPH ; that leads to the equation Y X ; sþ ¼g rþ H H so that, by Taylor’s Formula for g, sþ

Y X 1 ¼ gðrÞ þ g0 ðrÞ þo 2 ; H H H

and thus, as previously shown, stðY Þ ¼ g0 ðrÞ stð X Þ:

ð3Þ

Comparing formulas (2) and (3) leads to g0 ðrÞ ¼

d1 : d2

In order to compute g00 (r), we must distinguish between the curve C and its tangent T . For that, we use a stronger microscope, for example with a power H2, and direct it to another point that is infinitely close to P and belongs to T (otherwise we would see again C and T as equal). We can choose, for example, the point P1 r þ H1 ; s þ m H (note that would be equally convenient). point P2 r H1 ; s m H On the one hand, the use of the microscope MPH12 on the graph of g leads to

AUTHORS

......................................................................................................................................................... JACQUES BAIR is a professor at the uni-

VALE´RIE HENRY is a specialist in didactics

versities of Lie`ge and of Luxembourg. He began as a specialist in convex geometry (see, for example, Springer Lecture Notes in Mathematics, no. 489 by Bair and Rene´ Fourneau). His present interests are epistemology and the teaching of mathematics, especially mathematics applied to economics. He tries out his nonstandard approach to calculus on business engineering students. He is a tennis devotee, and he has written on the application of mathematics to sports.

and epistemology of science. Her doctoral thesis (Toulouse III, 2004) discussed the use of the nonstandard approach in teaching calculus to students of management. She is now a professor of mathematical education at the Universities of Namur (Belgium) and Luxembourg; at Lie`ge she teaches statistics. She is a horseback riding enthusiast.

HEC—Ecole de Gestion Universite´ de Lie`ge 4000 Lie`ge Belgium e-mail: [email protected]

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HEC—Ecole de Gestion Universite´ de Lie`ge 4000 Lie`ge, Belgium e-mail: [email protected] De´partement de mathe´matiques Faculte´s Universitaires Notre-Dame de la Paix, 5000 Namur Belgium e-mail: [email protected]

0

0 @ d1 d2

d1 d11 d21

1 d2 d12 A: d22

More generally, by induction, we can compute all the derivatives of the implicit function g at r. Indeed, let k be an integer greater than 1 and suppose that the numbers g0 (r), Then we consider the point g00 (r), …, g(k-1)(r) are known. Pk1 1 gðjÞ ðrÞ

1 Pk1 r þ H ; s þ j¼1 j! H j and we apply the microscope MPHk1 both to the graph of g and to the curve C: After some k elementary computations, we respectively obtain the two equations stðY Þ ¼ m stð X Þ þ

gðkÞ ðrÞ k!

and d1 stð X Þ þ d2 stðY Þ þ pk ¼ 0:

m Y 1 X þ ¼g rþ þ 2 H H2 H H 1 X g00 ðrÞ 1 X 2 1 þ 2 þ þ 2 þo 3 ; ¼ gðrÞ þ g0 ðrÞ H H 2 H H H

sþ

and thus, as previously, stðY Þ ¼ g0 ðrÞ stð X Þ þ

1 00 g ðrÞ: 2

ð4Þ

On the other hand, the application of MPH12 to the curve C gives 1 X m Y 0 ¼f r þ þ 2 ;s þ þ 2 H H H H 1 X m Y þ 2 þ d2 þ 2 ¼ f ðr; sÞ þ d1 H H H H " 2 1 1 X 1 X m Y þ d11 þ 2 þ2d12 þ 2 þ 2 2 H H H H H H 2 # m Y 1 þ þo 3 : þ d22 H H2 H

Then, from the last two formulas, gðkÞ ðrÞ ¼

k! pk d2

where pk can be calculated in terms of the partial derivatives of the given function f at P. For instance, we can so compute 3 jHf j d1 d d g000 ðrÞ ¼ 22 12 ðd2 Þ4 d2 ! 1 d1 ðd1 Þ2 ðd1 Þ3 d111 3 d112 þ 3 d122 d222 : d2 d2 ðd2 Þ2 ðd2 Þ3 In conclusion, we think that the use of microscopes provides a pleasant and novel way to compute systematically the derivatives of implicit functions.

REFERENCES

[1] A. Antibi, J. Bair, V. Henry, Une mode´lisation d’un zoom au moyen de microscopes virtuels, Teach. Math. Comput. Sci. 2/2

We easily get d1 stð X Þ þ d2 stðY Þ þ

1 d11 þ 2 m d12 þ m2 d22 ¼ 0: 2

(2004), 319–335. [2] R. Dossena, L. Magnani, Mathematics through Diagrams:

ð5Þ

Comparing formulas (4) and (5) gives g00 ðrÞ ¼

jHf j 1 d11 þ 2 m d12 þ m2 d22 ¼ ; d2 ðd2 Þ3

where jHf j denotes the bordered hessian associated with f at P, i.e., the determinant of the matrix

Microscopes in Non-Standard and Smooth Analysis, Studies in Computational Intelligence 64 (2007), 193–213. [3] H. J. Keisler, Elementary Calculus, Prindle, Weber & Schmidt, Boston, 1976. [4] A. Robinson, Non-Standard Analysis, North-Holland, Amsterdam, 1966. [5] D. Tall, Looking at graphs through infinitesimal microscopes, windows and telescopes, Math. Gazette 64 (1980), 22–46.

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Autism and Mathematical Talent IOAN JAMES

utism is a developmental or personality disorder, not an illness, but autism can coexist with mental illnesses such as schizophrenia and manic-depression. It shows itself in early childhood and is present throughout life; sometimes it becomes milder in old age. Nowadays it is recognised as a wide spectrum of disorders, with classical autism, where the individual is wrapped up in his or her own private world, at one extreme. It is estimated that in the United Kingdom slightly under one percent of the population, about half a million people, have a disorder on the autism spectrum The corresponding figure for other countries is not available, although it is unlikely to be very different. Autism is present in all cultures and, as far as we know, has existed for untold generations. Hans Asperger, a Viennese psychiatrist, found that some of his patients had a mild form of autism, with distinctive symptoms that later became known as Asperger’s syndrome. He was not the first to describe the syndrome but he may have been the first to recognise a connection with mathematical talent. As he observed (see Frith [13]): ‘‘to our own amazement, we have seen that autistic individuals, as long as they are intellectually intact, can almost always achieve professional success, usually in highly specialized academic professions, often in very high positions, with a preference for abstract content. We found a large number of people whose mathematical ability determines their professions.’’ Later he wrote, ‘‘It seems that for success in science or art a dash of autism is essential. For success the necessary ingredient may be an ability to turn away from the everyday world, from the simple practical, an ability to rethink a subject with originality so as to create in new untrodden ways, with all abilities canalised into the one speciality.’’ He went on to describe autistic intelligence—a kind of intelligence untouched by tradition and culture— unconventional, unorthodox, strangely pure and original. The ability to immerse oneself wholeheartedly in work or

A

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THE MATHEMATICAL INTELLIGENCER Ó 2009 Springer Science+Business Media, LLC

thought is something that is seen time and time again in the Asperger genius. Asperger syndrome is not the only form of autism with this connection. The Irish psychiatrist Michael Fitzgerald, for example, tells me that virtually all the people he diagnoses as autistic have an interest in mathematics. Their greatest wish, he says, is to bring the world under the control of pure reason, to create order and meaning out of the chaos that they experience around them, particularly in the puzzling social domain. Such people are naturally attracted to science, especially to the mathematical sciences, since mathematicians tend to create order where previously chaos seemed to reign. He attributes this attraction to a feeling of security that they find in the rational world of mathematics, which compensates for their inability to make sense of the mysterious social world. Much has been written about this, and the general public are now more aware of the presence of mildly autistic people in everyday life. Since I first wrote about autism in mathematicians in the Intelligencer [20] some years ago, more has been learnt about the disorder and more has been published. In this follow-up article I begin by describing research that places the link between autism and mathematical talent on a firmer footing. Then I describe some of the more recent case studies of Asperger geniuses in mathematics and associated subjects. Simon Baron-Cohen, director of the Autism Research Centre in Cambridge, has tried to put the connection on a more quantitative basis. For this purpose he devised a selfadministered questionnaire for measuring the degree to which an adult with normal intelligence has the traits associated with the autistic spectrum. From the answers to the questions a number is obtained, which he calls the autistic-spectrum quotient, providing an estimate of where a given individual is situated on the continuum from normality to autism. (Anyone who wishes to take the AQ test will easily find it by googling Simon Baron-Cohen.) When

the questionnaire was administered to students at Cambridge University, interesting results were obtained. Briefly, scientists scored higher than nonscientists; and within the sciences, mathematicians, physical scientists, computer scientists, and engineers scored higher than the more human or life-centred sciences of medicine and biology. Full statistical details are provided in [4] and [5]. This research was taken a step further in [6], where among 378 undergraduates reading mathematics at Cambridge there were seven who reported a formally diagnosed autism spectrum condition, whereas there was only one among 414 students in a control group of Cambridge undergraduates reading medicine, law, or social science. In the mathematical world, the establishment of a link between autism and mathematical talent will come as no surprise, but its recognition may have significant practical consequences for education and for choice of occupation. At school autism is regarded as a learning disability; its positive side should be recognised. Children with mild autism, who get on well in mathematics, may struggle with other subjects. They are likely to perform poorly at interviews, when they apply for a job, but they may be good at the right kind of work, for example in information technology, where their special abilities are appreciated. Although the disorder is a handicap in many ways, in others it is a great advantage. For the majority, life is a struggle, and only a minority make a success of it. There can be no doubt that gifted individuals with some degree of autism have contributed a great deal to research in mathematics. Not always, however; the tragic lives of Robert Amman [25] and William Sidis [28] show what can go wrong.

AUTHOR

......................................................................... Retiring after 25 years as Savilian Professor of Geometry at Oxford, Ioan James reinvented himself as a writer on nontechnical subjects. His latest book, ‘‘Driven to Innovate,’’ describes the lives of leading Jewish mathematicians and physicists born in the nineteenth century. He has also written a series of books about the lives of famous mathematicians, physicists, biologists, and (not yet published) engineers. His interest in autistic creativity has led him to write ‘‘Asperger’s Syndrome and High Achievement’’ and ‘‘The Mind of the Mathematician’’ (co-authored with psychiatrist Michael Fitzgerald). Among other distinctions Ioan James is a fellow of the Royal Society and an honorary fellow of two Oxford colleges.

IOAN JAMES

Mathematical Institute University of Oxford Oxford OX1 3LB England e-mail: [email protected]

When combined with high intelligence, as it often is, autism is associated with outstanding creativity, particularly in the arts and sciences. An enormous capacity for curiosity and a compulsion to understand are evident in those who have the syndrome, as is a tendency to reject received wisdom and the opinions of experts. They often suffer from depression, and mathematical work can have an antidepressant effect. Work is a form of self-expression for the autistic who finds other forms of expression difficult; it boosts their often low self esteem. The link with autism may throw fresh light on some aspects of mathematical creativity. More than a hundred years ago Henri Poincare´ addressed a conference of psychologists in Paris on Mathematical Creation (translated by Halsted [17]). Poincare´’s disciple Jacques Hadamard wrote a well-known monograph [16] on The Psychology of Invention in the Mathematical Field, which is mainly about mathematical creativity; a more recent discussion of this may be found in Changeux and Connes [7]. Much has been written about creativity in general, much of which applies to mathematical creativity, but Nettle [24] emphasizes that this differs from creativity in the arts. In a recent survey, comparing the psychology of a small sample of research mathematicians with poets and visual artists, Nettle finds that the cognitive style of the mathematicians was associated with convergent thinking and autism, whereas poetry and art are more associated with divergent thinking, schizophrenia, and affective disorders, such as manicdepression. (Divergent thinking means the ability to create new ideas based on a given topic; convergent thinking means the ability to find a simple principle behind a collection of information.) In the history of mathematics it is not difficult to find people who may have had Asperger syndrome, although without the right kind of biographical information we cannot say for sure whether each person had the syndrome or not. It is much less common among females than among males; it is difficult to find an example of an outstanding woman mathematician who was a clear case. It is not uncommon for individuals to have only a few features of the syndrome, not the full profile. Examples of well-known mathematicians who showed more than a trace of Asperger behaviour, without necessarily meeting all the diagnostic criteria, are Paul Erdo¨s, Ronald Fisher, G. H. Hardy, Alan Turing, Andre´ Weil, and Norbert Wiener. A detailed analysis for Srinavasa Ramanujan has been provided by Fitzgerald [10], for William Rowan Hamilton by Walker and Fitzgerald [27]. Some other cases are discussed by Fitzgerald and James [12]), whereas Baron-Cohen [1] describes one (who was, in fact, a Fields Medalist). Sheehan and Thurber [26] have suggested that John Couch Adams had the disorder and that this lay behind both his success in identifying the unknown planet Neptune as the cause of anomalies in the orbit of Uranus and also his failure to persuade the Astronomer Royal to search for it in the orbit he had calculated. Most of those who encountered the mathematical physicist Paul Dirac have a story to tell about his eccentricity. His recent biography by Farmelo [8] describes his aloofness, defensiveness, determination, lack of social sensitivity, literal-mindedness, Ó 2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

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obsessions, physical ineptitude, rigid pattern of activities, shyness, verbal economy, and much else. Some features of his complex personality can be attributed to his strange upbringing but most of it goes with some form of autism. Some people are critical of linking the syndrome with persons of genius. There is often strong resistance from the general public to any suggestion that a famous person might have had Asperger’s, but this is generally because of the popular misunderstanding of the nature of the disorder. People who are otherwise well informed find it difficult to believe what some of those with this disorder may be capable of achieving. Attempts at diagnoses of individuals no longer alive often result in controversy when experts differ and amateurs also become involved. Unless one is absolutely sure, it is advisable to be careful, for example, to say that someone displayed autistic traits rather than that person was autistic, even when the case is a strong one, since otherwise the diagnosis is liable to be questioned. Some of the standard books on the subject, notably Frith [14], discuss the problems of historical diagnosis. On the one hand, to know that there have been outstanding Asperger mathematicians impresses the rest of us and enhances the self-esteem of gifted people with the syndrome. On the other hand, those who are not so gifted may feel depressed that they cannot aspire to mathematical fame.

[8] Farmelo, Graham, The Strangest Man: the Hidden Life of Paul Dirac Quantum Genius. Faber and Feabre, London, 2009. [9] Fitzgerald, M., Is the cognitive style of persons with Asperger’s syndrome also a mathematical style? J. of Autism and Developmental Disorders, 30 (2000), 175–176. [10] Fitzgerald, M., Asperger’s disorder and mathematicians of genius. J. of Autism and Developmental Disorders 32 (2002), 59–60. [11] Fitzgerald, M., Autism and Creativity. Brunner Routledge, Hove, 2004. [12] Fitzgerald, M. and James I.M., The Mind of the Mathematician. Johns Hopkins University Press, Baltimore MD, 2007. [13] Frith, Uta (ed.), Autism and Asperger Syndrome. Cambridge University Press, Cambridge, 1991. [14] Frith, Uta, Autism: Explaining the Enigma. Basil Blackwell, Oxford, 2003. [15] Grandin, Temple, Thinking in Pictures. Vintage Books, New York, 1996. [16] Hadamard, J., The Psychology of Invention in the Mathematical Field. Princeton University Press, Princeton NJ, 1945. [17] Halsted, G.B., The Foundations of Science. Science Press, Philadelphia, PA, 1946. [18] Hermelin, Beate, Bright Splinters of the Mind. Jessica Kingsley, London and Philadelphia, 2001. [19] James, Ioan, Singular scientists. J. Royal Society of Medicine 96

[1] Baron-Cohen, S., The Essential Difference: men, women and the

(2003), 36–39. [20] James, Ioan, Autism in Mathematicians. Mathematical Intelli-

extreme male brain. Allen Lane, London, 2003. [2] Baron-Cohen, S. et al., Does autism occur more often in families

[21] James, Ioan, On Mathematics, Music and Autism. In Bridges

of physicists, engineers and mathematicians? Autism 2 (1998),

London (Reza Sarhangi and John Sharp, eds.), Tarquin Publica-

REFERENCES

296–301.

gencer 25, No. 4 (2003), 62–65.

tions, London, 2006.

[3] Baron-Cohen, S. et al., A mathematician, a physicist, and a

[22] Ledgin, Norm, Diagnosing Jefferson: Evidence of a Condition

computer scientist with Asperger syndrome: performance on folk

that Guided his Beliefs, Behaviour and Personal Associations. Future Horizons, Arlington TX, 2000. [23] Ledgin Norm, Asperger’s and Self-Esteem: Insight and Hope

psychology and folk physics test. Neurocase 5 (1999), 475–483. [4] Baron-Cohen, S. et al., The autism-spectrum quotient (AQ): evidence from Asperger syndrome/high-functioning autism, males and females, scientists and mathematicians. J. of Autism and Developmental Disorders 31 (2001), 5–17. [5] Baron-Cohen, S. et al., The systemizing quotient: an investigation of adults with Asperger syndrome or high-functioning autism, and

2002. [24] Nettle, Daniel, Schizotypy and mental health amongst poets, visual artists, and mathematicians. Journal of Research in Personality 40 (2006), 876–890.

normal sex differences. Philosophical Transactions of the Royal

[25] Senechal, M. The Mysterious Mr Ammann. The Mathematical

Society, Series B (special issue on autism mind and brain) 358

Intelligencer , 26(4) (2004), 10–21. [26] Sheehan, W. and Thurber, S., John Couch Adams’s Asperger

(2003), 361–740. [6] Baron-Cohen, S. et al., Mathematical talent is linked to autism. Human Nature 18 (2007), 125–131. [7] Changeux, J-P. and Connes, A., Conversations on Mind, Matter and Mathematics. Princeton University Press, Princeton N.J. 1995.

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through Famous Role Models: Future Horizons, Arlington TX,

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syndrome and the British non-discovery of Neptune. Notes Rec. R. Soc. 61 (2007), 285–299. [27] Walker, Antoinette, and Fitzgerald, Michael, Unstoppable Brilliance. Liberties Press, Dublin, 2006. [28] Wallace, A., The Prodigy. E.P. Dutton, New York, 1986.

Reviews

Osmo Pekonen, Editor

Naming Infinity by Loren Graham and Jean-Michel Kantor CAMBRIDGE, MA: HARVARD UNIVERSITY PRESS, 2009, 239 PP. ISBN:9780-674-03293-4 REVIEWED BY ROGER COOKE

They said, ‘‘Come, let us build for ourselves a city, and a tower whose top will reach into heaven, and let us make for ourselves a name, otherwise we will be scattered abroad over the face of the whole earth.’’ Genesis 11:4 (New American Standard Bible)

Feel like writing a review for The Mathematical Intelligence? You are welcome to submit an unsolicited review of a book of your choice; or, if you would welcome being assigned a book to review, please write us, telling us your expertise and your predilections

â Column Editor: Osmo Pekonen, Agora Centre, 40014 University of Jyva¨skyla¨, Finland e-mail: [email protected]

o say that Naming Infinity is the most unusual book written on the history of mathematics in recent years would be an understatement. This book is unique in showing how philosophical views can direct both the religious life and the mathematical research of one and the same person. Until I read it, I would not have said that a person’s religion would have any influence on his or her practice of mathematics. Looking at the mathematics written by Euler, Lagrange, Abel, Jacobi and Weierstrass, for example, if not for the gradual evolution of the mathematics itself in these writings, I could hardly tell who wrote what, much less whether the authors were Catholic, Protestant, Jewish or irreligious. While biographers have provided details of the religious backgrounds of Euler (Protestant), Abel (Protestant), Jacobi (Jewish) and Weierstrass (Catholic), those that I have read do not mention Lagrange’s religion, except to say that he was baptized in Italy, which strongly suggests that he was Catholic by upbringing. The history of mathematics has very little need to mention the religion of any of its major figures, except where—as in the case of Sylvester—it had some influence on their career opportunities or—as in the case of the legendary confrontation between Euler and Diderot that never occurred—it supposedly illustrates some aspect of their personalities. In his lectures on the development of mathematics in the nineteenth century, Klein mentioned religion only once, in his biographical sketch of Weierstrass, and here is all that he said: The majority of German mathematicians that we have discussed heretofore came from the Protestant community. With Jacobi comes the first of the Jewish mathematicians, whose numbers subsequently continued to grow. In contrast, Weierstrass came from a Catholic background. None of this was in any way relevant to the mathematics produced by these intellectual giants, and Klein rightly left

T

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the subject at that point. After his death, the Nazis illegitimately tried to use him as the symbol of ‘‘Aryan’’ mathematics, but as their own publication Deutsche Mathematik showed, if there was any difference between ‘‘Aryan’’ and ‘‘Jewish’’ mathematics, it was not to the credit of the former. One can, of course, recognize individual ethnic styles in the earliest mathematics, so that it is possible to distinguish the mathematics of China, India and Greece; and some of the distinctive characteristics of Indian mathematics can be related to the Hindu religion. But, in general, mathematics is mathematics, and one need not be a Hindu to stand in awe of the works of Brahmagupta and Bhaskara. In the book under review, however, a new nation—Russia—and a new religious tradition—that of the Orthodox Church—appear in the mathematical arena, and it turns out that there is a definite link between the mathematics produced by one school in Russia and the religious/ philosophical beliefs of the founders of that school. It is not a tight link, and the authors do not claim that it is extensive, but it should not be overlooked. I think almost any mathematician will find the story in this book fascinating both for the mathematics involved and the religious and political events in the lives of its three main protagonists, Egorov, Florenskii and Luzin. In the book itself, the mathematics is described only in general terms. Mathematicians who read it with some knowledge of the history of descriptive set theory will of course get more out of it than others, but even high-school students can read it with full comprehension, and I think many will wish to do so. For the benefit of the mathematically more sophisticated potential reader of the book, I am going to explain in more detail than the authors did where the mathematics comes in, balancing that emphasis with less detail on the personal and political side of the story.

The Mathematical Tower of Babel I have seldom read a book whose title was so aptly chosen. In two words, the authors have captured the two concepts that mathematics and religion have in common: Naming things and infinity. As the epigram above shows, both of these concepts appear early on in the Hebrew Scriptures, in the story of the Tower of Babel, the ‘‘gate of God.’’ One needs to update the cosmology slightly, since the original story seems to imply a belief that heaven is only a finite distance above the earth. Nowadays, both theologians and mathematicians struggle to find the best ways for finite beings (ourselves) to talk about infinity. I shall return to the problem of talking about infinity shortly. The problem of talking about anything at all is rather more fundamental. It seems slightly puzzling at first that the builders of the tower (the children of Noah) believed that they needed to ‘‘make a name’’ for themselves in order to keep from being ‘‘scattered abroad over the face of the whole earth.’’ But upon reflection, it is a striking fact that those who traced their religion to the people in this story actually were scattered abroad over the face of the whole earth, and they 60

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have managed to survive as a group to the present day because they have a common tradition to which a single name—Judaism—is applied. It appears, then, that ‘‘what’s in a name’’ is a matter of practical significance. The late Raul Hilberg, author of the definitive work on the Holocaust, often told me that an important aspect of the Nazi assault on the Jews was the macabre precision with which they defined a person to be a Jew. To give a less emotionally charged example, I note that the authors of the five Books of Moses—as a nonspecialist, I defer to the biblical scholars who ascribe these books to several authors—disagreed about the proper way to refer to the deity, whether the name should be Elohim, Yahweh or Adonai. The name used was not a trivial matter to those who took the scriptures seriously. Famously, when Moses asked in whose name he could demand that the Pharaoh release the Hebrew slaves, he was told simply to say ‘‘the one who is.’’

Talking About Infinity To those of a reflective temperament, the notion of infinity seems both indispensable to human discourse and fraught with paradoxes. The most famous and earliest of the latter are the paradoxes of Zeno, discussed by Aristotle. To take the simplest one, the dichotomy, and update it slightly, suppose you are driving from Baltimore to Philadelphia. It is clear that before you reach Philadelphia, you must reach the midpoint of the path you are following, and driving from Baltimore to that midpoint is a well-defined act that you can perform. Having reached that midpoint, before you reach Philadelphia, you will have to drive halfway from the midpoint to your destination, again a well-defined act that you can perform. The continuation of the argument should now be clear. Assuming your vehicle does not break down or have an accident, you will complete the entire journey in a finite time. Yet that journey can be divided in thought into an infinite number of distinct, welldefined human actions. Those actions can be arranged in a temporal sequence and described mathematically using the natural numbers as indices: A0 ; A1 ; . . .; An ; . . .: Given any particular one of these symbols, one has only to look at the index in order to say which of the actions it describes. Yet we cannot write down the whole sequence at once. We appear to have an actual, completed infinity in the physical world, but only a potential infinity in our description of it. How does this discrepancy occur? Topology comes to our rescue. Picture Baltimore at 0 and Philadelphia at 1. Then An represents the journey from 1-2-n to 1-2-n-1. If we carried out all of the actions described by these symbols, would we be at Philadelphia? The answer depends on whether one believes in an Archimedean line or a non-Archimedean one. The union of the segments [1-2-n, 1-2-n-1] is [0,1), not [0,1]. If we accept a standard Archimedean line, Philadelphia is the only place we could be, since [0,1] consists of the union of [0,1) and {1}, and every point of [0,1) corresponds to a point covered when carrying out some An and therefore a point

already behind when one is doing An+2. What a Weierstrassian analyst would say is that by carrying out a sufficiently large finite number of the actions An, we can get arbitrarily close to Philadelphia, and that fact is in complete accord with both physical reality and our imagination. Topology seems to have solved our paradox. However, we need not use a continuous model of physical space. A path consisting of a finite number of discrete pieces with instantaneous transitions from one piece to the next at a finite discrete set of times would model the physical journey just as well and avoid the paradox in the first place. Infinite discrete models present a further difficulty, however. If we wish to build our own Tower of Babel to ascend to mathematical heaven, the simplest and most natural way is to use the natural numbers 0; 1; 2; . . .; n; . . . : There is no end to this sequence. Even in thought, if we were to imagine ascending a Jacob’s Ladder to heaven, with each rung labeled by a natural number, we could not ever get to the halfway point. Even if we used a skyhook to give us a leg up, no matter what point we started from, there would always be incomparably more rungs ahead of us than behind us. That is different from the situation in Zeno’s paradox, since in that paradox one can at least imagine starting more than halfway to one’s destination. The god who, according to Kronecker, created the integers, made them such that angels can ascend and descend this ladder, but not human beings. The sequence of natural numbers is infinite, yet we can make meaningful statements that apply to every term in it. Each term after 1, for example, is either a positive power of a prime or a finite product of such powers. Thus, by using the incantation ‘‘For every n...,’’ we seem to gain the power to make infinitely many assertions simultaneously and to know that those assertions are true. Even better, we have a procedure—decimal or binary representations—that enables us to describe each integer in a perfectly definite way. To use the language of the book under review, we can name any one of them. Although we cannot grasp the integers as a whole, or all of them one at a time, we are not left wondering what the integers we will never think of are like: They are not qualitatively different from the ones we have thought of. Perhaps that is why Kronecker was so comfortable with them and so uncomfortable with the work of Georg Cantor. Set Theory and Topology Topology was inevitable from the moment Bolzano, Cauchy, and others began to study the notion of a limit point in connection with the study of continuity. Set theory, which meshes well with it, owes its existence (in my view) mostly to the need to study the convergence of series, particularly power series and trigonometric series. One can speak of ‘‘the points having such-and-such a property’’ or of geometric figures without too much inconvenience until the points involved or the figures become too complicated, as they soon do in discussions of the convergence of a general trigonometric series. At that point, set theory is a godsend.

Nowadays, we are all so steeped in set theory that would be terribly handicapped in nearly everything write if we couldn’t use its language. One consequence of Riemann’s 1854 work on uniqueness of trigonometric series (published in 1867, after his death) is that there is no trigonometric series SðxÞ ¼

we we the just

1 X ðan cos 2pnx þ bn sin 2pnxÞ n¼0

that fails to converge to zero at some finite positive number of points in [0,1], but does converge to zero everywhere else in [0,1]. To use set-theoretic language, let P be the exceptional set where convergence does not occur. This theorem says that a finite set P is a set of uniqueness, meaning that convergence to zero outside of P implies that all the coefficients in the series are zero. The case of an infinite exceptional set P appears to run up against the Bolzano–Weierstrass Theorem, which says that the derived set P 0 , consisting of the limit points of P, must be nonempty. However, due to two peculiar features of Riemann’s argument, one can still say that P is a set of uniqueness if P 0 is finite. The extension of this result is easy: If P 00 , the derived set of P 0 , is finite, then P is a set of uniqueness, and so on: As long as some derived set P(n) of finite rank is finite (and hence P(n+1) is empty), P is a set of uniqueness. That result represents the limit of Cantor’s work in the theory of trigonometric series, his point of departure into pure set theory. As is well known, the successive derived sets are nested: P 0 P 00 P 000 P ðnÞ P ðnþ1Þ : Once we grant that P 0 is determined (whether or not we can actually describe it) for any set P, we must grant that all the derived sets of finite rank P(n) are likewise determined. It therefore appears to be legitimate to define the derived set of infinite rank, which I will denote P(x), as the set of points belonging to all the derived sets of finite rank. Our power to use the universal ‘‘for every...’’ quantifier over an infinite collection has given us the ability to put a capstone on our Tower of Babel. Whether we can describe P(x) explicitly for a given set P is doubtful, but its definition does not appear to be less legitimate in principle than the proof that every positive integer is representable as a product of prime powers in one and only one way. We have reached an actual infinity, and the tower seems to be complete. Yet we can build it still higher. Given that P(x) is a well-defined set, it must have a derived set—indeed, it must contain that derived set, which we can denote P(x+1). We can now throw away the symbol P and concentrate on the indices we are using. We have created infinite ordinal numbers. Where will this construction lead us? Will logical difficulties arise and cause it to collapse like a house of cards, or can we go on building forever? The Axiom of Choice and the Continuum Hypothesis The axiom of choice implies that every set can be wellordered. In particular, the uncountable set [0,1] has, in theory, a well-ordering. But can we name that ordering? Ó 2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

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Can we describe it? I’m quite sure most of my readers have tried to do this, just as I did when I was an undergraduate. Probably, your experience resembles mine. What I wanted was some systematic way of writing down all the indices that one would have to use for such an ordering. It was easy to recognize that the initial segment of the ordinals must be isomorphic to the lexicographic ordering of the polynomials in x with natural-number coefficients, that is, all expressions of the form n0 xk þ n1 xk1 þ þ nk1 x þ nk : But these form only a countable set, and so, by the wellordering, there must be a first index a that is not of this form. It is at this point that discouragement begins to set in, since one must then proceed through all polynomials in a with coefficients that are polynomials in x with natural-number coefficients, only to produce yet another countable initial segment of the ordinals bounded above by the first countable ordinal b not of such a form. I soon realized that no matter how many times I repeated this procedure, I would have to start over again each time I reached a limit ordinal of higher rank. By definition, any proper initial segment of this set is countable. Even if the procedure is carried out a countable infinity of times, there will still be only a countable number of ordinal numbers ‘‘behind’’ while uncountably many remain ‘‘up ahead.’’ Describing all the countable ordinals using this approach is the perfect illustration of a Sisyphean task. It is as futile as standing on a chair to get a better view of the stars. This situation is worse than our previous predicament in relation to the natural numbers before. We are mired in countability as we were previously mired in finiteness, but we have no assurance that the ordinals we haven’t thought of are like the ones we have already described. Even if we assume the continuum hypothesis and postulate that each number in our hypothetical well-ordering of [0,1] has only countably many predecessors, we still have no way of naming all of the countable ordinal numbers. As we shall see, to Nikolai Nikolaevich Luzin, for reasons that were both mathematical and religious, that difficulty was fundamental and needed to be overcome.

Names in Mathematics and in Theology The set of all countable ordinals proved itself useful to Cantor in showing that every closed set of real numbers is the disjoint union of a perfect set (which, if nonempty, has cardinality of the continuum) and a countable set. This is because there is always a countable ordinal c such that P(c) is a perfect set and hence equal to all the derived sets from that point on. Borel Sets and Baire Functions When Borel, Lebesgue and Baire began to investigate integration and continuity in the years just before and after 1900, they found it necessary to use the countable ordinals for a different purpose. In order to define countably additive measures, Borel and Lebesgue needed to get a class of sets that contained all the intervals on the real line and was closed under countable unions and intersections. This was done by starting with any class of sets E containing the 62

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empty set and the entire real line and forming the class E* of all countable unions and intersections of sets in E. This operation allows us to carry out a finite induction, which becomes transfinite through the simple procedure of taking the union of all previous classes at any limit ordinal. The union of the classes indexed by all the countable ordinals is invariant under the operation just described, just as the intersection of all the derived sets of countable order is invariant under derivation. This union is precisely the sigma-field of Borel sets. Similarly, Baire defined functions of class 1 to be pointwise limits of sequences of continuous functions, then functions of class 2 to be pointwise limits of sequences of functions in class 1 and so on, producing by finite induction an increasing sequence of function classes. When limit ordinals are reached, one merely takes the union of all preceding classes. The collection of such classes indexed by all the countable ordinals is the Baire hierarchy. It is closed under pointwise limits of sequences. The authors correctly state that the French did not make a great fuss over these transfinite constructions. They were theoretically convenient, providing classes that were closed under the set-theoretic and limit operations needed in analysis. But as Borel remarked, the only sets one could effectively form belonged to some class of finite order. Lebesgue also liked the notion of effectiveness, procedures that produced what he called nameable (nommable) objects. The Moscow School and Onomatodoxy Although the most prominent Russian mathematicians of the nineteenth century—Chebyshev, Markov, Sonin, Bunyakovsky—were associated with Saint Petersburg, Moscow had its own school of talented people, among them Nikolai Egorovich Zhukovskii and Pavel Alekseevich Nekrasov. Nekrasov can be credited with working out an extension of the central limit theorem of probability theory to non-independent sequences of random variables. But his writing was deeply philosophical, and he could not confine himself to the clean mathematical style in which Markov, his rival in this area, wrote. Inevitably, it was Moscow rather than St. Petersburg where Cantor’s philosophical ideas were embraced, debated, and extended. And here, the French concern with naming things, with explicit constructions, found a surprising resonance with a movement in the Greek Orthodox Church known as onomatodoxy, which translates into Russian as imyaslavie: Name-glorification or nameworshipping. The essence of this movement was to find the correct name for God, to use just the right language, and even the right kind of lighting, in Orthodox services. Imyaslavie was regarded in some quarters as a heretical movement, and the authors open their account with a description of a 1913 raid by a Russian warship (ordered by the tsar) on the monastery on Mount Athos, which resulted in a number of Russian monks being brought back to Russia to face an ecclesiastical trial. Religion and Mathematics in the Moscow School In Moscow, real analysis got a good start in the first decade of the twentieth century under the leadership of Dmitrii

Fedorovich Egorov, who is nowadays remembered best for the theorem that on a finite interval almost-everywhere convergence is almost uniform (a result equivalent to remarks made by both Lebesgue and Borel eight years earlier, but whose importance neither of them emphasized). Egorov was a devout member of the Russian Orthodox Church. It was partly due to his influence that his two students Pavel Alekseevich Florenskii and Nikolai Nikolaevich Luzin, both of whom found a secular lifestyle unsatisfying, became Christians. The two students became close friends and both were attracted to imyaslavie. Florenskii declined to join Luzin in graduate study and instead took holy orders, although he continued to be involved in academic and scientific life. Luzin took his name-worshipping disposition with him into the academic debates over set theory. Like Lebesgue, who was one of his friends from an early visit to Paris, he wanted a mathematics that was based on naming things, that is, describing them unambiguously. Luzin very early saw the difficulties in conjuring objects into existence on the grounds that they were uniquely determined by known objects in some Platonic heaven. He even found fault with Cantor’s proof that the interval [0,1] is uncountable. As everyone knows, this proof proceeds by considering ‘‘any’’ list of numbers from that interval, say a1, a2,..., and then uses that list to ‘‘construct’’ a number a not in the list. Luzin was suspicious of this proof. Does the mere existence of the list entail the possibility of using it in this way to construct the number a? Is the procedure not tantamount to assuming that the sequence can be written out explicitly? He was willing to grant only that Cantor’s proof shows that [0,1] is not effectively enumerable. It did not follow, so he thought, that it wasn’t countable. To leave the set of countable ordinal numbers in the deplorably incomplete condition where we left it in the preceding section, was unacceptable to Luzin. His notes, which are now in the archives of the Russian Academy of Sciences, are full of attempts to get a complete description of all of these ordinals at once. In one valiant attempt, for example, he tried to imitate the method Dedekind had used to define real numbers as equivalence classes of cuts in the rational numbers. He took the class of subsets of the rational numbers between 0 and 1 that are well-ordered in their natural ordering, and introduced the natural orderequivalence relation on this class. Could one then define a countable ordinal number to be one of these equivalence classes? Certainly any such equivalence class will be orderisomorphic to a countable ordinal. But how could it be known that every countable ordinal is order-isomorphic to one of these equivalence classes? Luzin struggled with this problem of naming the countable ordinals for decades. One can almost hear him sigh as he wrote in one plaintive note that he left behind, ‘‘How many times must I write out the set of all ordinal numbers of types I and II?’’ Likewise, he tried with great diligence and the help of his student Lyudmila Vsevelodovna Keldysh to give an explicit description of the sets in the hierarchy that leads to the Borel sets. To give an explicit description of a set beyond the second class of sets (sets that are countable unions of countable intersections of open sets) is extremely

difficult, and the two of them never got beyond the fourth level in the hierarchy. I wonder why they even tried, given that no fundamental insight leading to a general description of the Borel hierarchy of sets was at all likely to emerge from the effort. Political Intrigues The rest of the story in Naming Infinity is about politics during the 1930s. Academic disputes between Luzin and Aleksandrov over the relative importance of descriptive set theory and topology mixed with the nature of the Soviet state to produce a very dangerous situation. Egorov was arrested in 1930 and died of self-inflicted starvation soon after his arrest and exile to Kazan. Florenskii was arrested in 1928, then again in 1933. Under torture, he implicated Luzin in a fantastic German plot against the Soviet regime. This charge, had it been brought in a Soviet court, would certainly have sealed Luzin’s fate. (Florenskii was shot in December 1937.) But for some reason, the charge was not brought. Luzin faced a tribunal in the Academy of Sciences in 1936 and was psychologically traumatized. But the matter was dropped fairly quickly, and he was allowed to remain a member of the Academy.

Conclusion I am enthusiastic about this book. It is free of technical mathematics and therefore readable by any educated person, yet it conveys a very good qualitative picture of the development of mathematical ideas in Germany, France and Russia over the half century from 1880 to 1930. The authors set out to present the whole picture of the Luzin school, especially the portion of it associated with imyaslavie. This, I must confess, is a picture that I did not see during my own more narrowly focused excursion in the archives of the Academy of Sciences 20 years ago. The role, and even the existence, of imyaslavie was a revelation to me, as I am sure it will be to nearly every reader. The authors have made a superb presentation. Laudably, they confine their conclusion to a modest statement justified by the facts: Name-worshipping was an inspiration to Luzin in the study of descriptive set theory, but perhaps not essential. He might, like Lebesgue, have preferred effective and constructive arguments simply as a matter of temperament. Thus, as the authors state, while mathematicians may not need religion to do their work, sometimes it can help. ‘‘What might have been’’ belongs to the subjunctive mood and is largely unknowable. What we know is what happened, and the link between religion and mathematics in the mind of Luzin is in the nature of a hard fact, in the declarative mood. Although in historical questions we can never fully illuminate that subjunctive mood, it seems unlikely that anything like imyaslavie could have taken root in a rationalist academic tradition such as that of France. As the example of Baire shows, French mathematicians also were concerned with naming things effectively. But Baire was largely without honor in his own country, and he sank into a deep depression. Does this mean, as the authors seem to

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imply, that the French culture itself was not receptive to the kind of intense study of particular classes of sets and functions that Luzin and his students engaged in? I don’t know. I’m rather inclined to say that the number of mathematicians active at a given time is finite, and therefore choices have to be made when research is pursued. Those choices will certainly be affected by what is fashionable in a given school or area. Whether those fashions are culturally

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determined or not is a question on which I remain a complete agnostic.

University of Vermont Burlington, VT 05405, USA e-mail: [email protected]

Emmy Noether: The Mother of Modern Algebra by M. B. W. Tent WELLESLEY, MA, A K PETERS, LTD., 2008, 184 PP., US $29 ISBN 978-1-56881-430-8 REVIEWED BY RENATE TOBIES

o comprehensive scientific biography of the (arguably) most famous female mathematician of the twentieth century exists. Emmy Noether (1882– 1935) can not only be considered as ‘‘the mother of modern algebra,’’ but also made a name for herself in theoretical physics with the two theorems that are named for her [1, 2, 3]. She was the first female scientist in Germany who was able to establish an outstanding international scientific school, though she held only an unofficial associate professorship [4]. M. B. W. Tent’s biography does not fulfill expectations even for a popular, not scientific, biography. After enthusiastically agreeing to review it, and then being disappointed after reading it, I questioned whether it was worthwhile to advertise it to a broad mathematical audience. However, the publisher promotes it with quotations by two well-known scientists at the Courant Institute in New York. I thus decided to review it within the context of recent research and to point out numerous inaccuracies and mistakes that easily could have been avoided. The book was written for the young adult reader, dedicated ‘‘To all bright mathematics students, past and present, whom I was privileged to teach at the Altamont School,’’ the middle school in Birmingham, Alabama, in which Tent taught before retiring in 2007. She uses imaginary dialogues, without quotation marks, between the protagonists. In the final chapter, ‘‘Tributes to the Mother of Modern Algebra,’’ the author only quotes extracts from well-known obituaries of Noether by Hermann Weyl, P. S. Alexandroff, Bartel L. van der Waerden and Albert Einstein. There is no bibliography. Moreover, there is no indication that Noether’s important mathematical works are available in her Collected Papers [5]. We learn from the author’s unusually long, five-page acknowledgements that she oriented herself on biographical details by consulting works that were published in the United States in 1981 [6, 7], and by visiting the archives in Go¨ttingen and Erlangen, writing that: ‘‘Dr. Tollmien’s excellent talk helped me as I was starting on this project.’’ Nonetheless, the mistakes in her biography reveal that she did not read Cordula Tollmien’s publications [8, 9] and that she did not understand her dissertation, which is available on the internet [10]. Tent explains mathematical concepts in simple words, thanks to the help of some mathematicians, and provides a glossary of the main words and concepts. Her bright mathematics students get to know how the board game

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‘‘Mu¨hle’’ (here ‘‘Nine Men’s Morris’’) is played, and how much Emmy Noether loved to eat Dr. Oetker’s pudding. But is this picture close to historical truth? Is the intention of the author to simplify, or is her lack of knowledge revealed when she presents something incorrectly or imprecisely? Tent writes in the preface: ‘‘Although this is a biography of Noether, it has an element of fiction as well.’’ But why is the story of the Noether family told incorrectly (p. 11)? The family name was Netter in the eighteenth century; they changed it to No¨ther (later Noether) after the anti-Jewish edict of Baden in 1809. Emmy Noether’s great grandfather had two given names, Elias Samuel. The second name becomes his family name in Tent’s biography. Furthermore, Emmy Noether’s grandfather never studied at a university (p. 11), but was a merchant. All of this is given correctly in Dick’s biography, including the history of the family name [6; German edition 1970, p. 4 f.]. The portrayal of an ahistorical, modern family (the children, for example, help wash the dishes) may correspond to Tent’s idea of good child rearing, but not to the social situation of a German professor’s family, in which Emmy Noether grew up. Do dialogues about clothes, hats and chocolate have to be invented to capture the interest of teenagers in a mathematician? Why do we need invented dialogues about whether Emmy wore a green or blue dress, bought dark instead of milk chocolate, or criticized the modern clothes and hats of Olga Taussky (1906-1995)? Women did wear hats at that time, including every European student. Emmy Noether is even pictured as an old woman with a hat on the dust jacket of this biography. This picture (with no explanation, and now colorized) is a 1933 photograph showing Noether at the central train station in Go¨ttingen. The well-known and beautiful picture of her in her youth would definitely have been better for Tent’s intended audience. I cringed when I read: ‘‘Emmy had never been considered beautiful’’ (p. 77). Do we say something like this for a male mathematician? Is it really a woman’s sole destiny to be beautiful and attractive? That Emmy Noether did not like to play the piano or supposedly was not interested in fashion suggests to teenagers that outstanding achievements in mathematics are incompatible with interest in other areas. The invented dialogue of Emmy’s father Max Noether (1844-1921) explaining Diophantine algebra and Diophantus’s equation to his daughter (even if Diophantus did not understand it as we do today) is acceptable, but why does the author say that ‘‘his papers have been lost’’ (p. 31) when Regiomontanus found six volumes of his main works in the Greek original in Italy in the fifteenth century? And why does each equation have to be formulated in words when Diophantus was the first Greek mathematician, so far as we know, who used symbols for powers of the unknown up to the sixth order? This would have been an ideal place to explain to students that scholars used different symbols, and that establishing a standardized system of symbols was a long process. It is misleading to introduce religious prejudice by citing Hypatia’s murder; Emmy Noether and her father resigned from the Jewish community in 1920 because they had

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ceased to practice their religion. They did not convert to Protestantism, as the author tells us (pp. 43, 145). Cordula Tollmien called attention to these unsubstantiated conclusions in her lecture in Go¨ttingen on June 13, 2006 [10], which Tent attended and to which she refers in her acknowledgements. When Emmy Noether matriculated at the University of Go¨ttingen in the winter term of 1903-1904, five women had already earned a doctoral degree there, including two Americans [11, 12]. Tent, however, omits this fact—a significant one for American students—and writes that ‘‘only Sophia Kovalevsky (a Russian) and Grace Chisholm Young (an Englishwoman) had been granted doctoral degrees in mathematics at Go¨ttingen’’ (p. 53). Sofja Kowalewskaja (1850-1890) was a student of the Berlin mathematician Karl Weierstraß (1815-1897); she received her doctoral degree from Go¨ttingen in 1874 because women were not allowed to enter the University of Berlin at that time. Grace Chisholm (1868-1944) and the American Mary F. Winston (married name Newson, 1869-1959), wrote their dissertations under Felix Klein (1849-1925) at Go¨ttingen, as is well known [13, 14]. A second American, Ann Lucy Bosworth (married name Focke, 1868-1907), was the first female doctoral student of David Hilbert (1862-1943) at Go¨ttingen in 1899. She wrote her dissertation on the principles of geometry. Before Emmy Noether arrived in Go¨ttingen in 1903, Hilbert helped another Russian on her way to her doctorate in the field of the calculus of variations in 1902: Nadjeschda von Gernet (1877-1943) became a university lecturer in St. Petersburg and returned to Go¨ttingen every summer until war broke out in August 1914. Max Noether was well informed about this situation in Go¨ttingen. Like Felix Klein, he came from the algebraicgeometrical school of Alfred Clebsch (1839-1872) and was on the editorial board of the Mathematische Annalen, founded by Clebsch in 1868 and edited by Klein. His extensive work on the editorial board was recognized by the inclusion of his name on its title page beginning in 1892. Emmy’s later and similarly extensive work for this same journal was not similarly recognized—but we learn nothing of this in Tent’s book. If Tent had investigated, she would have learned that Klein brought Max Noether to Erlangen when Klein moved to Munich in 1875. Moreover, although we read a lot about Klein’s bottle, we read little about his famous Erlanger program, only that: ‘‘After one year in Erlangen, Klein moved to Munich’’ (p. 56)—except that Klein stayed in Erlangen for three years. These inaccuracies are simply irritating. We read repeatedly about Emmy’s dresses, but we do not learn that she addressed her paper, ‘‘Ko¨rper und Systeme rationaler Funktionen,’’ Mathematischen Annalen 76 (1915), to tackle Hilbert’s 14th problem, which asks mathematicians ‘‘to decide whether it is always possible to find a finite system of relative functions of X1, …, Xm, through which every other relative function of X1, …, Xm can be made up rationally.’’ (M. Nagata finally answered this question in the negative.) Hilbert suggested to Noether that she hand in this paper as a Habilitation thesis (the absolute requirement for a professorship in Germany). This attempt in 1915 and another in 1917 at Habilitation failed 66

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(see [8, 9, 10]). Tent writes, however, that in 1919, ‘‘Noether made her second attempt at Habilitation’’ and ‘‘the scholars at the university decided to make an exception…’’ (p. 94). Tent’s lack of knowledge of the German academic system may be forgivable, but her statement that Emmy Noether’s work on the mathematical foundations of Einstein’s general theory of relativity was ‘‘unpublished’’ is not (p. 94). This was the work that Felix Klein presented, together with Noether’s two fundamental theorems, at the Go¨ttinger Gesellschaft der Wissenschaften on July 26, 1918 (that is, to the Go¨ttingen Academy, which could only elect a new member when a current member died). Einstein (1879-1955) was very enthusiastic about this work, and on his request Klein wrote to the Prussian Ministry of Education on January 19, 1919, to initiate Noether’s Habilitation, pointing out that Noether ‘‘has finished a couple of theoretical researches in the last year, which are superior to all other achievements (including those of the full professors)’’ [16]. Max Noether was proud of his daughter’s success and wrote to Klein in 1918: ‘‘I am very happy that we met each other due to the contact with my daughter in Go¨ttingen. Every day I see how her achievements increase and I get much pleasure from it.’’ [17] Klein, who had retired in 1913 and was not replaced by Richard Courant (1888-1972) at Go¨ttingen until 1920, nonetheless still carried weight there. Noether’s third attempt at Habilitation succeeded even before the official edict in Prussia on February 21, 1920 ordered that women no longer could be excluded from Habilitation because of their sex. Other noteworthy inaccuracies are: Heinrich Grell was not ‘‘her second doctoral student at Go¨ttingen’’—other students of Noether have been identified (cf.: [4]); Van der Waerden did not come to Noether as a postdoctoral student (p. 111), but wrote his dissertation using her methods and under her guidance [4]. Tent notes that Emmy Noether did not get a full professorship (p. 128); the reader should also be informed that no such position existed for women in Prussia, the largest German state with the most universities, until 1945. Throughout Germany, only two women had been appointed prior to 1945, both in 1923: One at the University of Jena in Thuringia, the other at the University of Stuttgart (Landwirtschaftliche Hochschule Hohenheim) in Wu¨rttemberg. No mathematician by the name of Anton Herglotz (p. 146) existed; Tent meant Gustav Ferdinand Maria Herglotz (1881-1953), who came to Go¨ttingen as the successor of Carl Runge (1856-1927), and who served as the second examiner of two of Noether’s doctoral students, Ludwig Schwarz and Ernst Witt. Moreover, Otto Schilling was not ‘‘Noether’s last doctoral candidate at Go¨ttingen’’ (p. 146); two more mathematicians completed their doctoral degrees there in 1936, thanking Noether for her encouragement and mentoring. Friedrich Karl Schmidt (1901-1977) undertook the oral examinations of these students after Emmy Noether had emigrated to the United States [4]. Tent’s claim that Edmund Landau (1877-1938), like Helmut Hasse (1898-1979), was able to stay at the University of Go¨ttingen (p. 146) even after 1933 is a regrettable mistake. The Jewish mathematician Landau, who was appointed professor in Go¨ttingen in 1909 and whose

dismissal was not required under the Nazi Civil Service Law, was forced out of his position by a boycott of his lectures by some Nazi students. It would also have been worth mentioning that, owing to Landau’s initiative, Noether was permitted to serve as an examiner of her doctoral candidates, starting with Margarethe Hermann (1901-1984) in 1925. Examiners normally were restricted to full professors, and, as an exception, she even received some payment. After her emigration and appointment at Bryn Mawr College, a women’s college in Pennsylvania, Emmy Noether came into close contact with Anna Johnson Pell Wheeler (1883-1966), who had studied in Go¨ttingen and had received her doctoral degree in Chicago in 1909 (not 1918, as Tent writes, p. 147). Just how enthusiastic Emmy Noether was about her trips with Anna Wheeler, how comfortable she already felt in America by March 1934, and how confidently she looked forward to receiving an appropriate appointment at Princeton University, can be seen in her letters to P. S. Alexandroff [18]. That someone should try to make Emmy Noether and her work accessible to teenagers is very welcome. It is also commendable to lighten her biography through dialogues. But if a biography purports to be about an actual historical person, factual errors must be avoided and the historical setting must be authentic. Such a biography of Emmy Noether could have been written based upon the available historical and scientific literature, especially by taking into account recent research, even a biography for teenagers. I hope that such a biography will be published soon.

[7] J. W. Brewer and M. K. Smith, Emmy Noether. A Tribute to Her Life and Work, Marcel Dekker, New York, 1981. [8] C. Tollmien, Eine Biographie der Mathematikerin Emmy Noether, zugleich ein Beitrag zur Geschichte der Habilitation von Frauen an der Universita¨t Go¨ttingen. Go¨ttinger Jahrbuch, 38 (1990), 153219. [9] C. Tollmien, Die Habilitation von Emmy Noether an der Universita¨t Go¨ttingen. NTM-Schriftenreihe fu¨r Geschichte der Naturwissenschaften, Technik und Medizin, 28 (1991), 1-11. [10] C. Tollmien, Emmy Noether, (1882-1935), http://www. tollmien.com/noetherlebensdaten.html. [11] R. Tobies, Felix Klein und David Hilbert als Fo¨rderer von Frauen in der Mathematik, Prague Studies in the History of Science and Technology, N.S., 3 (1999), 69-101. [12] A. Abele, H. Neunzert and R. Tobies, Traumjob Mathematik. Berufswege von Frauen und Ma¨nnern in der Mathematik. Birkha¨user: Basel, 2004. [13] L. S. Grinstein and P. J. Campbell (eds.), Women of Mathematics (A Biobibliographic Sourcebook), Greenwood Press: New York, Westport CT, London, 1987. [14] D. D. Fenster and K. H. Parshall, Women in the American Mathematical Research Community: 1891-1906. In: E. Knobloch and D. E. Rowe (eds.), The History of Modern Mathematics, Vol. III, Images, Ideas, and Communities, Academic Press: Boston, 229-261. [15] R. Tobies, Felix Klein in Erlangen und Mu¨nchen: Ein Beitrag zur Biographie. In: S. S. Demidov, M. Folkerts, D. E. Rowe and Ch. J. Scriba (eds.), Amphora. Festschrift fu¨r Hans Wußing zu seinem 65. Geburtstag. Birkha¨user: Basel, Boston, Berlin, 1992, 751772. [16] R. Tobies, Zum Beginn des mathematischen Frauenstudiums in

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[1] R. Tobies, Die Noether-Theoreme. In: D. Tyradellis and M. S. Friedlander (eds.), 10 + 5 = Gott. Die Macht der Zeichen, DuMont Literatur and Kunst: Leipzig, 2004, 283–284. [2] E. Noether, Invariante Variationsprobleme. In: Nachrichten von der Ko¨niglichen Gesellschaft der Wissenschaften zu Go¨ttingen, Math.-phys. Klasse, 1918, 235-257. [3] D. E. Rowe, The Go¨ttingen Response to General Relativity and Emmy Noether’s Theorems. In: J. Gray (ed.), Symbolic Universe, Oxford University Press, Oxford, 1999, 189-234. [4] M. Koreuber and R. Tobies, Emmy Noether—Begru¨nderin einer mathematischen Schule. Mitteilungen der Deutschen Mathematiker-Vereinigung, 10, No. 3 (2002), 8–21; revised in: R. Tobies (ed.), ,,Aller Ma¨nnerkultur zum Trotz?‘‘: Frauen in Mathematik, Naturwissenschaften und Technik. Campus: Frankfurt am Main,

Preußen. NTM-Schriftenreihe fu¨r Geschichte der Naturwissenschaften, Technik und Medizin, 28 (1991/1992) No. 2, 151172. [17] Niedersa¨chsische Staats- und Universita¨tsbibliothek Go¨ttingen, Cod. Ms. Klein. [18] R. Tobies, Briefe Emmy Noethers an P. S. Alexandroff. In: International Journal of History and Ethics of Natural Sciences, Technology and Medicine, N.S., 11 (2003), 100-115. [19] F. Lemmermeyer and P. Roquette (eds.), Helmut Hasse und Emmy Noether. Die Korrespondenz 1925–1935. Universita¨tsverlag: Go¨ttingen, 2006. [20] P. Roquette, The Brauer-Hasse-Noether Theorem in Historical Perspective (Schriften der Mathematisch-naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften, 15), Springer: Berlin, 2005.

New York, 2008, 149-176. [5] N. Jacobson (ed.), Emmy Noether, Gesammelte Abhandlungen— Collected Papers. Springer: Berlin, Heidelberg, New York, Tokyo, 1983. [6] A. Dick, Emmy Noether 1882-1935 (Beihefte zur Zeitschrift Elemente der Mathematik, 13), Birkha¨user: Basel, 1970; English Edition by Heidi I. Blocher, Boston, 1981.

Technical University of Braunschweig Historical Seminar Schleinitzstraße 13 D-38023 Braunschweig Germany e-mail: [email protected]

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Niels Henrik Abel and His Times, Called Too Soon by Flames Afar by A. Stubhaug, translated by R. Daly NEW YORK: SPRINGER, 2000, 580 PP. US $ 69.95 ISBN 3-540-66834-9 REVIEWED BY ULF PERSSON

n the 8th of April 1829, August Leopold Crelle had very good news for his friend and protege´ Nils Henrik Abel. Crelle’s year or so of frustrating efforts to secure a professorship at Berlin for Abel had finally borne fruit. The situation had become desperate, as Abel’s health had deteriorated. A frantic appeal to the university authorities had provoked a prompt and positive response. The situation now looked good indeed: A permanent and well-paid position would relieve Abel of the uncertainty and poverty that had distracted him from his mathematical work up to then. Furthermore, moving from the harsh environment of Norway to a more clement climate surely would spell a dramatic improvement to his failing health. At about the same time, Legendre in Paris, finally alerted to the existence of a memoir that Abel had submitted to the French Royal Academy of the Sciences in 1826, confronted a somewhat less than fully conscientious and responsible Cauchy. As a result, the lost memoir was retrieved from the piled up debris in the office of the latter. This memoir would vindicate Abel in the emerging priority conflict with the up-and-coming Carl Jacobi. Not that those two events were lucky breaks; rather, they were more or less inevitable outcomes of machinations behind the scenes. Abel was becoming known and appreciated all over the mathematical world. He had corresponded with Legendre, who was ecstatic over the recent accomplishments of those two young, ‘athletes’. Abel and Jacobi had revolutionized the field of elliptic functions to which Legendre had devoted the better part of his mathematical life, and with which he, almost 80 years old, was still making valiant efforts to keep up. But of the two, he clearly found Abel the superior: Abel’s work on the subject showed greater generality and depth and was far better organized, indicating a much better understanding and command of the subject than the rather sketchy approaches of Jacobi. Even Gauss concurred, disparaging Jacobi’s efforts as special consequences of results he himself had discovered in his youth. The work of Abel, on the other hand, generated nothing but praise, and besides, relieved Gauss of the onerous duty of editing and writing up his own results. Petitions to the King of Sweden and Norway—Karl XIV Johan, a former marshal under Napoleon—had been sent from Paris, calling his attention to the existence, among his subjects, of such a singularly talented mathematician. However, the king supposedly relegated the matter to his

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crown prince, who clearly did not consider it a prime priority. More to the point, the distinguished German explorer Alexander von Humboldt thought it of utmost importance to attach Abel and Dirichlet to a new institution in Berlin. With such backing, results were bound to emerge eventually. Abel’s genius was not unrecognized even though he was still very young, at the beginning of his career, and coming from an obscure part of Europe that most people at the time hardly knew existed. Things were indeed looking very good for Abel, but the timing was somewhat less than perfect. He had just died two days before. Had the good news come a few weeks earlier, it certainly would have made little difference. Abel was lethally sick, most likely stricken with tuberculosis, although no explicit authoritative diagnosis was made at the time. Tuberculosis was a well-known killer up to the middle of the twentieth century, a dark and ominous reminder of mortality, especially to the young, liable to have their glorious futures cut short. It seems that Abel showed the first signs of his condition three years earlier during his visit to Paris and that he might even have been tentatively diagnosed with the dreaded sickness. Such a diagnosis would spell doom, just as one of terminal cancer in our days, and for the rest of his short life Abel was in denial, preferring all kinds of alternative explanations such as colds, pneumonia, white pest, you name it. By January 1829, he was more or less bedridden, unable to work, on leave from his temporary position in Christiania (Oslo), and even too weak to carry on correspondence. He stayed at Froland, a small Norwegian mill town, where his fiance´e worked as a governess. His life slowly ebbed away over an extended winter. Not a single ray of the sun penetrated the small room to which he was confined. It is hard to conceive of a greater contrast between his actual condition and his potential. Biographies have always constituted a popular genre. The reasons are not hard to discern. A life provides a well-known narrative scheme with which we can all identify, yet allows a great variation in execution. We live but one life, but through biographies we can live a thousand vicariously. Biographies of artists, writers, politicians, and military men are legion. They tend to live exciting and colorful lives, and are constantly in the public eye. It is rarer to find biographies of scientists, although exceptions such as Einstein and Darwin readily come to mind. And rarest is the biography of a mathematician; so far only a handful have been published. Arild Stubhaug started out as a writer of literary fiction, in addition to being a published poet, but in the middle of his career he decided that people do not read novels as much as they used to, and that biographies purporting to tell true tales are far more attractive to the reading public. Why make up things when there is such wealth to be mined from what has actually happened? In 1996, his biography of Abel was published in Norway and attracted attention beyond the native mathematical scene: It won literary prizes in his home country for its laudable transgression of traditional boundaries between the humanities and the sciences. Later, it was translated internationally, making a stir in the mathematical world. The book may have generated momentum for the establishment by the Norwegian government of the Abel Prize, launched on the bicentennial of Abel’s birth. Since

then, the author has produced biographies of Sophus Lie and, most recently, of the father of Swedish mathematics, MittagLeffler (to be reviewed in a later issue of this journal). Abel is, of course, a gratifying subject for a writer. His life was both dramatic and tragic in its brevity. In his native land of Norway, he is something of a national monument, and he certainly belongs to a very select cadre of world-renowned mathematicians whose works and influences have been prophesied to endure for many a century. Of course, Abel has been written about before; many mathematicians know him from Bell’s collection of mini-biographies, Men of Mathematics. The modern classic authoritative text is that of Ø. Ore (incidentally, also first published in Norwegian). What does Stubhaug’s book add to what has already been published? It certainly is a much longer account than anything that has previously appeared, testifying to the modern trend of exhaustive biographies, the writing of which, not seldomly, take up the better part of the life of the biographers themselves. It certainly does not focus on the mathematics of Abel; that is, of course, not to be expected, given the motivation of the author. Thus, it makes no real contribution to mathematical history per se. Its focus is on the so-called ‘human interest’ catering to a wider audience. The writing of a standard modern biography appears, naively, to involve two very different activities. The first is ‘research’, i.e., the patient collection of all the documents that pertain to the hero of the work; in practice, this means letters and diary notes as well as secondary material. The second is the more or less artful articulation of all this amassed material as a readable account. Thus, a biographer is expected to be both a historian and a literary writer (because, as suggested initially, biography is expected to compete with novels), and to be judged on both counts. Thus, the writing of a biography involves obvious pitfalls. One is that what survives into posterity may give a skewed and misleading picture even if correct in factual details. Another is that the material so arduously collected (and in the case of Stubhaug an eightyear labor of love) makes a hostage of the author, who may be reluctant to reject material painstakingly ferreted out, even if marginal, resenting its return to oblivion. Such fidelity to the past (and invested effort) certainly runs the risk of making for tedious reading. In fact, the writing of a biography, as the biographer P. Ackroyd has noted, paradoxically makes more demands on the imagination than the writing of fiction. This is, in fact, the challenge to any historian, and as the British philosopher R. G. Collingwood scathingly remarks, any separation of the two activities (i.e., research and writing) is just a case of ‘scissors and paste’. The true historian tries to reconstruct the past into the present, and this can only be done through the systematic asking of questions, just as in any other scientific enterprise. The writing of history is a forensic exercise, according to Collingwood, in which theories have to be tested and refined through encounters with historical documents. The basic question one ought to pose is to what extent the biographer (in this case, Stubhaug) has avoided those pitfalls. The reviews of Stubhaug’s biography of Abel have, in general, been very sympathetic, perhaps even more so by the Norwegians who have taken great pride in both Abel, and the attention he has been given. Foreign reviewers have had to

contend with not altogether felicitous translations. (The translation of R. Daly abounds in peculiar translations of mathematical terminology, such as primary numbers instead of prime numbers—not intrinsically serious but bespeaking a lack of proper commitment from publishers.) And some have expressed exasperation at the presentation of irrelevant detail, in particular, the thorough documentation of what Abel may or may have not watched in the theatre, just because the documentation happens to be available. Yet the author maintains that Abel did have a deep interest in the theatre, the only commercial entertainment available at the time, and that he was, apart from his mathematical genius, a sweet and fun-loving young man of ordinary tastes. And when it comes to historical accuracy and interest, the casual reader is much less qualified to express an opinion than he is on readability and personal enjoyment, and, in fact, no reviewer has delved into the historical aspect, nor in any serious way challenged the presentation by the author. Clearly, nothing about Abel is controversial. However, certain sentimental myths about Abel are dispelled by this biography. Abel is often portrayed as the poor son of an obscure minister in the backwoods of Norway struck by mathematical brilliance. Norway may very well have been a poor and backward country at the time, but Abel was far from obscure: By birth he belonged to the elite of the nation. His paternal pedigree, with its long line of ministers, traced back to Germany of the Thirty Years War, while his maternal grandfather represented the world of business and wealth. Abel’s father was, in addition, a member of the parliament and active in the National Reawakening (a type of movement fashionable around Europe at the beginning of the nineteenth century) that gripped Norway as it was forcefully separated from Denmark and reluctantly attached to Sweden. He was clearly a man of independent interest as a representative of his period, and Stubhaug devotes much attention to him. Abel was given an education, and although at the time pedagogues were still enamored with corporal punishment (and as shown by an incident retold in the book, there could be too much of a good thing), the educational tradition was nevertheless flexible enough to accommodate a budding mathematical genius of the calibre of Abel, something it is not clear that our modern school system could manage. Abel was, of course, to a large extent an autodidact, but, like Gauss a quarter century earlier, he benefitted from the mathematical competence of a young man a few years his senior (in Abel’s case a future professor of mathematics, B. M. Holmboe). Abel did not show any remarkable talent outside mathematics and seems to have been of an altogether sympathetic, not to say angelic, nature. He was given a stipend and made the grand tour of Europe, ostensibly to meet the great lights of the time, such as Gauss. He wrote delightful letters about his travels and encounters with famous mathematicians, informal snapshots for posterity (his account of Legendre is particularly charming). But Abel was still a young man, and hence may be forgiven for sometimes choosing his itinerary less out of professional concern (he never sought out Gauss) than for companionship with his fellow student travelers. The mathematical fecundity and productivity he nevertheless enjoyed and achieved during those brief years is amazing. Ó 2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

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Abel and Crelle were both singularly fortunate in meeting each other. Abel got a most valuable connection to the world (networking as it is called nowadays) and an outlet for his articles, while Crelle’s mathematical journal (really the first of its kind) was jump-started through Abel’s contributions. (Incidentally, almost 200 years later Crelle’s journal has not run out of momentum.) Abel was, as Stubhaug points out in the introduction, not a misunderstood genius whose recognition was beyond that of his contemporaries. True, he suffered a series of severe setbacks, not just Cauchy’s temporary loss of his manuscript (how many of us elderly professors would not sympathize with Cauchy?), but more seriously, the protracted decline of his father due to a scandal and his subsequent early death hastened by alcohol abuse while Abel was still a teenager. In addition, his mother was also an alcoholic and of a promiscuous temperament, not above consorting with one of her servants during her husband’s funeral. Not surprisingly, she neglected her brood (including her oldest son who was clinically insane), leaving the responsibility to Abel: A responsibility so much more onerous after his rich maternal grandfather lost his wealth in the aftermath of the Napoleonic wars, rendering Abel impecunious. After his grand tour, he was not able to get a secure position in Norway, not due to hostility but out of a lack of imagination of the academic authorities. Norway was a poor country and its academic resources were geared towards the applied (understandable sentiments under the circumstances, and very much in vogue today as well). But this clearly was a temporary state of affairs, surely something would turn up sooner or later. It did, but—as we know—too late. Why is Abel great? If pressed, most people come up with his juvenile work on the impossibility of solving the fifth degree equation by radicals, the very aspect of his work that is most easily explained to the layman. But it constituted work later surpassed by Galois (an even more dramatic life, whose death really was an avoidable and pointless disaster) and retroactively shared by Ruffini. Abel’s mature work concerns elliptic functions and generalizations thereof, and his achievements in that regard have already been suggested in the introduction to this review. Furthermore, any mathematician may be assured that anything termed abelian (it is

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supposed to be the pinnacle of fame to become an adjective spelled without an initial capital) stems ultimately from the works and visions of Abel. What may be less appreciated was that he was one of the pioneers of rigor in mathematical analysis, lamenting the fact that no summing of infinite series had been done responsibly, expressing surprise that such frivolities had not produced more paradoxes, and subsequently setting the standards prevailing to this day. In particular, Abel presumably was the first to introduce absolute convergence. It can also be sobering to realize that Abel was also the first to explicitly formulate and elegantly disprove Pthe existence of a criterion of convergence to the effect that an converges if and only if lim /ðnÞan ! 0 for some function /(n). Something which could be an instructive exercise to the reader. Whether the sickness that did him in was the result of trying circumstances making him particularly vulnerable, or more likely a case of rotten luck, the deplorable result was the cutting short a most promising mathematical career which, if it had grown to fruition, certainly would have changed mathematical history, and perhaps also academic tradition in Norway. In any case, Abel lives on in posterity, the 200th anniversary of his birth being duly celebrated in Norway where he is widely known, one of the few examples of a mathematician present in the mind of a general public.

REFERENCES

Ø. Ore. Niels Henrik Abel. Et geni og hans samtid, Oslo, 1954. (English translation 1957: Niels Henrik Abel. Mathematician Extraordinary, Minneapolis, University of Minnesota Press.) E. T. Bell. Men of Mathematics, New York, 1937. (Chapter 17: Genius and Poverty. Abel) B. M. Holmboe. Oeuvres comple`tes de Niels Henrik Abel, Christiania, 2 Vols., 1839. See ‘‘The Works of Niels Henrik Abel.’’

Department of Mathematics Chalmers University of Technology Go¨teborg, Sweden e-mail: [email protected]

Geschichte des Husserl-Archivs/ History of the Husserl-Archives by Husserl-Archiv Leuven DORDRECHT: SPRINGER SCIENCE + BUSINESS MEDIA, 2007, 161 PP., € 19.95, $ 34.95, ISBN 978-1-4020-5726-7 REVIEWED BY HENNING PEUCKER

n 2009 many conferences on Edmund Husserl’s philosophy were held in honor of his 150th birthday. Husserl was born on the 8th of April in 1859 in the city of Prosnitz, Moravia, today in the Czech Republic. Before he became the famous founder of the phenomenological philosophy that piqued the interest of philosophers, sociologists and psychologists all over the world, he had studied mathematics and philosophy at the universities of Leipzig and Berlin. In Berlin he studied with Weierstrass and Kronecker and wrote his dissertation on the calculus of variations (Beitra¨ge zur Theorie der Variationsrechnung, 1883) under Weierstrass. He became Weierstrass’s assistant before moving to Vienna to study under Franz Brentano. Brentano advised him to finish his habilitation somewhere in Germany; Husserl completed ‘‘On the Concept of Number’’ at the University of Halle in 1887, staying on for 14 years as a Privatdozent. His ‘‘breakthrough to phenomenology’’ and the beginning of the phenomenological movement is connected with the publication of the Logical Investigations 1900–1901 [cf. 1–4]. From then on, Husserl was a well-known figure, first among German philosophers and later all over the world. The book under review is devoted to the history of the Husserl-Archives that were established after Husserl’s death in 1938 to save Husserl’s Nachlass and ensure its publication. This book is not a philosophical study, but rather a contribution to the history of the beginning of the phenomenological movement and its sources. It consists of two articles, printed both in German and in an English translation, together with photos and bibliographical material about the work of the Husserl-Archives. The two articles are Herman Leo Van Breda’s ‘‘Die Rettung von Husserls Nachlass und die Gru¨ndung des Husserl-Archivs’’/‘‘The Rescue of Husserl’s Nachlass and the Founding of the Husserl-Archives,’’ and Thomas Vongehr’s ‘‘Kurze Geschichte des Husserl-Archivs in Leuven und der Husserl-Edition’’/‘‘A Short History of the Husserl-Archives Leuven and the Husserliana.’’ As the title of the book suggests, it documents the history of the HusserlArchives. By doing just this, it reminds us of how important the initiative of a few courageous people can be for the history of science. Van Breda’s article, first published in German in 1959, is a breathtaking document that tells how, as a young Belgian Franciscan monk and student of philosophy, he rescued Husserl’s Nachlass from possible destruction by the Nazi regime and became the founder of the Husserl-Archive at the

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Catholic University of Leuven in Belgium. For the rest of his life, Van Breda spared no effort to support the publication of Husserl’s Nachlass. For the preservation of the biggest part of Husserl’s writings, we owe him a debt of gratitude, since without his assiduity, foresight and diplomacy, many of Husserl’s writings would have remained unknown. Husserl had spent the last 22 years of his life in Freiburg, where he was awarded a chair in philosophy at the local university in 1916. During his lifetime, he penned many lecture courses, research manuscripts, and letters to colleagues and friends, the majority written in a certain type of stenography. By the end of his life, he had accumulated almost 40,000 pages of stenographic material, sorted into folders, and around 10,000 additional pages of typed manuscripts. Van Breda, when he first visited Freiburg in 1938 to do research for his dissertation on Husserl’s philosophy, had not known so much material existed. He realized that Husserl’s manuscripts and his library were in danger of destruction because Husserl, of Jewish descent, had fallen victim to the so-called ‘‘non-Aryan’’ laws in the last years of his life, though he had been baptized in the Lutheran church in 1887. Van Breda wanted to make the Nachlass available in its entirety. In cooperation with the philosopher’s widow, Malvine Husserl, he managed to find a solution for protecting and rescuing the material. The Nachlass were first hidden in southwestern German cloisters. With the help of the Belgian diplomatic corps in Germany, Van Breda later brought the philosopher’s manuscripts in large suitcases by train to the Belgian embassy in Berlin, from where they were sent to Leuven. Thanks to Van Breda’s efforts, it was arranged in October 1938 for Ludwig Landgrebe and Eugen Fink, Husserl’s former assistants, to work for two years on the transcription of these manuscripts. Thus, the Husserl-Archive was established. During the war, young philosophers transcribed the manuscripts in secret, for fear of being discovered by the occupying German regime. The second article in the book provides a survey of the development of this Archive in subsequent years. Soon, Husserl’s work attracted great interest, especially that of young French philosophers who came to do research in Leuven. Among them were some of the most influential philosophers of the twentieth century, figures such as Maurice Merleau-Ponty, Paul Ricœur and Jacques Derrida. Particularly in its early years, the editing of Husserl’s writings confronted many difficulties, and it took years before the first volumes in the series of the collected writings, ‘‘Husserliana,’’ were published. Today, this series can boast 39 such volumes as well as some additional volumes of ‘‘Materialien.’’ In later years, branches of the Husserl-Archive in Leuven were established in Paris, Freiburg, Cologne, Buffalo, NY, and Pittsburgh. The systematic transcription of the entire Nachlass has not yet been completed and remains an ongoing project of the Archive. Husserl’s most important texts on mathematics—mostly from the beginning of his career—are already published in the ‘‘Husserliana’’ [5, 6], and, together with the other published writings, they contribute to our understanding of Husserl as an extremely rich, detailed, and multifaceted philosopher. This book provides fascinating insight into the history of the editing of the writings of a highly influential philosopher. Ó 2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

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It points to former difficulties as well as future projects of the Husserl-Archives and tells the story of its history. Although the book could have easily been enriched by including other articles on the same topic [7], it remains an informative document about an important chapter in the history of philosophy.

[4] J. Philip Miller: Numbers in Presence and Absence. A Study of Husserl’s Philosophy of Mathematics. The Hague, Nijhoff, 1982. [5] Edmund Husserl: Philosophie der Arithmetik. Mit erga¨nzenden Texten. Hrsg.: L. Eley. Den Haag, Nijhoff, 1970 (Husserliana XII). [6] Edmund Husserl: Studien zur Arithmetik und Geometrie. Texte aus dem Nachlass. Hrsg.: I. Strohmeyer. The Hague, Boston, Dordrecht: Nijhoff, 1983 (Husserliana XXI). [7] Walter Biemel: ‘‘Dank an Leuven. Erinnerungen an die Zeit von 1945–1952.’’ In: Profile der Pha¨nomenologie. Zum 50. Todestag

REFERENCES

von Edmund Husserl. Pha¨nomenologische Forschungen, 22, 236– 268, 1980.

[1] Herbert Spiegelberg: The Phenomenological Movement. A Historical Introduction. 3rd ed., The Hague, Nijhoff, 1981. [2] Jitendra Nath Mohanty: The Philosophy of Edmund Husserl. A Historical Development. New Haven, Yale University Press, 2008. [3] Henning Peucker: Von der Psychologie zur Pha¨nomenologie. Husserls Weg zur Pha¨nomenologie der ‘‘Logischen Untersuchungen.’’ Hamburg, Meiner, 2002.

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Fakulta¨t fu¨r Kulturwissenschaften Universita¨t Paderborn Philosophie, Warburger Str. 100, D-33098 Paderborn Germany e-mail: [email protected]

Digital Dice: Computational Solutions to Practical Probability Problems by Paul J. Nahin PRINCETON: PRINCETON UNIVERSITY PRESS, 276 PP., US $ 27.95, ISBN-13: 978-0-691-12698-2 REVIEWED BY DINO LORENZINI

ow! Few authors can hope to match the style of this book: clear, entertaining and witty. The reader is immediately motivated to dig deeper. Physicist Richard Feynman famously observed ‘‘What I cannot create, I do not understand.’’ In that spirit, Nahin proposes that the reader experiment with a series of 21 beautifully chosen problems in probability. All the problems arise from everyday life, and most can be solved with a Monte Carlo simulation. Nahin has three audiences in mind: ‘‘Teachers of either probability or computer science looking for supplementary material for use in their classes, students in those classes looking for additional study examples, and aficionados of recreational mathematics ... .’’ In a stimulating introduction, Nahin lays out the book’s philosophy; lets you check, using an amusing little anecdote, that you satisfy the prerequisites needed to understand this book; and then explains, using two geometric probability problems, what computer simulations are. Now, on to the fun stuff: The 21 problems. Nahin chose them not only for their pedagogical content, but also because of the engaging stories associated with many of them. As Nahin explains, Problem 4, ‘‘A curious coin-flipping game,’’ defied solution for a quarter-century. Problem 8, ‘‘A Toilet Paper Dilemma,’’ has achieved minor cult status as few Toilet Paper Problems ever have. The results of Problem 19, ‘‘Electing Emperors and Popes,’’ suggest that two reported events on historical elections of Popes are most unlikely actually to have occurred. Problem 16, ‘‘The Appeals Court Paradox,’’ lets you explore the probability that a court errs, and how this probability changes if the worst judge decides to always follow the lead of the best judge. Some problems can serve as an introduction to active fields of research, such as queueing theory in Problem 15, ‘‘How Long Is the Wait to Get the Potato Salad?’’ Here the reader is asked to simulate the operation of a deli counter: Customers present themselves randomly at the counter, at an

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easily measured average rate (say k customers per hour), and the deli clerk takes various amounts of time to fill the various orders. The service time for each customer is again a random quantity, but with an easily measured average rate of service (say l customers per hour). The reader is asked to help the store management in figuring out answers to such mathematical questions as what is the average total time at the deli counter for the customer (total time is the sum of the waiting time and the service time), and what is the maximum total time experienced by the unluckiest of the customers. What happens to these questions if a second, equally skilled deli clerk is hired? For this, the reader will need to write a computer simulation of a 10-hour day at the deli. This certainly involves getting down to the nitty-gritty of algorithm development, an important issue emphasized throughout the book. In contrast to most problems in the book, here a crucial question needs to be answered before the simulation can be done. What should one use in this problem to simulate the random time between the arrivals of Customer i and Customer i + 1? Nahin suggests -log(rand)/k, where rand is the uniform random variable; and he invites his readers to look up Poisson queues (not to be confused with the French queues de poisson) in any good book on stochastic processes or operations research to learn the theorerical underpinnings of this suggestion. Solutions to each problem include the complete computer code in MATLAB, and useful references to the literature. Several theoretical discussions are given in the appendices, such as Appendix 2 on evaluating the results of a Monte Carlo simulation. A glossary of terminology is also included, making this book very user-friendly. The reader will find in ‘‘Digital Dice’’ many examples of how much of mathematics really is done: Somebody gets an interesting idea and does some experimentation (here, a computer simulation), which is later followed by a theoretical confirmation (proof). Such examples are especially important to beginners in the subject. It is often very difficult to teach students how to experiment: More books such as this one, in other fields of mathematics, are waiting to be written. In the early nineteenth century, C. F. Gauss was able to gain deep insight into problems through his exceptional powers of computation. Nowadays, the power of computation is available to mathematicians who master the power of computers. No mathematics majors should graduate without a working knowledge of computer simulations. This delightful book provides ample incentive to gain that knowledge. Department of Mathematics University of Georgia Athens, GA 30602 USA e-mail: [email protected]

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Strange Attractors: Poems of Love and Mathematics Edited by Sarah Glaz and Joanne Growney WELLESLEY, MA: A. K. PETERS, 2008, US $39.00, 255 PP. ISBN 978-156881-341-7 REVIEWED BY SALLY I. LIPSEY

ll the poems in this unique anthology use mathematical allusions to express love. It is surprising to find that most of the poets represented here are not mathematicians, and, possibly more surprising, to see how many mathematicians (including the most famous) have written poetically about love. The source of the title, Strange Attractors, is a poem, Chaos Theory (p. 66), by an award-winning professor of English, Ronald Wallace. As we read this poem, we explore connections between seemingly random events in the universe and personal experiences (like love affairs). Wallace writes, ‘‘We are uniquely strange attractors, love’s/pendulum point or arc, time’s shape or fancy,/in a system with its own logic, be it/ the cool elegance of eternity, or/the subatomic matrix of creation and decay.’’ How does mathematics, famously cold and austere, and scary for so many, speak of love? How can it propose romance and describe sexual encounters? The writers are drawn to the sights, sounds and content of mathematical terms and symbols for the imagery and truths they wish to convey. The 150 poets represented in the anthology show us a variety of ways for mathematical imagery to portray love as well as tears, fascination and admiration. An example from early history: The Roman poet, Catullus (84–54 BC) wrote Let’s Live and Love: To Lesbia, asking her to ‘‘Give me a thousand kisses, a hundred more,/ another thousand, and another hundred,’’ (p. 12). In this century, Marion Cohen asks, ‘‘So how come there’s a discontinuity at the waistline?/How come, around there, Zeno whispers ‘halfway?’/How come that waistline is throbbing with infinity?/And my hand and heart throbbing/with zero?’’ (Scared and the Intermediate Value Theorem, p. 137). A high proportion of these poems use terminology common to everyday speech, requiring no special math education. Remember Elizabeth Barrett Browning’s answer to How Do I Love Thee? (p. 11): ‘‘Let me count the ways./I love thee to the depth and breadth and height/My soul can reach, …’’ In Paradiso: Canto XXXIII, Dante Alighieri speaks of ‘‘… the Love that moves the Sun and the other stars’’ and describes himself to be ‘‘Like a geometer wholly dedicated/to squaring the circle, but who cannot find,/think as he may, the principle indicated—’’ (p. 76). In a poem of yearning, On Your Imminent Departure: Considering the Relative Importance of Various Motions, Pattiann Rogers (p. 53), asks, ‘‘Which is more important, the motion of the wind/ …/or your hand in motion across my back, …’’. Another

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example is Parabolic Ballad by Andrei Voznesensky (p. 63), a poetic vision in which ‘‘Art, love and history race along recklessly/Over a parabolic trajectory.’’ (Parabolic Ballad is translated from Russian by W. H. Auden, one of the many distinguished translators cited in the book.) Human beings have a natural affinity for numbers, especially for the small ones of early childhood and the very large ones that are fun to say. Poets, both mathematical and not, exhibit this affinity. Harry Matthews ends each line of his sestina, Safety in Numbers (p. 44), with a number from 1 to 7, using a different permutation for each stanza. John Donne has a variety of numbers in The Computation (p. 21), where ‘‘Tears drowned 100, and sighs blew out two,’’ as does Carl Sandburg in Number Man, a poem dedicated to ‘‘the ghost of Johann Sebastian Bach’’ that gives human qualities to numbers (p. 185). The mysterious attributes of prime numbers, ‘‘the power, the peculiar glory …’’ are surveyed in Let Us Now Praise Prime Numbers (p. 190) by Helen Spalding. Beyond all numbers, infinity beckons: ‘‘Methinks I lied all winter, when I swore, /My love was infinite, if spring make it more.’’ (From Love’s Growth by John Donne, p. 20.) An intriguing poem, Yes, by David Brooks, (p. 10) refers indirectly to infinity by imagining ‘‘living this life/ over and over,’’ including ‘‘… that moment when our whole life/flashes before our eyes’’. Infinity was an important concept in the work of the distinguished mathematician Jakob Bernoulli (1654–1705), who shows his feelings about the idea in a poem inserted in his Treatise on Infinite Series: ‘‘What joy to discern the minute in infinity!/The vast to perceive in the small, what divinity!’’ (p. 130). Bernoulli is also one of many mathematicians who appear as references or characters in the poems. ‘‘Bernoulli would have been content to die,/Had he but known such a2cos 2U!’’ are the last two lines of the selection from Stanislaw Lem’s The Cyberiad (p. 39). Here, Lem also refers in an amusing way to Riemann, Hilbert, Banach, Cauchy, Fourier and others. A reference to Archimedes and a famous quote (‘‘Give me a place to stand and I can move the world’’) is used by Jean deSponde in building a portrait of ‘‘the love I feel for you, my dear’’ (Sonnet of Love XIII, p. 17). Fibonacci inspires Kathryn DeZur who longs ‘‘for the fertility/ of Fibonacci’s numbers, that mystical statistical world/where one plus one equals three’’ (Fibonacci Numbers, p. 85), and Bill Parry writes of friendship, relating it to Alexander’s horned sphere in a poem of the same name (p. 178). Benoit Mandelbrot is a character in Mandelbrot Set, song lyrics by Jonathan Coulton (p. 141). (Both the horned sphere and the Mandelbrot set are among the illustrations accompanying some of the poems in the book.) Poets who write of love and math often write with humor also. C. K. Stead applies a Venn diagram technique (with circular illustrations) to a survey of 19 love affairs in which ‘‘17 were over/7 were forgotten/and 13 irrelevant/ but only 2 were all three’’ (from Walking Westward, p. 58); and Haipeng Guo considers the difficulty of a love affair (When a P-Man Loves an NP-Woman, p. 161).

Classroom experiences inevitably generate humor and humorous poetry; teachers may find some perfect choices for a light moment in class. For instance, students will laugh appreciatively at Yehuda Amichai’s imaginative transformation of the problem ‘‘about a train that leaves from place A and another train/that leaves from place B. When will they meet?/…/None of the problems was about a man who leaves from place A/and a woman who leaves from place B. When will they meet,’’ (from Israeli Travel: Otherness Is All, Otherness Is Love, p. 5). Here is one originally written in the twelfth century for his daughter by Bhaskaracharya: ‘‘Whilst making love a necklace broke./A row of pearls mislaid./…/The young woman saved one third of them;/One tenth were caught by her lover./If six pearls remained upon the string/How many pearls were there altogether?’’ (from Lilavati, p. 131). For a little singing in the calculus classroom, try There’s a Delta for Every Epsilon, Tom Lehrer’s Lyrics for a Calypso Song (p. 167). The editors subdivided the anthology into three parts, namely: (1) ‘‘Romantic Love: from Heartaches to Celebrations’’ (pp. 3–71); (2) ‘‘Encircling Love: Of Family, Nature, Life and Spirit’’ (pp. 73–125); and (3) ‘‘Unbounded Love: For Mathematics and Mathematicians’’ (pp. 127– 198). Only in Part (3) do we find that the majority of poems are by mathematicians. I had fun doing my own special subsets also, classifying the poems according to varieties of love (sober, passionate, sexual, unrequited), varieties of math topics (from elementary school math to recent inventions), humor, history, use of symbols and diagrams, poetic structure based on math concepts (such as factorization or Fibonacci numbers), and quirkiness. One poem left me nonplussed—should I say it does not add up? It is a poem that actually sounds beautiful by

Becky Dennison Sakellariou called Math is Beautiful and So Are You (p. 54). It alternates between mathematical statements (in normal print) and personal statements (in italics), seemingly nonsequiturs, beginning with ‘‘If n is an even number/then I’ll kiss you goodnight right here,/but if the modulus k is the unique solution,/I’ll take you in my arms for the long night.’’ Perhaps what is required is a good imagination, a sense of humor and appreciation for music! Among the delights of this book, in addition to the poetry, are a substantive introduction, bibliographical resources, information about the poet-contributors, and about the mathematicians who are named in poems. From the introduction, I learned about Enheduanna, an ancient priestess who was responsible for mathematical survey calculations on the land and in the sky, and who also wrote poetry in the form of temple hymns. The introduction also gives information about other books that are likely to be of interest to readers of Strange Attractors. ‘‘Mathematical Poetry Resources for Further Exploration’’ extends the material given in the introduction. ‘‘Contributors’ Notes’’ and ‘‘About the Mathematicians Appearing in the Poems,’’ at the back of the book, provide details about the poets and the mathematicians, their degrees, careers, publications, and honors. Mathematicians and poetry lovers (with at least some feeling for math) will enjoy the many treasures in this anthology.

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2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010

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Stamp Corner

Robin Wilson

Recent Mathematical Stamps: 2002

Jovan Karamata (1902–1967) was a Serbian mathematician who spent most of his career in Belgrade and Geneva. His output was large and his main contributions were to topics in mathematical analysis. In particular, several results are named after him, such as the Karamata Tauberian theorems and the Karamata inequality for convex functions.

Mechanical calculator

Niels Henrik Abel (1802–1829) Abel’s greatest achievement was to prove that the general quintic equation has no solutions by means of radicals. During travels to Germany and France he obtained fundamental results on elliptic functions, the convergence of series, and ‘Abelian integrals’, many of which appeared in his 1826 ‘Paris Memoir’. Tragically, this memoir was lost for a time, and letters informing him that it had been found and offering him a prestigious job in Berlin arrived just two days after his early death from tuberculosis at the age of 26.

Ja´nos Bolyai (1802–1860) constructed a ‘non-Euclidean geometry’—a geometry that satisfies four of Euclid’s five basic postulates, but not the ‘parallel postulate’ that there is exactly one line through a given point and parallel to a given line; in Bolyai’s geometry there are infinitely many such lines. For almost 2,000 years it was generally believed that no such geometry can exist, yet the magnitude of Bolyai’s achievement was not fully recognised until after his death.

Niels Henrik Abel (Norway)

This cylindrical calculating machine was constructed around 1820 by Johann Christoph Schuster (1759–1823) of Ansbach in Bavaria. Schuster built several mechanical calculators based on designs by the inventor and priest Philipp Mattha¨us Hahn, with whom he served his apprenticeship. This machine is still in good working order and is now housed in the Arithmeum at the University of Bonn.

Pedro Nunes (1502–1594) a royal cosmographer, was the leading Portuguese nautical scientist of his day. He applied mathematical techniques to cartography and constructed a ‘nonius’ that measured fractions of a degree. His 1537 treatise on the sphere showed how to represent a rhumb line, the path of a ship on a fixed bearing, as a straight line.

Matteo Ricci (1552–1610) was an Italian Jesuit. The first missionary in China, towards the end of the Ming dynasty, he disseminated knowledge of Western science, especially in mathematics, astronomy and geography. His most important contribution was an oral Chinese translation of the first six books of Euclid’s Elements.

Mechanical calculator (Germany) Ja´nos Bolyai (Hungary) Jovan Karamata (Yugoslavia)

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Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics, Computing and Technology The Open University, Milton Keynes, MK7 6AA, England e-mail: [email protected]

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THE MATHEMATICAL INTELLIGENCER Ó 2009 Springer Science+Business Media, LLC

Pedro Nunes (Portugal)

Matteo Ricci (Italy)

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