Letters to the Editor
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.
Nonegenarian Fibonacci Devotee
Please let me take this opportunity to make one more obeisance to the Fi bonacci sequence.Fibonacci tended to take over my mathematical life from the time, many years ago, when I found that the occurrence of the numbers in leaf patterns needed more explaining. One thing led to another, decade after decade, paper after paper. 1 I lived com fortably among these numbers-until midnight of April 26, 2003. At that in stant, I ceased to be 89 years old; and there seems little prospect of my ever again having aFb i onacci number as my age. To be sure, my rural route address is now Box 532, Route 1, a concatena tion of Fibonacci numbers in reverse order, but that is small consolation. Something more is needed to re affirm my allegiance. Here is my offer ing. I will prove that theFb i onacci num bers with odd index can be generated iteratively from the quadratic equation x2+y2
(la)
=
3xy - 1
in the following way. Put x equal to any Fibonacci number with odd index;:::: 1, and solve (1a) for y; the larger root will be the Fb i onacci number with the next larger odd index. The Fibonacci num bers with even index are generated by
'For instance. 2V.
4
E.
my articles in J.
exactly the same procedure from the equation x2+y2=3xy+l.
(lb)
To prove these, I will use an imme diate consequence of the defining iter ation Fn+l = Fn+Fn-t: (2) I
Fn-2 + Fn+2
3F , .
will also use the identity
(3)
Fn-2Fn+2
=
Fn2+ (-1) n+l,
which is a special case of an identity in Hoggatt.2 Now I set x = Fn (n odd) in (la) (4)
Fr/ + y2
=
3FnY- 1,
and I am able to show that the larger root for y is F11+2 . Substituting (3) on the left and (2) on the right of (4) re duces it to
which does indeed have F,+2 as its larger root. Similarly for the assertion for even n. Irving Adler 297 Cold Spring Road North Bennington, VT 05257 USA e-mail:
[email protected]
Theor. Bioi. 45 (1g74), 1-7g: and J. Algebra 205 (1ggs), 227-243.
Hoggatt, Jr. Fibonacci and Lucas Numbers (Houghton Mifflin, 1969). See p. 59.
THE MATHEMATICAL INTELLIGENCER © 2006 Springer Science+ Bus1ness Med1a, Inc
=
BARRY KOREN
Computationa F uid Dynamics· Science and Too The year 2003 marked the 1 OOth anniversary of both the birth of John von Neumann and the first manned flight with a powered plane-both events of great importance for computational fluid dynamics.
he science of flows of gases and liquids is fluid dynamics, a subdiscipline of physics. No courses in fluid dynamics are given in high school, as it requires too much mathematical background. Fluid dynamics is taught at university and at engineering colleges, for one cannot ignore fluid dynamics if one wants to design an aircraft, a rocket, a combustion engine, or an artificial heart.
Particularly for aircraft design, knowledge and under standing of fluid dynamics-aerodynamics in this case-is of major importance. Except for the dangerous gravity, all forces acting on a flying plane are forces exerted by air. To fly an aircraft safely (tanked up with fuel and with pas sengers on board), a precise knowledge, understanding, and control of these aerodynamic forces is a matter of life and death. Moreover, flying must not only be safe but also fuel-efficient and quiet. For aerospace engineering, aero dynamics is indispensable. Nowadays, both experimental and theoretical means are available for investigating fluid flows. Wind tunnels are the canonical tool for experimental aerodynamics. The Wright brothers, who made the first manned flight with a powered plane (Fig. 1), had at their disposal a wind tunnel, one they themselves had made. Wind-tunnel testing has many dis advantages, but it is deemed trustworthy because real air is used and not the virtual air of theoretical aerodynamics. A Brief History of Computational Fluid Dynamics
Nowadays, the technological relevance of theoretical aero dynamics, of theoretical fluid dynamics in general, is widely appreciated. However, in the past it was mainly an acade-
mic activity, with results that strongly differed from ex perimental observations. The technical applications of fluid dynamics developed independent of theory. Theoretical and technological breakthroughs have since closed the gap between theory and practice, and today we see a fruitful interaction between the two. The airplane has played a very stimulating role in this development. I proceed by highlighting some key developments from the history of theoretical fluid dynamics with an eye toward computational fluid dynamics. Revolutionary innovations
Theoretical fluid dynamics has an illustrious history [1, 2, 3]. In the course of centuries, many great names have con tributed to the understanding of fluid flow and have helped in building up theoretical fluid dynamics, step by step. The oretical fluid dynamics goes back to Aristotle (384-322 BC), who introduced the concept of a continuous medium. In my opinion, though, it actually began 2000 years later, when Leonhard Euler published his equations of motion for the flow of liquids and gases, on the basis of Newton's second law of motion [4, 5]. Euler's idea to describe the motion of liquids and gases
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Trail-blazing ideas from Princeton
Figure 1 . Glass-plate photo of the first manned flight with a powered plane (flight distance: 37 meters, flight time: 12 seconds), Kitty Hawk, North Carolina, 1 0.35 h., December 17, 1 903. Prone on the lower wing: Orville Wright. Running along with the plane to balance it by hand if necessary: Wilbur Wright. Visible in the foreground: the rail from which the plane took off and the bench on which the wing rested. The very same day, the Wright brothers made a flight of 260 meters and 59 seconds!
in the form of partial differential equations was a revolu tionary innovation. However, his equations, known today as the Euler equations, were still unsuited for practical ap plications, because they neglect friction forces: only pres sure forces were taken into account. It was almost a cen tury later, in 1845, that George Stokes proposed fluid-flow equations which also consider friction [6]; equations which, for an incompressible flow, had already been found by Claude Navier [7] and are now known as the Navier-Stokes equations. With the introduction of the Navier-Stokes equa tions, the problem of understanding and controlling a large class of fluid flows seemed to be within reach, as it had been reduced to the integration of a handful of fundamen tal differential equations. Although formulating the Navier-Stokes equations con stituted great progress, the analytical solution of the com plete equations was not feasible. (It remains one of the out standing open mathematical problems of the 21st century.) One developed instead a large number of simplified equa tions, derived from the Navier-Stokes equations for special cases, equations that could be handled analytically. More over, a gap continued between experimental and theoreti cal fluid dynamics. The former developed greatly during the Industrial Revolution, independent of the latter. It was para doxical that the introduction of the Navier-Stokes equa tions led to a further fragmentation into different flow mod els, all of which described the flow of the same fluid (air in our case)-a theoretically highly undesirable situation. Theoretical fluid dynamics stagnated along a front of non linear problems. This barrier was finally broken in the sec ond half of the 20th century, with numerical mathematics, at the expense of much-often very much-computational work A key role was played in this by a Hungarian-born mathematician, John von Neumann.
6
THE MATHEMATICAL INTELLIGENCER
In the tens and twenties of the past century, Budapest was a fruitful breeding ground for scientific talent. It saw in 1903 the birth of John von Neumann (Fig. 2). In his early years von Neumann received a private education; at the age of 10 he went to school for the first time, directly to high school. There, his great talent for mathematics was discovered. He received extra lessons from mathematicians of the Univer sity of Budapest, among them Michael Fekete, with whom von Neumann wrote his first mathematics paper, at the age of 18. By then he was already a professional mathematician. Von Neumann studied at the ETH ZUrich and the University of Budapest; he obtained his PhD degree at the age of 22. Next he moved to Gem1any, where he lectured at universi ties in Berlin and Hamburg. There he was particularly active in pure mathematics: in set theory, algebra, measure theory, topology, and group theory. He contributed to existing theo ries: the sure way to quick recognition. From the mid 1930s, von Neumann chose a riskier way of working: breaking new ground. He turned to applied mathematics in the sense of mathematicians like Hilbert and Courant, i.e., not mathe matics applied to all kinds of ad hoc problems, but the sys tematic application of mathematics to other sciences, in par ticular to physics, with subdisciplines like aerodynamics. The rapidly deteriorating political situation in Europe, from which von Neun1ann had already emigrated to Princeton, 1 played a ·------- - -------
1Von Neumann was one of the many scientists who left Europe in the early 1 930s. For a description of the fall of Gottingen under Nazi pressures. see Richard Courant's biography [8].
Figure 2. John von Neumann, 1 903-1 957.
role in this decision. War was looming and brought in creasing demands for answers to questions related to mil itary engineering. Whereas von Neumann had worked on a mathematical basis for the equations of quantum mechanics before the war, during the war he "lowered" himself to developing nu merical solution methods for the Euler equations. His idea to compute possible discontinuities in solutions of the Euler equations without explicitly imposing jump relations was very original. Instead, von Neumann proposed the in troduction of artificial (numerical) diffusion, in such a way that the discontinuities automatically appear in a physically correct way: shock capturing, nowadays a standard tech nique. He also came up with an original method for ana lyzing the stability of numerical calculations: a Fourier method, now a standard technique as well. In 1944, the urgent need arose to apply von Neumann's numerical methods on automatic calculators, computers, beyond the scope of the machines of that time. This moti vated von Neumann to start working also on the develop ment of the computer. In 1944 and 1945 he did trail-blazing work, writing his numerical methods for computing a fluid flow problem in a set of instructions for a still non-existent computer. These instructions were not to be put into the computer by changing its hardware or its wiring. Instead, von Neumann proposed to equip computers with hardware as general as possible, and to store the computing instruc tions in the computer, together with the other data involved (input data, intermediate results, and output data). In 1949,
the first computer was realized which completely fulfilled von Neumann's internal programming and memory princi ples: the EDSAC (Electronic Delay Storage Automatic Cal culator), by M. V. Wilkes, at Cambridge University. Today, the two principles are still generally applied. L. F. Richardson and Richard Courant and colleagues had combined theoretical fluid dynamics and numerical mathe matics before von Neumann [9, 10], but still without clear ideas about computers-without computer science. Compu tational fluid dynamics (abbreviated CFD) is a combination of three disciplines: theoretical fluid dynamics, numerical mathematics, and computer science. Because von Neumann brought in this last discipline, he can be considered the found ing father of CFD. A detailed description of von Neumann's contributions to scientific computing is given by Aspray [11). A good overview of his other pioneering work can be found in the scientific biography written by Ulam [12]. Traveling from place to place as an honored mathe matician with many social and political obligations, von Neumann must have had very little time to write down his scientific ideas. He published only one paper about both shock capturing and the aforementioned stability analysis, and that not until 1950 [13]. On his many travels, von Neumann visited the Nether lands. In 1954, he was an invited speaker during the Inter national Congress of Mathematicians held in Amsterdam. A tea party with Queen Juliana of the Netherlands was arranged for a select group of participants, among them John von Neumann (Fig. 3).
Figure 3. John von Neumann and colleagues at Soestdijk Palace. Above: all together, John von Neumann front row, far left. Queen Juliana, with white handbag, is flanked by the two new recipients of the Fields Medal: Jean-Pierre Serre (with Herman Weyl's hands on his shoulders) and Kunihiko Kodaira. At the right of von Neumann: Mary Cartwright.
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Pioneering work in Amsterdam
In the third quarter of the 20th century, computer science was a new and growing discipline. Initially, the Netherlands played no significant role in the development of computer science, but the country was quickly moving forward. In 1946, the Mathematisch Centrum (MC) was founded in Am sterdam. The mission of this new institute was to do pure and applied mathematics research in order to increase "the level of prosperity and culture in the Netherlands and the contributions of the Netherlands to international culture." (Not at all the pure ivory tower.) The foundation of the MC did not proceed without struggle. The most prominent Dutch mathematician of the day, L. E. J. Brouwer of the University of Amsterdam, was of the opinion that mathe matics should be indifferent towards the physical sciences and even rejecting of technology; an odd point of view, con sidering the work of mathematicians like Hilbert, Courant, and von Neumann. With the MC, Brouwer wanted to tum Amsterdam into the new Gottingen of pure mathematics. It did not work out that way. His biographer feels that Brouwer was sacrificed to the foundation of the MC ([14], p. 479). The founders of the MC had heard about von Neumann's ideas about machines which should be able to perform a series of calculations as independently as possible. They wanted the MC to have a computing department in order to develop a computer and to execute advanced comput ing work. Aad van Wijngaarden (Fig. 4), former student of J. M. Burgers of the Delft University of Technology, was appointed as the first staff member of the MC, in 1947. That year he made a study tour to visit von Neumann in Prince ton. Van Wijngaarden and his co-workers designed and con structed the first Dutch computer: the ARRA I (Automa tische Relais Rekenmachine Amsterdam I, Fig. 5). New
Figure 4. Aad van Wijngaarden, 1 91 6- 1 987 (photo courtesy of CWI).
8
THE MATHEMATICAL INTELLIGENCER
computers were designed and built (one per design only), in 1955 exclusively for the Fokker aircraft industries: the FERTA (Fokker Elektronische Rekenmachine Type ARRA). Much human labor was required to perform com putations on these early computers. At the MC, this was done by young women (Fig. 6), schooled in mathematics by Van Wijngaarden. A highlight was the project for the de velopment of the Fokker Friendship airplane, a numerical project on which Van Wijngaarden and his "computing girls" worked from 1949 until 1951. The computations con cerned oscillations of the airplane's wing in subsonic flow: flutter. The first computing work was still very "external" and machine-dependent; for each computation, cables had to be plugged into the computer. With the accomplishment of internal programming as proposed by von Neumann, at tention shifted entirely to the invention of algorithms and their coding as computer programs. Edsger Dijkstra, a later Turing Award recipient, was appointed at the MC as the first Dutch computer programmer. Van Wijngaarden and Dijkstra left an international mark on computer science with their contributions to the development of the pro gramming language Algol 60 [15], and Van Wijngaarden later added to that reputation with Algol 68 [16]. In 1979, Van Wijngaarden was awarded an honorary doctorate for his pioneering work by the Delft University of Technology, and the MC grew into the present CWI (Centrum voor Wiskunde en Informatica), which celebrates its 60th an niversary in 2006. Computational fluid dynamics research on the basis of the Euler or Navier-Stokes equations, of the same funda mental character as that established by von Neumann, was not done in the Netherlands of 1945-1960. For this funda mental work, we have to go to the United States and the Soviet Union of the 1950s.
Figure 5. The first computer at the MC and in the Netherlands, the ARRA I in its final set-up. From left to right: power frame and the three arithmetic registers. On the table in the middle: the punch-tape reader. Some 1 200 relays are at the back of the machine {photo courtesy of CWI).
A continuous flow of CFD from New York
In early December 1941, a passenger ship carried a 15-year old Hungarian boy from Europe to the United States. The boy, along with his parents, was escaping the tragic fate threatening European Jews. (It was to be the last passen ger ship from Europe's mainland to the United States for years to come. During the voyage, the United States was drawn into the Second World War by the attack on Pearl
Harbor.) The young ship passenger carried with him two letters of recommendation from his teachers. It seems likely that von Neumann saw those letters brought by his young fellow-countryman, for the boy, Peter Lax (Fig. 7), had a meteroric rise to success. In 1945, while still a teenager, he became involved in the Manhattan Project. In 1949, he received his PhD degree from New York Univer sity, with Richard Courant as his thesis advisor, and in 1951,
Figure 6. Female arithmeticians at the MC, the "girls of Van Wijngaarden." In the foreground: Ria Debets, later the spouse of Edsger Dijk stra. {photo courtesy of CWI).
© 2CX>6 Springer Science+Business Media, Inc., Volume 28, Number 1, 2006
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from Lax: "The impact of computers on mathematics (both applied and pure) is comparable to the roles of telescopes in astronomy and microscopes in biology." Despite the Second World War and the Cold War, Lax has always had very good connections with scientists worldwide. One such relation is with a famous Russian mathematician, mentioned in the next section. A brilliant idea from Moscow
Figure 7. Peter D. Lax.
he became assistant professor there. His work in mathe matics continues to this day and has led to many honors and awards, among them the 2005 Abel Prize in mathe matics [17]. Like von Neumann, Lax is a homo universalis in math ematics. He has performed ground-breaking research, and has been a productive and versatile author of mathematics books. His books deal with such diverse topics as partial differential equations, scattering theory, linear algebra, and functional analysis. Above all, he is known for his research on numerical methods for partial differential equations, in particular for hyperbolic systems of conservation laws, such as those arising in fluid dynamics. Lax's name has been given to several mathematical discoveries of impor tance to CFD: •
•
•
•
•
the Lax equivalence theorem [18], stating that consis tency and stability of a finite-difference discretization of a well-posed initial-boundary-value problem are neces sary and sufficient for the convergence of that dis cretization, the Lax-Friedrichs scheme [ 19], a stabilized central finite difference scheme for hyperbolic partial differential equations, the Lax-Wendroff scheme [20], a more accurate but equally stable version of the Lax-Friedrichs scheme, the Lax entropy condition [21], a principle for selecting the unique physically correct shock-wave solution of nonlinear hyperbolic partial differential equations that al low multiple shock-wave solutions, and the Harten-Lax-Van Leer scheme [22], a very efficient nu merical method for solving the Riemann problem.
Like von Neumann, Lax was (and still is) a strong pro ponent of the use of computers in mathematics. A quote
10
A substantial part of the Euler and Navier-Stokes software used worldwide is based on a single journal paper [23], dis tilled by the then-young Russian mathematician Sergei Kon stantinovich Godunov (Fig. 8) from his PhD thesis. Godunov proposed the following. Suppose one has a tube and in it a membrane separating a gas on the left with uni formly constant pressure, from a gas on the right with a like wise uniformly constant but lower pressure (Fig. 9, top). If the membrane is instantaneously removed-the traffic light changes from red to green-then the yellow gas will push the blue gas to the right; the interface between the two gases, the contact discontinuity, runs to the right. At the same time, two pressure waves start running through the tube: a compression wave running ahead of the contact dis continuity and an expansion wave running to the left (Fig. 9, bottom). In the 19th century, the Euler flow in this tube, a shock tube, had already been computed by Riemann, with "pencil and paper" [24]. (For this old work of Riemann, Duivesteijn has written a nice, interactive Java applet [25].) For the computation of the flow in a tube in the case of an initial condition which has more spatial variation, Godunov proposed to decompose the tube into virtual cells (Fig. 10, above), with a uniformly constant gas state in each cell, and with each individual cell wall to be considered as the afore mentioned membrane (traffic light). To know the interac-
THE MATHEMATICAL INTELLIGENCER
Figure 8. Sergei Konstantinovich Godunov.
rarefaction wave
contact discontinuity .....
shockwave
Figure 9. Shock tube. Top: condition of rest in left and right part: high and low pressure, respectively. Bottom: condition of motion with shock wave and contact discontinuity running to the right and rarefaction wave running to the left. (drawing: Tobias Baanders, CWI).
tion between the gas states in two neighboring cells, one in stantaneously 'removes' the cell wall separating the two cells, and computes the Riemann solution locally there, and hence the local mass, momentum, and energy flux (Fig. 10, bottom). This is done at all cell faces. With this, the net transport for each cell is known and a time step can be made. A plain method and a very simple flow problem, so it seems. If one can do this well, the flow around a com plete aircraft or spacecraft can be computed. The remark able property of the method is that at the lowest discrete level, that of cell faces, a lot of physics has been built into it, not just numerical mathematics. The more cells, the better the accuracy, yet also the more expensive the computation. Godunov did not have ac cess to computers, but to "computing girls," who called Godunov and his fellow PhD students "that science," and who received payment on the basis of the number of com putations they performed, right or wrong. No real CFD there either! In 1997, Godunov received an honorary doctorate from the University of Michigan, and a symposium was organized
for him at the university's Department of Aerospace Engi neering. At that symposium, in a one-and-a-half-hour lec ture, Godunov gave insight into his earlier research, whose strategic importance was not appreciated in the Soviet Union at the time. This historic lecture has since been pub lished [26, 27]. A second important result in Godunov's classical paper from 1959 [23] is his proof that it is impossible to devise a linear method which is more than first-order accurate, with out being plagued by physically incorrect oscillations in the solution: wiggles (Fig. 11). With a first-order-accurate method, the solution becomes twice as accurate and re mains free of wiggles when the cells are taken twice as small. With a second-order-accurate method, the solution becomes four times more accurate then, but-unfortu nately-possibly wiggle-ridden. Wiggles can be very troublesome in practice. For ex ample, a simple speed-of-sound calculation in a single cell only may break down the entire flow computation, because of a possibly negative pressure. The wiggle problem does not occur only with Godunov's method; it is a general prob-
Figure 10. Shock tube divided into small cells. Top: cells. Bottom: wave propagation over all cell faces. (drawing: Tobias Baanders, CWI).
© 2006 Springer Science + Bus1ness Media, Inc., Volume 28, Number 1, 2006
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pressure
pressure
1
0
Figure 1 1 . Right and wrong pressure distribution. Left: without wiggles. Right: with wiggles. (drawing: Tobias Baanders, CWI).
lem. A drawback of Godunov's method is that it is com puting-intensive; at each cell face, the intricate Riemann problem is solved exactly. Technology pushes from Lelden
It took about two decades before good remedies were found for the wiggles of higher-order methods and the high cost of the Godunov algorithm. The aid came from an as tronomer. In space, large clouds of hydrogen are found. Simulation of the flow of this hydrogen provides models of the development of galaxies. The literally astronomical
Figure 12. Bram van Leer (photo: Michigan Engineering).
12
THE MATHEMATICAL INTELUGENCER
speeds and pressures which may arise in these computa tions impose high demands on the accuracy, and particu larly the robustness, of the computational methods to be applied. While still in Leiden, in the 1970s, the astronomer Bram van Leer (Fig. 12) published a series of papers in which he proposed methods which are second-order ac curate and do not allow wiggles. The fifth and last paper in this series is [28). Furthermore, Van Leer introduced a computationally efficient alternative to the Godunov algo rithm [29): two technology pushes, not only for astronomy but also for aerospace engineering, as well as for other dis-
ciplines.In 1990, Van Leer was awarded an honorary doc torate for this work by the Free University of Brussels. Efficient solution algorithms from Rehovot and other places
Broadly speaking, how does an Euler- or Navier-Stokes flow computation around an aircraft work? The airspace out to a large distance from the aircraft, may be divided into (say) small hexahedra, small 3D cells. Just as in the 1D shock tube example, one can then compute for each cell the net inflow of mass, momentum, and energy, using at each cell face the Godunov alternative ala Van Leer or other alternatives, like the Roe scheme [30] or the Osher scheme [31]. The finer the mesh of cells around the aircraft, the grid (Fig. 13), the more accurate the solution, but also the higher the computing cost. A grid of one million cells for an Euler- or Navier-Stokes-flow computation is not un usual. Suppose that we want to simulate a steady flow. We then have to solve, per cell, five coupled nonlinear partial differential equations. The cells themselves are coupled as well: what flows out of a cell flows into a neighboring cell (or across a boundary of the computational domain). In Navier-Stokes-flow computations, the flow solution in a sin gle cell may influence the flow solutions in all other cells. In our modest example, we may have to solve a system of five million coupled nonlinear algebraic equations. Effi cient solution of these millions of equations is an art in it self.Many efficient solution algorithms have been devel-
oped, the most efficient of which are the multigrid algo rithms.Multigrid methods were invented at different loca tions and by several people. A leading role has been played by Achi Brandt from the Weizmann Institute of Science in Rehovot, Israel [32].Multigrid algorithms have a linear in crease of the computing time with the number of cells. This may seem expensive-2, 3, or 4 times higher computing cost for a grid with 2, 3, or 4 times more cells, respectively but it is not. In numerical mathematics, no bulk discount is given. For many solution algorithms the rule is 22 ,32,42, . ..times higher computing cost for a grid with 2,3,4, . .. times more cells! For the interested reader, a book on multi grid methods is [33]. Present State of the Art in CFD An example
A quick impression will now be given of what can be done with CFD by looking at a standard flow problem. It con cerns the recent MSc work of Jeroen Wackers. From scratch, he developed 2D Euler software in which the grid is automatically adapted to the flow, and what follows de scribes one of his results. Consider the channel depicted in Figure 14, and in it a uniformly constant supersonic air flow (from left to right) at three times the speed of sound. One may consider the channel to be a stylized engine inlet of a supersonic aircraft. In fact it is just a benchmark geometry [34, 35]. The red vertical valve at the bottom of the chan nel instantaneously snaps up, so that, together with the red
Figure 1 3. Cross sections of a hexahedral grid around the Space SHuttle.
© 2006 Spnnger Science+ Business Med1a, Inc .. Volume 28, Number 1. 2006
13
'
' \_)
\ 7
flow speed
speed of sound
' ..J\
I
Figure 1 4. 20, parallel channel. In it, a parallel plate and a vertical valve which is still open.
horizontal plate, it forms a step which suddenly chokes part of the channel. Figure 15 shows a computational result. We see how the uniformly constant initial solution and the grid have de veloped after some time. The computational method highly satisfies the often conflicting requirements of numerical stability, accuracy, and monotonicity on the one hand, and computing and memory efficiency on the other [36].
also scientific journals dedicated to CFD. Moreover, off the-shelf CFD software can be purchased these days. Each issue of, e.g., the monthly Aerospace America contains col orful, full-page advertisements for CFD software. A practi cal overview of the CFD literature, software, and also va cancies can be found on the Web site of CFD Online [41]. Today, CFD's role is about as important as that of ex perimental fluid dynamics. And CFD continues to grow. It is fed by improvements in both computer science and nu merical mathematics. In addition, CFD itself stimulates re search in computer science and numerical mathematics: a fruitful interaction.
Books, journals, and software
Twenty years ago, textbooks on CFD were rare, but sev eral are available now (see, e.g., [37, 38, 39, 40]). There are
>
X
>
X Figure 15. Computational result some time after instanteously closing the lower part of the channel. Top: iso-lines of density. When the ver tical valve is still in the open position, the density in the entire channel is constant (everywhere the same blue color as at the inlet). Bottom: computational grid automatically adapted to flow solution.
14
THE MATHEMATICAL INTELLIGENCER
At present, CFD enters into full cooperation with other disciplines, such as structural mechanics (computational fluid-structure interactions) and electromagnetism (com putational magnetohydrodynamics).
[3] J. D . Anderson, A History of Aerodynamics, Cambridge University Press, Cambridge, 1997. [4] L. Euler, Principes gf!meraux du mouvement des fluides, Memoires de I'Academie des Sciences de Berlin, 11 (1755), pp. 274-31 5 . [5] M . D . Salas, Leonhard Euler and his contributions t o fluid me
Outlook
The fact that commercial CFD software is a success is proof of the practical importance of the theoretical fluid-dynam ics work since Euler. The growing availability of CFD soft ware may seem to be a threat for CFD research; CFD re searchers seem to make themselves redundant by their own success. Yet, this growing software availability may also be considered a good development. Not everyone has to write his/her own Euler or Navier-Stokes code. Coding such soft ware from scratch gives the best insight and is pleasing work, but it may easily take too much time.
chanics , AIM-paper 88-3564, AIM, Reston, VA, 1988. [6] G. G . Stokes, On the theories of the internal friction of fluids in mo
tion, and of the equilibrium and motion of elastic solids, Transac tions of the Cambridge Philosophical Society, 8 (1845), pp. 287. [7] C. L. M . H. Navier, Memoire sur les lois du mouvernent des f/uides , Memoires de I'Academie des Sciences, 6 (1822), pp. 389-440. [8] C. Reid, Courant in Gottingen and New York. The Story of an Im
probable Mathematician, Springer-Verlag, New York, 1976. [9] L. F. Richardson , Weather Prediction by Numencal Process, Cam bridge University Press, Cambridge, 1922. [10] R. Courant, K. 0. Friedrichs, and H . Lewy, Ober die partie/len Dif
ferenzgleichungen der mathematischen Physik, Mathematische Education
A new question arises: How to teach CFD, now that it has become more and more important as an easily available, au tomatic tool? Not just factual knowledge but also under standing of the mathematical and physical principles of CFD remains indispensable, not only when practicing it as a sci ence, but also when using it as a tool. The CFD-tool user must know and understand these principles well in (1) pos ing the computational problem, (2) choosing the numerical method to solve that problem, and (3) interpreting the com putational results. The user must know the possibilities and limitations of computational methods and should be able to assess whether the computational results obtained fulfill the expectations or not. If not, it should be found out why. Thus CFD is not solving flow problems by blind numerical force. On the contrary, stimulated by the growing potential of CFD, still more complicated flow problems will be considered, problems which will require even more knowledge and un derstanding of flow physics and numerical mathematics.
Annalen, 1 00 (1 928), pp. 32-74. [11] W. Aspray, John von Neumann and the Origins of Modern Com
puting, MIT Press, Cambridge, Massachusetts, 1990. [1 2] S. Ulam, John von Neumann, 1903-1957, Bulletin of the Ameri
can Mathematical Society, 64 (1958), pp. 1-49. [13] J. von Neumann and R. D. Richtmyer, A method for the numeri cal calculation of shocks, Journal of Applied Physics, 21 (1950), pp. 232-237. [14] D . van Dalen, L. E. J . Brouwer, 1 881-1966, Het Heldere Licht van
de Wiskunde, Bert Bakker, Amsterdam, 2002. [15] P. Nauer (ed .), Revised Report on the Algorithmic Language Algol
60
(available
for
download
from
http://www. masswerk.at/
algol60/report.htm). [16] A. van Wijngaarden et al., Revised Report on the Algorithmic Lan
guage Algol 68, Springer-Verlag , Berlin, 1 976. [17] http://www.abelprisen.no/en/. [18] P. D. Lax and R. D. Richtmyer, Survey of the stability of linear fi nite difference equations, Communications on Pure and Applied
Mathematics, 9 (1 956), pp. 267-293. [19] P. D . Lax, Weak solutions of nonlinear hyperbolic equations and
Research
their numerical computation, Communications on Pure and Ap
As
plied Mathematics, 7 (1954), pp. 159-1 93.
CFD becomes more and more mature, it also becomes more difficult to contribute fundamental research to it. In re cent decades a PhD student can hardly do such fundamental work as Godunov did. Students will have to acquire an ever growing knowledge and understanding of CFD before they can start working in it themselves. On the other hand, thanks to the availability of CFD tools, the possibilities for applica tion of CFD are far greater now than in Godunov's era. Just how CFD will develop remains unpredictable, and this is part of what makes it an exciting and attractive discipline. In CFD plenty of research questions remain. New fluid flow problems will continue to arise, and there will cer tainly be times when we may say with Orville Wright, "Isn't it astonishing that all these secrets have been preserved for so many years just so that we could discover them!"
[20] P. D. Lax and B. Wendroff, Systems of conservation laws, Commu
nications on Pure and Applied Mathematics, 13 (1960) , pp. 217-237. [21] P. D. Lax, Hyperbolic Systems of Conservation Laws and the Math
ematical Theory of Shock Waves, SIAM, Philadelphia, 1973. [22] A. Harten, P. D. Lax, and B. van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws,
SIAM Review, 25 (1 983), pp. 35-61. [23] S. K. Godunov, Finite difference method for the numerical com putation of discontinuous solutions of the equations of fluid dy namics, Mathemat1cheskfi"Sborn1k, 47 (1959), pp. 271-306. Trans lated from Russian at the Cornell Aeronautical Laboratory. [24] G. F. B. Riemann, O ber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite,
i n : Gesammelte Werke,
Leipzig,
1876. Reprint: Dover, New York, 1953. [25] G . F. Duivesteij n , Visual shock tube solver (to be downloaded from
REFERENCES
[1] T. von Karman, Aerodynamics, McGraw-Hill, New York, 1963. [2] H . Rouse and S. lnce, History of Hydraulics, Dover, New York, 1963.
http://www. piteon .ni/cfd/) . [26] B. van Leer, An Introduction to the article "Reminiscences about difference schemes", by S. K. Godunov, Journal of Computational
Physics, 153 (1999), pp. 1-5.
© 2006 Spn nger Science+Business Media,
Inc., Volume 28, Number
1, 2006
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[27] S. K. Gudunov, Reminiscences about difference schemes, Jour nal of Computational Physics, 1 53 (1 999), pp. 6-25.
AUTHOR
[28] B. van Leer, Towards the ultimate conseNative difference scheme. V. A second-order sequel to Godunov's method, Journal of Com
putational Physics, 32 (1 979), pp. 1 0 1 -1 36. Reprint: Journal of Computational Physics, 135 (1997}, pp. 229-248. [29] B. van Leer, Flux-vector splitting for the Euler equations, in Lec ture Notes in Physics, Vol. 170, Springer-Verlag , Berli n , 1 982, pp. 507-5 1 2 . [30] P . L. Roe, Approximate Riemann solvers, parameter vectors, and differences schemes, Journal of Computational Physics, 43 (1981) . pp. 357-372. BARRY KOREN
[31] S. Osher and F. Solomon, Upwind difference schemes for hyper
CWI
bolic systems of conseNation laws, Mathematics of Computation ,
P.O. Box 94079
38 (1 982), pp. 339-374
1 090 GB Amsterdam
[32] A Brandt Multi-level adaptive solutions to boundary-value prob ,
lems, Mathematics of Computation , 31 (1 977), pp. 333-390.
The Netherlands
[33] U. Trottenberg, C. W Oosterlee, and A Schuller, Multigrid, Aca
e-mail:
[email protected]
demic Press, New York, 2001 .
[34] A F. Emery, An evaluation of several differencing methods tor in
Barry Koren studied Aerospace Engineering at the Delft Insti
viscid fluid flow problems, Journal of Computational Physics, 2
tute of Technology, and Computational Fluid Dynamics at the
(1 968), pp. 306--331 .
Von Karman Institute for Fluid Dynamics in Belgium. He is now
[35] P. R . Woodward and P. Colella, The numerical simulation of two dimensional fluid flow with strong shocks, Journal of Computa
leader of the research group in Computing and Control at the Dutch Centre for Mathematics and Computer Science (CWI) in Amsterdam, and also professor of Computational Fluid Dy
tional Physics, 54 (1 984), pp. 1 1 5-1 73. [36] J. Wackers and B. Koren, A simple and efficient space-time adap tive grid technique for unsteady compressible flows, in Proceed
ings 1 6th AIM CFD Conference (CD-ROM), AIM-paper 2003-
namics at the Delft Institute of Techno l ogy . More information
can be found at http://homepages.cwi.nV-barry/. He is married and the father of three children.
3825, American Institute of Aeronautics and Astronautics. Reston , VA, 2003.
[37] P.
Wesseling,
Principles of Computational Fluid Dynamics.
Springer-Verlag, Berlin, 2001
tiona/ Methods for lnviscid and Viscous Flows, Wiley, Chichester, 1 988-1 990.
[38] P. J. Roache, Fundamentals of Computational Fluid Dynamics, Hermosa, Albuquerque. NM, 1 998.
[40] C. A J. Fletcher, Computational Techniques for Fluid Dynamics, Vol. 1 Fundamental and General Techniques, Vol. 2 Specific Tech
[39] Ch. Hirsch, Numerical Computation of Internal and External Flows. Vol. 1 Fundamentals of Numerical Discretization, Vol. 2 Computa-
niques for Different Flow Categories, Springer-Verlag, Berlin, 1 988. [41] http://www cfd-online.com
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16
THE MATHEMATICAL INTELLIGENCER
M a the m a tic a l l y B e n t
C o l i n Ad a m s , Editor
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18
THE MATHEMATICAL INTELLIGENCER
Dav i d E . R ow e , E d i t o r
Ancient Egyptian I Mathematics: New Perspectives on Old Sources
l
Pro captu lectoris habent suafata libelli (Terentianus Maurus)
f books, in general, have their own special fates-which depend on their readers-the same is true for the mathematical "books" from ancient Egypt. Indeed, modem editors and sub sequent readers have strongly influ enced the way we view them today. And even now, readers of the third mil lennium can alter the fate of these early texts by their careful (or careless) reading. 1 Sources and Early Historiography
Annette lmhausen
Send submissions to David E. Rowe, Fachbereich 1 7 - M athematik, Johannes Gutenberg University,
055099 Mainz, Germany.
For the past fifty years, the reputation of Egyptian mathematics has been rather poor. This has been due in part to the very limited number of available primary sources, particularly when compared with the vast collections of cuneiform mathematical texts pro duced in Mesopotamia. In ancient Egypt the production of mathematics (as well as literature) took place in cities. Then, as today, Egyptian cities were located along the Nile, and hence close to water. This circumstance has had significant consequences for con temporary Egyptological research. On the one hand, papyrus, the main writ ing material in this culture, was de pendent on absolute dryness for its preservation, a condition found in the Egyptian desert where most papyrus finds were made. However, in ancient Egyptian cities, where writings con cerned with the mundane affairs of daily life were discarded after use, this condition was usually not fulfilled. Therefore, most of the written evi dence documenting the role of mathe matics in Egyptian social, economic, and cultural life must be assumed lost forever. On the other hand, to the ex tent that such sources may still be re trievable some day, practical problems stand in the way. The locations of an cient Egyptian cities often coincide with those of modem urban centers.
This makes it next to impossible to ex cavate at a number of locations where extant remains might still be found. Among the few known (excavated) cities, the Middle Kingdom town of Lahun (also known as lllahun or Kahun) is exceptional, having yielded the rich est findings of Middle Kingdom papyri so far, among which incidentally are a number of mathematical fragments. 2 The two most significant sources, how ever, the famous Rhind and Moscow mathematical papyri, were bought on the antiquities market, making their prove nance uncertain. These and most of the other known mathematical sources were already published by 1930. The achievements of the earliest re searchers who studied these texts, es pecially those who worked during the first half of the twentieth century, were enormous. As editors, they managed to penetrate a foreign vocabulary of tech nical terms, which placed them in po sition to make a first attempt at un derstanding Egyptian mathematical methods. 3 As was common at that time, ancient sources and achieve ments were viewed and evaluated by means of direct comparison with mod em conventions and results. In many respects, it was found that Egyptian mathematics had little in common with the methods found in modem mathe matical textbooks. Nevertheless, with some effort the mathematical content of the ancient texts could be "decoded" and "translated" into modem mathe matics. Unfortunately, this type of reading often entailed a loss of the most strik ing characteristics of the original sources, a drawback that was little ap preciated at the time. Not surprisingly, the "achievements" of Egyptian math ematics, judged in terms of a different mathematical culture (from more than 3000 years later), looked rather crude and simple. One of the early leading au thorities on ancient mathematics was Otto Neugebauer, who wrote his dis sertation on Egyptian methods of cal-
© 2006 Springer Science+ Business Med1a. Inc., Volume 28, Number 1 , 2006
19
culating with fractions.4 Afterward, Neugebauer turned his attention away from Egyptian mathematics to study Mesopotamian mathematics and as tronomy, which he believed was a higher level of scientific achievement. As he once expressed this: Egypt provides us with the excep tional case of a highly sophisticated civilization which flourished for many centuries without making a sin gle contribution to the development of the exact sciences. [ . . . ] It is at this single center (Mesopotamia) that abstract mathematical thought first appeared, affecting, centuries later, neighbouring civilizations, and fi nally spreading like a contagious disease.5 It was surely in part due to the out standing quality of the early scholarly contributions that readers accepted so readily this kind of negative assess ment of Egyptian mathematics. As in dicated already, this situation was compounded by the lack of new source material which-had it been there would have required those capable of reading Egyptian texts to reflect upon the assessments of their predecessors. Thus, in the case of Mesopotamian mathematics, where new source mate rial is still being uncovered on an al most regular basis, readers' opinions have changed significantly over time. H Lacking this wealth of textual material, readers of the Egyptian texts seemed to have no basis for questioning the standard views of earlier experts like Neugebauer. Indeed, once the major Egyptian mathematical papyri became available in English or German trans lation, various historians of mathemat ics began contributing new ideas based on their own readings of these first translations. Often these involved mod em mathematical symbolism, leading to results that had almost nothing in com mon with the original source text. 7 This once common approach has now been recognized as both anachro nistic and misleading. Indeed, for the last 20 years historians of mathematics have started to take up and to rework the subject of ancient mathematics.8 It is now generally accepted that histori-
20
THE MATHEMATICAL INTELUGENCER
ans of mathematics cannot work on a source text without knowing the lan guage in which it is written or the cul tural background it comes from. At the same time, it has become obvious that mathematical knowledge is not uni versal. It is neither independent of the cultures in which it is produced and used, nor has it developed universally from basic beginnings to more and more advanced stages of knowledge. This dependence on cultural back ground begins already with number systems and number concepts, as has been demonstrated by various scholars working on ethnomathematicsY More advanced mathematical techniques and concepts have also been shown to be dependent on the culture that cre ated them. 10 Current Research
From this description of past research, it follows that the editions of Egyptian mathematical sources are by now outdated. It is to be hoped that new editions can be published before the current ones reach their centenary. Likewise, older studies of Egyptian mathematics, those written more than 30 years ago, must be read with cau tion, bearing in mind the kind of ap proach past researches typically took. For an up-to-date introduction to the subject, the reader should consult the articles by Jim Ritter. 11 In the follow ing sketch, I will attempt to give an overview of the state of current re search, illustrated with selected exam ples from the source material. Although there have been no spec tacular new finds of mathematical pa pyri, extant sources, including the much-studied Rhind and Moscow pa pyri, still offer many clues about the role of mathematics in Egyptian life. Alongside these, the Lahun mathemat ical fragments have just been re-edited, including several previously unpub lished fragments. 1 2 Other texts are still awaiting proper publication, such as the mathematical fragments of Papyrus Berlin 6619. The earlier publications from 1900 and 1902 only contain facsimiles of the two largest fragments. Moreover, the inter pretations of them then given are not without problems. n The Cairo wooden
boards are currently available in two very small and hardly legible photos with a discussion of some of their con tent. While a number of demotic math ematical texts have been published, no detailed study of Egyptian mathemat ics in the Graeco-Roman period is available yet. 14 Evidence from the Predynastic Period
Apart from the extant mathematical texts, however, there are further sources available throughout Egyptian history which inform us about aspects and uses of mathematics as it evolved in ancient Egypt in periods from which no mathematical texts are extant. Writ ten evidence exists from as early as around 3000 B.C., the oldest dating from shortly before the unification of Egypt. It comes from the tomb Uj at Abydos15 and consists of writing on pottery as well as on little tags of bone and ivory. These tags all reveal holes, suggesting they were probably once at tached to some perishable goods from this grave, thus indicating their prove nance and quantity. u; The quantities were rendered using elements and style familiar from the Egyptian num ber system in later times, i.e., a deci mal system without positional notation (see Figure 1). In this system, each power of 10 up to 1 million was repre sented by a different sign. In order to write any number, the respective signs, written as often as needed, were jux taposed in a symmetric way. Note that the hieroglyphic writing, which is what most people associated with ancient Egypt, was used mostly on stone mon uments. For daily life purposes, Egypt ian scribes wrote with a reed (dipped in ink) on papyrus or so-called ostraca (limestone or pottery shards). The
Figure 1 . Number representations on the tags from tomb Uj.
script used in this writing is more cur sive and abbreviated than hieroglyphic script. Several signs can be combined to form ligatures, whereas the writing itself can vary a great deal, depending on the individual scribe Gust like mod em handwriting).
'seqed' 7 p
Mathematics in the Old Kingdom
After the unification of Egypt under a single king (around 3000 B.C.), the Old Kingdom (OK; 2686-2160 B.C.) brought forth the first period of cultural bloom in Egyptian history. Extant architec tural remains, like the pyramids, as well as such artifacts as the scribal statues, demonstrate a high level of cultural attainment by this time. There can be little doubt that mathematical techniques lay at the heart of this de velopment as a significant tool for han dling organizational and administrative problems. To achieve something on the scale of the pyramids, mathematics was necessary not only for architec tural planning but also for the organi zation of labor. The scribal statues, which depict high officials from this period, demonstrate the importance of the administrative system. Despite this, there is practically no written evidence for mathematical practices extant from this time. Many of the monumental hi eroglyphic inscriptions are still ex tant-but these, of course, focus on eternity and tell us little about Egypt ian daily life and the affairs in which mathematics played an important part. Only very few papyri from this period have survived, some in a very frag mentary state. Nevertheless, there is other direct evidence of Egyptian mathematical techniques, for example from the plan ning and execution of building projects such as a mastaba from Meidum (see Figure 2). Around the comers of this mastaba, beneath the ground level, four 1-shaped mud-brick walls had been built. On these walls a series of diagrams can be found, which indicate the slope of the sides of the mastaba. This method of handling sloped sur faces points to the development of a concept which is well documented in the mathematical texts. 1 7 To express sloped surfaces, such as the sides of a pyramid, the Egyptians used the so-
a I m
s
Figure 2. Indication of a sloped surface at Meidum.
called sqd. This Egyptian term is de rived from the verb qd, meaning "to build." The sqd was used to measure the horizontal displacement of the sloped face for each vertical drop of one cubit, that is the length by which the sloped side had "moved" from the vertical at the height of one cubit. The sqd was always indicated in palms, and if necessary, digits. Although we have textual evidence for this concept only from the Middle Kingdom onward, sketches from the Old Kingdom indi cate that it was in use during this ear lier period. Note that the parallel lines drawn on the mud bricks are spaced at a distance of one cubit or seven palms. Furthermore there is early evidence for several metrological systems. While these units can also be found in later mathematical texts, their appearance in administrative papyri as well as in the inscriptions and depictions from tombs indicates that these systems go back at least to the Old Kingdom. Some of these systems changed over time, but the sources from the Old Kingdom suffice to trace these changes. Calculations with Unit Fractions
One of the most intriguing aspects of Egyptian mathematics concerns spe cial methods for calculating fractions, which were understood in ancient Egypt as inverses of integers. 1 8 Hence, the Egyptian notation for fractions did not consist of a numerator and de nominator, but rather a special symbol was used alongside an integer to des ignate the corresponding fraction. An exception was the fraction which had a special sign. The fractions and were also written by using spe-
%•
±
t. i•
cial signs (indicating that these may be older) rather than by using the general Egyptian notation. 1 9 In modem stud ies, Egyptian fractions are usually described as unit fractions, and it is often suggested that the Egyptians "restricted" themselves to calculations with fractions having a numerator of one.20 As explained in the paragraph above, however, this is a rather anachro nistic view. Moreover, seen from a mod em perspective, the Egyptian system inevitably appears awkward and un necessarily restrictive. One of the first to study Egyptian computations with fractions was Otto Neugebauer, who devised a notational system that parallels the Egyptian no tation. Fractions, as inverses of inte gers, are rendered by the value of the integer with an overbar: thus, would be written as as 6, etc. Th � exceptional fraction was rendered by Neugebauer as 3, whereas _1_, �. and _I_ 2 ,J 4 appeared as 2, etc. This notational system, which closely mirrors the Egyptian concept of fraction, has become the standard way of writing Egyptian frac tions in modem textbooks. Following this concept of fractions as inverses of integers, the next step consequently-was to express those parts that correspond to a multitude of inverses. This was done by (additive) juxtaposition of different inverses. Thus, � was written in the Egyptian sys4 - tern as 2 4, whereas a general fraction was given as a sum of different in verses written in descending order ac cording to their size. (Note that this no tation enables one to be as accurate as necessary by considering only ele ments up to a certain size.)
5, i %
© 2006 Springer Science+Business Media, Inc., Volume 28, Number 1, 2006
+.
21
Egyptian techniques of multiplica tion and division (see below for a more detailed description) frequently in volved the doubling of a number. This could be done very easily if the num ber to be doubled was an integer or the inverse of an even integer. However, to double an odd Egyptian fraction (when the result is supposed to be a series of different inverses only) can be quite difficult to accomplish. Consequently, it proved useful to prepare tables giv ing the results for doubling the inverses of odd numbers. These can be found in the so-called 2 7 N tables still extant in two sources: at the beginning of the Rhind Mathematical Papyrus (for odd N = 1 - 101) and in the Lahun frag ment UC 32159 (for odd N 1 - 21). Figure 3 shows the fragment UC 32159 in which the numbers are arranged in two columns. The first col umn shows (what we call) the divisor N, except for the first entry which shows both the dividend 2 and the di visor 3. This is followed by a second column that altematingly shows frac tions of the divisor and their value (as a series of inverses). Thus, the second line starts with the divisor 5 in the first column: it is 2 7 5 that shall be ex pressed as unit fractions. This � foJ lowed in the second column by 3, 1 3, 15, a.!ld 3. This has to be read � 3 of 5 is 1 3, and 15 of 5 is 3. Since 1 3 and 3 added equal 2, the series of unit frac tions needed to represent 2 7 5 is 3 15. =
The 2 7 N table in the Rhind papyrus shows the same arrangement of num bers; however, the solutions there are marked by the use of red ink. Obviously, the representation of 2 7 N as a series of unit fractions is not unique. However, the Egyptian 2 7 N Table uses for each N only one of the theoretically possible representations. Those we find in the Lahun fragment, for example, are identical to the ones found in the table of the Rhind papyrus. And whenever an odd fraction is dou bled within the mathematical texts, it is this same representation that we find used. This circumstance has fascinated a number of experts on additive number theory. In fact, there have been several attempts to crack the puzzle posed by the 2 7 N Table by finding the criteria that led the Egyptians to employ just these particular representations. Yet, while it is possible to describe some of the general tendencies-e. g. , represen tations with fewer elements are fa vored as are also representations with larger inverses, etc.-it has not been possible to establish strict mathemati cal rules that explain the choices the Egyptians mathematicians made. Rather than criticizing them for their lack of insight-or blaming them for not having followed strict rules that would comply with a different mathe matical concept of fractions devised by another culture several thousand years
2 3
3 2
5
3 1 3 15 3 - - - 4 1 2 4 28 4
7 9 11 13
19
10 1 2 30 2 - - - - - 12 1 3 12 51 3 69 4 - - - - 12 1 2 12 75 4 114 [6]
21
14 1 2 42 2
15 17
Figure 3. Fragment UC 321 59: 2 ogy, University College London).
22
THE MATHEMATICAL INTELLIGENCER
.;-
6 1 2 18 2 - - - 6 1 3 6 66 6 - - - - - 8 1 2 8 52 4 104 8
N table (Copyright Petrie Museum of Egyptian Archaeol-
later-it seems more appropriate to recognize that mathematics is, indeed, culturally dependent; our modem point of view may not afford us the best picture of past achievements. Thus, in stead of trying to concoct an explana tion of the Egyptian solutions by using modem mathematics, it may be more rewarding simply to "accept" the Egyptian table and examine its use and usefulness within the mathematical en vironment that employed it. Mathematical Problem Texts from the Middle Kingdom
Apart from tables, the mathematical texts also include special procedures articulated within problem texts. As these names indicate, such texts set out a problem and then give instruc tions showing how to solve it. Proce dure texts derive from an educational setting. They may have been written by a teacher, who was compiling a hand book, or perhaps by a student engaged in practicing mathematical techniques. An appreciation of this context is im portant for understanding these texts, which were intended to prepare scribes for the mathematical tasks they would later have to execute as part of their daily work. 21 Given that these texts were written for this type of mathematical education, it should not be expected that we can learn how the Egyptians developed their mathemati cal knowledge from sources of this na ture. The extant hieratic mathematical texts contain roughly one hundred problems. Furthermore, in the largest of these texts, the Rhind Mathematical Papyrus (see Figure 4), we can discern an arrangement of these problems ac cording to their rising level of diffi culty. This is not to be judged by purely mathematical aspects alone but also by additional knowledge (often from a practical background) which is neces sary to solve the problems. This can be seen, for example, in pRhind, problems 31-34 and those immediately follow ing, problems 35-38. Mathematically, both groups teach a procedure for de termining an "unknown" number if its sum with fractions of itself is given. The procedure for solving the prob-
lems in both groups is roughly the same. However, in the second group (pRhind, problems 35-38), the "un known" number is not an abstract num ber but a quantity of grain. Therefore the result, which is determined in the same way as in the preceding prob lems, needs to be transformed after wards into the respective metrological units.22 The style of Egyptian mathematical problem texts can best be appreciated by looking at an actual example, like problem 56 of the Rhind Mathematical Papyrus: Method of calculating a pyramid, 360 is its base, 250 is its height. You shall let me know its inclination. You calculate half of 360. It results as 180. You divide 180 by 250. 2 5 50 of a cubit results. 1 cubit is 7 palms. 23 You multiply with 7. \
\
\
2 5 50
7 32 1 3 15 10 25 1ts inclination: 5 25 palms
Problem 56, like the other four ex amples of pyramid problems found in the Rhind Papyrus (nos. 57, 58, 59, and 59b), teaches the relation between the base, height, and inclination of the sides. This example complements the OK sketch found on the walls around the mastaba with sloping sides, which was discussed above. In fact, the tech nical term sqd-the number of palms the slope of a slanted plane recedes per vertical difference of one cubit-is ex plicitly indicated in the problem text. Thus, the base, height, and inclination of a pyramid are linked by the relation: inclination
=
7 palms X
1 / base 2 height
The problem above presents a pyramid with base (360) and height (250); its in clination is to be calculated. The pro cedure calls for calculating half of the base and dividing this by the height. The result is then multiplied by 7 to ob tain the inclination in palms. Having grasped "what is going on" in this prob lem, let us now take a second, closer look at the Egyptian text and its means of structure. The text begins-as is typical for mathematical problem texts-with a title "Method of calculating a pyra mid." Note that the beginning of the ti tle is written in red ink (rendered in my translation in bold). This use of red ink helps the reader recognize at a glance the beginnings of individual problems. The title of mathematical problems is very often given as "Method of . . . " fol lowed by a key word which indicates the type of problem. In our example, the key word is the Egyptian mr, "pyramid." After this title, the given data are in troduced, and they are always specific numerical values. This statement of the data is generally followed by a question or command, outlining the problem that the scribe shall solve. In this ex ample: "You shall let me know its in clination." Next, we see a sequence of instructions, followed by intermediate results. This procedure then leads to the numerical solution of the problem. Each instruction usually consists of one arithmetic operation. The Egyptian mathematical language distinguishes addition, subtraction, multiplication, division, halving, inverting, squaring, and the extraction of square roots. These individual mathematical opera tions are expressed without any use of mathematical symbols. The instruc tions themselves are always given as complete sentences. Furthermore, in this part of the text, a special verb form is used, the so called sd.m.IJr=f. The name consists
Figure 4. Rhind Mathematical Papyrus, No. 56 Copyright The British Museum.
©
of the Egyptian verb "to hear" (sd.m), which is used in Egyptian grammars to demonstrate different conjugations, its characteristic morphological element (IJr) and the suffix pronoun of the third person singular (f). Its function is to express a "general truth" which results as a necessary sequence from previ ously stated conditions.24 In the math ematical texts, the sd.m.IJr=f is used for both instructions and announcing intermediate results. As for the latter, the verb form expresses "mathematical facts"-if 2 and 2 are added, the result will necessarily be 4. The use of the sd.m.IJr=f in the instructions under lines the specific procedural character of the text: the sequence of instruc tions necessarily has to be followed to solve the problem. The last i�s.!_ruction given, the multiplication of (2 5 50) by 7 is followed by a scheme of numbers. This carries out the actual multiplica tion in the Egyptian manner, which may now be described. Multiplication (and division) are ex ecuted following a scheme that uses two columns of numbers. 25 Each mul tiplication begins with the initialization which is found in the first line of the scheme: a dot is placed in the first col umn and the number to be multiplied in the second column. The multiplica tion is carried out by subsequent oper ations in both columns using a variety of techniques, depending on the nu merical values of the numbers that shall be multiplied. The aim is to find the multiplier as a combination of en tries in lines of the first column. The respective lines of the second column will then be the result of the multipli cation. Problem 56 of the Rhind Papyrus shows the notation used to compute 7 X 2 5 50. The initialization is followed by three more lines, each of which in dicates o�e !!_f the required fractional _ . parts (2, 5, 50) of the multiplier. How the individual entires of the second col umn were found is not obvious. It is possible that there may have been ta bles for fractional parts of 7, as this was a number that leads to compli cated calculations, but which came up frequently due to the metrological con ventions.26
2006 Springer Science+Business Media,
Inc., Volume
28,
Number I,
2006
23
Finally, the result of the problem is announced. Next to the text of the problem there is a sketch indicating characteristic measurements for this problem, i.e., the values of base and height (see Figure 5). This step-by-step layout can be found in virtually all Egyptian problem texts. This being the case, one can easily see that the formal aspect of phrasing mathematics in the form of procedures will be completely lost if a problem is "translated" into a modem algebraic equation (in this case: inclination = (� base/height) X 7 palms). While this formula has the ad vantage of informing a modem reader at a single glance how an ancient measure was defined, it conveys nothing what soever about the procedural character of Egyptian mathematics. Moreover, al gebraic formulae played no part in Egyptian mathematics so that the above formulation for the sqd is anachronistic, at best, as it is foreign to the methods actually found in Egyptian texts. Analyzing Egyptian Problem Texts
As it happens, a closer analysis of the problem texts reveals many hitherto unnoticed methodological features of Egyptian mathematics. Indeed, the procedural format can be used as a key to analyze not only individual problems but also various types of problems as found in the mathematical papyri. To get beyond a superficial understanding of Egyptian mathematics, however, a method was needed that enabled a reader to analyze and compare the Egyptian procedures. Such an ap proach was first proposed by Jim Rit ter.27 In my dissertation I have adapted this method to analyze the various pro cedures used in all hieratic mathemat ical problems. 28 The analysis of a specific problem text can be carried out by rewriting it in two stages. In the first, one keeps the numerical values indicated in the source text but rewrites the instruc tions by replacing the rhetoric for mulations with modem symbols that indicate the respective arithmetic op erations. The data are noted at the be ginning of the scheme by their numer ical values. Thus, for the example cited above (pRhind, problem 56), the text would be rewritten as follows:
24
THE MATHEMATICAL INTELLIGENCER
Figure 5. Sketch at the end of Rhind Mathe· matical Papyrus, No. 56.
Method of calculating a pyramid, 360 360 is its base, 250 250 is its height. You shall let me know its inclination. You calculate half of 360. (1) 2 X 360 = 180 It results as 180. You divide 180 by 250. (2) 180 -7- 250 2 5 50 of a cubit results. 2 5 50 1 cubit is 7 palms. You multiply with 7. (3) 2 5 50 X 7 =
The result allows one to see at a glance whether the arithmetic operations to be carried out were simplified by the choice of data. For example, in prob lem 43 of the Rhind papyrus, the cal culation of the volume of a granary with circular base, the diameter of the granary is given as 9. This greatly fa cilitates the calculational procedure, the first step of which is to determine � of the diameter. In the values of our 9 problem, the given data were 360 and 250. While the first step, halving 360, is fairly straightforward, the second, the division of the result of the first step by the second datum results in a frac tion of three parts, which then has to be multiplied by 7. Thus, by compari son, the data in problems 58 and 59 re sult in easier calculations. This first stage of rewriting is espe cially helpful when dealing with a cor rupt text, as the modem reader is forced to follow the source text and identify the procedure in a step-by-step fashion. It then becomes immediately apparent where specific difficulties arise in the source. To further analyze the text so as to reveal how its procedures are related
to those used in other problems, it is necessary to distinguish between dif ferent types of numbers that can ap pear throughout the procedure. The first numbers a reader encounters are the data of the given problem. From the second instruction on, three types of numbers are possible: data, intermedi ate results, and constants. To distin guish these, and also to get a clearer view of the structure of the procedure, a second stage of rewriting is required. In this stage the data are indicated by symbols D;, whereas intermediate re sults are specified by a number in paren theses (x) which specifies the step in the procedure that leads to the given re sult. The only actual numbers that now appear in the rewritten text are con stants. Thus, for our example, the result of this second rewriting is as follows: Method of calculating a pyramid, D1 360 is its base, D2 250 is its height. You shall let me know its inclination. You calculate half of 360. (1) 2 X D l It results as 180. You divide 180 by 250. (2) (1) -7- D2 2 5 50 of a cubit results. 1 cubit is 7 palms. You multiply with 7. (3) (2) X 7 In my dissertation I have analyzed the procedures of all hieratic math ematical problems by rewriting the procedure in the form of a symbolic algorithm. This makes it possible to compare the various procedures used and analyze their respective complex ity. The analysis of problems by means of their procedures or algorithms thus constitutes a powerful tool for com paring the structure of individual math ematical problem texts. From this, one can learn a great deal about Egyptian mathematical techniques. Within the Rhind Mathematical Papyrus, for exam ple, one fmds groups of problems with similar procedures (pRhind, No. 24-27), as well as a progression within one group from basic procedures to more elaborate ones (pRhind, No. 69-78). Identifying an unambiguous sym bolic algorithm can sometimes be
straightforward,
example
thority (state or temple). A quantity of
documents that prove the use of mathe
above. Unfortunately this is not the case
as
in
the
grain is taken from a granary and then
matical techniques. Thus Michel Guille
with all problems. Individual instruc
given to workers who produce bread
mot has used a ration text from Kahun
tions may be missing-sometimes they
and/or beer from it. Obviously it was
to
are replaced by a written calculation, or
necessary to ascertain the quantity-in
These can be linked to techniques taught
several steps are summarized in one in
loaves of bread or vessels of beer-of
in mathematical papyri.31 It is to be
struction only. These types of difficul
a given quality (in this case measured
hoped that this example can be followed
ties can sometimes be overcome by tak
by grain content) that was equivalent
for other texts as well.
ing into account all of the available
to the amount of grain initially given to
The most promising sources still to
source material. If-as in the Rhind Pa
the workers. The mathematical side of
be explored in this respect are the Reis
pyrus-several problems of the same
this control is represented by the bread
ner Papyri. This set of four papyrus
kind are available and their procedures
and beer problems. 29 The terminology
rolls
are identical insofar as they are explic
used in these problems is taken from
building of a sanctuary, including ra
itly stated, then those problems which
the respective technological language.
tion tables,
lack certain instructions can occasion
Thus the bread and beer problems
tions, as well as the administration of
ally be reconstructed by means of the
evolve around the psw, a unit which
workshops. 32 They not only enlarge the
more detailed problems.
measures how many loaves of bread
I would like to stress in this context that both types of rewriting are merely tools for analyzing specific aspects of
have been made from one
1}1;,3.t
of
grain. Apart from the psw, there are two additional standard phrases indicating
analyze
mathematical
contains
practices. 30
calculations
for
the
actual building calcula
meagre set of seven problems related to architecture which are known from the Moscow (problem (problems
14) and Rhind 56-60) papyri, but they also
the procedures found in the problem
the use of specific kinds of grain prod
demonstrate that the amount of work
texts, whereas the source texts them
ucts and their quantities. Obviously, this
done was linked to a specific number
selves remain central and should never
has further consequences for the re
of workers (and rations) per day.
be neglected in any analysis. Taking the
spective calculations. Similar observa
three versions of the procedure to
tions can be made for other groups of
Evidence of Mathematics
gether, however, enables one to form a
practical problems as well. These gen
in the New Kingdom
more complete analysis that includes
erally involve not only the "basic" math
While the mathematical texts date al
not only the various procedures but also
ematical terminology but also further
most exclusively from the Middle King
technical mathematical vocabulary, as
knowledge related to the technological
dom, other sources are available from
well as the relation of drawings and cal
or administrative background. This usu
all periods of Egyptian history. The
culations carried out in writing con
ally makes them not only more difficult
Wilbour Papyrus, a text from the New
nected with the procedure, and others.
to understand but also less likely to be
Kingdom, is an official record of mea
"mirrored" by a familiar problem in
surements and assessments of fields
Mathematics within the Context
modem mathematics. Thus, early his
over a distance of
of Egyptian Culture
torical research often neglected this
Nile. The fields are given by localization
Another integral part of the reassess
area of Egyptian mathematics.
90
miles along the
and acreage, their assessments referring
ment of Egyptian mathematics concerns
However, as is obvious from the or
its role within Egyptian culture. Mathe
dering of the problems found in the
to taxes specified in amounts of grain. Another major opportunity to find
matics was one of the key elements of
Rhind Papyrus, it was precisely these
relevant sources of mathematics for
scribal training in pharaonic Egypt. It
practical problems that were consid
the New Kingdom is provided by the
provided the scribes with a crucial tool
ered more advanced. After all, the aim
excavation of Deir el Medina. Deir el
they needed to fulfil their administrative
of the mathematical handbooks was to
Medina is the modem name of an an
tasks as well as to plan and carry out
prepare scribes for their daily admin
cient Egyptian village on the West
istrative work. So if we want to obtain
Bank of the Nile opposite Luxor. The
many of the mathematical problems
insights into Egyptian mathematics, we
village was inhabited by workmen who
they dealt with were related to practical
must consider these problems and try
were responsible for the construction
matters, e.g., the distribution of rations,
to understand them. The setting of the
and decoration of the tombs in the Val
the volume of granaries, or the amount
individual problem may help to point
ley of the Kings. Deir el Medina has
of produce to be delivered by a worker.
to further sources (not only textual)
yielded a huge quantity of artifacts and
Our
texts relating to daily life in the New
construction
projects.
Consequently,
mathematical
which may be useful to understand the
problems of this kind is at least partially
additional terminology and practice.
Kingdom-similar to the findings at
dependent on our appreciation for these
Furthermore, it is this type of problem
Lahun for the Middle Kingdom. Among
larger contexts.
that indicates other possibilities of
the sources are ration texts, building
gaining
Egyptian
plans, as well as texts for the educa tion of scribes. The ostracon in Figure
understanding
of
This can be demonstrated with the
knowledge
about
so-called bread and beer problems,
mathematics apart from the restricted
which appear against the background
corpus of mathematical texts. The ac
6
of economic activity, baking and brew
tual output of the scribes in doing their
the
ing, under the control of a local au-
daily work provides us with numerous
numbers. It shows in the first column
shows a fragment of an exercise in multiplicative
writing
of
large
© 2006 Springer Science+Bus1ness Media, Inc., Volume 28, Number I , 2006
25
ments d'histoire des sciences: 39-6 1 , Paris: Bordas 1 989: James Ritter, "Egyp tian Mathematics," in: Helaine Selin (ed.),
Mathematics Across Cultures: The History of Non- Western Mathematics : 1 1 5- 1 36, Dordrecht, Bosto n , London: Kluwer 2000, as well as Annette lmhausen, Agyptische
Algorithmen: Eine Untersuchung zu den mittelagyptischen
mathematischen
Auf
gabentexten , Wiesbaden: Otto Harras sowitz 2003. For Greek Mathematics, cf.
Figure 6. Deir el Medina: Remains and Ostracon with Number Exercise.
Serafina Cuomo, Ancient Mathematics , London,
(on the right) the numbers 600,000, 700,000, and 800,000 and in the second (middle) column the numbers 5,000,000, 6,000,000, and 7,000,000 written by the sign for the number 100,000 (or 1,000,000) with the respec tive multiplicative factors (6, 7, and 8 and 5, 6, and 7) below. The third col umn (left) shows again the sign for 1,000,000 and two illegible signs below.
New
York:
Routledge
200 1 ,
for permission to include photographs of
Michael N.
sources.
Apollonius of Perga 's Conica. Text, Con
2. See Annette lmhausen and Jim Ritter,
Fried and Sabetai Unguru,
text, Subtext, Leiden: Brill 2001 , as well as
"Mathematical Papyri , " in: Mark Collier and
David Fowler, The Mathematics of Plato's
Stephen Quirke (eds.), The UCL Lahun Pa
Academy: A New Reconstruction (Second
pyri: Religious, Literary, Legal, Mathematical
Edition), Oxford: Clarendon Press 1 999,
and Medical, Oxford : Arcaheopress 2004.
and Reviel Netz, The Shaping of Deduc
3. Among the early editions, the most note
tion in Greek Mathematics: A Study of
5 1 ),
worthy are still Thomas E. Peel, The Rhind
Cognitive History (Ideas in Context
Mathematical Papyrus. British Museum
Cambridge: Cambridge University Press
10057 and 1 0058, London: Hodder and
1999. For Mesopotamian mathematics,
Conclusions
Stoughton 1 923, and Wasili W. Struve,
see most recently Jens H0yrup, Lengths,
Although Egyptian mathematics will probably never have the vast number of sources that still can be found in other cultures like India or Mesopotamia, there is more available than has been used so far. 33 The analysis of all the available mathematical texts, taken along with the additional material from administrative economic and literary contexts related to Egyptian mathe matics, is certain to provide a better foundation for understanding its role within Egyptian culture. This inte grated approach represents an impor tant advance beyond the early studies that relied exclusively on an internal analysis of a small corpus of mathe matical texts, which served for several decades as the sole basis for assessing nearly three millennia of mathematical life in ancient Egypt. By carefully rereading these classical mathematical texts while according the new sources a serious first reading, we may antici pate that the fate of Egyptian mathe matics faces an exciting future.
Mathematischer Papyrus des Staatlichen
Widths, Surfaces. A Portrait of Old Baby
Museums der Sch6nen Kunste in Moskau
lonian Algebra and its Kin , New York:
NOTES
1 . I thank David Rowe for his comments on
26
(Quellen und Studien zur Geschichte der
Springer
Mathematik, Abteilung A: Quellen, Vol. 1 ) ,
Mesopotamian Mathematics, 2 1 00-1 600
2002,
and
Eleanor
Robson,
BC: Technical Constants in Bureaucracy
Heidelberg: Springer 1 930. 4 . Otto Neugebauer, Die Grundlagen der
agyptischen Bruchrechnung, Berlin: Julius
and
Education
(Oxford
Editions
of
Cuneiform Texts XIV), Oxford: Clarendon Press 1 999.
Springer 1 926.
5 . Otto Neugebauer, A History of Ancient
9 . See Gary Urton, The Social Life of Numbers.
Mathematical Astronomy (Part Two). Berlin,
A Quechua Ontology of Numbers and Phi
Heidelberg, New York: Springer 1 975: 559.
losophy of Arithmetic, Austin , Texas: Uni
6. See for example the interpretations of
versity of Texas Press 1 997, and Marcia As
An
Plimpton 322, e.g., compare Joran Friberg ,
cher,
"Methods and traditions of Babylonian
Exploration of Ideas across Cultures, Prince
mathematics: Plimpton 322 , Pythagorean
Mathematics
Elsewhere.
ton, N . J . : Princeton University Press 2002.
triples and the Babylonian triangle param
1 0. See, for example, for Mesopotamia Jens
eter equations , " Historia Mathematica 8
H0yrup, Lengths, Widths, Surfaces. A Por
(1 98 1 ): 277-318 and the recent reassess
trait of Old Babylonian Algebra and its Kin ,
ment by Eleanor Robson (Eleanor Robson,
New York: Springer 2002.
"Neither Sherlock Hol mes nor Babylon : a
1 1 See note 8.
reassessment of Plimpton 322 , " Historia
1 2 . See Annette lmhausen and Jim Ritter,
1 67-206 and
"Mathematical Papyri , " in: Mark Collier and
Eleanor Robson, "Words and pictures: new
Stephen Quirke (eds ) , The UCL Lahun Pa
light on Plimpton 322 , " American Mathe
pyri: Religious, Literary, Legal, Mathematical
Mathematica 28 (200 1 ) ;
matical Monthly 1 09 (2002): 1 05-1 20).
and Medical, Oxford: Arcaheopress 2004.
7. An extreme example of this is Richard
Another mathematical fragment will be pub
Gillings, "The Volume of a Truncated Pyra
lished in the next volume of that series.
mid in Ancient Egyptian Papyri , " The Math
1 3 . See Oleg Berlev, "Review of William Kelly
ematics Teacher 5 7 (1964): 552-555.
Simpson: Papyrus Reisner Ill: The Records
previous versions of this article and for his
8. For Egyptian mathematics, see for exam
of a Building Project in the Early Twelfth
corrections of my English. I also thank
ple James Ritter, "Chacun sa verite: les
Dynasty, Boston : Museum of Fine Arts
Richard Parkinson of the British Museum
mathematiques
and Stephen Quirke of the Petrie Museum
sopotamie , " i n : Michel Serres (ed .),
THE MATHEMATICAL INTELLIGENCER
en
Egypte
et
en
Me
1 969," Bibliotheca Orienta/is 28 (1 97 1 ):
Ele-
324-326, esp. p. 325.
1 4 . Richard Parker, "A Demotic Mathematical
one exception)." (Richard J. Gillings, Math
Mathematical Texts and their Contexts, "
Papyrus Fragment," Journal of Near East
ematics in the Time of the Pharaohs, Cam
Science in Context 1 6, 2003: 367-389.
ern Studies 1 8 (1 959): 275-279; Richard Parker,
bridge, Mass . : MIT Press 1 972, p. 20).
30. Michel Guillemot, "Les notations et les pra tiques operatoires permettent-elles de par
Mathematical Papyri,
2 1 . See Jim Ritter, "Egyptian Mathematics , " in:
Providence, R . I . : Brown University Press
Helaine Selin (ed.), Mathematics across
ler de 'fractions egyptiennes'?", in:
1 972; Richard Parker, "A Mathematical Ex
Cultures.
The History of Non- Western
Benoit, Karina Chemla, Jim Ritter (eds.), His
ercise-P. Dem. Heidelberg 663 , " Journal
Mathematics, Dordrecht, Boston, London:
loire de fractions, fractions d'histoire, Basel,
of
Demotic
Egyptian
Archaeology
61
(1 975):
1 89-1 96. A list of Demotic mathematical
Boston, Berlin: Birkhauser 1 992: 53-69.
Kluwer 2000, p. 1 20. 22. For a discussion of the use of an abstract
ostraca can be found in Jim R itter, "Egypt
number
ian Mathematics , " in: Helaine Selin (ed.),
metrological
system
Paul
and
conversions
systems,
see
Jim
into
Ritter,
3 1 . Annette lmhausen, "Calculating the Daily Bread: Rations in Theory and Practice,"
Historia Mathernatica 30 (2003): 3-1 6.
Mathematics across Cultures. The History
"Egyptian Mathematics," in: Helaine Selin
32. A first attempt to analyze the mathemati
of Non- Western Mathematics , Dordrecht:
(ed.), Mathematics across Cultures. The
cal content of some parts of the Reisner
Kluwer 2000: 1 34 , note 27.
History
1 5 . See Gunter Dreyer, Umm ei-Qaab I. Das
pradynastische Konigsgrab U-j und seine fruhen Schriftzeugnisse, Mainz: Von Zabern
of Non- Western
1 6 . For a discussion of the inscriptions on these
Papyri
has been made by Richard J .
Dordrecht, Boston, London: Kluwer 2000,
Gillings, Mathematics in the Time of the
pp, 1 2 1 - 1 22.
Pharaohs, Cambridge, Mass. MIT Press
23. The cubit was the Egyptian standard mea sure of length.
1 998
Mathematics,
1
cubit consisted of 7
1 972, pp. 2 1 8-23 1 . 33. A variety of architectural sources (with mathematical implications) can be found in
palms; each palm, of 4 digits.
tags, see Gunter Dreyer, Umm ei-Qaab I.
24. An English example for its use would be
Corinna Rossi, Architecture and Mathe
Das pradynastische Konigsgrab U-j und
the statement "If I have a stone in my hand,
matics in Ancient Egypt, Cambridge: Cam
seine fruhen Schriftzeugnisse, Mainz: Von
and let it drop, then the stone falls to the
bridge University Press 2004.
Zabern
John
groun d . " The last part of this statement
Baines, "The Earliest Egyptian Writing: De
"then the stone falls to the ground" is
velopment, Context, Purpose , " in: Stephen
where the
1 998,
pp.
1 37-145,
and
D. Houston, The First Writing. Scrip t Inven
tion as History and Process, Cambridge: Cambridge University Press 2004: 1 50-1 89. 1 7 . See problems 56-60 of the Rhind Mathe
ian text. 25. Examples
sqm.l)r==f is can
be
used in an Egypt
found
in
Annette
l rnhausen and Jim Ritter, "Mathematical Fragments: UC321 1 4, UC321 1 8, UC321 34 , UC321 59-UC32 1 62 , " in: Mark Collier and
matical Papyrus. "Mathematics in Egypt, " in:
Stephen Quirke, The UCL Lahun Papyri:
Helaine Selin (ed.), Encyclopedia of the
Religious, Literary, Legal, Mathematical
History of Science, Technology and Med
and Medical (British Archaeological Re
icine in Non-Western Cultures, Dordrecht,
ports I nternational Series 1 209): 7 1 -96,
1 8. Jim
Ritter,
Boston, London: Kluwer 1 99 7 , p. 631 . 1 9. For the prehistory of Egyptian fractions and
Oxford; Archaeopress 2004, esp. pp. 85-86. 26. The New Kingdom Ostracon Senmut 1 53
+ in this
their development see Jim Ritter, "Metrol
may be interpreted as a table of
ogy and the Prehistory of Fractions," in:
way, see David Fowler, The Mathematics
Paul Benoit, Karine Chernla, Jim Ritter
of Plato's Academy: A New Reconstruc
(eds.), Histoire de fractions, fractions d'his
tion (Second Edition), Oxford: Clarendon
toire:
1 9-34,
AUTHOR
Basel,
Boston ,
Berlin:
Birkhauser 1 992.
ANNETTE I MHAUSEN
Trinity Hall Cambridge University Trinity Lane, Cambridge CB2 1 TJ
UK
e-mail:
[email protected]
Press 1 999, p. 269. 27. Jim Ritter, "Chacun sa verite: les rnathe
Annette lmhausen studied mathe
20. See, for example, the description of Cou
rnathiques en Egypte et en Mesopotamie,"
matics, history of mathematics, and
choud: " . . . il ne semble avoir connu que
in: Michel Serres (ed.), Elements d'histoire
les fractions unitaires, c'est a dire celles
Egyptology and received her PhD
des sciences: 39-6 1 , Paris: Bordas 1 989
from Mainz University (Germany) un
dans lesquelles le numerateur est toujours
(English edition: Jim Ritter, "Measure for
der David E. Rowe. She has held fel
equivalent a !'unite, . . . " (Sylvia Couchoud,
Measure:
and
lowships at the Dibner Institute for the
Mathematiques Egyptiennes. Recherches
Mesopotamia," in: Michel Serres (ed .), A
History of Science and Technology
sur les connaissances mathematiques de
History of Scientific Thought. Elements of
(Cambridge, Mass.) and at Trinity Hall
I 'Egypte pharaonique, Paris: Le Leopard
a History of Science: 44-72, Oxford:
(Cambridge University, England). She
Blackwell 1 995).
is currently completing a book that
d'Or 1 993, p. 2 1 ) or that of Gillings: "When
Mathematics
in
Egypt
the Egyptian scribe needed to compute
28. Annette lrnhausen, A gyptische Algorith
outlines the historical development of
with fractions he was confronted with
men. Eine Untersuchung zu den mittel
ancient Egyptian mathematics, situ
many difficulties arising from the restriction
agyptischen mathematischen Aufgaben
ating it within the larger social, eco
of his notation. His method of writing num
texten (Agyptologische Abhandlungen 65).
nomic, and cultural background. In
bers did not allow him to write such sim
Wiesbaden: Otto Harrassowitz 2003.
her spare time she enjoys running 1 /2
ple fractions as
% or % because all fractions
had to have unity for their numerators (with
29. For a detailed discussion of these prob lems see Annette lmhausen,
and 2/3 marathons.
"Egyptian
© 2006 Springer Science t Business Media. Inc., Volume 28. Number 1, 2006
27
M?tffi i •i§u6hl%1i@i§4fii.J t§.id ..
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, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.
M i c h ael Kleber a n d Ravi Vaki l , Ed itors
The Locker Puzzle Eugene Curtin and Max Warshauer
S
Please send all submissions to the Mathematical Entertainments Editor, Ravi Vakil, Stanford University, Department of Mathematics, Bldg. 380, Stanford , CA 94305-21 25, USA e-mail:
[email protected] .edu
28
uppose I take the wallets from you and ninety-nine of your closest friends. We play the following game with them: I randomly place the wal lets inside one hundred lockers in a locker room, one wallet in each locker, and then I let you and your friends in side, one at a time. Each of you is al lowed to open and look inside of up to fifty of the lockers. You may inspect the wallets you find there, even check ing the driver's license to see whose it is, in an attempt to find your wallet. Whether you succeed or not, you leave all hundred wallets exactly where you found them, and leave all hundred lockers closed, just as they were when you entered the room. You exit through a different door, and never communi cate in any way with the other people waiting to enter the room. Your team of 100 players wins only if every team member finds his or her own wallet. If you discuss your strategy beforehand, can you win with a probability that isn't vanishingly small? We develop a more mathematical formulation to facilitate a precise dis cussion of the problem. This consists of numbering our players, and replac ing wallets by player numbers! Our game is played between a single Player A against a Team B with 100 members, B 1 , B2, . . . , B 1 00. Player A places the numbers 1, 2, . . . , 100 randomly in lock ers 1, 2, . . . , 100 with one number per locker. The members of Team B are ad mitted to the locker room one at a time. Each team member is allowed to open and examine the contents of exactly 50 lockers. Team B wins if every team member discovers the locker contain ing his own number. Team B is allowed
THE MATHEMATICAL INTELLIGENCER © 2006 Spnnger Sc1ence+Bus1ness Med1a. Inc.
an initial strategy meeting. No com munication is allowed after the initial meeting, and each team member must leave the locker room exactly as he found it. It is important to realize that the solution does not involve some trick to pass information from one player to another. We could equally well make 100 copies of the room and make an identical distribution of num bers into lockers for each room, then ask the members of Team B to perform their searches simultaneously, with one person per room. Each individual will succeed in find ing his own number with probability 1/2. If they act independently, they must get lucky 100 times in a row, and the team will win with probability only e/z) 100. Team B needs some help! Amaz ingly there is a strategy which gives sig nificant probability of success for Team B. Even if we give the problem with 2n players on Team B each of whom can examine n out of 2n lock ers, Team B can apply the strategy to succeed with probability over 30% regardless of how large a value we take for n. Your problem is to find this strategy. Searching For Ideas
Let's play with some ideas using a more manageable number of players. To be as concrete as possible, let's switch to the case of 10 players on Team B, each of whom can examine 5 out of 10 lock ers. Here random guessing by each player is already somewhat hopeless and succeeds with probability (l/z)10 1�24 . A first try to improve the probability of success is to search for a clever way to assign a set of lockers for each person to examine. Certainly we can improve over random guessing in this manner. For example if team members 1-5 ex amined lockers 1-5, and team mem bers 6-10 examined lockers 6-10, they would succeed provided numbers 1-5 are placed in lockers 1-5. Number 1 is placed somewhere in the first 5 lock ers with probability 5/10, then given
=
that number 1 is so placed, number 2 is also in the first 5 with probability 4/9 and so on. Following this plan, Team B will succeed with probability
5 4 3 2 1_ 1 242 "
10 9 8 7 6 -
While this is an improvement over ran dom guessing, it still leaves Team B with slim chances. Although the scheme fails, it is worth noticing that if B1 finds his number in this scheme, then B6 will find his number with prob ability 5/9 (as he will look in 5 lockers not including the one containing the number 1 ), but B2 will find his with probability only 4/9. The success or failure of B1 can influence the proba bilities of success of the other mem bers. This is the first clue! An ideal strategy would be one where if B1 succeeds then everyone else does too. Note that this would al low the whole team to succeed half the time even though each individual member fails half the time. This ideal is not attainable, but perhaps you can find a strategy where if B1 succeeds, then everyone else is more likely to succeed. No method of preassigning lockers will accomplish this, as if B1 finds his number in locker k anyone with locker k in their preassigned set has his chances reduced. This suggests that the locker choices will have to de pend on information not available at the initial meeting. The only such in formation available is the numbers a player finds inside the lockers he opens. With this further hint try one more time to find a good strategy be fore we proceed to the solution! Developing the Solution
Once we realize that the locker B1 opens at any stage can depend on what he has found inside the lockers he has already opened, the number of possi ble strategies to consider is enormous, even in the 10-player case. The strategy must tell B1 which locker to open first (10 choices), which locker to open next if he is not lucky on the first try (9 choices for each of the possible 9 numbers he may see), which to open third if he is not lucky on his second attempt either (8 choices for each of the 9 X 8 possible sequences of 2 numbers he has seen so far), and so
on. So
79 xsx7
B1 alone has 10 X 99 X 89 X8 X x 69 xsx7x 6 possible strategies.
To compute the number of strategy choices for the whole team, we raise this to the lOth power and get a num ber 28,537 digits long! How are we to choose one? In this section we will show that one very simple strategy lets the team win with remarkably high probability. The strategy for any one player is entirely un remarkable; the magic arises from the fact that the chances of the different players winning are highly correlated. Moreover, in the next section, we will show that the strategy is in fact optimal. Fortunately the good strategy is simple to implement and the choice of the next locker does not depend on the entire sequence of numbers seen but only on the most recent number. The good strategy has player Bi start by opening locker i. Then if he finds num ber k at any stage and k =I= i, he opens locker k next. Notice that player Bi , never opens a locker (other than locker i) without first finding its number, so each time he opens a new locker he must find either his own number or the number of another unopened locker. Again let's look at a particular case with 10 players and suppose, for exam ple, that the numbers are distributed in the order 6,8,9,7,2,4, 1,5, 10,3. Player B1 first examines locker 1 and finds the number 6. So he looks in locker 6 fmd ing the number 4, then locker 4 finding the number 7, then finally in locker 7 finding his number. When he finds his number, B1 will now know that B6, B4, and B7 will look in exactly the same lockers in the same cyclic order, each finding his number on the 4th try! He also knows that none of the other play ers will waste any tries on these lockers. We may represent any permutation of numbers into lockers by listing the cycles. The permutation 6,8,9, 7,2,4, 1 ,5, 10,3 gives the cycles (6, 4, 7, 1) (8, 5, 2) (9, 10, 3), and Team B succeeds be cause there is no long cycle. To find the probability that Team B wins, we count the number of permutations of 10 num bers with a cycle of length 6 or longer. First let's count how many have a 6-cy cle. Choose which 6 elements go into the 6-cycle, arrange them in cyclic or der, and then pick an arbitrary permu-
tation of the remaining 4 elements. The number of ways to do this is
( ) 10 6
10!5!4! 10! ' ' 5.4. 6!4! - 6
So 116 of the 10! permutations have a 6-cycle, and a random permutation has a 6-cycle with probability 1/6. The same argument can be used to find the probability of a permutation of 1-10 having a cycle of any length longer than 6. (We warn that the argument does not work for counting the number of per mutations of 1-10 with a 5-cycle (or shorter) as the permutation could have two 5-cycles.) A permutation of 10 numbers has a 7-cycle with probability 117 and so on, and the probability of a cycle of length 6 or larger is 1/6 + 117 + 1/8 + 1/9 + 1 / 1 0 1 6 2 7/2 5 2 0 0.645635. This gives the probability that Team B will fail, so of course Team B wins with probability 1 - 1627/ 2520 893/2520 0.354365. Over 35% of the time, all 10 members of Team B find their own wallets! Will this idea be good enough for the initial version with 100 players? We can do the analogous computation and see that this pointer-following strategy works with probability 1 - ( 1151 + 1152 + . . . + 1/100) .31 1828. Notice that while our strategy has still performed remarkably well for 100 players, the probability of success was still less than in the 10-player version. As we increase the number of players, does the success rate decrease to zero, or does it always stay above a certain positive number? With 2n players and 2n lockers, Team B will win provided that the permutation of numbers in lockers has no cycle of length n + 1 or longer. The probability of such a long 1 cycle is IJ:�1 --. By viewing this ex-
=
=
=
=
=
n+k
pression as an upper Riemann sum for
1- dx and a lower Riemann sum fn2n x+1 for f2n l dx we obtain n X
(
1 ln 2 n+ 1 s
n L
k�l
) = l2n
1 n+k
n
-- s
1 dx x+ 1
12n 1 dx n X -
=
ln 2.
So II:� 1 -1- ---+ ln 2 as n ---+ oo; moreover n+k the sum increases monotonically with n. So the expression 1 - II:� 1 -1- giv n+k ing the probability of success is
© 2006 Springer Science+Business Media, Inc., Volume 28, Number 1 , 2006
29
=
monotonically decreasing to 1 - ln 2 0.306853. Team B wins with the pointer following strategy with probability ex ceeding 30%, regardless of the number of players and lockers. Now that we have found a good strategy, we turn our attention to whether it provides the best possible solution. Is Pointer-Following Optimal?
We establish the optimality of pointer following by comparing the game con sidered above (Game 1) with a new game (Game 2) between the same ad versaries, Player A and Team B. For simplicity we give the argument in terms of the 10-player versions. Recall that in Game 1 we are allowing each player to examine 5 lockers. We first modify this rule and say that each player must continue examining lockers until he has opened the locker containing his number, and then he is not allowed to open any further lockers. Team B wins if no player opens more than 5 lockers. This change makes no difference to who wins in Game 1, but it will clarifY the comparison with Game 2. In Game 2, Player A again distrib utes the 10 numbers at random in the 10 lockers. Then all of team B is invited into the locker room together. Team member B 1 is required by the rules to start opening lockers and continue un til she reveals the number 1 . Once she has opened the locker containing the number 1 , she may not open any fur ther lockers; then, the lowest-num bered member of Team B whose num ber has not yet been revealed is required to take over opening lockers until he finds his number and so on. Team B continues until all lockers are opened. Again Team B wins if no indi vidual team member opens more than 5 lockers. Before proceeding, we invite you to consider the following ques tions: With what probability can Team B win Game 2? What strategy should the team members employ? Does their choice of strategy even matter? Let's sit in the locker room and ob serve Team B in the process of playing Game 2. We record the progress, list ing the numbers in the order in which they are revealed. Our list of numbers is sufficient to determine how many lockers were opened by each player.
30
THE MATHEMATICAL INTELLIGENCER
For example, if we record the list 2,6, 1 , 4,9,7, 10,8,3,5, we know that player B1 revealed the numbers 2, 6, and 1. Then player B3 was required to take over, and he opened the lockers containing the numbers 4, 9, 7, 10, 8, and 3, in that order. Then player BR opened the re maining locker containing the number 5. In this example Team B lost, as player B3 opened 6 lockers. Notice that we will record any given ordering of the numbers 1-10 with probability 1110!. The first number revealed is 2 with probability 1110, no matter which locker is opened, given that the first is 2 the second will be 6 with probability 119, and so on. What strategy is Team B following here? It makes absolutely no difference! Team B can choose lock ers at random or follow the most so phisticated plan; we still get probabil ity 1/10! for each of the 10! possible orders in which the numbers could be revealed. In Game 2 Team B's proba bility of success is completely inde pendent of strategy. To find the probability that Team B wins, we must count how many of the 10! possible orders of the numbers 1-10 represent wins. We employ a ver sion of the classical records-to-cycles bijection [6, p 1 7] to assign a permuta tion written in cycle notation to each ordering. The first cycle of our permu tation consists of the numbers opened by B1 in order; the second cycle, the numbers opened by the second locker opener; and so on. So, for example, 2,6, 1 ,4,9, 7, 1 0,8,3,5 ----> (2,6, 1)( 4,9, 7, 1 0,8,3) (5). Furthermore we see that each per mutation arises in this manner from a unique ordering of the numbers 1-10. We first write the permutation in cycle notation, rotate each cycle so that the lowest number in the cycle is written last, and then order the cycles so that their last numbers are in ascend ing order. For example (9, 7,8)(1,3, 10,5) (2,4,6) (3, 1 0 , 5 , 1 ) (4 , 6, 2 ) (8 , 9 , 7) ----> 3, 10,5, 1,4,6,2,8,9,7. We have established a one-to-one correspondence between lists for which Team B wins and the permutations of 1-10 with no cycles of length greater than 5. Thus the proba bility that Team B wins Game 2 is the probability that a random permutation of 1-10 has no cycle of length greater than 5, and we have already computed
=
=
this as 893/2520 0.354365. This is ex actly the probability of success for Team B in Game 1 using pointer fol lowing! Our analysis has a significant con sequence for Game 1. Team B can take any Game 1 strategy and adapt it to Game 2 as follows: If player Bi is open ing lockers in Game 2, he can use his Game 1 strategy for choosing lockers to open, simply observing the contents without wasting a turn if the indicated locker is already open. Thus if a strat egy succeeds in Game 1 for a particu lar distribution of numbers into lock ers it will also succeed in Game 2. If there were a better strategy for Game 1 we could apply it in Game 2 and get a better chance to win this game also. But this is impossible, as all strategies for Game 2 lead to the same probabil ity of success. We have one final small puzzle: Happy with their optimal strategy for Game 1 , Team B began a sequence of matches with Player A, but they soon found themselves down 10 to 0. What do you suspect Player A is doing? (It seems that Player A subscribes to the Intelligencer and has devised a plan to defeat Team B.) What can Team B do to counter Player A's plan? History of The Locker Puzzle
Our problem was initially considered by Peter Bro Miltersen, and it appeared in his paper [4] with Anna Gil, which won a best paper award at the ICALP conference in 2003. Miltersen says of the problem, "I think it started spread ing when I presented it to several peo ple at Complexity 2003, which was held in Aarhus, where I was a local orga nizer." In their version there is one numbered slip of paper for each player on the team. Player A then colors each slip either red or blue. Each member of Team B may examine up to half the lockers. He is then required to state or guess the color of the slip of paper with his number. Again every team member must state or guess his color correctly for the team to win. Initially Miltersen expected that Team B's probability of success would approach zero rapidly as the number of players increased. However, Sven Skyum, a colleague of Miltersen's at the University of Aarhus,
brought his attention to the beautiful pointer-following strategy. Finding this is left as an exercise in the paper. Miltersen and G:il originally consid ered the case where there are n team members and 2n lockers, half of them empty; each team member still gets to open up to half of the lockers. This is a more difficult problem. Clearly sim ple pointer-following will not work as empty lockers do not point anywhere. It is an open question whether the win ning probability must tend to zero for large n. In [5] Navin Goyal and Michael Saks build on Skyum's pointer-following to devise a strategy for Team B in a more general setting, varying both the pro portion of empty lockers and the frac tion of lockers each team member may open. As the number of players in creases, their probability of success for Team B approaches zero less rapidly than conjectured in [4]. And fixing the number of players and fraction of lock ers each may open, their probability of
winning remains nonzero even as more empty lockers are added. The problem also appeared in Joe Buhler and Elwyn Berlekamp's puzzle column in the Spring, 2004 issue of The Emissary [3] , with lockers replaced by ROM locations and colored numbers re placed by signed numbers. Here it is pointed out that the team benefits from the members carefully planning their guessing strategy as well as their locker searching strategy. For example, if there are 2n lockers and the longest cy cle has length n + 1 , the team members caught in the n + 1 cycle can guess in such a manner that they all guess cor rectly or all guess incorrectly. The trick is the same as that employed in the hat problem of Todd Ebert [2]. Variations of the hat problem are described in Joe Buhler's article in this column [ 1 ] and in Peter Winkler's book [7, p66, p 120]. The locker problem will be discussed in a fu ture edition of Winkler's book also. We thank Joel Spencer for intro ducing us to the problem, and we thank
MAX WARSHAUER
Ravi Vakil and Michael Kleber for en couraging us to write this note and pro viding many useful suggestions. REFERENCES
[ 1 } Joe Buhler, Hat tricks. Math. lntelligencer 24 (2002), no. 4, 44-49. [2} Todd
Ebert,
http://www.cecs.csulb.edu/
�ebert/ [3} The Emissary,
http ://www.msri .org/publi
cations/emissary/ [4] Anna Gal and Peter Bro Miltersen. The Cell Probe Complexity of Succinct Data Struc tures, Proceedings of 30th International
Colloquium on Automata, Languages and Programming (ICALP) 2003, 332-344. [5} Navin Goyal and Michael Saks, A Parallel Search Game, to appear in Random Struc
tures and Algorithms. [6} Richard
Stanley,
Enumerative Combina
torics Vol 1 . Cambridge Studies in Ad vanced Mathematics, 49, Cambridge Uni versity Press, Cambridge, 1 999 [7} Peter Winkler,
Mathematical Puzzles: A
Connoisseur's Collection, A K Peters, Ltd . , Natick, MA, 2004.
EUGENE CURTIN
Department of Mathematics
Department of Mathematics
Texas State Un ive rsity-San Marcos
Texas State University-8an Marcos
San Marcos,
TX 78666-4945
San Marcos,
TX 78666-4945
e- mail :
[email protected]
e-mail: ec01 @txstate.edu
Max Wars hau er received his Ph.D. at LSU under Pierre Conner
Eugene Curtin, a native of Ireland , received a Ph . D . from Brown
and is now Professor in the Mathematics Department at Texas
State, where he founded and directs Mathworks" a center for Math ematics and Mathematics Education, one of five programs in Texas
University in 1 988
un der
Thomas F. Banchoff. He is curren t ly a
p rofessor at Texas Stat e University. His research interests have in
cluded differential geometry, ring theory,
combi natorics ,
and graph
to receive t h e 2001 Texas Higher Education Star Award for Clos
theory. Eugene has been a faculty member with the Honors Sum
ing the Gaps. Max was one of 10 individuals in the country to re
mer Math Camp
ceive the 2001 Presidential Award for Excellence in Science, Math
mer since 1 992. He enjoys mathematical games a nd puzzles, and
ematics, and Engin eering Mentoring. His hobbies include playing chess, go, Ping - Pong , and bicycling. His research interests have
included quadratic forms, analysis of alg orith ms, and mathematics
u n der
Max Warshauer ' s directorship every sum
likes to play chess, go, and backgammon. He was chess cham
pion of Ireland in 1 984 and 1 985, and chess champ ion of Texas in 1 99 1 , 1 992, and 1 998
education.
© 2006 Springer Science +Business Media, Inc., Volume 28, Number 1 , 2006
31
RAJENDRA BHATIA AND JOHN HOLBROOK
N o n co m m utative G eo m etri c M ean s For, in fact, what is man in nature? A Nothing in comparison with the Infinite, an All in comparison with the Nothing, a mean between nothing and everything. -Blaise Pascal
A veraging operations entered mathematics rather early. .rt.Fascinated as they were by geometric proportions, the
ancient Greeks defined as many as eleven different means. The arithmetic, geometric, and harmonic means are the three best-known ones. If Pascal had one of these in mind when he composed his Pensees [P] , he would soon have realised that mixing zero and infinity is a source of as many problems as mixing mathematics and divinity. For centuries, mathematicians perfom1ed their opera tions either on numbers or on geometrical figures. Then in 1855 Arthur Cayley introduced new objects called ma t i ces, and soon afterwards he gave the laws of their algebra. Seventy years later, Wemer Heisenberg found that the non commutativity of matrix multiplication offers just the right conceptual framework for describing the laws of atomic mechanics. Matrices were found to be useful in the de scription of classical vibrating systems and electrical net works as well. For mathematicians, analysis of linear op erators was a subject of intense study throughout the twentieth century and into the twenty-first century. Many quantities of basic interest such as states of quan tum mechanical systems and impedances of electrical net works are defined in terms of matrices. Mixing of the un derlying systems in various ways leads to corresponding operations on the matrices representing the systems. Not surprisingly, some of these are averaging operations or means. Of the three most familiar means, the geometric mean combines the operations of multiplication and square roots. When we replace positive numbers by positive definite ma trices, both of these operations involve new subtleties. In this article we introduce the reader to some of them.
r
0 0 0 32
THE MATHEMATICAL INTELLIGENCt:R © 2006 Spnnger Sc1ence
t-
Bus1ness Media, Inc
Let IR+ be the set of all positive real numbers. Given a and b in IR + a mean m(a,b) could be defined in different ways. It is reasonable to expect that the binary operation m on IR + has the following properties: (i) (ii) (iii) (iv) (v)
m(a,b) = m(b,a). min(a,b) :S m(a,b) :S max(a,b). m(aa,ab) am(a,b) for all a > 0. m(a,b) is an increasing function of a and b. m(a,b) is a continuous function of a and b. =
The three familiar means, arithmetic, geometric, and har monic, satisfy all these requirements. Other examples of means include the binomial means, also called the power means, defined as mp(a,b)
(
=
aP
)
+ bJ! lip , 2
- oc
:S p
:s; x.
Here, it is understood that for the special values p ±: oc we define mp(a,b) as the limits mo(a,b) = limp mx( a,b) = limp m-x(a,b) = limp
__, __,
-->
= 0 and
omp(a,b) = v;;:b, xmp(a,b) = max( a,b ), -xmp(a,b) = min(a,b).
The arithmetic and the harmonic means correspond to the cases p = 1 and - 1 , respectively. Inequalities between means have been studied for a long time. See the classic [HLP], and the more recent [BMV]. A sample result here is that for fixed a and b, mp(a,b) is an increasing function of p. This includes, as a special case, the inequality between the three familiar means. There exists a fairly well-developed theory of means for positive definite matrices. Let MnCO be the set of all n X
n complex matrices, §n the collection of all self-adjoint el ements of Mn(IC), and iJ=Dn that of all positive definite ma trices. The space §n is a real vector space and iJ=Dn is an
open cone within it. This gives rise to a natural order on §n. We say that A 2: B if A - B is positive definite or pos itive semidefinite. Two elements of §n are not always com parable in this order. Every element X of GLn (the group of invertible matrices) has a natural action on iJ=Dn· This is given by the map rx(A) = X*AX. We say that A and B are congruent if B = rx(A) for some X E GLn. In the special case when X is unitary, we say that A and B are unitarily equivalent. The group of unitary matrices is denoted by QJn· Now we have enough structure to lay down conditions that a mean M(A,B) of two positive definite matrices A and B should satisfy. Imitate the properties (i)-(v) for means of numbers. This suggests the following natural conditions: (I) M(A,B) = M(B,A). (II) If A :::::: B, then A :::::: M(A,B) :::::: B. (III) M(X*AX,X*BX) = X*M(A,B)X, for all X E GLn. (IV) M(A,B) is an increasing function of A and B; i.e., if A 1 2: A2 and B 1 2: B2 , then M(A 1 ,B1) 2: M(A2,B2). (V) M(A,B) is a continuous function of A and B. The monotonicity condition (IV) is a source of many in triguing problems in constructing matrix means. This is be cause the order A 2: B is somewhat subtle. For example, if A =
[� �1
and
then A 2: B but A2 ;t B2. What functions of positive numbers, when lifted to pos itive definite matrices, preserve order? This is the subject of an elegant and richly applicable theory developed by Charles Loewner. Letf be a real-valued function on IR+ . If A is a positive definite matrix and A = 'i.Aiuiui is its spec tral resolution, then f(A) is the self-adjoint matrix defined as j{A) = 'i.J{Ai)uiui. We say that j is a matrix monotone junction if for all n = 1 , 2, . . . , the inequality A 2: B in IJ=Dn implies j{A) 2:j{B). One of the theorems of Loewner says thatfis matrix monotone if and only if it has an analytic con tinuation to a mapping of the upper half-plane into itself. As a consequence, the functionj{x) = xP is matrix monotone if and only if 0 :::::: p :::::: 1. The function f(x) = log x is matrix monotone, but j{x) = exp x is not. We refer the reader to Chapter V of [B] for an exposition of Loewner's theory. Returning to means, the arithmetic and the harmonic means of A and B are defined, in the obvious way, as eA + B) and [ - 1 + B- 1 ) ] - 1 , respectively. It is easy to see that they satisfy the conditions (I)-(V) above. The notion of geometric mean in this context is more elusive, even treacherous. Every positive definite matrix A has a unique positive definite square root A 112. However, if A and B are positive definite, then unless A and B com mute, the product A 112B 112 is not self-adjoint, let alone pos itive definite. This rules out using A 112B l12 as our geomet ric mean of A and B, except in the trivial case when AB = EA. We should look for other good expressions in A and B
t
tCA
that reduce to A 112B 112 when A and B commute. One plau sible choice is the quantity
( 1)
exp
(
log A + log B
2
)_
. - hmp
__.
o
(
)
AP + BP lip
2
.
The equality of the two sides of (1) was noted by Bhag wat and Subramanian [BS], who studied in detail the "power means" occurring on the right-hand side. This too is not monotone in A and B, as can be seen by choos ing positive defmite matrices X and Y, for which X 2: Y but exp X ;t exp Y, and then choosing A and B such that X = (log A + log B) and Y = log B. The condition (III), sometimes called the transformer equation, is not innocuous either. Our failed candidates fail on this count too. The noncommutative analogue of v;;J; with all desirable properties turns out to be the expression
t
t
(2)
A#B = A 112 (A - 1/2 BA - 112) v2 A vz,
that was introduced by Pusz and Woronowicz [PW] in 1975. At the outset it does not appear to be symmetric in A and B; but it is, as we will soon see. The monotonicity in B is assured by the facts that congruence preserves order (B1 2: B2 implies X*B1X 2: X*B,y() and the square root function is matrix monotone. Symmetry in A and B is apparent more easily from an al ternative characterisation of A#B due to T. Ando [A]. We have
(3)
{
A#B = max x : X = X* and
[� �1 o}. 2:
Among its other characterisations, one describes A#B as the unique positive definite solution of the Riccati equation X A - 1X = B.
(4)
We call A#B the geometric mean of A and B. It has the de sired properties (I)-(V) expected of a mean M(A,B) : prop erty (III) may be verified easily from (3) or (4). It satisfies the expected inequality
(5)
(
)
A - 1 + B- 1 - 1
2
:::::: A#B ::::::
A +B
-
2
,
and has other pleasing properties. Many of these were de rived by Ando [A]. Two positive definite matrices A and B can be diago nalised simultaneously by a unitary conjugation r u if and only if they commute. In the absence of commutativity, A and B can be diagonalised simultaneously by a congruence in two steps: 112 112 (A,B) rA-"' (I,A - BA - )
� (I,D),
where U is a unitary such that U* (A - 112BA - 112) U is a di agonal matrix D. This takes some of the mystery out of the formula (2). In fact, any mean m(a,b) of positive numbers leads to a mean M(A,B) of positive definite matrices by the procedure M(A,B) = rAJt2(m(I,D)). To ensure that M is an increasing function of A and B, we have to assume that the functionf(x) = m(l,x) is matrix monotone. The formula (2) corresponds to the case when m(a,b) = (ab) 112 .
© 2006 Spnnger Science+ Business Media, Inc., Volume 28, Number 1, 2006
33
The indirect argument we have used to deduce the sym metry of the geometric mean is not necessary. Let m(a,b) be any mean, let f(x) m(l,x), and =
(6)
M(A,B) = A112j (A - 112BA - 112) A112 .
Though this expression seems to be asymmetric in A and B, in fact M(A,B) = M(B,A). For this we need to prove
f(A - 112BA - 112) = A - 112B112j (B- 112AB - li2)B112A - 112.
Using the polar decomposition A - 112B112 = PU, where P is positive definite and U unitary, this statement reduces to
This, in tum, is equivalent to saying that for every eigen value A of P, we have
But that is a consequence of properties (i) and (iii) of the mean m. A similar argument verifies (III). A simple corollary of this construction is the persistence of inequalities like (5) when one passes from positive num bers to positive definite matrices. Kubo and Ando [KA] de veloped a general theory of matrix means and established a correspondence between such means and matrix mono tone functions. What happens when we have three positive definite ma trices instead of two? The arithmetic and the harmonic means present no problems. Plainly, they should be defined as �(A + B + C) and [�(A - 1 + B - 1 + c- 1)] - 1, respectively. The geometric mean, once again, raises interesting problems. We would like to have a geometric mean G(A,B,C) that reduces to A 113B113C113 when A, B, and C commute with each other. In addition it should have the following properties. (a) G(A,B,C) = G(1r(A,B,C)) for any permutation 1r of the triple (A,B,C). (/3) G(X*AX,X*BX,X*CX) = X*G(A,B,C)X for all X E GLn. ( y) G(A,B,C) is an increasing function of A, B, and C. (8) G(A,B,C) is a continuous function of A, B, and C. None of the procedures presented above for two matrices ex tends readily to three. The expressions (2), (3), and (4) have no obvious generalisations that work The idea of simulta neous diagonalisation does not help either: while two posi tive definite matrices can be diagonalised simultaneously by a congruence, generally three can not be. Defining a suitable geometric mean of three positive definite matrices has been a ticklish problem for many years. Recently some progress has been made in this direction, and we describe it now.
standing of it is achieved by linking it with some standard constructions in Riemannian geometry. The space Mn(C) has a natural inner product (A,B) = tr A*B. The associated norm IIAib = (tr A*A)112 is called the Frobenius, or the Hilbert-Schmidt, norm. If A is a matrix with eigenvalues A , . . . , An, we write A (A) for the vector 1 (A 1, . . . , An) or for the diagonal matrix diag(A 1 , . . . , An)· The set IP n is an open subset of §n and thus is a differ entiable manifold. The exponential is a bijection from §n onto IPn· The Riemannian metric on the manifold IPn is con structed as follows. The element of arc length is the dif ferential ds = IIA - 112 dA A - 112 llz .
(7)
This gives the prescription for computing the length of a differentiable curve in IPn· If y : [a,b] � IPn is such a curve, then its length, obtained by integrating the formula (7), is
L(y) = r ��y - 112(t)y'(t)y- 112(t)lb dt.
(8)
a
If A and B are two elements of IPn, then among all curves y joining A and B there is a unique one of minimum length. This is called the geodesic joining A and B. We write this curve as [A ,B], and denote its length, as defined by (8), by the symbol 82 (A,B). This gives a metric on 1Pn called the Riemannian metric. From the invariance of trace under similarities, it is easy to see that for every X in GLn the map rX : IPn � IPn is a bi jective isometry on the metric space (IPn,8 2). An important feature of this metric is the exponential met ric increasing property (EMI). This says that the map exp from the metric space (§11,ll·llz) to (IPnh) increases distances. More precisely, if H and K are Hermitian matrices, then (9) To prove this, one uses the formula (8) and an infinitesi mal version of (9): (10) for all H, K E §n· Here Defi(I() is the derivative of the map exp at the point H evaluated at K, i.e.,
Defi(K) = limt--.o
(11)
While the geometric mean A#B has been much studied in connection with problems of matrix analysis, mathemati cal physics, and electrical engineering, a deeper under-
34
THE MATHEMATICAL INTELLIGENCER
t
There is a well-known formula due to Daleckii and Krein (see [B], chapter V, for example) giving an expression for this derivative. Choose an orthonormal basis in which H = diag(A b . . . , An). Then
Defi(K) =
0 0 0 One geometry cannot be more true than another; it can only be more convenient. -Henri Poincare [Po}
efi+tK - elf
[
fl'; - eAi i A '·· _ AJ· k j .
J
(The notation here is that [Xij] stands for a matrix with en tries Xij·) From this, one sees that the (iJ) entry of e -H12Defi(K)e -H12 is sinh(A i - Ai)/2 · (A.; - Aj)/2 kij
( 12) Since
sinh X X
2: 1, the inequality ( 10) follows from this.
In the special case when H and K commute, a calcula tion shows that there is equality in (9). In this case the func tion exp maps the line segment [H,K] in the Euclidean space §n isometrically onto the geodesic segment [ef/,ef<] in !f1>n· If A eH and B eK, this says that the geodesic seg ment joining A and B is the path
=
y(t)
= eC1 - t)H+tK
=
eC 1 - t)He tK
=
A 1 - t Bt ,
0 :::; t :::; 1 .
=
(13)
=
y(t)
=
=
Further, o2 (A,y(t)) to2 (A,B) for each t in [0, 1 ] . The case of noncommuting A and B can b e reduced to the commuting case using the fact that rA-lt2 is an isom etry on the space (ifl>n,82). The geodesic segment [I,A - 1!2 BA - 112 ) is parametrised by y0(t) (A - 112BA - 112)1, by what we said about the commuting case. So, the geodesic (A,B) !fA'"{J),fAv.{A- 112BA- 112)) is parametrised by
=
lllog I - log (A - 112BA - 112)112 lllog(A - 112BA - 112)llz. The ma trices A - 112BA - l/2 and A - 1B have the same eigenvalues. So, this can be expressed as
= A 112 (A- 112 BA - 112)1 A 112 , 0 :::; t :::; 1.
-
This shows that the geometric mean A#B defined by the fonnula (2) is nothing but the midpoint of the geodesic join ing A and B in the Riemannian manifold !f1>n· Thus while (2), (3), and (4) might have ap,Reared as over-imaginative non commutative variants of �. very natural geometric con siderations lead to the same notion of mean as is given by (2). Note that for each t, y(t) defmed by ( 13) is a mean of A and B corresponding to the functionf(x) :xf in the for mula (6). Those means are not symmetric, however: (I) fails unless t 1/2. This discussion also gives an explicit formula for the metric 8 2 . We have o2 (A,B) 8 2 (I,A - 112BA - 112)
=
=
=
The inequality (9) captures an essential feature of lfl>n : it is a manifold of nonpositive curvature. To understand this, consider a triangle with three vertices 0, H, and K in §n· Under the exponential map, this is mapped to a "triangle" with vertices I, exp H and exp K in !f1>n· The lengths of the two sides [O,H) and [O,.K] measured by the norm ll· llz are equal to the lengths of their images [I, exp H) and [I, exp K] measured by the metric 82 . By the EMI (9), the length of the third side [ exp H, exp K] of the triangle in !f1>n is larger than (or equal to) IIH Kllz. The general case of a geodesic triangle with vertices exp A, exp B, exp C in !f1> n may be re duced to the special case by applying the congruence fexp(-A/2) to all points and thus changing one of the ver tices to I. This is often described by saying that two geo desics emanating from a point in !f1>n spread out faster than their pre-images (under the exponential map) in §n· It is instructive here to compare the situation with that of IUn, a compact manifold of non-negative curvature (Fig ure 1 ). In this case the real vector space i§n consisting of skew-Hermitian matrices is mapped by the exponential onto IUn. The map is not injective; it is a local diffeomor phism. Using the formula ( 1 1) with H and K in i§n, we reduce
=
exp(iA)
2
2. 5
2
1. 5
0. 5
0
0.5
1 .5
2
2.5
Figure 1. Three curvatures, showing a comparison of a Euclidean (curvature zero) triangle in §2 with its images under exp(-) in P2 (nonposi tive curvature) and exp(i·) in Q.J2 (non-negative curvature). The colours indicate matching vertices. Note that the geodesics emanating from exp(A) spread out faster than Euclidean ones (compare the straight lines at A), whereas those emanating from exp(iA) spread more slowly.
© 2006 Springer Science+Business Media, Inc., Volume 28. Number 1 , 2006
35
Returning to IPn and the geometric mean, it is not diffi cult to derive from the information at our disposal the fact that given any three points A, B, and C in 1Pn we have (15)
A#C
B
c
B#C
Figure 2. Geodesic distance from A#B to A#C is no more than half that from B to C. Joining the midpoints of the sides of a geodesic triangle in IP'n results in a triangle with sides no more than half as long. Iterating this procedure leads to the construction of Ando, Li, and Mathias, described in the text.
H to diag(iA 1 o now
•
•
•
, iAn) with A1 real. Instead of (12) we have
sin(A i - AD/2 i· (Ai - AJ)/2 k i
Since [sin x[ :S 1, the inequality ( 10) is reversed in this case, X as is its consequence (9), provided elf and eK are close to each other.
o 2 (A#B,A#C)
1
:S 2 o2 (B, C).
This inequality says that in every geodesic triangle in IPn with vertices A, B, and C, the length of the geodesic join ing the midpoints of two sides is at most half the length of the third side. (If the geometry were Euclidean, the two sides of (15) would have been equal.) Figure 2 illustrates ( 15). We saw that the geometric mean A#B is the midpoint of the geodesic [A,B]. This suggests that we may possibly de fine the geometric mean of three positive definite matrices A, B, and C as the "centroid" of the geodesic triangle Ll(A,B,C) in 1Pn. In a Euclidean space �. the centroid x of a triangle with vertices x1 , x2 , X3 is the point x �(x1 + Xz + x3). This is the arithmetic mean of the vectors x1 , x2 , and x3. This point may be characterised by several other properties. Three of them are:
=
(M 1) x is the unique point of intersection of the three medians of the triangle ll(x1 ,x2,x3), as in Figure 3; (M2) x is the unique point in � at which the function attains its minimum; (M3) x is the unique point of intersection of the nested sequence of triangles [lln} in which ll 1 Ll and ll1+ 1 is the triangle obtained by joining the mid=
expA
�--__,.
A
expC
� c
Figure 3. In the hyperbolic geometry medians may not meet. While the medians of a Euclidean triangle intersect at the centroid, the corre sponding median geodesics of a triangle in IP'n may not intersect at all. A 3-D wire model would make it clear that, generically, the medians do not even intersect in pairs.
36
THE MATHEMATICAL INTELLIGENCER
points of the three sides of t:.J (Figure 2 mimics this construction in the non-Euclidean setting of IP'n). To define a geometric mean of A, B, and C in IP' n we may try to imitate one of these definitions, now modified to suit the geometry of IP'n· Here fundamental differences between Euclidean and hyperbolic geometry come to the fore, and (Ml), (M2), and (M3) lead to three different results. The first definition using (Ml) fails. The triangle t:.(A,B,C) may be defined as the "convex set" generated by A, B, and C. (It is clear what that should mean: replace line segments in the definition of convexity by geodesic seg ments.) It turns out that this is not a 2-dimensional object as in ordinary Euclidean geometry (see Figure 4). So, the medians of a triangle may not intersect at all in some cases (again, see Figure 3). With (M2) as our motivation, we may ask whether there exists a point X0 in IP'n at which the function
J(X)
= 8�(A,X)
+
8 �(B,X) + 8 � (C,X)
attains a minimum. It was shown by E lie Cartan (see, for example, section 6. 1.5 of [Be]) that given A, B, and C in IP'n• there is a unique point Xo at which f has a minimum. Let G2(A,B,C) X0, and think of it as a geometric mean of A, B, and C. This mean has been studied in two recent papers by Bhatia and Holbrook [BH] and Moakher [M]. In another recent paper [ALM], Ando, Li, and Mathias define a geometric mean G3(A,B,C) by an iterative proce dure. This iterative procedure has a nice geometric inter pretation: it amounts to reaching the centroid of the geo desic triangle 11(A,B,C) in IP'n by a process akin to (M3).
=
Starting with /1 1 as the triangle 11(A,B,C) one defines /12 to be 11(A#B,A#C,B#C), and then iterates this process. Figure 2 shows the beginning of this process. The inequality (15) guarantees that the diameters of these nested triangles de n scend to zero as 112 . It can then be seen that there is a unique point in the intersection of this decreasing sequence of triangles. This point, represented by G3(A,B,C), is the geometric mean proposed by Ando, Li, and Mathias. It turns out that the two objects G2(A,B,C) and G3(A,B,C) are not always equal (Figure 5 illustrates this phenome non). Thus we have (at least) two competing notions of the centroid of 11(A,B,C). How do they do as geometric means? The mean G3(A,B,C) has all of the four desirable properties (a)-(8) that we listed for a mean G(A,B,C). Properties (a) , (/3), and (8) are almost obvious from the construction. Prop erty ( y)-monotonicity-is a consequence of the fact that the geometric mean A#B is monotone in A and B. So mo notonicity is preserved at each iteration step. The mean G2(A,B,C) does have the desirable properties (a) , (/3), and (8). Property (/3) follows from the fact that rX is an isome try of (IP' n,82) for every X in GLn. However, we have not been able to prove that G2(A,B,C) is monotone in A, B, and C. We have an unresolved question: Given positive definite matri ces A, B, C, and A' with A 2: A', is G2(A,B,C) 2: G2(A ' ,B,C)? An answer to this question may lead to better under standing of the geometry of IP'n, the best-known example of a manifold of nonpositive curvature. Certainly this is of interest in matrix analysis. Computer experiments suggest an affirmative answer to the question. Finally, we make a brief mention of two related matters. The Frobenius norm is one of a large class of norms called
0.25 0.2 0.15 0.1 0.05
0.2 0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 4. Conv (A,B,C) is not two-dimensional. In the hyperbolic (nonpositive curvature) geometry of l?m the convex hull of a triangle (formed by successively adjoining the geodesics between points that are already in the object) is not a surface but rather a "fatter" object.
© 2006 Springer Science+Business Media, Inc., Volume 28, Number 1 , 2006
37
unitarily invariant norms or Schatten-von Neumann norms. These norms 11 · 11"' have the invariance property IIU A VII"' I AII"' for all unitary U and V. Each of these norms corresponds to a symmetric norm
=
=
=
=
=
0 0 0 But the whole wondrous complications of interference, waves, and all, result from the little fact that :i:p - px is not quite zero. -Richard Feynman [FLSj The generalised version of EMI has a fascinating connection with yet another subject: inequalities for �he matrix exponential function discovered by physicists and mathematicians. Many such in equalities compare eigenvalues of the matri ces ef!+K and eHeK, and are much used in 1uantum statistical mechanics and lately in quantum information theory. In [S] I. Segal proved for any two Hermitian matrices H and K the inequality
Here A 1 (X) is the largest eigenvalue of a matrix X with real eigenvalues. In a similar vein, we have the famous Golden Thompson inequality (17) The matrices ef!+K and ef112eKeH12 are positive definite. So, the inequalities ( 16) and (17) say
l efl+K] IP ::::: l lefll2eKeH12IIP, for p = 1 , oc .
The EMI (9) generalised to all unitarily invariant norms is the inequality
By well-known properties of the matrix exponential, this implies (19) This inequality, called the generalised Golden-Thompson inequality, includes in it the inequalities ( 16) and ( 1 7) . The origins of these inequalities and their connections with quantum statistical mechanics are explained in Simon [Si] (page 94). Still more general versions have been discovered by Lieb and Thirring, and by Araki, again in connection with problems of quantum physics. See Chapter IX of [B] . Gen eralisations in a different direction were opened up by Kostant [K], where the matrix exponential is replaced by the exponential map in more abstract Lie groups. A common thread running between matrix analysis, Rie mannian and Finsler geometry, and physics! Pascal would have approved. REFERENCES
We have included some articles that are related to our theme but not specifically mentioned in the text. [A] T. Ando, Topics on Operator Inequalities, Lecture Notes, Hokkaido University, Sapporo, 1 978. [ALM] T. Ando, C . -K. Li, and R . Mathias, Geometric means, Linear Al
gebra Appl. 385(2004), 305-334. [Be] M. Berger, A Panoramic View of Riemannian Geometry, Springer Verlag, 2003. [B] R. Bhatia, Matrix Analysis , Springer-Verlag, 1 997. [B2] R . Bhatia, On the exponential metric increasing property, Linear
Algebra Appl. 375(2003), 2 1 1 -220. [BH] R . Bhatia and J . Holbrook, Riemannian geometry and matrix geometric means, to appear in Linear Algebra Appl. [BrHa] M . Bridson and A. Haefliger, Metric Spaces of Non
positive Curvature, Springer-Verlag, 1 999. [BMV] P S. Bullen, D . S. Mitrinovic, and P. M . Vasic, Means and Their
Figure 5. The "Cartan surface" contains G2(A,8,C) but not G3(A,8,C). The Cartan surface consists of points minimizing the convex combi nations all �(A.xJ
+
bll �(B,x)
+
cll �(C,x); here the colours of the points
Inequalities, D . Reidel, Dordrecht, 1 988. [BS] K. V. Bhagwat and R. Subramanian, Inequalities between means of positive operators, Math. Proc. Camb. Phil. Soc. 83(1 978), 393-40 1 .
shown are chosen to reflect the relative strengths of the weights a,b,c.
[CPR] G . Corach, H . Porta, and L. Recht, Geodesics and operator
Thus G2(A,8,C) corresponds to 1 /3, 1 /3, 1/3 (see yellow dot on sur
means in the space of positive operators, Int. J. Math. 4(1 993),
face). The small black circle locates G3(A,8,C), which is not on the surface in general. Thanks to J.-P. Shoch for computing this picture of a Cartan surface.
38
THE MATHEMATICAL INTELLIGENCER
1 93-202. [FLS] R. Feynman, R. Leighton, and M. Sands, The Feynman Lectures
on Physics, volume 3, page 20-1 7, Addison -Wesley, 1 965.
A U T H OR S
JOHN HOLBROOK
RAJENDRA BHATIA
Department of Mathematics and Statistics
Indian Statistical Institute
University of Guelph
7, S. J. S. Sansanwal Marg New
Guelph, Ontario N 1 G 2W1
Canada
Del hi 1 1 00 1 6 India
e-mail:
[email protected]
e-mail:
[email protected]
Rajendra Bhatia did his doctoral studies at lSI Delhi with Kalyan M ukherjee. He has been based there most of the quarter-century since, along with his wife lrpinder and their son Gautam. John Holbrook is now Professor Emeritus at Guelph. He and his wife Catherine divide their time between Guelph and Fowke Lake (farther north), generally in the company of children, g randc hil dren, and cats. This photograph of the authors (courtesy of Peter Semrl) shows them in yet another continent, Europe: in the beautiful Alps of Slove nia. The photo may also serve as encouraging evidence that it is possible to collaborate on mathematical projects and remain on good terms !
[ H LP] G. H. Hardy, J. E. Littlewood, and G. P61ya, Inequalities, Cam bridge University Press, 1 934 . [K] B. Kostant, On convexity, the Weyl group and the lwasawa de composition, Ann. Sc. E. N. S. 6(1 973), 4 1 3-455. [KA] F. Kubo and T. Ando, Means of positive linear operators, Math.
Ann. 246(1 980), 205-224. [LL] J . D. Lawson and Y. Lim, The geometric mean, matrices, metrics, and more, Amer. Math. Monthly 1 08(200 1 ), 797-81 2 . [M] M . Moakher, A differential geometric approach t o the geometric mean of symmetric positive-definite matrices, SIAM J. Matrix Anal.
App/. 26(2005) , 735-747.
[P] B. Pascal, Pensees , translation by W. F. Trotter, excerpt from item
72, Encyclopaedia Britannica, Great Books 33, 1 952. [Po] H . Poincare, Science and Hypothesis , from page 50 of the Dover reprint, Dover Publications, 1 952. [PW) W. Pusz and S. L. Woronowicz, Functional calculus for sesquilin ear forms and the purification map, Reports Math. Phys. 8(1 975),
1 59-170 [S] I . Segal. Notes towards the construction of nonlinear relativistic quan tum fields I l l , Bull. Amer. Math. Soc. 75(1 969), 1 390-1 395. [Si] B . Simon, Trace Ideals and Their Applications, Cambridge Univer sity Press, 1 979.
© 2006 Springer Science+ Business Media, I n c . , Volume 2 8 , Number 1 , 2006
39
Numerical Properties of Rudolph M ichael Schindler's Houses in the Los Angeles Area
T
he architecture of Los Angeles mir
Schindler's primary concern in ar
rors the diversity of cultures rep
chitecture is the assembly and delin
resented within the city and includes
eation of spaces. He crystallized his
the work of many eminent contempo
idea of space in his mind, rather than
0.
by visualizing it through a physical
Gehry. Any serious study of Los Ange
model, or bodily movement combined
les
name
with the perceptual process, or other
rary
architects, architecture
such
as
invokes
Frank the
As
of Rudolph Michael Schindler as a ma
methods.
jor inspiration. Schindler's numerous
Schindler house, you find yourself in a
a result, once you enter a
buildings in Los Angeles, built over
space-form that provides a full array of
thirty years, from the 1920s to the
new spatial experiences, and, at times,
1950s, are recognized as icons of twen
surprises that arise from the complex
tieth-century design.
ity of the spatial flow.
Mathematicians interested in ex
It is hard to predict Schindler's
ploring and discovering Los Angeles ar
space-forms by examining his plans,
chitecture will find Schindler's build
elevations, and sections. For Schindler,
ings (Fig. 1) of particular interest. To
these were just two-dimensional nota
make the most of your visit, you should
tional forms, like scales in music. His
understand the underlying principle of
approach was not one of figure and
Schindler's architecture: his propor
ground. Rather, the interior and the ex
tional system (he called it "reference
terior spaces are intertwined. In the in
frames in space") and its unique nu
terior, the spaces flow into each other
merical properties.
and, on the exterior, simple and recti-
Jin-Ho Park
Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the caje 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? Jj so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.
Please send all submissions to Mathematical Tourist Editor,
Pacific Ocean
Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende, Belgium e-mail:
[email protected]
40
Figure 1 . Map of Los Angeles, and Schindler's buildings discussed in the present paper.
THE MATHEMATICAL INTELLIGENCER © 2006 Springer Sc>ence- Bus>ness Media, Inc
Figure 2. R. M. Schindler: The McAimon House, 1 935, 271 7-2721 Waverly Drive, Los Angeles; The Mackey Duplex, 1 939, 1 1 37 South Cochran Avenue, Los Angeles.
linear space-forms are highly inter
in 48-inch units, but after ± 12", ±24" re
understood in the mind as a continu
locked. Their union creates a rhythm
fmements, results always lie within a 12-
ous quantity in three dimensions. Aris
of complex combinations. Thus it is of
inch grid. A variety of L-shapes and other
totle bisects "quantity" into two parts:
ten hard to distinguish where the build
arrangements
magnitude and multitude. For Aristo
ings start or end; frequently they lack
dated. Such a unit is useful to measure
tle, magnitude is that which can be
a clear main fa<;ade. Therefore, to ap
dimensional relationship of a build
measured: it has indivisible, continu
preciate Schindler's space-forms, one
ing's heights, widths, and lengths. Verti
ous attributes. Multitude, which can be
must visit the houses in person, and ex
cally, s11ch rectangles are elevated in 16inch steps to create spatial elements.
discrete
perience them through the movement
There
of the body in space.
are
are
two
easily
accommo
reasons
for
his
counted with number, has divisible or attributes.
Schindler's
unit
system can be interpreted as measured
Architecture, for Schindler, began
choice of this unit. First, it had to be
space
with thinking about and reasoning with
related to the human figure to satisfy
Thus, for Schindler, a space can be
the building, shaping and feeling it as
all
for
measured numerically by the unit sys
a structure in the mind (Schindler,
rooms, doors, and ceiling heights; sec
tem in the mind-with subdivisions
the
necessary requirements
replaced
by
counted
It
space.
1946). Here, the proportional system
ond, for practical reasons, the 48-inch
and multiples of the unit.
plays a significant role in composing a
module fit the standard dimensions of
rationalized method of creating a spa
is a fully
tial structure. He wrote:
specific compositional sensibility with
materials and common construction
the practicalities of physical construc
methods available in California at that
[The space architect] must establish
tion.
time.
a unit system which he can easily
The purposes of his system were,
the
carry in his mind and which gives
first, to provide a mental structure for
strategic transfer of the architect's in
him the size values of his forms di
conceptualization before pencil met pa
tention and vision, accurately and pre
rectly without having to resort to
per and second, to communicate to the
cisely. Schindler utilized his space ref
mathematical computations.
builders a map locating elements, al
erence
surements in figures express ab
lowing them to easily scale off dimen
three-dimensional space. Numbers and
stract size relations void of any hu
sions. By doing so, the location of every
letters are laid out on the grid on the
man connotation and offering no
Architectural
frame
drawings
in a grid
are
pattern in
Mea
identified ac
floor plans in sequence, and the verti
assistance for the imagination.
curately in the convenience of composi
cal module is identified with an eleva
Thus, the unit modules replace mea-
tion and construction. Thus, no obscure
tion grade. This pattern was original to
surements. The designer can play with
element of the building
is
or arbitrarily unrelated measurements
Schindler. In his earlier designs, the
them, in his mind, freely and accu
are involved in the system.
grid was present on drawings and in
rately, without any mechnical measur
Usually, Schindler recommended a
the house; later, the grids disappeared
ing devices. Accordingly, space forms
basic unit of 4 feet (48 inches), to be
from the house and, at times, from the
are successively conceived from the
drawings. This doesn't mean he aban
mental play of the unit system.
1l2, 1h and 114 subdivisions. In many cases, he used 114 and 1l2 subdivisions for plans and 113 for vertical subdivision. The 114 and 112 adjustments "humanize"
used with simple multiples and with
doned his system; on the contrary, it
Divisibility is another essential fea
remains embedded in the designs as
ture of the unit system. The number 48
the underlying principle.
is divisible by 10 factors ( 1 , 2,
Schindler's choice of 48-inch units
3,
4, 6,
8, 12, 16, 24, 48), which form a sub
the coarseness of the space reference
has two significant mathematical as
module group and relate organically to
frame itself. The architect must think
pects. With the unit system, a space is
the unit system. This group forms a
© 2006 Springer Science+ Business Media. Inc., Volume 28, Number 1 , 2006
41
--------..,..
-
Figure 3. Schindler's three-dimensional grid of space cubes.
family, so that the measurements of the whole building are in relationship to one another. Thus, all the parts of the dimensions harmonize through unifor mity of scale and rhythm. In other words, unrelated dimensions in the whole scale can be excluded. In Schindler's words, " . . . only coarse ness allows him to break that rhythm by introducing arbitrary unrelated di mensions into his layout." The 48-inch unit is characterized as a "highly composite number" by Lionel March (1994a), who accounts for various composite numbers, such as 36, 48, 60, 180, with a lattice diagram, showing a hierarchical structure. Through a process of abstraction, Schindler began to construct and ex plore a variety of spaces and space forms in his mind. He defined spaces as abstract geometric forms whose po sitions could be uniquely determined by a given number of coordinates. Here, Schindler's space reference frame helps to measure the exact dis tance between points in space, and to identify space-forms of a design in precise locations, allowing for formal and spatial coherence in his architec ture. It is easy to see how Schindler worked with those unit dimensions in his spatial computation. Because the space architect can conceive of 4 X 4 x 4 foot "units" of space, it seems quite reasonable to imagine such a space element as being slightly less or slightly more, :±:: 1/4, or simply halfway
42
THE MATHEMATICAL INTELLIGENCER
between, :±:: 1h in the mind. In the de sign process, Schindler is able to ad just the dimension bit by bit. Schindler roughed in his designs us ing the 4 X 4 X 4 foot space reference frame and located spatial elements rel ative to one another, introducing re finements between elements-taking a little here, adding a little there. As he explained, "It is not necessary that the designer be completely enslaved by the grid. I have found that occasionally a space-form may be improved by devi ating slightly from the unit." Nor was Schindler obsessed with particu lar numbers, such as the Fibonacci se quence or ratios. Design can be imag ined in the mind without recourse to pencil and paper. This represents a ma jor distinction from the architect Le Corbusier, who used regulating lines in his early work, transforming them into Le Modular, and was noted for the per sistence of the golden section in his work after World War II. Schindler also used the space refer ence frame to measure room sizes. In his preliminary sketches, room sizes with
whole numbers on drawings are com monly presented. These numbers are in crements of unit multiples with subdivi sions. At times, various rooms are not rectangular in shape but interlocked, overlapped, and even zigzagged. Despite their shape, Schindler labeled room di mensions in terms of a X b on the draw ings. It is certain that he approximated measurements of the room in a rational manner with his space reference frame. The notation would be the standard room and component sizing notation: a X b with a, the width, greater than the length, b. Here there is no temptation to reduce the terms by common factors or ratios, since the repetition of the same ratio throughout the parts is seen as pro ducing a confusion of scale. Schindler is concerned about scale precisely be cause it would imply a change of the unit of measurement within the same work He wrote, " 'Scale' denotes a consistent dimensional relationship of parts of a structure to each other and to a basic unit." Thus, with such a notation, scale is always retained. Among Schindler's 200 houses in the Los Angeles area, three are dis cussed in this article-the Kings Road House of 1920; the Lovell House of 1923; and the How House of 1925. They represent exquisite examples of Schindler's finest work Most of his houses are private residences and are not accessible to the public. However, the Kings Road house is open; since 1994, the house has been used as the MAK Center for Art and Architecture in Los Angeles, a satellite of the Mu seum for Applied Art in Vienna. Kings Road House, 1 92 1 -22
835 North Kings Road,
West Hol lywood
The Kings Road house is the best "representation of Schindler's "or ganic" type of dwelling. 180
Figure 4. The lattice diagram of highly composite numbers such as 36, 48, 60, and 180 are shown (after Lionel March).
6
B ath b
�
Living R'M Kilclk!=n l:>our
0 0
8 J
0
L?gr,...•._...._ r---.,..IA
IB
K::
Patio Front Elevation As
opposed to imposing modem structures, the simplicity of its volume is in tune with its environment, creating a refined refuge from the bustle of Los Angeles. The house lies in a densely wooded grove. Three L-shape units are arranged in a pattern on a 200 X 100foot lot, forming a courtyard with en closed patios and outdoor fireplaces. Each pair of units opens to outdoor liv ing patios. The third one is formed by the kitchen, guest studio, and garage. The impression is of primitive boxes resting in their natural place. Schindler wrote, "The shape of rooms, their relation to the patios and the alternating roof levels, create an entirely new spatial interlock ing between the interior and the garden." There is almost no difference in level be tween the ground floor and the garden, suggesting an infinite extension to the open ground in accordance with the character of the land. Schindler's "or ganic" building type is fully realized in this house; the house and the outdoors unite in perfect rapport to embrace their extraordinary surroundings.
r
B
Living RM
I
I
I
L.iving RM
(
Patio
o
Living R�l
B §
ID
�
IF
L.iving RM
B
B
IG
Figure 5. Pueblo Ribera Court. Plan: elevations, window, and furniture.
The structural components of the house are simple: concrete walls on one side and two wooden posts from the other side support all ceilings. "All parti tions and patio walls are non-supporting screens composed of a wooden skeleton filled in with glass or with removable 'in sulate' partition. These basic materials are used in a lucid way to form "the cave-
tent shelter of concrete, wood and can vas" which relates the project to the cli mate, the region, and the surroundings. Schindler appears to have erected a mod ern hut, expressing the immemorial re lationship between humanity and nature. It is a true sustainable aesthetic. On the drawings, the dimensions and placements of various spatial
Figure 6. The Kings Road House, 1 921-22. 1 /4 scale model constructed with basswood.
© 2006 Springer Science+Business Media, Inc., Volume 28, Number 1 , 20C>6
43
Figure 7. The Kings Road House, 1 921-22. Courtyard view.
forms and details of the house are con trolled by Schindler's unit system. The 48-inch unit system is clearly identi fied in the plan. A 12-inch ( 1/4 unit module) vertical module is employed. Thus the heights of all window and door mullions are based on a 12-inch module. The rafters of the horizontal wooden structures are regulated ac cording to a 24-inch distance-half the unit module. Most of the major rooms in the con struction drawings are approximately measured in whole numbers, although rooms are frequently not simple rec tangles. Unlike the complexity of the space-forms, these involve surprisingly few dimensions and corresponding ra tios. In the house there are six differ ent ratios of dimensions: 2 : 1 , 3:2, 4:3, 6:5, and 9:8. The noteworthy character of the ratio is that most of the fractions vary by adding 1 to both numerator and denominator. This is equivalent to the classical sequence of subsuperparticu lar numbers (March, 1998). Interest ingly, these room ratios are also found in musical ratios within an octave: oc tave (2: 1 ), second (9:8), fourth (4:3), a minor third (3:2), and a major third (6:5). Schindler himself briefly referred to musical ratios in his early discussion of proportion in his unpublished 1917 Emma Church School lecture notes (Park 2003; March 2003). In addition, March (1994b) used a musical analogy to examine the proportional design of the Schindler house: in his paper "Dr. How's Magical Musical Box" March re calls "architecture as frozen music."
44
T H E MATHEMATICAL INTELLIGENCER
According to March, any analysis of ra tios and proportions of Schindler's houses will inevitably correspond to musical intervals. It does not mean the musicality is intentional but it is a nec essary consequence of the relations of small numbers. Lovell House, 1 922-26 1 242 Ocean Avenue,
Newport Beach, California
Built for Dr. Philip Lovell, this house is the most revealing project in Schind ler's oeuvre, displaying dramatic spa tial complexity as well as structural resolution. Schindler uniquely devised ingenious structures to control and
raise the upper-level sleeping quarters above a flat beach site. In the house, he used five reinforced concrete columns, opening the ground level for play and recreation. Liberating the ground floor recalls one of Le Corbusier's five points, plan libre. Piloti, or columns, in con crete and steel, carry the structural load, lifting the box into space; interior walls are then freely arranged according to cir culation or other functional require ments (Park and March 2003). The 48-inch unit system is clearly identified in the plan with numbers and letters, and the 16-inch vertical module in elevation with grades. The grid is marked along the bottom with num-
2 : 1
9 : 8
6 : 5
4 : 3
3 : 2
2 : 1
2 : 1 Figure 8. The Kings Road House, 1 921-22. Ratios of each room.
bers from 1 to 16, and up the right-hand side with letters A to J. Its vertical mod ule is based on a 16-inch unit system, which controls not only the height of the room but other elements as well, including built-in furniture, chairs, ta bles, windows, doors, and clerestory, providing uniformity of scale and pro portional beauty. In Schindler's words, "Floor plans and elevations were de signed by a scheme of unit lines to as sure uniformity of scale. All woodwork including concrete forms and furni ture, was built of eight inch boards with wide joints." Sarnitz (1986) provides a simple but interesting proportional study of the house. His analysis of the house plans and elevations is based on the belief that subdivisions of square and double square determine the overall propor tional system of the house. How House 1 925
931 North Gainsborough Drive,
South Pasadena
The How House stands out from Schindler's other works of this period in its conspicuous and transparent play about a diagonal axis that overlays a 48-inch unit system. The use of both the modular and symmetrical systems in the How House is subtle. Schindler increases the significance of the diag onal axis by setting the orthogonal lines of the ground plan at a 45° angle to the boundaries of the lot and the road frontage (Park 2000). The lower portion of the building's volume is built in concrete with Schindler's own "slab cast" construction system, while the structure of the upper portion is red wood. The horizontal stratification of the continuous concrete course as well as the battened boards on the wall of the house were laid to coincide with a 16-inch vertical module. The module incorporates the heights of all ele ments of the main structure, as well as built-in furniture, chairs, windows, and doors. The How House is mainly deter mined by reflective symmetry about a diagonal axis. As Schindler writes "the rooms form a series of right angle shapes placed above each other and facing alternately north and south." The analysis focuses on the piano no-
Figure 9. The Lovell House, 1 922-26. Five-layered vertical structures of the Lovell house in tegrate all layered stairs and corridors.
bile-the living room, the dining room, and Dr How's study-where the spatial and structural setting of the whole composition is based on the diagonal axis. However, the floor plan of the pi ano nobile does not conform to sym metry along the diagonal axis in a strict manner: the symmetry is broken by ad ditional spaces such as the kitchen and the entrance hall, and also in certain details such as the principal fireplace and the stairway to the lower, bedroom floor. This asymmetry does not depend on an arbitrariness of personal taste or a rejection of the principles of symme try in the cause of modernism. Instead, it is generated from a disciplined un derstanding of symmetry. The final de sign displays an abundance of symme tries within the parts while negating the strict symmetry of the whole.
The locations of the rooms are clearly arranged according to his sub divisions and multiplications of the unit module. The floor plan of the house demonstrates an interesting propor tional relation with regard to his system. The house is set within a 48-inch unit module, numbered from 10-24 (a 14unit module) and lettered from H-W (again a 14-unit module). All architec tural elements are disposed within this 14-module square along the diagonal axis. In the house, Schindler employs a 16-inch (1/3 unit) vertical module. In RM Schindler-Composition and Construction, Lionel March provided an
in-depth analysis of this house. Accord ing to March, its principal features fol low the classical arrangement known as ad quadratum. First of all, the layout of the house is placed within a 14-module
Figure 1 0. R. M. Schindler, the Lovell House, 1 922-26. Facade.
© 2006 Springer Science+Business Media, Inc., Volume 28, Number 1, 2006
45
this square is the center of the 60-inch by 60-inch open shaft which provides light and ventilation for the bedroom floor from the roof terrace above. The large
5 1/2
X
51/2
module square
is
the
terrace, measured to the outer edge of the planters. The small
%
X
% module
square locates a built-in light fixture in the cantilevered porch for the terrace. The three diagrams, taken together and superimposed, show a sequence of cen tering points along the diagonal axis .
There are two structural layers at the
gallery level
% module apart; the lower
layer, which has the same herringbone pattern as the ceiling of the porch ex tending over the terrace as the pattern on the living room ceiling, and the upper Figure 1 1 . R. M. Schindler, the Lovell House, 1 922-26. Corner view.
layer, which splits into two wings over the dining room and the study, respec tively. Both L-shaped layers are set along
square. Another square covers the prin
suggests that the layout of the house
cipal volumetric elements of the house:
may be ordered from a series of
the living room, dining room, the study,
quadratum arrangements. It also turns out that three
and the roof terrace. This square with comers at V-24 and L-14 is a 1 0 X 1 0
the diagonal axis facing each other in op
ad
posing south and north orientations. In
pairs of
the ingenuity of Schindler's method, al
particular, the lower part demonstrates
concentric squares define conceptually
lowing two partially cantilevered beams
module. From the external comer o f the
the principal "blocks" of the house. First
to cross above the open shaft area with
living room a 7 X 7 square defines the
of all, the living room of the
piano no
out any support in the center.
edge of the dropped ceilings in the din
bile is located on the center point of the
The exposed ceiling structure of the
ing room and study. This is clearly visi
14 X 14 square. It lies in 5 X 5 modules.
living room is particularly worth men tioning. It fills a square plan with a span
ble at the gallery level. The major space
In the plan of the piano nobile, the main
divisions generate a series of propor
volume of the house is planned within a
of 20 feet, or five 48-inch modules. Two
tions, such as 7:5, 10:7, and 14:10, which
9 1/4
chimney stacks on adjacent sides em-
X
91/4 module square. The center of
Figure 1 2. The Lovell House, 1 922-26. 1/4 scale study model.
Figure 1 3. The Lovell House, 1 922-26. Analysis of plan and elevation of the Lovell house (after August Sarnitz).
46
THE MATHEMATICAL INTELLIGENCER
Figure 1 4. R. M. Schindler, The How House, 1 925. Living room roof structure.
Figure 1 5. R. M. Schindler, The How House, 1 925. Interior and furniture designs.
Figure 1 6. R. M. Schindler, The How House, 1 925. Exterior view.
© 2006 Springer Science+ Business Media, Inc . . Volume 28, Number 1 , 2006
47
March (1994b), the ceiling of the How House living room is set out in the man ner of the Greek gnomon: the sequence of squares (1 x 1, 2 x 2, 3 x 3, . . . , 9 x 9) and the sequence of oblongs (2 X 1 , 3 X 2, 4 X 3 , . . . , 9 X 8 ) (Figure 19).
H
0 p
0 R
REFERENCES
March, Lionel and Judith Sheine, eds. 1 994.
RM Schindler-composition and construction . London: Academy Editions.
\Q
11
12
13
14
15
16
11
18
18
20
21
22
2'�
/
u
March, Lionel. 1 994a. "Proportion is an alive and expressive tool. . . . " pp. 88-1 01 in RM
Figure 1 7. The How House, 1 925. Plan analysis of the How House showing the ad quadratum arrangement.
Schindler: Composition and Construction, L March and J Sheine, eds. London, Academy Editions. March, Lionel. 1 994b. "Dr How's Magical Mu
phasize the diagonal axis (Figure 18). The structure of this ceiling is a system of redwood beams at 1/2 module (24inch) intervals: each joist is secured in a wall at one end while the other end is nailed to a longer joist in a her-
ringbone pattern along the diagonal axis. With this arrangement, only one beam is required to span the whole space. The structure of the living room ceiling provides an interesting series of proportional relations. According to
sic Box." Pp. 1 24-1 45 in RM Schindler:
Composition and Construction, L March and J Sheine, eds. London, Academy Editions. March, Lionel. 1 998. Architectonics of Human ism: Essays on Number in Architecture. Lon don: Academy Editions, John Wiley & Sons.
ceiling st ructm·c of the Jh·ing room
The
lliJ I= i= I==
The upper structure
of tlw gaUery floor
ill n
JL
I oo I
a
The lower structure
ur the gallery floor
The pi ano
hl
nobile
b
Il l c
Figure 1 9. The How House, 1 925. Ceiling de Figure 1 8. The How House, 1 925. Axonometric.
48
THE MATHEMATICAL INTELLIGENCER
sign.
March, Lionel. 2003. "Rudolph M. Schindler: Space Reference Frame, Modular Coordina
AUT H OR
tion, and the 'Row ' , " Nexus Network Jour
nal 5, 2: 48-59. Park, Jin-Ho. 2003. "Rudolph M. Schindler: Proportion, Scale and the 'Row'," Nexus
Network Journal 5, 2 : 60-72. Park, Jin - H o. 2000. "Subsymmetry Analysis of Architectural Designs: Some Examples, " En
vironment and Planning 8- Planning & De sign 27, 1 : 1 2 1 -1 36. Park, Jin-Ho and Lionel March. 2003. "Space Architecture:
Schindler's
1 930
Braxton
JIN-HO PARK
Shore project," Architectural Research Quar
Department of Architecture
terly 7 , 1 : 5 1 -62.
lnha University
Park, Jin-Ho. 2004. "Symmetry and Subsym
Nam-gu, lncheon
metry of Form-Making: The Schindler Shel
Korea
ters of 1 933-42 ," Journal of Architectural
e-mail:
[email protected]
and Planning Research 2 1 : 24-37. Sarnitz, August. 1 986. R M Schindler: Architekt
Jin-Ho Park earned his BS in architecture from lnha University, Korea, and his MA and
1887-1 953. Vienna: Edition Christian Brand
PhD Degrees. also in architecture, from the University of Califo rnia at Los Angeles. He
statler.
has taught at the University of Hawaii (Manoa), where he received the University of Hawaii Board of Regent's Medal for Excellence in Teaching in 2002 and the ACSA/AIAS New
Schindler, R. M. 1 946. "Reference Frames in
Faculty Teaching Award in 2003. He is presently associate professor at lnha University.
Space. " Architect and Engineer 1 65 : 1 0 , 40,
His articles have appeared in numerous journals, and he currently serves as corre
44-45.
sponding editor of the online Nexus Network Journal: Architecture and Mathematics.
N E W
I N
P A P E R B A C K
With an introduction by Thomas Banchoff
F LAT LA N D A Romance of Many Dimensions
Edwin Abbott Abbott
Flatland has fascinated generations of readers, becoming a
peren n ial science-fiction favorile. A first-rate ficlional guide
to the concept of multiple dimensions of space, the book will also a ppeal to those who are interested in computer graph ics. In his inlroduction, Thomas Banchoff po i n ts out that there i s no beHer way to begin exploring the problem of understand ing higher-di mensional slicing phenomena than readi ng lhis classic novel of the Victorian era.
Praise for Princeton 's previous edition:
"One of the most imaginative, delightful and, yes, touching works of mathematics, this slender 1 884 boo k purports to
be the memoir of A. Square, a citizen of an entirely two
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(0800) 243407 U.K. 800-777-4726 u.s. math.pupress.princeton.edu
© 2006 Springer Science+ Business Media, Inc . . Volume 28, Number 1 , 2006
49
The Wedd i ng Reuben Hersh Philosophy:
The bride and groom are here, and have
agreed to be wed today, in the presence of several wit ness
s.
Let the wedding begin. Logic, speak to Geometry.
Logic: I come to you, Geometry, my beautiful bride-to
be, as your guide and your elder, your counselor and corrector, your only lord and master, to have and to hold,
for good or for ill, in siclmess and in health, till death do us part.
Geometry:
You come to me, Logic, my groom and my
husband, as my guide and my junior, my counselor and intem1pter, my self-named lord and master, to hold, for goo d
or for
to have and
ill, in siclmess and in health, till
the unlmown and unknowable, whatever is to come at as
your guide and your
elder, your counselor and corrector, your only lord and master, to have and to hold, for good or for ill, in sick ness and in health, till death do us prut?
G: I do, with a rese1vation.
Philosophy: We will hear your explanation. G: I am the elder, you are the younger, Lord
L: Easy for you to say. Who will pay the bills?
G:
Oh, Logic, you are hopeless. The bills! Is this love or
business?
L: G:
It'
the
business
of love and the love of business.
Another paradox! I hate your silly little paradoxes! I
hate them! Why did I ever agree to do this?
L: Because you need me.
G: Yes. It's true. I ne d you. But why and for what?
L: Without me to look after you, what would become of
G: I
don't know. What would become of me?
L: Look at your wretched, lo t cousins. Weather and Financial Prediction. great-grand-children. \\lhere ar
umerology.
Old Pythagoras's
they now?
L: Yes. In the Gutter.
Logic.
your only reservation?
L:
Don't worry about them. They don't understand any
thing, anyhow.
G: In the Gutter.
P: Let it be so recorded, Geometry is the elder. Is that
G: It could be.
G:
you?
the end of our joint dominion.
L: Do you, Geometry, take me
Census Bmeau, and the Atomic Energy Commission are here.
G: Why are they all in the Gutter?
L: Because they tried to live without Logic. G: Will you be good to me? Wi ll you be kind? ever be kind?
nless you wish to hear more.
You may go on.
L: I will be good for you. I will be as kind as I
Will you am
able
G: You bring me strength and control. You bring me your
to be.
beautiful nun1bers. I an1 happy to be yow· bride.
place, in the presence of witnesses both honorable and
L: This is well explained.
dishonorable, Logic and Geometry were lawfully wed
darling and precious offspring,
G:
1, 2, 3,
and many other
I bring you shape, fom1, and continuity. The possi
bility of fruitful intercourse, and offspring uncountabl
L: You lmow I love you, Geometry.
G:
I know you love me, Logic. I do not know if I
you. You
ru·e
.
love
hru·d. You are unforgiving. You think you
L: Look, is this a wedding, or what'?
It was announced and scheduled as a wedding. TI1e
guests
ar
here, with
Let it be ordered and written that on this
gifts.
L: Yes, beloved friends and frunily have come. Mechan
ics and Statistics, of course. Even the Stock Market, the
day, in th is
ded, to have and to hold, for good or for ill, in sickness and in health, till death do them part. And may God some day
can
can be my lord and master.
G:
P:
�
forgive
believe
� was
us for what we do here today. Hermann Weyl who once
said that the angel Algebra and the
devil Topology were struggling for the soul of Mathematics.)
Department of Mathematics and Statistics University of New Mexico Albuquerque, NM 871 3 1 USA e-mail :
[email protected]
© 2006 Springer Science+Business Media, Inc., Volume 28, Number 1 , 2006
51
lil§iit§'MJ
O s m o Peko n e n , Ed itor
I
Erich Kahler. Mathematische Werke. Mathematical Works edited by Rolf Berndt and Oswald Riemenschneider BERLIN, WALTER DE GRUYTER, 2003. 971 PP. €228.00 ISBN 3- 1 1 -0 1 7 1 1 8-X REVIEWED BY ERNST KUNZ
Feel like writing a review for The Mathematical Intelligencer? 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.
Column Editor: Osmo Pekonen, Agora Center, P . O . Box 35, University of Jyvaskyla, Fl-4001 4 Finland e-mail:
[email protected]
52
T
his volume of almost one thousand pages aims at describing the life, activities, and convictions of Erich Kahler ( 1906-2000), one of the great and many-sided mathematicians of the
20th century, and a strong and fasci nating personality. It also deals with the developments that have grown out of his work Nearly 700 pages contain most of Kahler's original publications in math ematics and mathematical physics and three of his philosophical essays. The remaining 300 pages describe his life and contain surveys by experts in the fields in which Kahler was active, de scribing the impact of his work on present-day mathematics, physics, eco nomics, and other areas of intellectual life. That this part of the book is so large shows the great influence that Kahler's work still has. The editors de serve high appreciation for having col lected the pieces to compose a rich pic ture of modem science. Of Kahler's 399-page book Geome tria aritmetica, volume 45 of Annali di Matematica, which was considered by many of his students, by many math ematicians, and maybe by Kahler him self as his main work in mathematics, only the introduction is reproduced. But its content is described in the long article "Kahler Differentials and Some Applications in Arithmetic Geometry" by R. Berndt, who has also translated much of the text from Kahler's unus ual philosophically motivated language into modem mathematical terminal-
THE MATHEMATICAL INTELLIGENCER © 2006 Springer Scrence+Business Media, Inc
ogy. Kahler had anticipated the publi cation of the book by surveys given at conferences in various languages. The book shows Kahler as a forerunner of the theory of schemes, as was recog nized by Grothendieck in his Elements
de geometrie algebrique ("parmi les travaux qui se rapprochent de notre point de vue en Geometrie algebrique, signalons l'important travail de E. Kahler"). At a memorial colloquium in honor of Kahler at the University of Leipzig, Kahler's former collaborator H. Schumann reported that Kahler con sidered the Geometria aritmetica a first step towards a more comprehen sive theory but abandoned the project when the first volumes of Groth endieck's work appeared. A. Weil ex pressed concern in Mathematical Re views that the unconventional style and the unusual language would pre vent the book from receiving the at tention it deserved ("the author seems to have done everything in his power to discourage prospective readers and is only too likely to have succeeded"), and so it turned out. History has fol lowed the French school. Kahler's main objective in Geome tria aritmetica was to unite number theory and geometry. He emphasized that for this purpose it was necessary to study arbitrary local rings, not only those containing a field, as was done in classical algebraic geometry. Con trary to Grothendieck's more general point of view, Kahler's schemes (called ''varieties") are contained in "algebraic" fields, that is, fields which are finitely generated over their prime field. He also tried to embed both disciplines into a philosophical system which was con ceived by him but not well understood by others, and in fact hardly under standable for the reader without further instruction. An explanation of his system, a mathematical model of the world, the universe of all existing things, is given by him in the essay "Wesen und Erscheinung als mathematische
Prinzipien der Philosophie" (Essence
colleagues for a long time considered
equations
and Appearance as Mathematical Prin
Kahler as a figure of the first third of
Maxwell equations and the Dirac equa
ciples of Philosophy) and in other pub
the 20th century and that they were not
tion and some of their solutions in
lications which are not contained in
aware that he was still alive and active
terms of differential forms.
of
physics
such
as
the
the collected works. In his paper "In
in the 1980s. In a "Tribute to Berm
As a rule, the sciences select or de
fmitesimal-Arithmetik" which appeared
Erich Kahler" S. S. Chern appreciates
velop the tools from mathematics they
Geometria aritmetica,
the influence that Kahler had on his
need to describe and to solve their
own work
problems. Kahler's thinking went also
four years after
maybe as a reaction to previous critics, Kahler gave an outline of its contents
In recent decades Kahler manifolds
in the other direction. He was con
in ordinary mathematical language (ex
have entered theoretical physics again
vinced that the powers of highly de
cept that the local rings having a given
in the construction of models to unify
veloped disciplines such as arithmetic
faces
the fundamental laws of physics. In his
and algebra were not sufficiently used
(Seiten) of the field and valuation rings the perfect faces). The mathematical
article "Supersymmetry, Kahler Geom
in physics ("Number theory is deter
etry and Beyond" H. Nicolai reports on
mined to become the leading power of
discipline to which the book belongs
these developments.
natural sciences"; "What is the mean
quotient
field
are
called
the
ing for natural sciences of the re
and in which some of the most spec
Since early in his career Kahler's
tacular successes were achieved in re
work made extensive use of Cartan's
sources collected in algebra and num
cent years has adopted the name of the
theory of exterior differential forms,
ber
book: Arithmetic Geometry. R. Berndt,
which he refined and recreated and
specific, he supported his views some
and in another essay J. B. Bost ("A Ne
generalized in purely algebraic terms.
times
glected Aspect of Kahler's Work in
Many of his publications on the so
speculations. It seems that theoretical
Arithmetic Geometry: Birational Invari
called Kahler differential forms are de
physicists have adopted few of these,
ants of Algebraic Varieties over Number
an exception being Kahler's (quater
connection with the mathematical con
voted to this goal. His book Ein fiihrung in die Theorie der Systeme von Differentialgleichungen (Introduc
cepts which were introduced by Kahler
tion to the theory of systems of differ
Lorentz group of classical relativity
ential equations) of 1934, which is not
theory.
reproduced in the
collected works,
Nicolai's contribution that the new Poincare group is identical with the de
Fields") describe some developments in
in what was called by some his
magnum. there is a
opus
In another article by Berndt
theory?"). with
Without
bold
being
very
conjectures
and
nionic) "new Poincare group" which according to him should replace the
It is mentioned in a footnote
of
discussion of "Kahler's Zeta
uses differential forms. It became the
Function," and A Deitmar ("A Panorama
origin of what is now called Cartan
Sitter group from theoretical physics.
of Zeta-Functions") gives an overview of
Kahler theory, with the theorem of Car
The mathematical properties of the
results and conjectures about zeta func
tan-Kahler as its fundamental result.
new Poincare group are discussed in
tions in general.
This part of Kahler's work is thor
an article by A Krieg.
Much better known and much more influential than
Geometria aritmetica " Ober
oughly discussed in a report by R. Berndt and 0. Riemenschneider, which
Kahler,
who
was
influenced
by
Plato's concept of "ideas," Leibniz's monadology, Hegel, and Nietzsche, gave
be
contains a much more detailed survey
Metrik"
of all of Kahler's mathematical results
university courses in philiosophy. His
(On a remarkable Hermitian metric)
than can be given here. In another es
friends report that he considered his
was
Kahler's
merkenswerte
note
eine
Hermitesche
from 1932, which developed into a
say by I. Ekeland a problem of eco
texts combining mathematics with phi
large and important discipline-Kah
nomics is explained which leads to a
losophy and theology as more impor
lerian geometry, with "Kahler mani
system of partial differential equations
tant than his mathematical achieve
fold"
concept-which
of first order and which can at present
ments, and that he suffered from the
had strong influence on many areas of
be solved only by means of the Cartan
fact that they did not receive the ap
mathematics
Kahler theorem.
preciation he had hoped for. For him, at
as
its
main
(differential
geometry,
complex analysis, algebraic geometry)
Kahler wrote influential papers on
least in his later years, mathematics was
and mathematical physics (relativity,
the singularities of complex functions
a language designed to formulate and to
field theory). Here many notions have
in two variables. W. D. Neumann re
solve not only problems of the sciences
their origin in this paper of 14 pages,
views these publications and puts them
but much more ("mathematics as a
written by him at the age of 27, and
in the perspective of the topological
thread of Ariadne which guides us out
bear his name as in Kahler metric, Kah
and algebraic
classification first of
of the labyrinth of modem sciences"). I
ler-Einstein metric, Kahler potential,
plane curve singularities and then of
quote also from the article "The Life of
Kahler class, etc. In his report "The Un
isolated hypersurface singularities.
Erich Kahler" by R. Berndt and A
abated Vitality of Kahlerian Geometry"
J.
Kahler's
work
in
mathematical
Bohm: "The ingenious mathematician
P. Bourguignon describes Kahler's
physics started with three papers on
became a mathematical dreamer who
paper and gives a detailed survey of all
the 3-body and n-body problem and
thought he could solve all problems of
one on fluid dynamics. Later work is
this world by mathematical methods."
developments
since
the paper was
written. He confesses that he and many
devoted to
expressing
fundamental
I do not feel competent to comment
© 2006 Spnnger Science+Bus1ness Media, Inc., Volume 28, Number 1 , 2006
53
on the philosophy of Kahler. He calls mathematics "an infinite refmement of language." In the collected works there are several attempts to guide the reader to his philosophical thinking. First there are three essays by Kahler himself, two in Italian, and the one which was men tioned above. References to his other philosophical texts are given. Moreover two articles "Kahler's Vision of Mathe matics as a Universal Language" by R. Berndt and "An Approach to the Phi losophy of Erich Kahler" by K. Maurin serve the same purpose. Kahler, who had a strong sense of mission, started his independent sci entific production at the age of 17 and remained active until shortly before his death at 94. The principal stages of his academic career were: Doctor's degree at the Uni versity of Leipzig under Leon Lichtenstein 1930 Habilitation in Hamburg 1931- As a Rockefeller fellow in Rome, he met Enriques, Cas1932 telnuovo, Levi-Civita, Severi. B. Segre, and A. Weil 1936 Ordinary professor in Konigs berg 1948 Successor to Koebe in Leipzig 1958 After serious political tensions with the East German administration Kahler ac cepted a position at the Technical University of West Berlin 1964 Successor to Emil Artin in Hamburg 1974- Professor Emeritus, work2000 ing mainly on his Mathe matical Philosophy and occasionally lecturing at physics and mathematics conferences. 1928
The book under review does not con a list of Kahler's former PhD-stu dents. Here is one, maybe incomplete: Walter Thimm (Konigsberg 1939), Gtin ter Hauslein (Leipzig 1955), Gerhard Lustig (Leipzig 1955), Armin Uhlmann (Leipzig 1957), Rolf Berndt (Hamburg 1969). Kahler also initiated the theses of Gtinther Eisenreich (Leipzig 1963) and Horst Schumann (Leipzig 1968), but tain
54
THE MATHEMATICAL INTELLIGENCER
since he had left East Germany he could not act formally as advisor. It seems that Kahler's philosophical texts have been widely ignored. As mathematicians, we admire his numer ous contributions to our science which have come to bear so much fruit inside and outside mathematics. With his col lected works, the editors have given him a worthy monument. Fakultat fUr Mathematik U niversitat Regensburg D-93040 Regensburg Germany e-mail:
[email protected]
History and Science of Knots edited by J. C. Turner and P.
van de Griend
SINGAPORE, WORLD SCIENTIFIC, 1 996. 464 PP. US$78, ISBN 981 -02-2469-9 REVIEWED BY KISHORE B. MARATHE
T
he book History and Science of Knots consists of 18 chapters grouped into 5 parts: I. Prehistory and antiquity, II. Non-European traditions, III. Working knots, IV. Towards a sci ence of knots?, V. Decorative knots and other aspects. The editors, Turner and van de Griend, have chosen a wide va riety of experts as authors for the in dividual chapters; more information about them may be found in "About the Authors" on page 419. The con struction and use of knots, links, and braids from string-like objects pre dates known human history. They oc cur in diverse areas of human activity, from magic tricks and decorative arts to shipping, fishing, and religious and medical practice. Such structures also occur in nature, for example, in poly mer chemistry and biology. In spite of the vast range of topics and the time frame, the book gives a representative sampling of knots and knot applica tions. The historical aspects of each topic are also discussed. Knots are dis-
cussed from this broad perspective in all but one chapter. Only chapter 1 1 is devoted to knots as defined in mathe matics. We will discuss the contents and highlights of each part and give a more detailed look at the part dealing with the mathematical theory of knots. Part I deals with evidence for the use of knots from prehistoric times to the Egyptian civilization. The chronol ogy of the knot technology over this vast period is summarized in Chapter 1 by studying various aspects of human and animal life which may have re quired the use of knots. Chapter 2 is devoted to some speculations regard ing the earliest forms of knots and their origin. Chapters 3 and 4 are written by archaeologists who examine the knot ted structures found at many excava tion sites in Europe and Egypt. The ear liest such structure is a Mesolithic fish-net fragment found in 1913 on the Karelian isthmus (formerly in Finland) which uses a knot type used in Estonia and in the Finnish settlement. It is now called the "Antrea Russian Knot." The evidence of rock art and artifacts using knots in all major ancient civilizations, extends over thousands of years. This and other finds lead us to conclude that our prehistoric ancestors had both the materials and the skills for making complicated knots. Part II has three chapters which give a sample of knot history in non-Euro pean civilizations. The use of knotted cords or quipus by the Incas is detailed in Chapter 5. There is an interesting discussion here of the construction of knotted cords and their use in repre senting numbers and storing numerical data. Chapter 6 traces the rich history of knots in China covering a period of some eighteen thousand years. The main theme in this chapter is the de scription of the decorative use of knots which began in ancient China and con tinues to this day. Beautiful examples of such knots were much in evidence during ICM 2002 held in Beijing. Knots used by Inuit Eskimos are the subject of Chapter 7. I would like to note that knots and braids made by using strings and organic materials have been (and are now) used in religious functions in India since before the Vedic period. It
would be very interesting to study this, as excellent written sources are read ily available. Part III presents a historical account of two fields where knots have been extensively used. Chapter 8 deals with knots and ropes used by seamen and fishermen. The knowledge and use of knots was substantially affected in the transition of man from a land-dweller to a mariner. There is vast literature dealing with the use of knots and ropes at sea. The author has managed to give a very good presentation of this sub ject in just a few pages. Chapter 9 dis cusses what the author calls life-sup port knots. It covers the use of knots in rock climbing, rescue work etc., with particular attention to their prop erties in life-support tasks. Part N is the longest, with five chap ters. Behaviour, under load, of single stranded knots tied in fiber rope is studied in Chapter 10. Chapter 12 is de voted to classification of knots by methods different from those used in topology. Various encyclopedias of knots are also briefly described. The work of Mandeville on trambling (i.e. producing sequences of knots by alter ing one tuck at a time) is dealt with in Chapter 13. The last chapter is devoted to the history and techniques of cro chet work. I found the discussion of the CADD (Computer Aided Doily Design) system and its applications developed by the author and her coworkers at the University of Waikato, New Zealand, quite interesting. Chapter 1 1 should be of greatest in terest to the readers of this magazine. To a mathematician, a knot is an em bedding of a circle in the three-di mensional Euclidean space R3 or its compactification, the 3-sphere S3. This definition is easily modified to obtain knots in any manifold. In particular, embedding of the standard unknotted circle is called the unknot. A system atic study of knots was begun in the second half of the 19th century by Tait and his followers. They were moti vated by Kelvin's theory of atoms mod eled on knotted vortex tubes of ether. It was expected that physical and chemical properties of various atoms could be expressed in terms of prop-
erties of knots, such as the knot in variants. Though Kelvin's theory did not work, the theory of knots grew as a subfield of combinatorial topology. Tait classified the knots in terms of the minimal crossing number of a regular projection. Recall that a r·egular pro jection of a knot on a plane is an or thogonal projection of the knot such that at any crossing in the projection exactly two strands intersect trans versely. Tait made a number of obser vations about some general properties of knots which have come to be known as the "Tait conjectures." In its sim plest form the classification problem for knots can be stated as follows: Given a projection of a knot, is it pos sible to decide in infinitely many steps if it is equivalent to an unknot. This question was answered affirmatively by W. Haken in 196 1 , who proposed an algorithm which could decide if a given projection corresponds to an unknot. However, because of its complexity it has not been implemented on a com puter even after 40 years. The simplest combinatorial invariant of a knot K is the crossing number c(K), defmed as the minimum number of crossings in any regular projection of the knot K. The classification of knots up to crossing number 16 is now known [2]. The crossing numbers for some spe cial families of knots are known, but the question of fmding the crossing number of an arbitrary knot is still unanswered. Another combinatorial invariant of a knot K that is easy to define is the un knotting number u(K), the minimum number of crossing changes in any pro jection of the knot K which makes it into a projection of the unknot. Upper and lower bounds for u( K) are known for any knot K. An explicit formula for u( K) for a family of knots called torus knots, conjectured by Milnor nearly 40 years ago, has been proved recently by a num ber of different methods. One of the earliest investigations in combinatorial knot theory is contained in several unpublished notes written by Gauss between 1825 and 1844 and pub lished posthumously as part of his estate. They deal mostly with his attempts to classify Tracifiguren or
plane closed curves with a finite num ber of transverse self-intersections. Such figures arise as regular plane pro jections of knots in R3. However, one fragment deals with a pair of linked knots. In this fragment of a note dated January 22, 1833, Gauss gives an ana lytic formula for the linking number of a pair of knots. This number is a com binatorial topological invariant. As is quite common in Gauss's work, there is no indication of how he obtained this formula. The title of the note "Zur Elec trodynamik" ("On Electrodynamics") and his continuing work with Weber on the properties of electric and magnetic fields leads us to guess that it origi nated in the study of the magnetic field generated by an electric current flow ing in a curved wire. Maxwell knew Gauss's formula for the linking number and its topological significance and its origin in electromagnetic theory. In ob taining a topological invariant by using a physical field theory, Gauss had an ticipated Topological Field Theory by almost 150 years. Even the term topol ogy was not used in his era. It was in troduced in 184 7 by J. B. Listing, a stu dent and protege of Gauss, in his essay "Vorstudien zur Topologie" ("Preliminary Studies on Topology"). Gauss's linking number formula can also be interpreted as the equality of topological and analytic degree of a suitable function. Thus it can be con sidered as an example of an index the orem. Starting with this, a far-reaching generalization of the Gauss integral to higher self-linking integrals can be ob tained. This forms a small part of the program initiated by Kontsevich [3] to relate topology of low-dimensional manifolds, homotopical algebras, and non-commutative geometry with topo logical field theories and Feynman di agrams in physics. The Alexander polynomial provided a new type of knot invariant. There was an interval of nearly 60 years between the discovery of the Alexander polyno mial and the Jones polynomial. Since then, a number of polynomial and other invariants of knots and links have been found. A particularly interesting one is the two-variable polynomial generaliz ing both the Alexander polynomial and
© 2006 Springer Science+ Business Media, Inc., Volume 28, Number � , 2006
55
the Jones polynomial. This polynomial is called the HOMFLY polynomial (a name formed from the initials of au thors of the article [ 1 ]). The author of this chapter gives an excellent account of the history of topological knot theory and related theory of braids, bringing the readers up to the mid-1990s. Some important recent develop ments are not included in this chapter. Jones's work in the 1980s was a major advance in knot theory, leading to the resolution of several of the longstand ing Tait conjectures. However, it did not resolve the chirality conjecture: If the crossing number of a knot is odd, then it is chiral (i. e. , not equivalent to it,s mirror image). A 15-crossing knot which provides a counter-exam ple to the chirality conjecture is given in [2]. Jones did not provide a geomet rical or topological interpretation of his polynomial invariants. Such an in terpretation was provided by Witten [6], who applied ideas from Quantum Field Theory (QFT) to the Chem Simons Lagrangian. In fact, Witten's model allows us to consider the knot and link invariants in any compact 3manifold M. Witten's ideas have led to the creation of a new area called Topo logical Quantum Field Theory (TQFT) which, at least formally, allows us to express topological invariants of man ifolds by considering a QFT with a suit able Lagrangian. An account of several aspects of the geometry and physics of knots may be found in [4] and [5] . Recently, several topological and geometric invariants of knots and links have been used in polymer chemistry and in studying the mathematical struc ture of DNA. These early results have led molecular biologists to believe that knot theory may play an increasingly signifi cant role in understanding the geomet ric and topological properties of DNA and that these in turn may help in re solving some of the riddles encoded in these basic building blocks of life. Un derstanding the structure and dynamics of DNA, RNA, and proteins, in general, may very well require the forging of new mathematical tools. The last part, V, deals with decora tive knots and their use as symbols in heraldry and love. The history of
two widely practiced crafts, namely macrame and lace is given in Chapters 15 and 16, respectively. Chapter 17 de scribes the various ways knots have been used in European Heraldry. The final chapter deals with the concept of love knot, its occurrence in literary works and the various forms that it has taken over the last five centuries or so. Indeed the most widely used synonym for marriage is "tying the knot." I enjoyed browsing through all the chapters. They contain material that a mathematician would not normally come across in his work There is a well-known story about Alexander the Great unraveling the Gordian knot with his sword. Today's problems in knot theory, mathematical or otherwise, will require tools far more sophisti cated than a sword. BIBLIOGRAPHY
[1]
R.
Freyd, et a/., A new polynomial in
variant of knots and links. Bulletin of Ameri
can Mathematical Society (New Series, 1 2 :239-246, 1 985. [2] J . Haste, M . Thistlethwaite, and J . Weeks. The First 1 , 701 ,936 Knots. Mathematical ln
telligencer, 20(4):33-48, 1 998. [3] M . Kontsevich.
Feynman Diagrams and
Low-Dimensional Topology. In First European
Congress of Mathematics, vol. I I Progress in Mathematics, 1 20, pages 97-1 2 1 , Berlin, 1 994. Birkhauser. [4] K. B. Marathe, G. Martucci, and M . Fran caviglia.
Gauge
Theory,
Geometry
and
Topology. Seminario di Matematica deii'Uni
versita di Bari, 262 : 1 -90. [5] Kishore B. Marathe. A Chapter in Physical Mathematics: Theory of Knots in the Sci ences. Mathematics Un/imited-2001 and
Beyond, pages 873-888. Springer-Verlag , Berlin, 2001 . [6] E. Witten. Quantum Field Theory and the Jones Polynomial. Communications in Math
ematical Physics, 1 2 1 :359-399, 1 989. Max Planck Institute for Mathematics in the Sciences, Leipzig, and Department of Mathematics 1 3 1 7b Ingersoll Hall City University of New York Brooklyn College 2900 Bedford Ave. Brooklyn, NY 1 1 2 1 0-2889, USA e-mail: kbm@sci. brooklyn.cuny.edu
© 2006
Codebreakers: Arne Beurling and the Swedish Crypto Program during World War II
b y Bengt Beckman
PROVIDENCE, Rl, AMERICAN MATHEMATICAL SOCIETY, 2003. 259 PP
,
US $39, ISBN 0·821 8-2889-4
REVIEWED BY HAKAN HEDENMALM
T
he author of this book, Bengt Beck man, is one of early members of the Swedish cipher bureau FRA of Forsvarsstaben (Defense Staff Head quarters), which was operational in 194 1 but officially founded a year later. By the time Beckman came to FRA as a conscript in 1946, there was much discussion of Arne Beurling (1905-1986), the mathematics professor who had made himself famous at FRA by break ing the German code of the Geheim schreiber (developed by Siemens in the 1930s) at a time pivotal for Sweden during World War II. This feat is of the same order of magnitude as the British effort to break the Enigma code dur ing the same war. In England as well as in Sweden, mathematicians played a vital role for the intelligence deci phering part of the war effort; perhaps the most famous mathematician work ing for the British at Bletchley Park was Alan Turing. The Swedish effort to keep a low profile with regard to this intelligence gathering was quite suc cessful; indeed, the importance of Beurling's contribution to Sweden's ability to keep out of the war was, un til recently, known only to narrow cir cles inside Sweden. By now, more than sixty years have passed, and the veil of secrecy has been lifted; Beckman, who stayed with FRA until 1991, is able to tell the story as he remembers it, from what he picked up a long time ago, as well as from recent in-depth interviews with people more closely involved. This story first aired in a 1993 Swedish Television documentary G sam i hemlig (G as in secret), pro duced by Beckman and Olle Hager. In the book, Beckman is able to tell a
Springer Sc1ence+Business Media, Inc., Volume
28,
Number
1, 2006
57
much more detailed story, of course. With the translation to English, com missioned by the American Mathemat ical Society, the material is now avail able to a considerably wider audience. The translator, Kjell-Ove Widman, who himself has been involved in cryptol ogy at the Swiss company Krypto AG, has done a very thorough job and pro duced a very readable text. However, the style differs a little from the origi nal: Beckman tells a story by the camp fire, but Widman's translation sets higher literary standards. Having for the moment assumed a slightly critical stance, let me also mention that the map of Stockholm on the page follow ing xviii places Karlaplan a bit off, somewhere in the forest area north of the Royal Institute of Technology (KTH); the mark should be placed a lit tle bit further southeast. It is also un fortunate that the suggested explana tion of Beurling's analysis is marred by cipher typos (on pp. 80, 82). The book takes a historical per spective on ciphers and begins with an exercise to decipher a coded message from the 18th century. This is quite en joyable; the cipher is of simple substi tution type, and it is just a matter of running a frequency analysis to guess the most common letters of the cipher key. Then the perspective changes a lit tle, and automatic ciphering machines enter the picture, along with the Swedish names Damm, Hagelin, and Glyden. Then, a description of radio signal interception and cryptanalysis before 1939 follows. The Kingdom of Sweden was poorly prepared for the war that broke out on the European continent on September 1, 1939. The situation became particu larly dire on April 9, 1940, when Ger many invaded neighboring Denmark and Norway. As Sweden could not af ford a massive military build-up, it be came imperative to be able to second guess the German intentions regarding Sweden. Then Sweden's foreign policy could be modified to be more palatable for the Germans, and hence avoid ac tual invasion. This was done quite suc cessfully-German shipments of mate rials and supplies as well as of troops were permitted in sealed transit trains through Sweden-and it is commonly
58
THE MATHEMATICAL INTELLIGENCER
believed that this is the reason Sweden was able to remain outside the war. But this is not the whole story. The invasion of Norway offered Sweden the chance not to second-guess but to actually read the potential enemy's cards. The diplomatic traffic between occupied Oslo and Berlin was trans mitted along Swedish telegraph lines, and the Swedes were able to tap the messages. The only problem was that the messages were not in plain text; moreover, the encryption was not of simple substitution type, as could be seen from a simple frequency analysis. Given the sheer volume of encrypted traffic, it was suspected that a machine was doing the encryption automati cally. One day in 1940 Beurling, who had already been involved with some simpler cryptanalysis tasks for the Swedish Defense Department, col lected the tapped telegraph traffic at the Karlaplan office dated May 25 and May 27, 1940, which he believed to be essentially free of transcription errors. After a couple of weeks, he had more or less cracked the code. This was an impressive feat, especially compared with the British Enigma effort, which was based on the physical capture of an encryption machine from the Ger mans. If we think of the Geheim schreiber encryption as a kind of sub stitution cipher, then the cipher key apparently was changing with each new letter of the message. Also, the ini tial key settings were altered every few days. The way the Geheimschreiber was made, it would not begin cyclically repeating its cipher on any given mes sage, for the number of possible en codings was much much bigger than the total quantity of information ex changed over the entire war. Beurling never revealed how he per formed his feat; he would say that a magician never reveals his tricks. Nev ertheless, Beckman offers a possible explanation, based on a reconstruction attributed to Carl-Gosta Borelius. The encryption may have been perfect in theory, but in practice telegraph lines were not 100 percent reliable in those days, so the German telegraphers would frequently rerun parts of the message, using the same code. This al lowed Beurling to get a foot in the
door, and using some sound hypothe ses regarding the nature of possible codes on teleprinters (where each let ter corresponds to a sequence of five Os and Is )-essentially combinations of permutations and transpositions he was able to complete his task It should be noted that Beurling did this with a rather small data sample, and without actually having seen a Geheimschreiber. Today a Geheim schreiber is on display in the Beurling library of the Mathematics Department at Uppsala University in Sweden, where Beurling worked in the 1940s. At first, the Swedes carried out the deciphering manually, in accordance with Beurling's instructions, but later, and certainly by 1942, machines called Apps-were doing the job. The value of having cracked the Geheim schreiber depreciated toward the end of the war. The Germans sensed that their transmissions were being read, and reacted to it. By that time, how ever, the risk that Sweden would get dragged into the war was much re duced. Beurling was a deep mathematician equipped with a difficult temperament. The stories about his disagreements, rows, or even outright fights with col leagues are widely known in mathe matical circles in Scandinavia. Some of these stories are retold in this book Beurling was apparently quite charm ing to the ladies, and this aspect of his life, based on interviews with Anne Marie Yxkull Gyllenband, takes up a chapter. He married twice. His first wife is not mentioned by name in the book, but it is known that she worked as a physician, and Beurling had two children with her. Later, in 1950, he met his second wife, Karin Lindblad, at the party his student Lennart Carleson hosted to celebrate his thesis defense at Varmlands nation, one of the student clubs in Uppsala. As far as I know, Karin was a friend of Carleson's, and was Forste Kurator at Viirmlands at the time, the highest post a student could assume at a student club in Uppsala. Karin and Arne remained together for the rest of their lives. Beurling worked in three areas of mathematical analysis: potential the ory, harmonic analysis, and complex
analysis. His collaborators were few
planted genetics. After a 1948 national
but well chosen: Ahlfors, Deny, Helson,
conference
and Malliavin. Probably it was his im
koism the only true Soviet biology,
pressive collaboration and deep friend
dozens of geneticists lost their jobs
ship with Lars Ahlfors that landed him
and were exiled from the laboratories
the position of permanent member at
to sheep-breeding farms at the end of
the Institute for Advanced Study in
the empire; and genetics was purged
Princeton, New Jersey, in
1954.
In
Princeton, he took over the office pre
that
proclaimed
Lysen
from university curricula. In physics, too,
several scientists joined
ideo
viously occupied by Albert Einstein. It
logues in an attack on quantum me
is unfortunate that Beurling was not
chanics and relativity theory. The at
EM I N ENT MATH EMAT I C IAN S T HE ART OF CONJ ECTU R I N G,
TOGETH ER WITH " LETTER TO A F RI E N D ON S ETS IN COU RT TE N N I S" J acob Bernoulli translated with an introduction ond notes by Edith Dudley Syllo
able to continue on American soil the
tack nearly had disastrous results for
extremely productive period he en
physicists, many of whom were forced
joyed in Uppsala in the 1940s, with
to abandon research. In both of these
students such as Goran Borg, Lennart
cases, questions of epistemology, class
"Bernoulli's The Art of Conjecturing
Carleson, Yngve Domar, Carl-Gustav
struggle, and other issues of impor
of the outstanding texts in the history
hos always been recognized as one
of probability, marking a d ramatic
Esseen, and Bo I\jellberg. Had he been
tance in the official Soviet philosophy
able to wield more influence in Prince
of
ton, the period of abstraction in math
played a role, as did Cold War pres
ematical analysis, which was such a
sures to promote a new, Soviet science
it was."
dominant theme
and
that was different from, and better than
-James Franklin,
1960s, might have been balanced by a
that in the West, particularly in the
deep and elegant approach that fo
United States.
$70.00 hardcover
in the
1950s
cused not on form but on content.
science,
dialectical
materialism,
In a superb contribution to the his
Bengt Beckman has produced a fas
tory and philosophy of science, Slava
cinating book that acquaints us with
Gerovitch considers the place of cy
some of the stars of 20th-century Scan
bernetics in Soviet philosophical dis
dinavian mathematics. In addition, we
putes, and the development of what
gain some insights into basic cryptol
he calls "cyberspeak" in the postwar
ogy.
S-1 0044 St ockhol m , Sweden
From
Newspeak
to
Cyberspeak,
he defines
REVIEWED BY PAUL JOSEPHSON
M
is now
focusing on human-machine interaction in the Soviet space program.
US $42 ISBN 0-262-07232-7
ARTH U R CAYLEY
Mathematician Laureate of the Victorian Age Tony Crilly Arthur Cayley
( 1 82 1 - 1 895)
the most
bernetics for some time, and beyond
CAMBRIDGE, MA, THE MIT PRESS, 2002. xiv + 369 PP
U niversity of South Wales
was one of
The Royal Institute of Technology
by Slava Gerovitch
comprehensive and revolutionary work
USSR. Gerovitch, a research associate history of Soviet computing and cy
From Newspeak to Cyberspeak. A History of Soviet Cybernetics
translation, it becomes clear what a
at the Dibner Institute, has studied the Department of Mathematics
e-mail: haakanh@mat h . kth.se
development in the theory. With Syllo's
as
Cyberspeak
a universal language of
man-machine metaphors described in
prolific and important mathemati cians of the Victorian era. His influence sti II pervades modern mathematics,
such terms as information and feedback
in group theory (Cayley's theorem),
and control, e.g., the organism as an en
matrix algebra (the Cayley-Hamilton
tropy-reducing machine, the computer as a brain, the brain as a computer.
theorem), and invariant theory, where he mode his most significant contribu tions. Tony Cril ly, the world's leading
Gerovitch joins several other scholars in
a uthority on Cayley, provides the first
rejecting the notion that cybernetics was
definitive account of his life. $69.95 hardcover
severely damaged by ideological in terference in the form of such official pronouncements that cybernetics was a "reactionary pseudo-science. " That in terference
was relatively short-lived,
ost readers know of the impact
and scientists learned how to manage it.
of ideological interference in the
Yet the fact that attacks were short
practice of Soviet scientists. In the case
lived and ignorant should not lessen
of biology,
the peasant agronomist
our appreciation of the way in which
Trofun Lysenko rose to the top of the
they reflected the dangers of doing sci
scientific establishment with Stalin's
ence in the USSR generally. Those who
personal endorsement. He advanced
carried on the anti-cybernetics campaign
Lamarckian notions of the influence
employed accepted ways of discourse
of acquired characteristics that sup-
and dispute to rescue their careers and
S I R WI LLIAM ROWAN HAM I LTO N
Thomas
L.
Hankins
"This is an interesting, well-written biography of the great nineteenth century mathematician."
-Mathematical Reviews $22.50 paperback
TH E JOH N S H O P K I N S U N IVERS ITY PRESS 1 -800-537-5487 www. press.jhu.edu •
© 2006 Springer Science+ Business Media, Inc., Volume 28, Number 1, 2006
59
grind their axes. These activities surely are seen in the West, but not with the fury and lasting costs to people and fields of research. Perhaps, as Gerovitch implies, because cybernet ics was needed for radar and rocketry, the Cold War saved it from further at tacks-in the way that research on the atomic and hydrogen bombs saved rel ativity theory and quantum mechanics from ideological interference. Gerovitch provides a detailed dis cussion of what made attacks on cy bernetics possible in the first place in a discourse that took place between the language of cybernetics and the language of Soviet ideology. He asserts that the ideological language was much more flexible than many historians have gathered. Soviet scholars learned to play by the rules of the establish ment-the government bureaucracy and Party apparatus-in their rhetori cal styles, modes of thought, and argu ments to defend cybernetics. Like their opponents, who saw in cybernetics idealism, "kow-towing" to the West, among other dangers, they turned to quotation-mongering and label sticking (what others have called the "quote and club" method) to put them on the defensive. Gerovitch shows that the postwar ideological campaigns lacked coordination and coherence; they were rarely orchestrated from the top down. In fact, Soviet cybernetics devel oped along many of the same lines along which cybernetics developed in the West. Such scientists as Andrei Kol mogorov and later Alexei Liapunov contributed to the foundations of cybernetics. Gerovitch discusses the work of these scientists against the background of comprehensive analysis of the contributions to the field of Nor bert Wiener, Arturo Rosenblueth, and Claude Shannon. Although cyberneticians were adept at disputation, there were significant pitfalls that awaited them until after Stalin's death. Their contributions were measured against the standards of the West, but they were required to avoid showing contamination with Western ideas. Soviet scholars had to avoid falling behind the West in com puting and simultaneously following
60
THE MATHEMATICAL INTELLIGENCER
Western trends too closely. At the same time, a Cold War ideological battle against the West led to the cutting off of international contacts. Intellectuals were forced to toe the Party line. In this environment, the boundaries between academic and political disputes disap peared. An important facet of the history of computer science and the first Soviet computing machines was the connec tion between cyberneticians and their powerful military and Communist Party patrons. One such example is Mikhail Lavrent'ev, who, as director of the In stitute of Computer Technology, re ceived facilities and protection from the new Moscow City Party Chief, Nikita Khrushchev. Later, as Party leader, Khrushchev enabled Lavrent'ev to found the Siberian city of science, Akadem gorodok, with its own new Computer Center in the late 1950s. When it came to military purposes, computers were a technology without ideological devia tions. After the death of Stalin, in the mid1950s Soviet computers were deified and entered the public realm, no longer to be held under wraps of military se crecy. Cyberneticians touted comput ers as paragons of objectivity based on quantitative knowledge, precise lan guage, and precise concepts. They also commenced an attack against past ide ological interferences of the philoso phers and their allies among scientists and Party officials. Gerovitch's discus sion of the battle against dogmatism and calcification of philosophical dis course under Stalin at this time is en gaging. Ultimately, Gerovitch writes, cyberneticians "overturned earlier ideological criticisms of mathematical methods in various disciplines and put forward the goal of the 'cybernetiza tion' of the entire science enterprise" in search of objectivity in the life sci ences and social sciences alike. (p. 199) This conclusion reinforces the sense that an excessive scientism de veloped in the USSR. In the 1960s cybernetics became a full-fledged science in the Soviet es tablishment. It found an institutional home in a council of the Academy of Sciences of the USSR, experienced
rapid institutional growth, and saw a new publication, Problemy kiberni tiki, which was one of the most influ ential such publications in the world. Its promoters claimed that cybernetics had become a universal, objective lan guage, and as such would break inter disciplinary barriers and legitimize the use of mathematical methods in other sciences. Its power was evident in the fact that even before Lysenko was de posed, cyberneticians promoted ge netic research in physics and chem istry institutes, speaking about genes as units of hereditary information. Sup porters thought cybernetics might be a panacea for reforming the Soviet economy through the creation of "opti mal" models for planning and manage ment. But scientists ultimately realized that central mainframes in large-scale systems to centralize input and output calculations for such a huge economy were simply unfeasible. Once institutionalized, Gerovitch concludes, cybernetics became part of the Soviet establishment in service of the nation's management, administra tion, direction, and government pur poses. The goal was to control the en tire national economy, technological processes, and so on, to ensure opti mal governance. An alliance between cybernetics and dialectical materialism followed in the early 1960s. In the end, cyberspeak became so much a part of establishment thinking, so much the mode of dominant discourse, that its supporters grew disillusioned with ef forts to apply it willy-nilly. Fissures in the cybernetics community as in Soviet society itself created new disputes. Some scientists had grown increasingly conservative and anti-Semitic, while others joined the dissident movement to protest increasing violations of human rights under Leonid Brezhnev. This sug gests that scientism or not, cybernetics, like other sciences in other countries, could not avoid reflecting the social, po litical, and cultural norms of the nation in which it developed. Gerovitch only touches on reasons why the USSR failed to embrace the computer revolution, some of which have roots in the debates over "think ing machines" that occurred from 1950
to 1965. He does not consider the de cision to build computers based on reverse engineering after so many decades of success in building indige nous machines. In addition, because his focus is on philosophy and intel lectual history, some readers will need to seek other sources to gather the im pact of the broader context of Soviet history and politics on cybernetics. Gerovitch's study is based on a thor ough use of archival materials and unpublished memoirs and interviews. This book will be of interest to ad vanced undergraduate students, gradu ate students, and teachers, as well as to computer scientists, historians, and philosophers. I recommend it highly. Program in Science, Technology and Society Colby College 5320 Mayflower Hill Waterville, ME 04901 -8853 USA e-mail:
[email protected]
Traditional Japanese Mathematics Problems of the 1 8th and 1 9th Centuries by Hidetoshi Fukagawa and John F. Rigby SINGAPORE, SCT PUBLISHING, 2002. 1 91 PP. US$50.00 ISBN 981 -04-2759-X
Japanese Temple Geometry Problems San Gaku by Hidetoshi Fukagawa and Dan Pedoe WINNIPEG, THE CHARLES BABBAGE RESEARCH CENTRE, 1 989. 206 PP. US$40.00 ISBN 0-91961 1 -2 1 -4 REVIEWED BY CLARK KIMBERLING
T
he best thing about these books is their content, which is based on problem proposals carved and drawn on Japanese wooden tablets dating from a span of isolation from the West. During that time Japanese mathemati cians developed their own "traditional
mathematics," which, in the 1850s, be gan giving way to Western methods. There were also changes in the script in which mathematics was written, and as a result, few people now living know how to interpret the historic tablets. One of these is Hidetoshi Fukagawa, the Japanese author of the two books. The 1989 book opens with these words: A selection from the hundreds of problems in Euclidean geometry displayed on devotional mathemati cal tablets (SANGAKU) which were hung under the roofs of shrines or temples in Japan during two cen turies of schism from the west, with solutions and answers. Implicit in this description is the def inition of sangaku (often written San gaku and Sangaku). The phrase "with solutions and answers" applies to the books, not the sangaku. Dan Pedoe, co-author of the 1989 book, explains in the Preface: There were few colleges or univer sities in Japan during the period of separation from the west, but there were many private schools, and obviously many skilled geometers who wished to thank the god or gods for the discovery of a particularly lovely theorem, and also, it may be guessed, who were not averse to dis playing their discoveries to other geometers . . . with the implicit chal lenge: "See if you can prove this!" The 2002 book continues the collection with additional problems and solutions. For both books, many of the solutions use modem methods. With admirably little overlap in content, the two books give historical descriptions, photographs, figures, calligraphy, solu tions, and references, all well focused on sangaku. For a broader context, one may cite Chapter 22 of Yoshio Mikani's The Development of Mathe matics in China and Japan, second edition, Chelsea, 1974 (originally pub lished in German, 1913). Mikani places sangaku in the per spective of Seki Kowa, who has been called the Japanese Newton and father of Japanese mathematics. Although Seki's lifetime ( 1642-1708) preceded sangaku, his influence in algebraic and analytic methods set the stage for san-
©
gaku. Mikani writes, "The highest de velopment of the Japanese mathemat ics must of course be looked upon as the invention of . . . 'circular theory' " and it is precisely the enchantment of circle-problems that pervades san gaku. Indeed, a majority of the prob lems in the 1989 and 2002 books in volve circles. One of the foremost mathemati cians represented in sangaku was Ajima Chokuen (1732-1798). (The fam ily name is Ajima. The given name Chokuen is used in the 1989 book, but the more formal given name Naonobu is used in the 2002 book) The two books contain a number of spinoffs from Ajima's famous problem about three pairwise tangent circles in scribed in a triangle. A view of Ajima's place in the books provides insights into the organization and mathematical tone of the two books and also gives insights into the work of one of the leading representatives of Japanese "traditional mathematics" (as it is called in the 2002 book). On pages 28-30 of the 1989 book, Ex ample 2.3 and Problems 2.3. 1 to 2.3. 7 are presented under the heading "Three Cir cles and Triangles," followed by Exam ples 2.4(1) and 2.4(2) and Problems 2.4.1 to 2.4. 7 under "Four Circles and Trian gles." The presentation is in two-column format with figures in the right column. Each problem proposal is labeled "Ex ample" or "Problem." Here are three items from the 1989 book: Example 2.3: The three circles, 01Cr1), Oz(rz), and 03(r3) have ex ternal contact with each other. The triangle ABC is formed by the com mon tangents to the circles. Find the radius of the incircle of triangle ABC in terms of r1 . r2, and r3. Example 2.4( 1 ): I(r) is the incircle of triangle ABC, and the circles 01Cr1) , Oz(rz), and 03(r3) respec tively touch AB and AC, BA and BC, and CA and CB, and all touch I(r) externally. Show that r=
Vr)r; + y:;;; + v;;;:;-.
Problem 2.4. 1: ABC is a triangle, I(r) its incircle. The circle 01(r1) touches AB and AC produced and
2006 Springer Science+Business Media, Inc., Volume 28, Number 1 ,
2006
61
also I(r) externally, and 02(r2), and 03(r3) are defined similarly. Show that 1 r
-
1
=
v;v;
--
+
1
y,;;;
--
+
1
�.
--
A footnote refers to page 106, in Part I, titled "Solutions to Selected Problems and Answers." There, under the heading "The Malfatti Problem," solutions and historical comments are given. The con struction of the circles as in Example 2.3 is often attributed to Malfatti, but the historical comments note that Ajima posed and solved the problem about 30 years before Malfatti did. In modem times, this Ajima-Malfatti construction has received consider able attention. For example, let A' be the touchpoint of circles 02(r2), and 03(r3), and cyclically, let B' = 03(r3) n 01(r1) and C' = 01Cr1) n 02Cr2). Then M 'B'C' is perspective to MEG, and the perspector (John Conway's im provement over "center of perspec tive") is a point known as the Ajima Malfatti point. For a discussion, visit the Encyclopedia of Triangle Cen ters-ETC: http://facu1ty.evansvil1e.edu/ck6/ encyclopedia/ETC.htm1 and scroll down to X(179). Peter Yff found remarkable trilinear coordinates for the Ajima-Malfatti point:
(These are respectively proportional to the distances from the point to the sidelines BC, CA, AB.) The 2002 book states "Ajima's Theo rem" at the beginning of Chapter 4 and "Ajima's second theorem" at the begin ning of Chapter 5. The second theorem is elsewhere cited in the book on sev eral pages, as is a third theorem attrib uted to Ajima, labeled "Boushajutu". Another traditional mathematician represented in both books was Shoto Kenmotu (1790-1871). His configura tion in Problem 3.2. 1 of the 2002 book is the earliest known (1840) construc tion of a point now known as the Ken motu point. The configuration, involv ing three congruent isosceles right triangles, extends easily to the one de picted here.
62
THE MATHEMATICAL INTELLIGENCER
A
The three congruent squares meet in the Kenmotu point, indexed as X(371) in ETC. Trilinears were found by John Rigby: cos(A - 7T/4) : cos(B - 7T/4) : cos(C - 7T/4). Both books are partitioned into Part 1 (problems) and Part 2 (essentially, solutions and comments). The 1989 book has chapter headings ( 1) Circles, (2) Circles and Triangles, (3) Circles and Polygons, (4) Polygons, (5) El lipses (and One Hyperbola), (6) El lipses and Circles, (7) Ellipses and Polygons, (8) Ellipses, Circles and Quadrilaterals, and (9) Spheres. The to tal number of problems is 249. The 2002 book chapter headings are (1) Number theory, (2) Numerical Analysis, (3) Geometry of Polygons, (4) Geometry of Circles, (5) Geometry of Circles and Triangles, (6) Geometry of Ellipses, (7) Solid Geometry, and (8) Maxima and Minima. There are 287 problems. In both books, the organization of material is indeed wonderful, when you stop to think that consecutive closely related problems started out in temples scattered across Japan. For example, in the 1989 book, for the nicely sequenced Problems 2.3. 1 to 2.3.5, we read in Part 2 that these orig inated in various prefectures at various times: Fukusima in 1891, Iwate un dated, Iwate in 1842, Miyagi in 1857, and Fukusima in 1901. Regarding such places and dates, Dr. Hiroshi Kotera of fers a very attractive website: http://www.wasan.jp/english/ (There is also a Japanese version.) There you will find a Clickable SAN GAKU Map of Japan, showing 24 la beled prefectures in which sangaku are
found. By clicking any of them, you will be able to examine individual tablets on which problems are posed. In par ticular, you can click Tokyo, then se lect English, and see not only a very well preserved tablet, but also markers indicating Ohkunitama Shrine, a date of March 1885, the text of the problem, and so on. Another website is also rec ommended: Tony Rothman's (with the cooperation of Hidetoshi Fukagawa) Japanese Temple Geometry, http://www2.gol.com/users/ coynerhm/0598rothman.html Both books have extensive bibli ographies. The 1989 book gives 78 items, and the 2002 book gives 109. The 2002 book has an Index. Fukagawa's coauthor of the 1989 book, Dan Pedoe, is well known as the author of Circles (Pergamon, 1957), in which he wrote, regarding the nine point circle, "This circle is the first really exciting one to appear in any course on elementary geometry." (This famous line is quoted at the beginning of a section in Coxeter's Introduction to Geometry.) The coauthor of the 2002 book, John F. Rigby, is well-known in geomet ric circles. For example, Ross Hons berger's Episodes in Nineteenth and Twentieth Century Euclidean Geometry (Mathematical Association of America, 1995) reserves a page on which special acknowledgment is given to John Rigby for his contributions to the book. Pages 132-136 introduce a Rigby point which serves as a seed for families of points in ETC beginning at X(2677). Two other Rigby points are indexed in ETC as X(1371) and X(1372). For more on both kinds of Rigby points, visit Eric Weisstein's MathWorld: http://mathworld.wolfram. com/ RigbyPoints.html. Physicist Freeman Dyson's Fore word (or "Forward", as it not inappro priately appears) to the 2002 book is a noteworthy piece, reminiscent of his Imagined Worlds (Harvard University Press, 1997). The Forward opens with these words: "One of the most impor tant scientific enterprises of the twen tieth century is the search for ex traterrestrial intelligent species." Once
contact is made, there will be the prob lem of communication. Dyson writes, "Eighteenth-century Japan is stranger to me, in language and in historical tra dition, than any other past or present culture on this planet. To my delight, I see in Fukagawa's books a collection of mathematical messages that are profoundly strange but none-the-less intelligible. Fukagawa has collected and arranged these messages so that their strange beauty is now accessible to everyone, eastern and western alike." The 1989 book can be ordered di rectly from The Charles Babbage Re search Centre, P. 0. Box 272, St. Nor bert Postal Station, Winnipeg, Canada R3V 1L6. To order the 2002 book, send a check payable to Mathematics and Informatics to Susan Wildstrom, 10300 Parkwood Drive, Kensington, MD 20895-4040, USA. Include a letter telling whether you wish to receive the book (from Singapore) by surface mail (total $50.00) or by air mail (total $60.00). Department of Mathematics University of Evansville 1 899 Lincoln Ave Evansville, IN 47722-0001 USA e-mail:
[email protected]
Mathematics: A Very Short Introduction by Timothy Gowers NEW YORK, OXFORD UNIVERSITY PRESS, 1 56 PP. US $9.95, ISBN 0·1 9·285361 ·9 REVIEWED BY JEAN-MICHEL KANTOR
T
his is a very short review of Math ematics: A Very Short Introduc tion, by Timothy Gowers, a very smart mathematician and an excellent com municator of mathematics (the two concepts are distinct). Gowers's 1998 Fields Medal might be considered a proof of the first statement; this book, together with his already famous Clay Lecture on the importance of mathe matics [ 1], an argument for the second. He has done a wonderlul job in pro ducing this rich little book on the es sentials of mathematics.
The main part consists of three chapters: •
•
•
Models (turning practical problems into mathematical ones). The abstract method (or axiomatic method), for which he gives good ar guments showing its power and sug gesting its pedagogical usefulness (I will not open here the Pandora's box of discussing didactics). Proofs, which some people consider to be at the heart of mathematics. Gowers gives examples of proofs, and of seemingly obvious statements that need proofs.
The rest of this charming book is made up of stories, illustrations, ex amples, and well-illustrated concepts such as limits, estimates, dimension, infinity. . . . Gowers has chosen a list of simple concepts of great mathemat ical importance, which he presents to the curious reader. Finally, the chapter "Some frequently asked questions" re sponds to what people generally ask about mathematicians (or so mathe maticians imagine). Among the ques tions, "Why do so many people posi tively dislike mathematics?" When people asked Henri Poincare why they never understood mathemat ics at school, he answered them, "What I don't understand is that people don't understand mathematics!". Gowers sug gests using the abstract approach. We wish him all the best! I would strongly recommend this book to first-year undergraduates, al though professional mathematicians will also find it a useful introduction to many beautiful examples on Gowers's Web site, such as the astonishing re sults that won him the Fields Medal [2]. The general audience too will fmd much of interest on the Web site, in cluding an article about what is defin able in mathematics, and the text ver sion [3] of the Clay Lecture, in which Gowers shows how mathematics is a subject where importance and beauty are connected. REFERENCES
[1 ] The Importance of Mathematics. A Lecture by Timothy Gowers, Springer VideoMATH , 2002. VHS/NTSC video tape. ISBN 3-54092652·6.
©
[2] http ://www . d p m m s . cam . a c . uk/�wtg1 0 /vsipage.html [3] http://www . d pmms. cam . ac . uk/�wtg 1 0/ importance.pdf
ADDED IN PROOF
See also The Princeton Companion to Mathe
matics, under the supervision of T. Gowers, to appear in 2006, Princeton University Press.
Universite de Paris VII 75251 Paris Cedex 75005 France e-mail:
[email protected]
Baseball's All-Time Best Hitters by Michael Schell PRINCETON, NJ: PRINCETON UNIVERSITY PRESS, 1 999. xxi + 295 PP. US $1 7.95 ISBN 069 1 - 1 2343·8 (PAPER)
REVIEWED BY JIM ALBERT
B
aseball is one of the most popular tean1 games in the United States. Professional baseball started near the end of the 19th century. Currently in the United States and Canada, there are 30 professional teams in the Amer ican and National Leagues, and mil lions of people watch games in ball parks and on television. Baseball is a game between two teams of nine players each, played on an enclosed field. A game consists of nine innings. Each inning is divided into two halves; in the top half of the inning, one team plays defense in the field and the second team plays of fense, and in the bottom half, the teams reverse roles. The team that is batting during a particular half-inning, the of fensive team, is trying to score runs. A player from the offensive team begins by batting at home base. A run is the score made by this player who ad vances from batter to runner and touches first, second, third, and home bases in that order. A team wins a game by scoring more runs than its opponent at the end of nine innings. A basic play in baseball consists of a player on the defensive team, called a pitcher, throwing a spherical ball (called a pitch) toward the batter. This
2006 Springer Science+ Business Media, Inc., Volume 28, Number 1 , 2006
63
Rank
Player
Batting
Career At-
Average
bats
Career
Ty Cobb
0.3664
11434
1905-1928
2
Rogers Hornsby
0.3585
8173
1915-1937
3
Joe Jackson
0.3558
4981
1908-1920
4
Lefty O 'Doul
0.3493
3264
1919-1934
5
Ed Delahanty
0.3459
7505
1888-1 903
6
Tris Speaker
0.3447
1 0195
1907-1928 1939-1960
7
Ted Williams
0.3444
7706
8
Billy Hamilton
0.3443
6268
1888-1901
9
Dan Brouthers
0.3421
6711
1879-1904
10
Babe Ruth
0.3421
8399
191 4-1935
of the AVG as a measure of hitting performance, I believe that this is a reasonable aim. The definition of batting average is carefully explained, and Schell can focus his efforts on the proper adjustment of this measure to make comparisons of players. (Schell is currently completing a second book, Premier Batsmen, that makes similar adjustments to other measures of hitting performance.) What Adjustments Should
confrontation is called the batter's at bat. The batter is attempting to make contact with the pitch using a smooth round stick called a bat. A strike is a pitch that is struck at by the batter and missed, or is not struck by the batter and passes through a region called the strike zone. A ball is a pitch that is not struck at by the batter and does not en ter the strike zone in flight. After a number of thrown pitches, the batter will either be put out or be come a runner on one of the bases. The batter may be put out in several ways: (1) he hits a fly ball (a ball in the air) that is caught by one of the field ers, (2) he hits a ball in "fair" territory, and first base is tagged before the bat ter reaches first base, (3) a third strike is caught by the catcher. A hitter can advance to become a runner and reach base safely by: (1) Receiving four pitches that are balls (outside of the strike zone). In this case, the batter receives a walk or base-on balls and can advance to first base. (2) Hitting a ball in fair territory that is not caught by a fielder or thrown to first base before the runner reaches first base. There are different types of hits depending on the advancement of the runner on the play. A single is a hit when the runner reaches first base, a double is a hit when the runner reaches second base, a triple is a hit when a runner reaches third base, and a home run is a "big" hit (usually over the out field fence) when the runner advances around all bases safely. Rating Players
Baseball players are rated with respect to their ability to hit and their ability to field. The classical measure of a
64
THE MATHEMATICAL INTELLIGENCER
Be Made?
player's ability to hit is the batting average (AVG), defined as the propor tion of "official at-bats" (essentially a player's at-bats that are not walks) when the player gets a base hit. Above is a table listing the players since the beginning of professional baseball (1876) that have the ten highest batting averages. From this table, it appears that Ty Cobb is the best baseball hitter of all time. But was he really the best hitter? This book provides a serious at tempt to make the proper adjustments to batting average so that one can make a fair comparison of players who played during different time periods. Why Consider Batting Average?
It should be clarified what this book is about and what this book is not about. Although batting average (AVG) has been the standard way of measuring a player's batting ability since the begin ning of professional baseball in 1876, it is well known that AVG is a relatively poor measure. The objective of a hitter is to help create runs for his team, and there are alternative measures of a player's hitting success that are much more positively associated with runs than AVG. Albert and Bennett (2003) give an overview of the many superior alternatives to AVG. The book under review has been criticized by many people for compar ing hitters with respect to batting average instead of alternative "good" measures of batting performance. I think this criticism is unfair. Schell makes it clear in the introduction that his analysis is not finding the best "bat ters," but rather the best "hitters"-the ones who have the best chance of get ting a hit. Given the current popularity
To illustrate some of the issues dis cussed in this book, let's compare two great baseball hitters, Ty Cobb and Ted Williams. On the surface, Cobb appears to be the better hitter, for his career batting average was 0.366 compared to Williams's 0.344. But this superfi cial comparison ignores some relevant concerns. First, we question whether 0.366 and 0.344 are representative mea sures of the ability of the two players to get hits. Ty Cobb played baseball for 24 seasons. Figure 1 plots his season batting averages against his age. In it, a smooth-fitting (loess) curve is placed on top of the plot. We see a general pattern in this plot that is typ ical for many players. Baseball players, like many other athletes, mature and become more proficient in performance until a particular age, after which their performance deteriorates until retire ment. A player's career batting average includes the periods of maturation and deterioration during which the player has a low batting average. A better measure of hitting performance might be the batting average for a player dur ing the middle years of his career. A second general concern in com paring Cobb and Williams is that they played baseball during different eras and their careers did not overlap. The basic rules of baseball have remained the same over the years, but the play ing conditions that affect the effective ness of the pitchers have changed, and these changes have had a significant ef fect on the batting averages of players. It is difficult to say that Cobb was a bet ter hitter than Williams, because they played against different pitchers and the game was different in the two eras. For these reasons, a reported batting
•
0.4
careers. In the "Second Base" chapter,
•
Schell makes an adjustment for seasons
•
during which the batters had general ad vantages or disadvantages versus the pitchers-he calls these periods hitting
0.35
feasts and famines. A mean adjusted bat
(!)
�
ting average is computed, which
is
the
ratio of the player's batting average to
0 .3
the average league batting average for that particular season. This adjustment diminishes the achievement of players
0.25
that had high batting averages during pe
•
Figure 1
20
25
AGE
35
30
riods where hitting was relatively easy.
40
More subtle adjustments to batting average are made in the "Third Base" and "Home" chapters. The third ad justment accounts for differences in
average of, say, .320 is not very mean
from which the pitcher throws), and
the talent pool during different eras.
ingful unless we understand when this
these ballpark characteristics can have
Schell claims that the standard devia
average was achieved. Schell makes
a significant effect on hitting statistics.
tion of players' batting averages is a
the plausible assumption that the num
If one player hits for a 0.300 batting av
useful measure of the talent pool of
ber of great hitters has stayed constant
erage in a "hitter's ballpark" and a sec
players, and small standard deviation
over the years, and it is not meaning
ond player hits for an average of 0.300
values indicate more talent.
ful to look at a player's absolute bat
in a "pitcher's ballpark,"
one
justed batting average is proposed that
ting average. Rather, he believes, a
would rate the second hitting perfor
takes into account the difference in tal
player's
be
mance as superior because it was more
ent pools. This method essentially as
judged relative to the batting averages
difficult to get hits in his ballpark
sumes a percentile ranking for each
of other players who played during the
Thus, one needs to adjust a player's
player for the era in which he played,
same seasons.
batting average for his ballpark
and then combines the percentile rank
Making the Adjustments
Base" chapter, Schell makes the final
batting
average
should
then
A third concern is related to the pool of talented players during different
A
new ad
ings into a single list. In the "Home
A player's
eras of baseball. Schell describes how
Schell carefully describes how to make
adjustment.
the recruiting pool of potential ball
the
de
is adjusted for the ballpark where he
players has changed over the years.
scribed above to find the 100 best base
played half of his games. This is a te
Most of the players in the early years
ball hitters of all time. He starts by only
dious calculation, for adjustments have
of baseball were whites from the North
considering "qualifying" players-the
to be made for each of the ballparks
four
types
of adjustments
batting average
east United States. In later years, base
retired players who had at least 4000
where Major League baseball has been
ball included players from the South,
at-bats and the current players who
played over the past 130 years. This
later still, African-Americans, and now
have had at least 8000 at-bats. Note that
method multiplies a batting average by
players from Latin America and Asia.
one of the players in the above list,
a specific "ballpark effect."
The change in U.S. population and the
Lefty O'Doul, would be excluded from
source of eligible players has had a sig
the top list because he didn't have the
And the Best Hitter Is?
nificant impact on the general talent of
required 4000 at-bats. With this exclu
After performing all of these adjust
the group of professional players. Gen
sion, Schell displays the "traditional"
ments to batting average, who are the
erally, it can be said that the range of
list of 100 best hitters, and in the four
best hitters of all time? As expected,
talent has decreased over the years
chapters
"First
many of the players who had high bat
that
follow
(called
(players are currently more homoge
Base," "Second Base," "Third Base,"
ting averages during the feast period of
nous in terms of ability), and this
and "Home"), adjusts the traditional
the 1920s and 1 930s get devalued, and
change should affect the comparison
list using the four concerns.
many of the current players rise in the
In the "First Base" chapter, Schell
ratings. For example, Roger Hornsby,
final concern is that all ballplay
adjusts a player's batting average for
who played in the 1920s, drops from
ers play half of their games in their
late career declines. Essentially, this
second on the traditional list to fifth on
home ballpark and half of their games
method adjusts a career
by con
Schell's list, and Rod Carew, a 1970s
in ballparks away from their home
sidering only the proportion of hits in
player, rose from 28th (traditional) to
town. There are significant differences
the first 8000 at-bats. This adjustment
3rd (Schell). Surprisingly, the best hit
in the dimensions of the ballparks, the
has the effect of raising the batting av
ter of all time is Tony Gwynn, who re
weather, the type of grass, and the con
erages of players whose averages de
tired in the 200 1 season. Schell's re
dition of the pitcher's mound (the hill
teriorated in the closing years of their
sults received much media attention,
of batting averages from different eras.
A
AVG
©
2006 Springer Science+ Business Media, Inc., Volume 28, Number I , 2006
65
and the author had an opportunity to
ball can provide insights. The author,
Chain and uses transition probability
meet with Gwynn at the San Diego ball
Jim Albert, suggests that TSUB "can be
matrices to derive conclusions.
park to discuss his findings.
used as the framework for a one-se mester introductory statistics
class
The case studies and exercises pose interesting baseball questions, and the
General Assessment
that is focused on baseball" or "as a re
book and accompanying Web site pro
This book provides a nice historical
source for instructors who wish to in
vide easy access to much baseball data.
view
fuse their present course in probability
Here are a couple of my favorites:
of baseball from a statistical
perspective. There have been notable
or statistics with applications from
Case Study 2-5 presents an interest
changes in the game over the years,
baseball." Regarding the first sugges
ing look at managerial strategy, by
and one can view these changes by use
tion, I would call such a course "sta
studying the use of the sacrifice bunt.
of time series plots of the appropri
tistics appreciation," for TSUB does
When the dotplot of sacrifice attempts
ate statistics. Through this historical
not
statistics
is displayed in parallel (separately) for
introductory
the American and National Leagues,
cover the
traditional
study, one learns a lot about the great
material
hitters of all time. Also, although many
courses, and very little formal infer
there are noticeably fewer bunts in the
baseball fans may disagree about the
ence is done.
AL. The designated hitter rule provides
contained
in
final ranking of best hitters, the author
The topics of TSUB are closely
uses scientifically sound methods to
linked with those of [2] . The book is or
make the necessary adjustments in bat
ganized into 9 chapters. The first chap
Case Study 9-4 is built on previous
ting average. Although I might choose
ter is an introduction to the book; the
case studies in Chapter 9 and uses
different methods to make my adjust
remaining chapters are composed of
Markov Chain concepts for its devel opment. Within an inning, there are 24
a "simple explanation for this discrep ancy."
ments, I suspect that I would reach
an average of 5 case studies and 19 ex
similar conclusions. My only concern
ercises, including "leadoff exercises"
possible "states," based on the number
in reading this book is the relatively
featuring Rickey Henderson, probably
of outs (0, 1, or
high technical level of the presentation;
baseball's greatest leadoff hitter.
figurations (e.g., bases loaded). From
2
introduces
and the
runner
con
stemplots
this model, Albert estimates the average
(also called stem-and-leaf diagrams) ,
value of a home run to be 1.40 runs to
yet most fans would be interested in
the "five-number summary" of a dis
the team. This corroborates similar esti
the conclusions. I also believe that the
tribution (minimum, maximum, and
mates obtained by others by regressing
adjustment procedures described here
three quartiles) , histograms, and graph
wins on various team batting statistics.
would be helpful in constructing simi
ing of time series data. Chapter 3 com
Albert has adroitly kept the statistical
lar types of adjustments to measures of
pares batches of data, principally using
development work to the minimum re
performance in other sports.
parallel stemplots, boxplots, or stan
quired to solve the question.
the book can be read only by baseball fans with a quantitative background,
Chapter
2)
dardization. Chapter 4 examines rela
In other case studies, though, the
REFERENCE
tionships between variables using scat
author simplifies
Albert J. and Bennett, J . , Curve Ball, Springer
terplots, correlation coefficients, and
doubtedly for pedagogical reasons).
regression, using the RMSE to evaluate
Students should know that these analy
(2003).
Department of Mathematics and Statistics
the
analyses
(un
how well alternative measures (like
ses lack the rigor needed for more de
batting average and slugging average)
finitive answers to the questions posed.
Bowling Green State University
predict run-scoring. In Chapter 5, prob
I now provide several such cases as an
Bowling Green, OH 43403
ability concepts are introduced and
aid to the potential instructor.
USA
applied to the design of baseball table
Case Study 3-5 is close to my heart,
e-mail: albert@math .bgsu.edu
top games. Plots of cumulative rates of
for it involves comparison of four of the
the
best single-season batting averages in
earned run average of a pitcher over
baseball history. Albert standardizes the
the course of a season) are also fea
raw averages by using the means and
tured. In Chapter 6, the binomial dis
standard deviations of "regular players"
tribution and goodness-of-fit Pearson
(those with 2:: 400 at bats) from the four
some
Teaching Statistics Using Baseball b y Jim Albert ---.. -- ·-----
WASHINGTON. DC, MATHEMATICAL ASSOCIATION OF AMERICA, 2003. 304 PP US $46.95. ISBN 0·88385-727-8 (PAPER) REVIEWED BY MICHAEL J. SCHELL
baseball
statistics
(like
residuals are presented in the case
seasons. For Rod Carew's 1977 and Tony
studies, and the Poisson and geomet
Gwynn's 1994 seasons, Albert obtains
ric distributions are introduced in the
batting average z-scores of 4.07 and 3.15,
exercises. Chapters 7 and
8
focus on
respectively. Given that Gwynn's season
statistical inference. Typically, data are
ranks first in my book, [3] (Carew's sea
simulated according to various distri
son ranks fourth), I was surprised that
Obutional assumptions, and inference
the z-score differential in TSUB was so
T
among alternatives is conducted using
disparate. Two simplifications of the
eaching Statistics Using Baseball
Bayes's rule, probability intervals, and
problem led to a substantial shift in the
(TSUB hereafter) addresses vari
visual inspection of graphs. Chapter 9
findings. First, batting averages were
ous issues in statistics for which base-
models a baseball game as a Markov
not adjusted for the players' home ball-
66
THE MATHEMATICAL INTELLIGENCER
park in TSUB. Second, the definition of a "regular player" biased the standard ization, because a players' strike short ened the 1994 season. Consequently, only 63 players (2.2 players per team) from 1994 were "regular players" com pared to 168 (6. 5 players per team) from 1977. These simplifications gave Carew 13 and 16 net batting points over Gwynn, respectively, compared to the more detailed analysis given in my book, which yielded z-scores of 4.24 for Gwynn and 4.03 for Carew. For Case Study 7-3, Albert shows re sults from 150 simulations but notes that they "aren't too precise," so he re simulates the example 1000 times. Then, in actually solving the problem, he uses about 333 simulations. If 150 isn't precise, and 1000 is, what about 333? I would have preferred that 10,000 simulations be used. Sometimes TSUB relies heavily on the reader's baseball knowledge. Having discussed leadoff hitter Rickey Hender son's distribution of plate appearances in the 24 runner and out situations for 1990 (p. 260), Albert asks the reader whether Barry Bonds's 1990 profile should be similar. The question seems to spin on whether Bonds was a leadoff hit ter or not. Well . . . early in his career, Bonds WAS a leadoff hitter. Bonds led off only 13 games in 1990, but in 1989 he led off 106 of the 159 games he played. The casual fan might not know anything about Bonds's lineup position at all. The intermediate fan might think of him as a cleanup hitter. The advanced fan wor ries about when he made the transition from being a leadoff hitter. In Chapter 8, TSUB proposes as sessing the "streakiness" of players by visual comparison of player moving average plots to simulated ones. How ever, most of the plots based on simu lated data are of different size, render ing comparison very difficult. In summary, Jim Albert is to be ap plauded for providing a host of baseball data and examples for use in learning statistics. Sports examples provide one of the best avenues for helping stu dents develop an interest in statistical concepts. In 1997, Frederick Mosteller [ 1] sounded the call that "statisticians should do more about getting statistical material intended for the public written
in a more digestible fashion" and relate the interests of sports enthusiasts "to the more general problems of statistics." Through Curve Ball and Teaching Sta tistics Using Baseball, Jim Albert has responded well to that call. REFERENCES 1 . Mosteller, Frederick. Lessons from sports
statistics. American Statistician,
5 1 :305-
3 1 0, 1 997. 2 . Albert, Jim and Bennett, Jay, Curve Ball:
Baseball Statistics and the Role of Chance in the Game, Copernicus Books. New York, 2001 . 3. Schell, Michael J. Baseball's All-Time Best
Hitters: How Statistics Can Level the Play ing Field, Princeton University Press, Prince ton, 1 999.
Department of Biostatistics University of North Carolina Chapel Hill, NC 27599 USA e-mail:
[email protected]
few years. The number of celestial ob jects, from asteroids to remote galax ies, recorded this way is in the billions. Hence, classifying each of them must be done automatically. Mathematically speaking, a classi fier is a function that takes some inputs (the characteristics of the customer in the credit rating problem, or the posi tion and image of the object if one wants to classify astronomic observa tions) and outputs a label, e.g. ::!:: 1. For example, a linear classifier is de fined by
Machine Learning: Discriminative and Generative by Tony Jebara BOSTON, KLUWER,THE KLUWER INTERNATIONAL SERIES IN ENGINEERING AND COMPUTER SCIENCE, VOL. 755. 2004. 224 PP. HARDCOVER US$82, ISBN 1 -400-7647-9 REVIEWED BY MARINA MElLA
I
f "probability and statistics [ . . . ] have separated, then divorced" [2], machine learning is the place where they coexist happily and occasionally get up to interesting things together. This book is the refreshing result of just such an encounter. To pursue another metaphor of the same flavor, machine learning (ML) is the estranged child of Artificial Intelli gence, which gave up on high-level, rule-based representations of knowl edge for low-level, often stochastic models estimated from data. It em braced probability from its inception, discovered statistics in the 1990s, all the while maintaining its roots in com putation. This was fortunate, as some
© 2006
of machine learning's most famous successes include an algorithmic in gredient. One of the oldest and most studied problems in ML is classification. Auto mated classification is often desired in practical applications of computers. For example, credit card companies detect when a card is stolen from the pattern of activity in the account on the past few days; banks give credit ratings "good = will pay = + 1" or "bad = will default = - 1" to potential customers based on a set of a dozen or so ob served characteristics (usually kept se cret by the banks). Recently, with the advent of the massive scientific data bases, automatic classification is in demand in the sciences as well. In biological sciences, automatic protein classifiers are being developed that predict the functional class of a protein from its chemical formula. Astronomy has introduced massive sky surveys, with a telescope continually photo graphing the night sky for as long as a
f(x)
=
sign(81x1 + 8zxz + . . . 8mxm),
where x1, x2, . . . Xm are the inputs and el, 8z, . . . em are the parameters of the classifier. This classifier defines a hy perplane in m-dimensional space. All examples (x1, Xz, . . . Xm) falling on one side of the hyperplane are labeled + 1 (e.g., "good credit rating"), while the points on its other side are labeled - 1 ("bad rating"). The hyperplanef(x) = 0 itself is called the decision surface. The values of the parameters, i.e., the choice of the particular hyperplane, is made based on a set of hand-labeled examples called the training data. Hence, training a classifier consists of picking a function! and a set of val-
Springer Science+ Business
Media, Inc.,
Volume
28,
Number
1 . 2006
67
ues for its parameters e. More ab stractly, one chooses a family '!fo of functions parametrized by 81, . . . Om, a criterion for selecting the values of the
question by choosing a family of gen erative models rich enough to approx imate well any distribution the data may have. But this way one may fall
parameters J(O; training data), and some algorithm for obtaining the opti mal O's according to J. The first task, choosing '!fo, is sometimes left to the "application expert," but the second and third are at the core of ML. One way of practically approaching classification was pioneered by Sir Ronald Fisher and, to no surprise, typ ifies the point of view of statistics: As sume that one knows that the good credit applicants have a distribution p+ (x) (for example a multivariate nor mal) and the bad credit applicants are distributed according to p-(x). Then, to decide whether xnew deserves a good or a bad credit rating we compare the probabilities that the two mod els assign to xnew. For instance, if p+ (xnew)IP- (xnew) > 1, xnew receives a good rating, otherwise it receives a bad one. This approach can also be put
prey to overfitting, which is exaggerat ing the importance of the inputs in pre dicting the data, when some of them may be noisy or irrelevant. For exam ple, if a student sees a cat on the day he fails an exam, and concludes that seeing a cat on future exam days will predict failure, then he is probably overfitting the data. The discriminative way to look at classification is to pragmatically pick a classifierf E '!fo that classifies the train ing set best, doing away with the gen
in aj(x) form, with
f(x)
=
sign[log p+ (x) - log p-(x) ]
(1)
Because p+ , p- are referred to as the generative models of the two classes, this approach is called the generative approach to classification. By contrast, directly choosing a decision surface f(x) without resorting to generative models is called discriminative classi fication. The statistical approach induces one to think of how the examples of each class were generated, which is more intuitive than directly selecting a class '!fo of decision surfaces, especially in high dimensions and when one deals with more than 2 classes. Statistics also offers principles-like Fisher's Maximum Likelihood-and algorithms for estimating the parameters of each model from data. It was also proved that, when the true model classes for p± are known and the number of train ing examples tends to infinity, the gen erative classifier is optimal. The problems appear in real condi tions, when one knows little about the true distributions p± . What P to choose? One can seemingly bypass this
68
THE MATHEMATICAL INTELLIGENCER
erative model, i.e., with the assumption that we know something about the form of the "true" sources that gener ated the data. A now classic theory [ 1 ] allows one to predict the classification error rate on future data based on just three numbers: the number of training examples, the number of errors off on these examples, and the capacity of '!fo the class of functions we are optimiz ing over. To understand the im portance of the last factor, think of over fitting. If the class '!fo is richer, something measured by its capacity, then it is more likely that the chosenfis overfitting the observed examples but will do poorly on new examples Gust like not seeing a cat the next exam day will be a poor predictor of success). Discriminative methods have the advantage of directly optimizing the quantity of interest (the classification error) instead of the description of the classes separately through p+ , p-. In practice they have been shown to be more robust to overfitting, especially when few training examples are avail able. However, their output cannot be converted easily into a confidence measure, and incorporating other knowledge about the domain, some thing easily done with generative mod els, was only achieved on a case-by case basis. How to combine the advantages of generative models-ideal for incorpo rating knowledge about the domain and returning a probabilistically mean ingful answer-with the principles of discriminative classification that seem
to govern problems with finite data? "Machine Learning: Discriminative and Generative" offers a way of doing just that: it is Maximum Entropy Discrimi nation (MED for short). While introducing the core of the MED method to a machine learning ed ucated audience can be done in one re search paper, this monograph has a more ambitious goal. It aims to situate MED in the context of its sources, to motivate it, and to exhibit its many ties with other old or recent advances in machine learning. The book opens with an overview of machine learning, stressing the fun damental trade-off between choosing a model expressive enough to solve the problem and avoiding overfitting the observed data. The next chapter weaves in the precursor ideas for MED. The generative and discriminative ap proaches are introduced. Each is illus trated by a notable success story in teresting in itself, but which also supports the understanding of the rest ofthe book For generative models, the jewel of the crown are the "graphical models," featured on the cover of the book itself. Graphical models have be come a language for expressing prob abilistic dependencies between many variables by way of graphs. Like any good language, the concise represen tations are precise in a probabilistic sense, and translate into very efficient algorithms both for making inferences from the model and for estimating the models themselves. The discriminative methods' reso nant success is the Support Vector Ma chine (SVM), also symbolized on the book cover by a decision surface. SVM demonstrates that important inventions need more than one good idea; in the present case three ideas converged from domains as different as convex opti mization, reproducing kernel Hilbert spaces, and statistical learning theory. Chapter 3 introduces the MED core ideas: Similarly to SVM, a family '!fo of discriminative functions is chosen and one explicitly enforces good classifica tion of the training examples. But, in stead of choosing just one f E '!fo, one hedges the bets by averaging over all the options. It is a weighted average, in
which an f gets a higher weight if it makes fewer mistakes on the training data. Beautifully and surprisingly, it turns out that for many classes 21', solv ing this apparently more complex prob lem (an average over an infinite set of functions) is no harder than choosing one optimal! Even better, the task can be cast into a convex optimization problem with unique solution, that can be solved by a generic algorithm. One nice consequence of learning a distribution instead of a single f is that now one can naturally achieve most of the goals listed above: One can construct thefs as in (1) from generative models, yet be explicitly optimizing the classifi cation error. One can interpret the clas sifier's output both discriminatively and probabilistically. Prior knowledge of var ious kinds can be introduced. Extensions to the main method are presented in chapters 4 and 5. The for mer extends the MED approach to a wider range of problems, like classifi cation with 3 or more classes (multi way classification), predicting a real valued "label" (regression), selecting which inputs are relevant for classifi cation and which are not (feature se lection). Experimental results also highlight the behavior of MED com pared with other methods. Chapter 5 discusses the case when some relevant inputs are not observed (these are called latent variables). For instance, two stars that are close together in an image may be part of a binary system or may appear close only seen from the Earth (much less likely!). Which is the case is something that may perhaps be inferred given other data, but cannot be observed directly; it is thus a latent variable. Finally, the discussion in chapter 6 summarizes the ideas in the book and outlines some directions for further research. Experiments punctuate most sec tions and the illustrations are very help ful (one wishes though that the Matlab plots reproduced better). As the book spans domains so varied as statistics, optimization and functional analysis, each with its own jargon, it is a challenge for any author to fuse these into a co herent and precise language. The text does a good job of it, but readers must
expect to do their part too. Those who prefer a fully rigorous mathematical pre sentation will sometimes have to follow the references. This is especially so in the background chapters where not everything is formalized, partly to keep the book a manageable size. For the ML researcher, understand ing MED and its sources of inspiration is highly recommended, as the method is at the center of a rapidly expanding area of research. Chances are that, even after reading this monograph, one will see possible ways of expanding the MED framework to new situations. For someone wanting to apply ML solu tions to engineering or science, the book offers a very flexible one-a so lution that can be adapted to a variety of tasks and of problem characteris tics. And for the reader interested in mathematical ideas at work in another field, the topics in this book rate five stars. The fundamental issues of ma chine learning are the mathematical expression of problems that every sci entist has to deal with. The ideas MED is based on (like maximum entropy, exponential models, VC theory [ 1]) are each on its own very powerful statisti cal and mathematical tools, lying at the core of most notable successes of ma chine learning in the last decade. MED itself is the result of combining them in an elegant and surprising way. Overall, the text is as an enthralling presentation of ideas that move machine learning these days. Let us remember that ML is ultimately a field with practi cal significance, which currently is ex periencing substantial growth. It's excit ing to be there, and this book takes you to the heart of the matter. Department of Statistics University of Washington Seattle, Washington 981 95-4322 USA e-mail:
[email protected]
REFERENCES
[ 1 ] V. Vapnik. Statistical Learning Theory. Wi ley, 1 998. [2] Marc Yor. Review of "Weighting the Odds : A Course in Probability and Statistics" by David Williams. The Mathematical lntelli
gencer, 25(2): 7 7-78, 2003.
The Politics of Large Numbers. A History of Statistical Reasoning by Alain Desrosieres Translated by Camille Naish CAMBRIDGE, MA, HARVARD UNIVERSITY PRESS, PAPERBACK, 2002. 384 PP. US $20.50 ISBN 0-674-00969-X REVIEWED BY PATTI W. HUNTER
T
oday, the term statistics carries a double meaning. On the one hand, it refers to collections of numerical facts about (among other things) polit ical, medical, or agricultural conditions. On the other, statistics is a scientific dis cipline, a method of analyzing data. For most of the nineteenth century and be fore, only the first meaning had wide spread use-a statistician was a col lector of numerical data. Not until the twentieth century did any statisticians think of themselves as mathematical scientists. How administrative and mathemat ical statistics came together is the sub ject of Alain Desrosieres's book, The Politics ofLarge Numbers. Drawing on historical research of the last few decades, Desrosieres examines the in teraction of these two domains, ana lyzing their development from the point of view of the sociology of sci ence. He is particularly interested in how the evolution of statistics (admin istrative and scientific) sheds light on the process by which such phenomena as unemployment, poverty, and infla tion acquire objective status. Covering the seventeenth through the early twentieth centuries, Desrosieres alternately considers the two faces of statistics. Three chapters (1, 5, 6) ex amine the origins and development of administrative statistics and their role in the state, comparing Germany, Eng land, France, and the United States. Chapters 2, 3, and 4 deal with subjects more closely connected to the mathe matical face of statistics: seventeenth and eighteenth-century probability, averages, correlation, and regression.
© 2006 Springer Science+ Business Med1a, Inc., Volume 28, Number I , 2006
69
Chapter 7 examines the social condi tions in which sampling techniques originated. Chapter 8 considers prob lems associated with choosing cate gories into which to classify people and things. Moving into the twentieth cen tury, chapter 9 traces the history of modem econometrics. The book has a number of interest ing and informative sections. Chapter 1 includes a detailed discussion of how the French Revolution shaped people's understanding of which aspects of social and demographic information were important and of what sets of categories most effectively classified that information. The chapter explor ing Adolphe Quetelet's "average man" (chapter 3) provides an engaging glimpse into nineteenth-century debates among French medical scientists over the value of data about public health, par ticularly in the context of efforts to identify the causes of the cholera epi demic of the 1830s. In the second half of chapter 6, Desrosieres traces the professionalization of what became the United States Census Bureau. Here we see the influence of antebellum de bates about the productivity of North em versus Southern states, of dis agreements about immigration policy, and of Franklin D. Roosevelt's New Deal. Readers not familiar with the his torical or mathematical details of the topics treated by Desrosieres may find his sociological analysis more accessi ble after first reading for themselves the sources from which the author draws his material (for example, [ 1 , 2, 3] discuss the history of probability be fore the twentieth century). In these works, they will find the writings of the historical figures analyzed carefully; they will learn something of the bi ographies of people who play impor tant roles in Desrosieres's discussion and of the professional and scientific contexts in which they worked. Math ematicians will need no introduction to Blaise Pascal or Simeon Poisson. But what about Wesley Mitchell? The au thor introduces him only as a "former census statistician" (p. 198), whose work took on new importance during World War I when Woodrow Wilson centralized national data collection.
70
THE MATHEMATICAL INTELLIGENCER
Some 80 pages further in, we learn that Mitchell founded the National Bureau of Economic Research in 1920, but the first discussion of this agency oc curs thirty pages later. About Tjalling Koopmans, whose ideas figure promi nently in the history of econometrics covered in chapter 9, some readers may find it helpful to be reminded of something more than the fact that he had a degree in physics. From where? What impact did this and his other training have on his economic thought? Desrosieres notes in his introduction that he is addressing readers from di verse cultural backgrounds. He seems to treat the household names of each culture (and sometimes their ideas) as familiar to all. This lack of attention to readers' di verse knowledge of history would not alone make The Politics of Large Numbers difficult for scholars new to Desrosieres's interests. Particularly in discussions of the broader questions about the sociological implications of the history of statistics, the exposi tion relies on specialized vocabulary and rather complicated sentences. Set ting out the purpose of the book, Desrosieres explains that he wants to link the technical history of statistics with its social history. "The thread that binds them," he writes, "is the develop ment-through a costly investment of technical and social forms. This en ables us to make disparate things hold together, thus generating things of an other order" (p. 9, italics in the origi nal). Desrosieres thus seeks to under stand the development of these "forms," as well as how social phenomena like unemployment, and technical phenom ena such as correlation managed to achieve objective status. As he puts it, The amplitude of the investment 'in forms realized in the past is what con ditions the solidity, durability, and space of validity of objects thus con structed: this idea is interesting pre cisely in that it connects the two di mensions, economic and cognitive, of the construction of a system of equivalences. The stability and per manence of the cognitive forms are in relation to the breadth of invest ment (in a general sense) that pro duced them. This relationship is of
prime importance in following the creation of a statistical system (p. 1 1). The introductory and concluding chapters are most densely populated with such statements, but they can be found throughout the text. From a comparison with the original 1993 French edition, this style does not seem to be a consequence of the trans lation. Desrosieres's exploration of the re lationship between statistics and the public sphere raises some intriguing questions about the mutual impact of mathematical ideas and the functions of the state. Readers with an interest in those questions and the willingness to fill in some historical detail and work their way through the exposition may find some thought-provoking an swers. Department of Mathematics Westmont College Santa Barbara, CA 931 08 USA e-mail:
[email protected]
REFERENCES
[ I ] Lorraine Dasto n , Classical Probability in the
Enlightenment, Princeton University Press, 1 988. [2] Theodore Porter, The Rise of Statistical
Thinking: 1 820- 1900, Princeton University Press, 1 986. [3] Stephen M. Stigler, The History of Statis
tics: The Measurement of Uncertainty be fore 1 900, Harvard University Press, 1 986.
Mathematical Circles, vol. I, II, Ill b y Howard Eves WASHINGTON, DC, THE MATHEMATICAL ASSOCIATION OF AMERICA, 2003. US $98.00. ISBN 0-88385-542-9, 088385-543-7, and 0-88385-544-5 REVIEWED BY STEVEN G. KRANTZ
G
ian-Carlo Rota observed that we mathematicians are more likely to be remembered for our expository work than for our research. (The ex ceptions are figures like Gauss, Rie mann, and Cauchy.) While only a hand ful of research mathematicians from the 1950s and 1960s are worth even a
mention, the name of Howard Eves (191 1-2004) stands tall. Everyone has heard of Howard Eves. Why? Eves did little research, but he wrote texts and articles on a vast array of subjects, ranging from combinatorial topology to geometry to complex variables to number theory. He is best remembered for his many wonderful expository books. Notable among these is the six volume Mathematical Circles collec tion. Using Google, I found no fewer than 2400 hits for Howard Eves. He is spoken of in glowing terms-"eminent mathematician and educator Howard Eves." Sounds perhaps like Saunders MacLane or Steve Smale. But it is Howard Eves, because Eves spoke to all of us with authority, with credibil ity, and with sincerity, in language that we can all understand. What was his secret? Eves was a considerable authority on mathematical history. That is a rather recondite subject area, and had he limited his publications to math his tory journals then he would probably languish in well-deserved obscurity. But he chose to share his erudition by way of anecdotes and aphorisms about our collective heroes. He wrote well, he wrote accurately, and he wrote with conviction. Eves's Mathematical Cir cles volumes constitute one of our col lective treasures. The original Prindle, Weber, and Schmidt editions are now
out of print, and we are fortunate that the Mathematical Association of Amer ica has seen fit to republish the six books in three elegant volumes. What is so special about these books? These days, with the great vogue in popular mathematical writing, we are beset with volumes about An drew Wiles, about the Riemann hy pothesis, about the history of zero, about chaos and fractals, and about any other quasi-mathematical topic that the public has a hope in hell of un derstanding. It should be noted that Eves was one of the pioneers of math ematical popularization and of mathe matical story-telling. And his books are serious. He does not tell-at least in his first five volumes-of the digestive quirks of Descartes or the romantic peccadilloes of Galois. He really wants to tell us about the mathematical en terprise, about the people who do mathematics and why they do it. Eves's purpose is serious, and his result-his written record-is substantive and valu able. We are fortunate to have these col lections of stories. Among the more charming anec dotes are • The story of why Thales never mar ried. • Four versions of the death of Archi medes. We have all heard the story of how Archimedes told one of Mar cellus's troops to get out of his light-the soldier was disturbing his
circles. The irate soldier ran the great scholar through with his lance. Eves offers at least three other pos sible versions of the story. • The story of how Napier identified his servant who was stealing. • The story of how l'Hopital obtained the rule named after him from Jo hann Bernoulli. • A determination of who were the second and third most prolific math ematicians in history (after Leon hard Euler). • The story of how the Indiana legis lature passed a law to declare it pos sible to square the circle (this may be the first and primary source for the story). • The story of how Walter Koppelman (in 1970) at the University of Penn sylvania was shot and killed by his graduate student Robert H. Cantor. Thus we see a range of events, from the historical to the current. The stories are told with a compelling accuracy and authority, rendered in concise and lively prose. Reading these stories is like eating Fritos: you cannot stop with just one. There is always a danger with se quels: You have told all your best stuff in the first volume. When your public or your publisher comes to you with demands for more, then you must cook something up. And it may not be up to the standard of the "stories of a life time" that you set forth in Volume I.
A CALCU LUS BOOK WORTH READING •
Clear narrative style
•
Thorough explanations and accurate proofs
•
Physical i nterpretations and appl ications
" U nlike any other calculus book I have seen . . . Meticulously written for the i ntelligent person who wants to u nderstand the subject. . . Not only more intuitive i n its approach to calculus, but also more logically rigorous in its discussion of the theoretical side than is usual. . . This style of explanation is well chosen to guide the serious
begin ner . . . A course based on it wou ld in my opinion definitely have a much greater
chance of prod ucing students who understand the structure, uses, and arguments of calculus, than is usually the case . . . Many recent and popular works on the topic will appear intellectually sterile after exposure to this one." -Roy Smith, Professor of
Calculus: The Elements MICHAEL COMEN ETZ
537 pp $46 softcover (981 -02-4904-7) $82 hardcover (981 -02-4903-9) Both editions have sewn bindings
Mathematics, U n iversity of Georgia (complete review at publ isher's website) "One has the feeling that it is a work by a mathematician still in close touch with physics . . . The author succeeds well in giving an excel lent i ntuitive introduction while ultimately maintaining a healthy respect for rigor." -Zentralblatt MA TH (online) A selection of the Scientific American Book Club
World Scientific Publishing Company 1 -800-227-7562
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©
2006 Spnnger Sc1ence+Business Media, Inc., Volume 28, Number 1, 2006
71
Howard Eves stood up pretty well to
exposition and popularization were ex
cal community are many modem-and
5
ecuted by those who are qualified to do
often much more sophisticated and
one can perceive some dissipation.
it. Of course the cognoscenti are all too
complex-developments that thus far
1
contained stories
busy proving theorems and going to
have not had many applications be
like the invention of the slide rule and
conferences. You will not catch them
yond their area of origin.
the
writing books like
this challenge. But around Volume Whereas Volume mathematical
treasures
Rhind papyrus, Volume
of
the
5 contains trite
cles.
Mathematical Cir
The encyclopedia under review tries
And we are all the poorer for it.
to cover the basic as well as many of
stories about the addition of vectors
Howard Eves was a virtuoso at his
the more specialized and new develop
and the formulation of IOU notes. Vol
craft. He knew exactly what he knew,
ments in the field of topology. It ad
and he knew his limitations. He wrote
dresses both established mathemati
about things that he had mastered and
cians and students, independent of their
ume
6
degenerates to the boringly
anecdotal:
A
student once threw a book at a
understood. He set a marvelous exam
area of specialization, and it is designed
mathematics instructor.
ple, to which we could all aspire. And
to lead them quickly to the terminology
"I wouldn't have done that," re
I hope that some of us will.
and results that might be useful to their
marked another student. "Why
not?"
asked
the
own investigations in other areas. culprit.
"Because that's no way to treat a book," was the reply. "Must we first take all these prelim
It explains terms like "Hedgehog",
Department of Mathematics Washington University in St. Louis
"Weak P-Point", "Cliquish Function",
St. Louis, MO 631 30
"Talagrand Compactness", "Thick Cov
e-mail:
[email protected]
ers", and "Resolutions". Similarly, it discusses
http://www. math.wustl.edu/�sk
concepts
like
"the
Sous
inary courses?" asked a mathemat
lin Hypothesis", "Hit-and-Miss Topolo
ics student.
gies", "the Normal Moore Space Con
"There's only one endeavor in which one can start at the top, and that's dig ging a hole, " replied the instructor. The trouble with humor is that it can
jecture", "the Blumberg Property", "the
Encyclopedia of General Topology
Is the class of meta-Lindelof spaces
rapidly become dated. I imagine that
edited by K.
someone thought these last two stories
Jun-iti Nagata, and J. E. Vaughan
were amusing at some time or another, but they seem pretty pointless at the moment. Still and all, Howard Eves's magnum
opus
is a noble one, and an important
part of our literature. It is arguably the first source for many important and
A
P.
Class MOBI", or "Bing's Example G". It also deals with difficult open problems:
Hart,
----- · --------- -- ----
preserved under perfect maps, or is .
-----
AMSTERDAM, ELSEVIER, 2004. 526 PP € 1 45, ISBN
0-444-50355-2.
there a Dowker space with a u-disjoint base? Finally, a wealth of known re sults and methods are treated: What is the Dugundji Extension Theorem?
REVIEWED BY HANS-PETER A. KUNZI
When is the completion of a topologi cal ring a field? Does the Sorgenfrey
A
s a student you may have been
line have a connected Hausdorff com
taught that normality is neither
pactification? How can one prove with
productive.
the method of elementary submodels
many of us cut our teeth on these books,
Later on in your career you may have
that the cardinality of a first-countable
and we re-read them with pleasure.
encountered intricate problems in var
compact Hausdorff space is at most
Eves has a great spirit, and a great sense
ious contexts that could be reduced to
that of the continuum?
of mathematical culture. He makes us
seemingly plain topological questions
The encyclopedia includes about
feel good about ourselves and what we
which, nevertheless, you could not im
articles contributed by a similar number
do. He shows us as erudite and human
mediately solve. For instance, in my
of topologists from all over the world.
and charming all at the same time. His
case, I could not find answers to some
Mostly written by experts in the spe
stories are never mean-spirited or criti
questions about the behaviour of nor
cialized fields, these articles outline, in
cal.
They show mathematicians for
mality in box products, or to the prob
a short sequence of definitions and re
what they are, in the strongest and most
lem of whether each compact topology
sults, many basic aspects of the treated
positive sense of the word. They make
is coarser than a compact topology in
topics. The average length of a contri
us feel good about our profession and
which all compact subsets are closed.
bution is about four pages. Very few
It is difficult to be a mathematician
proofs are given. The articles are col
and not use some basic concepts and
lected under ten headings that imitate
popular mathematical stories.
great
our enterprise. They leave us thirsting for more.
hereditary
nor
finitely
120
methods from General Topology every
Section 54 of the
now and then. Over the past decades a
ject Classification as used by Mathe
Fermat's last theorem and the lore of
few polished
some
matical Reviews and Zentralblatt MATH.
wavelets-mostly written by people
fundamental basic terminology of this
The key words listed below will give
who do not know what they are talk
field have become so well known and
the reader an idea about the contents of
ing about, just anxious for a quick
efficient that they now belong to the
the work. They roughly follow the titles
buck-I wish that Howard Eves were
folklore of today's university mathe
of the articles as they are listed in the
still writing. I wish that mathematical
matics. Less known to the mathemati-
table of contents of the book. Some
When I read modem inept volumes about the history of
72
7T
and the proof of
THE MATHEMATICAL INTELLIGENCER
techniques
and
2000 Mathematics Sub
readers might want to skip the list be low. I decided to include it in spite of a frowning referee, because it gives the reader a thorough and comprehensive first impression of the book In light of the high concentration of the presented results in the volume, the full flavour of the book cannot be grasped by merely glancing through a few examples of the orems and techniques, as they are dis cussed in the present review. GENERALITIES: Topological Spaces, Modified Open and Closed Sets, Cardinal Functions, Conver gence, Several Topologies on One Set (Minimal and Maximal Topologies). BASIC CONSTRUCTIONS: Sub spaces, Relative Properties, Product (Quotient, Adjunction, and Cleav able) Spaces, Hyperspaces, Inverse and Direct Systems, Covering Prop erties, Locally (P)- and Rim(P) Spaces, Categorical Topology, and Special Spaces. MAPS AND GENERAL TYPES OF SPACES DEFINED BY MAPS: Continuous (Open, Closed, Perfect, and Cell-Like) Maps, Extensions of Maps, Topological Embeddings (Universal Spaces), Continuous Selections, Multivalued Functions, The Baire Category Theorem, Ab solute Retracts, Extensors, Gener alized Continuities, Spaces of Func tions in Pointwise Convergence, Radon-Nikod:Ym (Corson, Rosenthal, and Eberlein) Compacta, Topologi cal Entropy, and Function Spaces. FAIRLY GENERAL PROPERTIES: Separation Axioms, Frechet (Se quential, and Pseudoradial) Spaces, Compactness (Local Compactness, Sigma-Compactness, Countable Com pactness, and Pseudocompactness), The LindelOf Property, Realcom pactness, k-Spaces, Dyadic Com pacta, Paracompact Spaces (Gener alizations and Countable Variants), Extensions of Topological Spaces, Remainders, The Cech-Stone Com pactification (in Particular of N and �), Wallman-Shanin Compactifica tion, H-Closed Spaces, Connected ness, Connectifications, and Special Constructions. SPACES WITH RICHER STRUC TURES: Metric Spaces, Metriza tion, Special Metrics, Completeness,
Baire Spaces, Uniform and Quasi Uniform Spaces, Proximity Spaces, Generalized Metric Spaces, Mono tone Normality, Probabilistic Metric Spaces, and Approach Spaces. SPECIAL PROPERTIES: Contin uum Theory, Dimension Theory (General, and of Metrizable Spaces), Infinite Dimension, Dimension Zero, Linearly Ordered and Gener alized Ordered Spaces, Unicoher ence and Multicoherence, Topolog ical Characterizations of Spaces, and Higher-Dimensional Local Con nectedness. SPECIAL SPACES: Extremally Disconnected (Scattered, and Dow ker) Spaces. CONNECTIONS WITH OTHER STRUCTURES: Topological Groups (Rings, Division Rings, Fields, and Lattices), Free Topological Groups, Homogeneous Spaces, Transforma tion Groups and Semigroups, Topo logical Discrete Dynamical Systems, Fixed Point Theorems, and Topo logical Representations of Algebraic Systems. OF INFLUENCES OTHER FIELDS: Descriptive Set Theory, Consistency Results in Topology (Quotable Principles, Forcing, and Large Cardinals), Digital Topology, Computer Science, Non Standard Topology, Topological Games, and Fuzzy Topological Spaces. CONNECTIONS WITH OTHER FIELDS: Banach Spaces, Measure Theory, Polyhedra and Complexes, Homology, Homotopy, Shape The ory, Manifolds, and Infinite-Dimen sional Topology. Throughout the book it is assumed that the reader has some basic knowl edge of set theory, algebra, and analysis. It is always easy to criticize various shortcomings of a book of this kind: Without doubt, each expert in the area will find some subject that in his or her opinion is lacking or is dealt with in sufficiently. Similarly, it is easy to feel that some topics treated are of minor importance for the field and could have been less stressed or even completely neglected. Thus some readers of this encyclopedia might miss sections on Constructive Topology, Topological Ordered Spaces, or Frame Theory,
©
while others might wonder whether the concept of Cleavability or the the ory of Approach Spaces are already so well established that they deserve their own sections in this volume. Also, in light of the large number of authors, the nature of the articles is un even in many respects, even after some unifying work was done by the editors. In particular the articles about the more specialized topics often and un avoidably have to assume a certain fa miliarity of the reader with basic con cepts. This familiarity cannot be gained by reading a few introductory articles in the encyclopedia. Thus while the en cyclopedia will certainly look impres sive on the shelves of a library or on the desk of a mathematician, the ques tion remains whether it is of practical use. As can be guessed from the short de scription above, the book will be very helpful to those who want to get a first overview of specific areas in the field and are looking for some references. However the work surely cannot re place the usual text books, mono graphs, or original research papers. In some sense, the value of encyclopedias in mathematics is quite limited: Essen tially they can only provide quick ori entation for informed readers. But they can hardly be relied on to teach com plete novices, and they are generally too superficial for the working spe cialist. Those who want to study some of the sketched theories in any depth will necessarily have to learn more from the references at the end of each arti cle, and from a general list of basic ref erences that is used throughout the book Persons who are simply inter ested in the solution to an isolated problem, for instance, need to know at once the basic facts of "the theory of bomological convergences", might prefer finding the latest literature deal ing with their question with the help of the Internet. Indeed, the book will be most useful to those who already know a lot about topology and now still want to deepen their understanding of vari ous areas closely related to their own field of interest. Certainly some discussions in topol ogy seminars could be based on parts
2006 Springer Science+ Business Media, Inc., Volume 28,
Number 1 ,
2006
73
of the more elementary articles. The students would be asked to fill in the missing arguments and appropriately expand the presentation of the material by consulting the pertinent literature. Nevertheless the Encyclopedia of General Topology is a remarkable book, and one to which the editors have contributed a huge amount of work Such efforts are important in a time when the value of specialized original research is often overvalued at the cost of a systematization of the natural historic developments of mathematical knowledge. The book represents a significant attempt to classify and order much of the work done in general topology and related areas in recent decades. It will inform future generations of mathematicians about the research already conducted and may thus avoid unnecessary du plications and subsequent disappointments. Without such books, complex theories cannot develop properly, and even excellent mathe matical concepts and results are soon forgotten and do not survive their discoverers or inven tors. Let us finally mention that the historical background of many of the theories presented is discussed in some detail in the three volumes thus far published by C. E. Aull and R. Lowen (eds.) under the title "Handbook of the History of General Topology", Kluwer Academic Pub lishers, 1997-2001.
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74
THE MATHEMATICAL INTELLIGENCER
Intelli
kj@ij,j.iq.U.I§i
Robin Wilson
The Philamath' s Alphabet-L Lagrange: Joseph-Louis Lagrange (1736-1813) wrote the first 'theory of functions', using the idea of a power se ries to make the calculus more rigor ous, and his mechanics text Mechani que analytique was highly influential. In number theory he proved that every positive integer can be written as the sum of four perfect squares. Laplace: Pierre-Simon Laplace ( 1749-1827) wrote a fundamental text on the analytical theory of probability and is also remembered for the
I
areas of regular polygons with 96 and 192 sides and deduced that 7T lies be tween 3.1410 and 3. 1427. Logarithmic spiral: The Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, . . . occurs throughout nature, and the ratios of successive terms 1/1, 2/1, 3/z, 5/3, . . . tend to the 'golden ratio' 1/2(1 + Y5) 1.618. . . . A 'golden rectangle' with sides in this ratio has the prope1ty that the removal of a square from one end leaves another golden rectangle; this process is shown on a Swiss stamp, which also features the closely related logarithmic spiral, found on snail shells and ammonites. Lyapunov: The Russian mathemati cian Aleksandr Lyapunov (1857-1918) was much influenced by Chebyshev. He worked on the stability of rotating liquids and the theory of probability. In 1918 his wife died of tuberculosis; Lya punov shot himself the same day and died shortly after.
'Laplace transform' of a function. His monumental five-volume work on ce lestial mechanics, Traite de mechani que celeste, earned him the title of 'the Newton of France'. Leibniz: Although Newton could claim priority for the calculus, Gottfried Wil helm Leibniz (1646-17 16), who devel oped it independently, was the first to publish it. His notation, including dyldx and the integral sign, was more versatile than Newton's and is still used. Leibniz's calculus was different from Newton's, being based on sums and differences rather than velocity and motion. Liu Hui: An ancient Chinese work, the Jiuzhang suanshu (Nine chapters on the mathematical art), contains the cal culation of areas and volumes, the eval uation of roots, and the systematic so lution of simultaneous equations. Around 260 AD, while revising the Jiu zhang suanshu, Liu Hui calculated the
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Robin Wilson, Faculty of Mathematics,
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THE MATHEMATICAL INTELLIGENCER © 2006 Springer Science+ Business Media. Inc.
