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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 Fibonacci 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 comfortably among these numbers-until midnight of April 26, 2003. At that instant, I ceased to be 89 years old; and there seems little prospect of my ever again having a Fibonacci number as my age. To be sure, my rural route address is now Box 532, Route 1, a concatenation of Fibonacci numbers in reverse order, but that is small consolation.
Something more is needed to reaffirm my allegiance. Here is my offering.
I will prove that the Fibonacci numbers with odd index can be generated iteratively from the quadratic equation
(la) x2 + y2 = 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 Fibonacci number with the next larger odd index. The Fibonacci numbers with even index are generated by
exactly the same procedure from the equation
(lb) x2+y2=3xy+l.
To prove these, I will use an immediate consequence of the defining iter
ation Fn+l = Fn + Fn-t:
(2) Fn-2 + Fn+2 = 3F,.
I 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) reduces it to
which does indeed have F,+2 as its larger root. Similarly for the assertion for even n.
he became assistant professor there. His work in mathematics continues to this day and has led to many honors and awards, among them the 2005 Abel Prize in mathematics [ 17].
Like von Neumann, Lax is a homo universalis in mathematics. 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 importance to CFD:
• the Lax equivalence theorem [18], stating that consistency and stability of a finite-difference discretization of a well-posed initial-boundary-value problem are necessary and sufficient for the convergence of that discretization,
• the Lax-Friedrichs scheme [ 19], a stabilized central finitedifference 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 allow multiple shock-wave solutions, and
• the Harten-Lax-Van Leer scheme [22], a very efficient numerical method for solving the Riemann problem.
Like von Neumann, Lax was (and still is) a strong proponent of the use of computers in mathematics. A quote
1 0 THE MATHEMATICAL INTELLIGENCER
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
A substantial part of the Euler and Navier-Stokes software used worldwide is based on a single journal paper [23], distilled by the then-young Russian mathematician Sergei Konstantinovich 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 uniformly constant pressure, from a gas on the right with a likewise 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 discontinuity 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 aforementioned membrane (traffic light). To know the interac-
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 instantaneously '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 complete aircraft or spacecraft can be computed. The remarkable 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 access 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 computations 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 Engineering. At that symposium, in a one-and-a-half-hour lecture, 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 published [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, without 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 remains 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-unfortunately-possibly wiggle-ridden.
Wiggles can be very troublesome in practice. For example, 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).
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 computing-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 astronomer. 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).
1 2 THE MATHEMATICAL INTELUGENCER
speeds and pressures which may arise in these computations impose high demands on the accuracy, and particularly 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 accurate 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 algorithm [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 doctorate 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-Stokesflow 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 unusual. 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 single 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. Efficient solution of these millions of equations is an art in itself. Many efficient solution algorithms have been devel-
oped, the most efficient of which are the multigrid algorithms. Multigrid methods were invented at different locations 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 increase 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, respectivelybut 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 multigrid 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 concerns 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 describes 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 channel instantaneously snaps up, so that, together with the red
Figure 13. Cross sections of a hexahedral grid around the Space SHuttle.
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 developed 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].
Books, journals, and software
Twenty years ago, textbooks on CFD were rare, but several are available now (see, e.g., [37, 38, 39, 40]). There are
>
>
X
X
also scientific journals dedicated to CFD. Moreover, offthe-shelf CFD software can be purchased these days. Each issue of, e.g., the monthly Aerospace America contains colorful, full-page advertisements for CFD software. A practical overview of the CFD literature, software, and also vacancies can be found on the Web site of CFD Online [41].
Today, CFD's role is about as important as that of experimental fluid dynamics. And CFD continues to grow. It is fed by improvements in both computer science and numerical mathematics. In addition, CFD itself stimulates research in computer science and numerical mathematics: a fruitful interaction.
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 (computational magnetohydrodynamics ).
Outlook
The fact that commercial CFD software is a success is proof of the practical importance of the theoretical fluid-dynamics work since Euler. The growing availability of CFD software may seem to be a threat for CFD research; CFD researchers 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 software from scratch gives the best insight and is pleasing work, but it may easily take too much time.
Education
A new question arises: How to teach CFD, now that it has become more and more important as an easily available, automatic tool? Not just factual knowledge but also understanding of the mathematical and physical principles of CFD remains indispensable, not only when practicing it as a science, but also when using it as a tool. The CFD-tool user must know and understand these principles well in (1) posing the computational problem, (2) choosing the numerical method to solve that problem, and (3) interpreting the computational 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 understanding of flow physics and numerical mathematics.
Research
As CFD becomes more and more mature, it also becomes more difficult to contribute fundamental research to it. In recent decades a PhD student can hardly do such fundamental work as Godunov did. Students will have to acquire an evergrowing 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 application 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 fluidflow problems will continue to arise, and there will certainly 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!"
REFERENCES
[1] T. von Karman, Aerodynamics, McGraw-Hi l l , New York, 1963.
[2] H . Rouse and S. lnce, History of Hydraulics, Dover, New York,
1963.
[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 to fluid me
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.
[1 0] R. Courant, K. 0. Friedrichs, and H. Lewy, Ober die partie/len Dif
ferenzgleichungen der mathematischen Physik, Mathematische
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, Berl in, 1 976.
[17] http://www.abelprisen.no/en/.
[18] P. D. Lax and R. D. Richtmyer, Survey of the stability of l inear 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
their numerical computation, Communications on Pure and Ap
plied Mathematics, 7 (1954), pp. 159-1 93.
[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, Ober die Fortpflanzung ebener Luftwellen von
endlicher Schwingungsweite, in : Gesammelte Werke, Leipzig,
1876. Reprint: Dover, New York, 1953.
[25] G. F. Duivesteijn , Visual shock tube solver (to be downloaded from
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
Barry Koren studied Aerospace Engineering at the Delft Insti
tute of Technology, and Computational Fluid Dynamics at the
Von Karman Institute for Fluid Dynamics in Belgium. He is now
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
namics at the Delft Institute of Techno logy. More information
can be found at http://homepages.cwi.nV-barry/.
He is married and the father of three children.
tiona/ Methods for lnviscid and Viscous Flows, Wiley, Chichester,
1 988-1 990.
[40] C. A J. Fletcher, Computational Techniques for Fluid Dynamics,
Vol. 1 Fundamental and General Techniques, Vol. 2 Specific Tech
niques for Different Flow Categories, Springer-Verlag, Berl in, 1 988.
[41] http://www cfd-online.com
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Ancient Egypt ian Mathemat ics : New Perspectives on O ld Sources Annette lmhausen
Send submissions to David E. Rowe,
Fachbereich 1 7 -Mathematik,
Johannes Gutenberg University,
055099 Mainz, Germany.
Pro captu lectoris
habent suafata libelli
(Terentianus Maurus)
I 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 subsequent readers have strongly influenced the way we view them today. And even now, readers of the third millennium can alter the fate of these early texts by their careful (or careless) reading. 1
Sources and Early Historiography
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 produced 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 contemporary Egyptological research. On the one hand, papyrus, the main writing material in this culture, was dependent 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 concerned with the mundane affairs of daily life were discarded after use, this condition was usually not fulfilled. Therefore, most of the written evidence documenting the role of mathematics in Egyptian social, economic, and cultural life must be assumed lost forever. On the other hand, to the extent that such sources may still be retrievable some day, practical problems stand in the way. The locations of ancient Egyptian cities often coincide with those of modem urban centers.
This makes it next to impossible to excavate 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 richest findings of Middle Kingdom papyri so far, among which incidentally are a number of mathematical fragments. 2
The two most significant sources, however, the famous Rhind and Moscow mathematical papyri, were bought on the antiquities market, making their provenance uncertain. These and most of the other known mathematical sources were already published by 1930.
The achievements of the earliest researchers who studied these texts, especially those who worked during the first half of the twentieth century, were enormous. As editors, they managed to penetrate a foreign vocabulary of technical terms, which placed them in position to make a first attempt at understanding Egyptian mathematical methods. 3 As was common at that time, ancient sources and achievements were viewed and evaluated by means of direct comparison with modem conventions and results. In many respects, it was found that Egyptian mathematics had little in common with the methods found in modem mathematical textbooks. Nevertheless, with some effort the mathematical content of the ancient texts could be "decoded" and "translated" into modem mathematics.
Unfortunately, this type of reading often entailed a loss of the most striking characteristics of the original sources, a drawback that was little appreciated at the time. Not surprisingly, the "achievements" of Egyptian mathematics, judged in terms of a different mathematical culture (from more than 3000 years later), looked rather crude and simple. One of the early leading authorities on ancient mathematics was Otto Neugebauer, who wrote his dissertation on Egyptian methods of cal-
culating with fractions.4 Afterward, Neugebauer turned his attention away from Egyptian mathematics to study Mesopotamian mathematics and astronomy, which he believed was a higher level of scientific achievement. As he once expressed this:
Egypt provides us with the exceptional case of a highly sophisticated civilization which flourished for many centuries without making a single 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 finally spreading like a contagious disease.5
It was surely in part due to the outstanding quality of the early scholarly contributions that readers accepted so readily this kind of negative assessment of Egyptian mathematics. As indicated already, this situation was compounded by the lack of new source material which-had it been therewould 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 material is still being uncovered on an almost 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 translation, various historians of mathematics began contributing new ideas based on their own readings of these first translations. Often these involved modem mathematical symbolism, leading to results that had almost nothing in common with the original source text. 7
This once common approach has now been recognized as both anachronistic 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 language in which it is written or the cultural background it comes from. At the same time, it has become obvious that mathematical knowledge is not universal. 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 background 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 created 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 caution, bearing in mind the kind of approach 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 following sketch, I will attempt to give an overview of the state of current research, illustrated with selected examples from the source material.
Although there have been no spectacular new finds of mathematical papyri, extant sources, including the much-studied Rhind and Moscow papyri, still offer many clues about the role of mathematics in Egyptian life. Alongside these, the Lahun mathematical fragments have just been re-edited, including several previously unpublished 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 interpretations 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 content. While a number of demotic mathematical texts have been published, no detailed study of Egyptian mathematics 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. Written 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 attached to some perishable goods from this grave, thus indicating their provenance and quantity. u; The quantities were rendered using elements and style familiar from the Egyptian number system in later times, i.e., a decimal system without positional notation (see Figure 1). In this system, each power of 10 up to 1 million was represented by a different sign. In order to write any number, the respective signs, written as often as needed, were juxtaposed in a symmetric way. Note that the hieroglyphic writing, which is what most people associated with ancient Egypt, was used mostly on stone monuments. For daily life purposes, Egyptian 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).
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 handling 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 surfaces points to the development of a
concept which is well documented in
the mathematical texts. 17 To express
sloped surfaces, such as the sides of a pyramid, the Egyptians used the so-
7 p a I
m s
'seqed'
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. 18 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 t. i• and ± were also written by using spe-
cial signs (indicating that these may be
older) rather than by using the general
Egyptian notation. 19 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 5, i 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 sys-
tem, which closely mirrors the Egyptian
concept of fraction, has become the
standard way of writing Egyptian fractions 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 sys-4 - -
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
Egyptian techniques of multiplication and division (see below for a more detailed description) frequently involved the doubling of a number. This could be done very easily if the number 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 giving 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 fragment 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 column shows (what we call) the divisor N, except for the first entry which shows both the dividend 2 and the divisor 3. This is followed by a second column that altematingly shows fractions 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 expressed 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 fractions needed to represent 2 7 5 is 3 15.
The 2 7 N table in the Rhind papyrus shows the same arrangement of numbers; 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 doubled 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. , representations with fewer elements are favored as are also representations with larger inverses, etc.-it has not been possible to establish strict mathematical 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 mathematical concept of fractions devised by another culture several thousand years
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, instead of trying to concoct an explanation 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 environment 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 instructions showing how to solve it. Procedure texts derive from an educational setting. They may have been written by a teacher, who was compiling a handbook, or perhaps by a student engaged in practicing mathematical techniques. An appreciation of this context is important 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 mathematical knowledge from sources of this nature.
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 according to their rising level of difficulty. This is not to be judged by purely mathematical aspects alone but also by additional knowledge (often from a practical background) which is necessary to solve the problems. This can be seen, for example, in pRhind, problems 31-34 and those immediately following, problems 35-38. Mathematically, both groups teach a procedure for determining an "unknown" number if its
Figure 3. Fragment UC 32159: 2 .;- N table (Copyright Petrie Museum of Egyptian Archaeol- sum with fractions of itself is given. ogy, University College London). The procedure for solving the prob-
22 THE MATHEMATICAL INTELLIGENCER
lems in both groups is roughly the same. However, in the second group (pRhind, problems 35-38), the "unknown" number is not an abstract number but a quantity of grain. Therefore the result, which is determined in the same way as in the preceding problems, needs to be transformed afterwards 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.
Problem 56, like the other four examples 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 technical term sqd-the number of palms the slope of a slanted plane recedes per vertical difference of one cubit-is explicitly indicated in the problem text. Thus, the base, height, and inclination of a pyramid are linked by the relation:
1 /2 base inclination = 7 palms X
height
The problem above presents a pyramid with base (360) and height (250); its inclination is to be calculated. The procedure calls for calculating half of the base and dividing this by the height. The result is then multiplied by 7 to obtain the inclination in palms. Having grasped "what is going on" in this problem, 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 pyramid." Note that the beginning of the title 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 . . . " followed 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 introduced, 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 example: "You shall let me know its inclination." 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 operations are expressed without any use of mathematical symbols. The instructions themselves are always given as complete sentences.
Furthermore, in this part of the text, a special verb form is used, the socalled sd.m.IJr=f. The name consists
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 previously stated conditions.24 In the mathematical 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 underlines the specific procedural character of the text: the sequence of instructions 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 multiplication in the Egyptian manner, which may now be described.
Multiplication (and division) are executed following a scheme that uses two columns of numbers. 25 Each multiplication begins with the initialization which is found in the first line of the scheme: a dot is placed in the first column and the number to be multiplied in the second column. The multiplication is carried out by subsequent operations in both columns using a variety of techniques, depending on the numerical values of the numbers that shall be multiplied. The aim is to find the multiplier as a combination of entries in lines of the first column. The respective lines of the second column will then be the result of the multiplication.
Figure 4. Rhind Mathematical Papyrus, No. 56 Copyright The British Museum.
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 indicates
_ o�e
.!!_f the required fractional
parts (2, 5, 50) of the multiplier. How the individual entires of the second column were found is not obvious. It is possible that there may have been tables for fractional parts of 7, as this was a number that leads to complicated calculations, but which came up frequently due to the metrological conventions.26
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 advantage of informing a modem reader at a single glance how an ancient measure was defined, it conveys nothing whatsoever about the procedural character of Egyptian mathematics. Moreover, algebraic 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 approach was first proposed by Jim Ritter.27 In my dissertation I have adapted this method to analyze the various procedures used in all hieratic mathematical 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 instructions by replacing the rhetoric formulations with modem symbols that indicate the respective arithmetic operations. The data are noted at the beginning of the scheme by their numerical 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 is its base, 360 250 is its height. 250
You shall let me know its inclination. You calculate half of 360.
(1) 2 X 360 It results as 180. = 180 You divide 180 by 250.
(2) 180 -7- 250 2 5 50 of a cubit results.
1 cubit is 7 palms. You multiply with 7.
= 2 5 50
(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 problem 43 of the Rhind papyrus, the calculation of the volume of a granary with circular base, the diameter of the granary is given as 9. This greatly facilitates 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 fraction of three parts, which then has to be multiplied by 7. Thus, by comparison, the data in problems 58 and 59 result in easier calculations.
This first stage of rewriting is especially helpful when dealing with a corrupt 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 different types of numbers that can appear 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, intermediate results, and constants. To distinguish 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 results are specified by a number in parentheses (x) which specifies the step in the procedure that leads to the given result. The only actual numbers that now appear in the rewritten text are constants. Thus, for our example, the result of this second rewriting is as follows:
Method of calculating a pyramid, 360 is its base, D1
250 is its height. D2 You shall let me know its inclination. You calculate half of 360.
(1) 2 X Dl 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 mathematical 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 complexity. The analysis of problems by means of their procedures or algorithms thus constitutes a powerful tool for comparing the structure of individual mathematical problem texts. From this, one can learn a great deal about Egyptian mathematical techniques. Within the Rhind Mathematical Papyrus, for example, 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 symbolic algorithm can sometimes be
straightforward, as in the example above. Unfortunately this is not the case with all problems. Individual instructions may be missing-sometimes they are replaced by a written calculation, or several steps are summarized in one instruction only. These types of difficulties can sometimes be overcome by taking into account all of the available source material. If-as in the Rhind Papyrus-several problems of the same kind are available and their procedures are identical insofar as they are explicitly stated, then those problems which lack certain instructions can occasionally be reconstructed by means of the more detailed problems.
I would like to stress in this context that both types of rewriting are merely tools for analyzing specific aspects of the procedures found in the problem texts, whereas the source texts themselves remain central and should never be neglected in any analysis. Taking the three versions of the procedure together, however, enables one to form a more complete analysis that includes not only the various procedures but also technical mathematical vocabulary, as well as the relation of drawings and calculations carried out in writing connected with the procedure, and others.
Mathematics within the Context
of Egyptian Culture
Another integral part of the reassessment of Egyptian mathematics concerns its role within Egyptian culture. Mathematics was one of the key elements of scribal training in pharaonic Egypt. It provided the scribes with a crucial tool they needed to fulfil their administrative tasks as well as to plan and carry out construction projects. Consequently, many of the mathematical problems they dealt with were related to practical matters, e.g., the distribution of rations, the volume of granaries, or the amount of produce to be delivered by a worker. Our understanding of mathematical problems of this kind is at least partially dependent on our appreciation for these larger contexts.
This can be demonstrated with the so-called bread and beer problems, which appear against the background of economic activity, baking and brewing, under the control of a local au-
thority (state or temple). A quantity of grain is taken from a granary and then given to workers who produce bread and/or beer from it. Obviously it was necessary to ascertain the quantity-in loaves of bread or vessels of beer-of a given quality (in this case measured by grain content) that was equivalent to the amount of grain initially given to the workers. The mathematical side of this control is represented by the bread and beer problems. 29 The terminology used in these problems is taken from the respective technological language. Thus the bread and beer problems evolve around the psw, a unit which measures how many loaves of bread have been made from one 1}1;,3.t of grain. Apart from the psw, there are two additional standard phrases indicating the use of specific kinds of grain products and their quantities. Obviously, this has further consequences for the respective calculations. Similar observations can be made for other groups of practical problems as well. These generally involve not only the "basic" mathematical terminology but also further knowledge related to the technological or administrative background. This usually makes them not only more difficult to understand but also less likely to be "mirrored" by a familiar problem in modem mathematics. Thus, early historical research often neglected this area of Egyptian mathematics.
However, as is obvious from the ordering of the problems found in the Rhind Papyrus, it was precisely these practical problems that were considered more advanced. After all, the aim of the mathematical handbooks was to prepare scribes for their daily administrative work. So if we want to obtain insights into Egyptian mathematics, we must consider these problems and try to understand them. The setting of the individual problem may help to point to further sources (not only textual) which may be useful to understand the additional terminology and practice. Furthermore, it is this type of problem that indicates other possibilities of gaining knowledge about Egyptian mathematics apart from the restricted corpus of mathematical texts. The actual output of the scribes in doing their daily work provides us with numerous
documents that prove the use of mathematical techniques. Thus Michel Guillemot has used a ration text from Kahun to analyze mathematical practices. 30 These can be linked to techniques taught in mathematical papyri.31 It is to be hoped that this example can be followed for other texts as well.
The most promising sources still to be explored in this respect are the Reisner Papyri. This set of four papyrus rolls contains calculations for the building of a sanctuary, including ration tables, actual building calculations, as well as the administration of workshops. 32 They not only enlarge the meagre set of seven problems related to architecture which are known from the Moscow (problem 14) and Rhind (problems 56-60) papyri, but they also demonstrate that the amount of work done was linked to a specific number of workers (and rations) per day.
Evidence of Mathematics
in the New Kingdom
While the mathematical texts date almost exclusively from the Middle Kingdom, other sources are available from all periods of Egyptian history. The Wilbour Papyrus, a text from the New Kingdom, is an official record of measurements and assessments of fields over a distance of 90 miles along the Nile. The fields are given by localization and acreage, their assessments referring to taxes specified in amounts of grain.
Another major opportunity to find relevant sources of mathematics for the New Kingdom is provided by the excavation of Deir el Medina. Deir el Medina is the modem name of an ancient Egyptian village on the West Bank of the Nile opposite Luxor. The village was inhabited by workmen who were responsible for the construction and decoration of the tombs in the Valley of the Kings. Deir el Medina has
yielded a huge quantity of artifacts and texts relating to daily life in the New Kingdom-similar to the findings at Lahun for the Middle Kingdom. Among the sources are ration texts, building plans, as well as texts for the education of scribes. The ostracon in Figure 6 shows a fragment of an exercise in the multiplicative writing of large numbers. It shows in the first column
Figure 6. Deir el Medina: Remains and Ostracon with Number Exercise.
(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 respective multiplicative factors (6, 7, and 8 and 5, 6, and 7) below. The third column (left) shows again the sign for 1,000,000 and two illegible signs below.
Conclusions
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 mathematics, is certain to provide a better foundation for understanding its role within Egyptian culture. This integrated approach represents an important advance beyond the early studies that relied exclusively on an internal analysis of a small corpus of mathematical 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 anticipate that the fate of Egyptian mathematics faces an exciting future.
NOTES
1 . I thank David Rowe for his comments on
previous versions of this article and for his
corrections of my English. I also thank
Richard Parkinson of the British Museum
and Stephen Quirke of the Petrie Museum
26 THE MATHEMATICAL INTELLIGENCER
for permission to include photographs of
sources.
2. See Annette lmhausen and Jim Ritter,
"Mathematical Papyri , " in : Mark Collier and
Stephen Quirke (eds.), The UCL Lahun Pa
pyri: Religious, Literary, Legal, Mathematical
and Medical, Oxford: Arcaheopress 2004.
3. Among the early editions, the most note
worthy are still Thomas E. Peel, The Rhind
Mathematical Papyrus. British Museum
10057 and 10058, London: Hodder and
Stoughton 1 923, and Wasili W. Struve,
Mathematischer Papyrus des Staatlichen
Museums der Sch6nen Kunste in Moskau
(Quellen und Studien zur Geschichte der
Mathematik, Abteilung A: Quellen, Vol. 1 ) ,
Heidelberg: Springer 1 930.
4. Otto Neugebauer, Die Grundlagen der
agyptischen Bruchrechnung, Berl in: Julius
Springer 1 926.
5 . Otto Neugebauer, A History of Ancient
Mathematical Astronomy (Part Two). Berlin,
Heidelberg, New York: Springer 1 975: 559.
6. See for example the interpretations of
Plimpton 322, e.g. , compare Joran Friberg ,
"Methods and traditions of Babylonian
mathematics: Plimpton 322 , Pythagorean
triples and the Babylonian triangle param
eter equations," Historia Mathematica 8
(1 98 1 ): 277-318 and the recent reassess
ment by Eleanor Robson (Eleanor Robson,
"Neither Sherlock Holmes nor Babylon : a
reassessment of Plimpton 322 , " Historia
Mathematica 28 (200 1 ) ; 1 67-206 and
Eleanor Robson, "Words and pictures: new
light on Plimpton 322," American Mathe
matical Monthly 1 09 (2002): 1 05-1 20).
7. An extreme example of this is Richard
Gill ings, "The Volume of a Truncated Pyra
mid in Ancient Egyptian Papyri , " The Math
ematics Teacher 57 (1964): 552-555.
8. For Egyptian mathematics, see for exam
ple James Ritter, "Chacun sa verite: les
mathematiques en Egypte et en Me
sopotamie , " in : Michel Serres (ed.) , Ele-
ments d'histoire des sciences: 39-61 ,
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, Boston, 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.
Serafina Cuomo, Ancient Mathematics,
London, New York: Routledge 2001 ,
Michael N. Fried and Sabetai Unguru,
Apollonius of Perga 's Conica. Text, Con
text, Subtext, Leiden: Brill 2001 , as well as
David Fowler, The Mathematics of Plato's
Academy: A New Reconstruction (Second
Edition), Oxford: Clarendon Press 1 999,
and Reviel Netz, The Shaping of Deduc
tion in Greek Mathematics: A Study of
Cognitive History (Ideas in Context 5 1 ) , Cambridge: Cambridge University Press
1999. For Mesopotamian mathematics,
see most recently Jens H0yrup, Lengths,
Widths, Surfaces. A Portrait of Old Baby
lonian Algebra and its Kin, New York:
Springer 2002, and Eleanor Robson,
Mesopotamian Mathematics, 2 1 00-1 600
BC: Technical Constants in Bureaucracy
and Education (Oxford Editions of
Cuneiform Texts XIV), Oxford: Clarendon
Press 1 999.
9 . See Gary Urton, The Social Life of Numbers.
A Quechua Ontology of Numbers and Phi
losophy of Arithmetic, Austin , Texas: Uni
versity of Texas Press 1 997, and Marcia As
cher, Mathematics Elsewhere. An
Exploration of Ideas across Cultures, Prince
ton, N.J . : Princeton University Press 2002.
1 0. See, for example, for Mesopotamia Jens
H0yrup, Lengths, Widths, Surfaces. A Por
trait of Old Babylonian Algebra and its Kin,
New York: Springer 2002.
1 1 See note 8.
1 2 . See Annette lmhausen and Jim Ritter,
"Mathematical Papyri , " in: Mark Collier and
Stephen Quirke (eds ) , The UCL Lahun Pa
pyri: Religious, Literary, Legal, Mathematical
and Medical, Oxford: Arcaheopress 2004.
Another mathematical fragment will be pub
lished in the next volume of that series.
1 3 . See Oleg Berlev, "Review of William Kelly
Simpson: Papyrus Reisner I l l : The Records
of a Building Project in the Early Twelfth
Dynasty, Boston : Museum of Fine Arts
1 969," Bibliotheca Orienta/is 28 (1 97 1 ):
324-326, esp. p. 325.
1 4 . Richard Parker, "A Demotic Mathematical
Papyrus Fragment," Journal of Near East
ern Studies 1 8 (1 959): 275-279; Richard
Parker, Demotic Mathematical Papyri,
Providence, R . I . : Brown University Press
1 972; Richard Parker, "A Mathematical Ex
ercise-P. Dem. Heidelberg 663 , " Journal
of Egyptian Archaeology 61 (1 975):
1 89-1 96. A list of Demotic mathematical
ostraca can be found in Jim R itter, "Egypt
ian Mathematics , " in: Helaine Selin (ed.),
Mathematics across Cultures. The History
of Non-Western Mathematics , Dordrecht:
Kluwer 2000: 1 34, note 27.
1 5 . See Gunter Dreyer, Umm ei-Qaab I . Das
pradynastische Konigsgrab U-j und seine
fruhen Schriftzeugnisse, Mainz: Von Zabern
1 998
1 6 . For a discussion of the inscriptions on these
tags, see Gunter Dreyer, Umm ei-Qaab I.
Das pradynastische Konigsgrab U-j und
seine fruhen Schriftzeugnisse, Mainz: Von
Zabern 1 998, pp. 1 37-145, and John
Baines, "The Earliest Egyptian Writing: De
velopment, Context, Purpose," in: Stephen
D. Houston, The First Writing. Script 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
matical Papyrus.
1 8. Jim Ritter, "Mathematics in Egypt, " in :
Helaine Selin (ed.), Encyclopedia of the
History of Science, Technology and Med
icine in Non-Western Cultures, Dordrecht,
Boston, London: Kluwer 1 997, p. 631 .
1 9. For the prehistory of Egyptian fractions and
their development see Jim Ritter, "Metrol
ogy and the Prehistory of Fractions," in:
Paul Benoit, Karine Chernla, Jim Ritter
(eds.) , Histoire de fractions, fractions d'his
toire: 1 9-34, Basel, Boston, Berlin:
Birkhauser 1 992.
20. See, for example, the description of Cou
choud: " . . . i l ne semble avoir connu que
les fractions unitaires, c'est a dire celles
dans lesquelles le numerateur est toujours
equivalent a ! 'unite, . . . " (Sylvia Couchoud,
Mathematiques Egyptiennes. Recherches
sur les connaissances mathematiques de
I 'Egypte pharaonique, Paris: Le Leopard
d'Or 1 993, p. 2 1 ) or that of Gill ings: "When
the Egyptian scribe needed to compute
with fractions he was confronted with
many difficulties arising from the restriction
of his notation. His method of writing num
bers did not allow him to write such sim
ple fractions as % or % because all fractions
had to have unity for their numerators (with
one exception)." (Richard J. Gill ings, Math
ematics in the Time of the Pharaohs, Cam
bridge, Mass . : MIT Press 1 972, p. 20).
2 1 . See Jim Ritter, "Egyptian Mathematics , " in:
Suppose I take the wallets from you and ninety-nine of your closest
friends. We play the following game with them: I randomly place the wallets inside one hundred lockers in a locker room, one wallet in each locker, and then I let you and your friends inside, one at a time. Each of you is allowed to open and look inside of up to fifty of the lockers. You may inspect the wallets you find there, even checking 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 communicate 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 discussion of the problem. This consists of numbering our players, and replacing wallets by player numbers! Our game is played between a single Player A against a Team B with 100 members, B1, B2, . . . , B100. Player A places the numbers 1, 2, . . . , 100 randomly in lockers 1, 2, . . . , 100 with one number per locker. The members of Team B are admitted 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 containing his own number. Team B is allowed
an initial strategy meeting. No communication 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 numbers 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 finding 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! Amazingly there is a strategy which gives significant 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 lockers, 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 lockers. 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 examined lockers 1-5, and team members 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 lockers 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 10 9 8 7 6 - 242 "
While this is an improvement over random 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 probability 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 probabilities of success of the other members. This is the first clue!
An ideal strategy would be one where if B1 succeeds then everyone else does too. Note that this would allow 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 depend on information not available at the initial meeting. The only such information 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 before 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 possible 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 B1 alone has 10 X 99 X 89X8 X 79xsx7 x 69xsx7x6 possible strategies. To compute the number of strategy choices for the whole team, we raise this to the lOth power and get a number 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 unremarkable; 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 number 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 example, 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 fmding 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 players 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 because there is no long cycle. To find the probability that Team B wins, we count the number of permutations of 10 numbers with a cycle of length 6 or longer. First let's count how many have a 6-cycle. Choose which 6 elements go into the 6-cycle, arrange them in cyclic order, and then pick an arbitrary permu-
tation of the remaining 4 elements. The number of ways to do this is (10) ' ' -
10!5!4! -
10! 6
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 permutations 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 oflength 6 or larger is 1/6 + 117 + 1/8 + 1 /9 + 1 / 1 0 = 1 62 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 cycle is IJ:�1 -1-. By viewing this ex-
n+k pression as an upper Riemann sum for
f2n -1- dx and a lower Riemann sum n x + 1 for f2n l dx we obtain n X ( 1 ) l2n 1 ln 2 - n + 1 = n x + 1
dx
n 1 12n 1 s L -- s - dx = ln 2.
k � l n + k n X 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- givn+k ing the probability of success is
monotonically decreasing to 1 - ln 2 = 0.306853. Team B wins with the pointerfollowing strategy with probability exceeding 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 pointerfollowing by comparing the game considered above (Game 1) with a new game (Game 2) between the same adversaries, 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 distributes the 10 numbers at random in the 10 lockers. Then all of team B is invited into the locker room together. Team member B1 is required by the rules to start opening lockers and continue until she reveals the number 1 . Once she has opened the locker containing the number 1 , she may not open any further lockers; then, the lowest-numbered member of Team B whose number 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 individual team member opens more than 5 lockers. Before proceeding, we invite you to consider the following questions: 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 observe Team B in the process of playing Game 2. We record the progress, listing 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 remaining 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 lockers at random or follow the most sophisticated plan; we still get probability 1/10! for each of the 10! possible orders in which the numbers could be revealed. In Game 2 Team B's probability of success is completely independent 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 version of the classical records-to-cycles bijection [6, p17] to assign a permutation written in cycle notation to each ordering. The first cycle of our permutation 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, 10,8,3,5 ----> (2,6, 1)( 4,9, 7, 10,8,3) (5). Furthermore we see that each permutation 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 ascending order. For example (9, 7,8)(1,3, 10,5) (2 ,4 ,6) = (3, 10 ,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 probability 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 exactly the probability of success for Team B in Game 1 using pointer following!
Our analysis has a significant consequence for Game 1. Team B can take any Game 1 strategy and adapt it to Game 2 as follows: If player Bi is opening 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 strategy succeeds in Game 1 for a particular distribution of numbers into lockers 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 probability 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 spreading when I presented it to several people at Complexity 2003, which was held in Aarhus, where I was a local organizer." 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 considered 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 simple pointer-following will not work as empty lockers do not point anywhere. It is an open question whether the winning 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 proportion of empty lockers and the fraction of lockers each team member may open. As the number of players increases, their probability of success for Team B approaches zero less rapidly than conjectured in [4]. And fixing the number of players and fraction of lockers 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 replaced 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 cycle has length n + 1 , the team members caught in the n + 1 cycle can guess in such a manner that they all guess correctly 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, p120]. The locker problem will be discussed in a future edition of Winkler's book also.
We thank Joel Spencer for introducing us to the problem, and we thank
Ravi Vakil and Michael Kleber for encouraging us to write this note and providing many useful suggestions.
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 operations either on numbers or on geometrical figures. Then in 1855 Arthur Cayley introduced new objects called ma trices, and soon afterwards he gave the laws of their algebra. Seventy years later, W emer Heisenberg found that the noncommutativity of matrix multiplication offers just the right conceptual framework for describing the laws of atomic mechanics. Matrices were found to be useful in the description of classical vibrating systems and electrical networks as well. For mathematicians, analysis of linear operators 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 quantum mechanical systems and impedances of electrical networks are defined in terms of matrices. Mixing of the underlying 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 matrices, both of these operations involve new subtleties. In this article we introduce the reader to some of them.
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) m(a,b) = m(b,a).
(ii) min(a,b) :S m(a,b) :S max(a,b).
(iii) m( aa,ab) = am( a,b) for all a > 0. (iv) m(a,b) is an increasing function of a and b. (v) m(a,b) is a continuous function of a and b.
The three familiar means, arithmetic, geometric, and harmonic, satisfy all these requirements. Other examples of means include the binomial means, also called the power means, defined as ( aP + bJ! )lip
mp(a,b) = 2 , - oc :S p :s; x.
Here, it is understood that for the special values p = 0 and ±: oc we define mp(a,b) as the limits
mo(a,b) = limp __, omp(a,b) = v;;:b, mx( a,b) = limp --> xmp( a,b) = max( a,b ) ,
m-x(a,b) = limp __, -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 elements of Mn(IC), and iJ=Dn that of all positive definite matrices. The space §n is a real vector space and iJ=D n 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=D n· This is
given by the map r x(A) = X* AX. We say that A and B are
congruent if B = r x(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
A1 2: A2 and B1 2: B2, then M(A1,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, thenf(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
teA + B) and [ tCA - 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 112 B112 is not self-adjoint, let alone pos
itive definite. This rules out using A112Bl12 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
that reduce to A 112B112 when A and B commute. One plau
sible choice is the quantity
(1) ( log A + log B ) _ . ( AP + BP )lip exp 2 - hmp __. o 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 = t (log A + log B) and Y = t 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
(2) A#B = A 112 (A - 1/2 BA - 112)v2 Avz,
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 ofA#B due to T. Ando [A]. We have
(3) A#B = max{x : X = X* and [� �1 2: o}. Among its other characterisations, one describes A#B as
the unique positive definite solution of the Riccati equation
(4) XA- 1X = B.
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) : property (III) may be verified easily from (3) or ( 4). It satisfies the expected inequality
( A - 1 + B- 1 ) - 1 A + B (5) 2
:::::: 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:
(A,B) rA-"' (I ,A - 112 BA - 112) � (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) = r AJt2(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.
The indirect argument we have used to deduce the symmetry of the geometric mean is not necessary. Let m( a,b) be any mean, letf(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 eigenvalue 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 numbers to positive definite matrices. Kubo and Ando [KA] developed a general theory of matrix means and established a correspondence between such means and matrix monotone functions.
What happens when we have three positive definite matrices 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 113 B113C113 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 extends readily to three. The expressions (2), (3), and ( 4) have no obvious generalisations that work The idea of simultaneous diagonalisation does not help either: while two positive 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.
0 0 0
One geometry cannot be more true than another; it can only be more convenient.
-Henri Poincare [Po}
While the geometric mean A#B has been much studied in connection with problems of matrix analysis, mathematical physics, and electrical engineering, a deeper under-
34 THE MATHEMATICAL INTELLIGENCER
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 I IAib = (tr A*A)112 is called the Frobenius, or the Hilbert-Schmidt, norm. If A is a matrix with eigenvalues A 1, . . . , An, we write A (A) for the vector (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 differentiable manifold. The exponential is a bijection from §n onto IP n· The Riemannian metric on the manifold IP n is constructed as follows. The element of arc length is the differential
(7) ds = IIA - 112 dA A - 112l lz.
This gives the prescription for computing the length of a differentiable curve in IP n· If y : [ a,b] � IP n is such a curve, then its length, obtained by integrating the formula (7), is
(8) L(y) = r � �y- 112(t)y'(t)y- 112(t)lb dt. a
If A and B are two elements of IP n, 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 r X : IP n � IP n is a bijective isometry on the metric space (IP n,82).
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,l l · l lz) 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 infinitesimal 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.,
efi+tK - elf (11) Defi(K) = limt--.o 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 [ fl'; - eAi J Defi(K) =
A · _ A · kij . '· J
(The notation here is that [Xij] stands for a matrix with entries Xij·) From this, one sees that the (iJ) entry of e-H12Defi(K)e-H12 is
( 12) sinh(Ai - Ai)/2
(A.; - Aj)/2 kij·
Since sinh X 2: 1, the inequality ( 10) follows from this. X
In the special case when H and K commute, a calculation shows that there is equality in (9). In this case the function 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 segment joining A and B is the path
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 r A-lt2 is an isometry 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
This shows that the geometric mean A#B defined by the fonnula (2) is nothing but the midpoint of the geodesic joining A and B in the Riemannian manifold !f1> n· Thus while (2), (3), and (4) might have ap,Reared as over-imaginative noncommutative variants of �. very natural geometric considerations 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 formula (6). Those means are not symmetric, however: (I) fails unless t = 1/2.
This discussion also gives an explicit formula for the metric 82. We have o2(A,B) = 82 (I ,A - 112 BA - 112) =
2 2. 5 2 1 . 5 0. 5
lllog I - log (A - 112BA - 112) 1 12 = lllog(A - 112BA - 112) l lz. The matrices A - 112 BA - l/2 and A - 1 B have the same eigenvalues. So, this can be expressed as
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 l l · l lz 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 reduced to the special case by applying the congruence fexp(-A/2) to all points and thus changing one of the vertices to I. This is often described by saying that two geodesics 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 (Figure 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 diffeomorphism.
Using the formula (11) with H and K in i§n, we reduce
exp(iA)
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.
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 • • • , iAn) with A1 real. Instead of (12) we have now
sin(Ai - AD/2 (Ai - AJ)/2 kii·
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.
expA
Returning to IP n and the geometric mean, it is not difficult to derive from the information at our disposal the fact that given any three points A, B, and C in 1Pn we have
1 (15) o2(A#B,A#C) :S 2 o2(B,C).
This inequality says that in every geodesic triangle in IP n with vertices A, B, and C, the length of the geodesic joining 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 define 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:
(M1) 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 ll1 = Ll and ll1+ 1 is the triangle obtained by joining the mid-
�--__,. expC
A � 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 Elie 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).
0.25
0.2
0 . 15
0.1
0.05
0.2
0 0.2 0.3 0.4
Starting with /11 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
scend to zero as 112n. 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 r X 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 matrices 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 understanding 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
0.5 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.
norms. These norms 1 1 · 1 1"' have the invariance property I IU A VII"' = I IAII"' for all unitary U and V. Each of these norms corresponds to a symmetric norm <P on IR11; that is, a norm <P that is invariant under permutations and sign changes of coordinates. The correspondence is given by I IA I I''' = <P (s,(A), . . . , Sn(A)), where s,(A) 2: · · · 2: sn(A) are the singular values of A. Common examples are the Holder norms <Pp(X) = o:lxj:V)11P and the corresponding Schatten
norms I IAIIv = (� s}(A))11P, 1 ::::: p ::::: oc. The Frobenius norm is the special case p = 2.
For each of these norms we may define a metric o,v on 1P'11 as in the formula (14). The EMI in the form (9) or (10) remains true (see [B2]). The import of this remark is that, with any of these metrics, IP'n is a Finsler manifold of nonpositive curvature; the special Frobenius norm arises from an inner product and gives rise to a Riemannian structure. In recent years metric spaces of non positive curvature have been studied in great detail; see the comprehensive book by Bridson and Haefliger [BrHa] . The spaces IP' n with norms 1 1 · 11''' are interesting and natural examples of such spaces.
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 [ FLS j
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 matrices 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
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
shown are chosen to reflect the relative strengths of the weights a,b,c.
Thus G2(A,8,C) corresponds to 1 /3, 1/3, 1/3 (see yellow dot on sur
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
Here A 1 (X) is the largest eigenvalue of a matrix X with real eigenvalues. In a similar vein, we have the famous GoldenThompson inequality
(17)
The matrices ef!+K and ef112eKeH12 are positive definite. So, the inequalities ( 16) and (17) say
llefl+K]IP ::::: l lefll2eKeH12I IP, 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 (17) . 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] . Generalisations 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, Riemannian 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. Haefl iger, Metric Spaces of Non
positive Curvature, Springer-Verlag, 1 999.
[BMV] P S. Bullen, D . S. Mitrinovic, and P. M . Vasic, Means and Their
Inequalities, D. Reidel, Dordrecht, 1 988.
[BS] K. V. Bhagwat and R. Subramanian, Inequalities between means of
Figure 4. The lattice diagram of highly composite numbers such as 36, 48, 60, and 180 are
shown (after Lionel March).
6 Bath
b B � B I Living R'M §
I I
Kilclk!=n l:>our Living R�l Living RM L.iving RM
0 0
8 ( o Patio r
J 0 L?gr,...._...._•r-- -.,..-
L.iving RM B<dR ·I B<dRM
IA IB K:: ID � IF IG
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 100-
foot lot, forming a courtyard with enclosed patios and outdoor fireplaces. Each pair of units opens to outdoor living 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 interlocking between the interior and the garden." There is almost no difference in level between the ground floor and the garden, suggesting an infinite extension to the open ground in accordance with the character of the land. Schindler's "organic" building type is fully realized in this house; the house and the outdoors unite in perfect rapport to embrace their
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 partitions and patio walls are non-supporting screens composed of a wooden skeleton filled in with glass or with removable 'insulate' partition. These basic materials are used in a lucid way to form "the cave-
tent shelter of concrete, wood and canvas" which relates the project to the climate, the region, and the surroundings. Schindler appears to have erected a modern hut, expressing the immemorial relationship between humanity and nature. It is a true sustainable aesthetic.
On the drawings, the dimensions and placements of various spatial
extraordinary surroundings. Figure 6. The Kings Road House, 1 921-22. 1/4 scale model constructed with basswood.
Jin-Ho Park earned his BS in architecture from lnha University, Korea, and his MA and
PhD Degrees. also in architecture, from the University of California at Los Angeles. He
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
Faculty Teaching Award in 2003. He is presently associate professor at lnha University.
His articles have appeared in numerous journals, and he currently serves as corre
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 nial science-fiction favorile. A first-rate ficlional guide to the concept of multiple dimensions of space, the book wil l also appeal to those who are interested in computer graphics. In his inlroduction, Thomas Banchoff points out that there is 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 book purports to be the memoir of A. Square, a citizen of an entirely twodimensional world."- Washington Post Book World Princeton Science librory Paper $9.95 0-69 1 - 1 2366-7
Celebrating 100 Years of Excellence PRINCETON (0800) 243407 U.K.
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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 mentioned above. References to his other philosophical texts are given. Moreover two articles "Kahler's Vision of Mathematics as a Universal Language" by R. Berndt and "An Approach to the Philosophy of Erich Kahler" by K. Maurin serve the same purpose.
Kahler, who had a strong sense of mission, started his independent scientific production at the age of 17 and remained active until shortly before his death at 94. The principal stages of his academic career were:
1928 Doctor's degree at the University of Leipzig under Leon Lichtenstein
1930 Habilitation in Hamburg 1931- As a Rockefeller fellow in 1932 Rome, he met Enriques, Cas-
1936
1948
1958
1964
telnuovo, Levi-Civita, Severi. B. Segre, and A. W eil Ordinary professor in Konigsberg Successor to Koebe in Leipzig After serious political ten-sions with the East German administration Kahler accepted a position at the Technical University of West Berlin Successor to Emil Artin in Hamburg
1974- Professor Emeritus, work-2000 ing mainly on his Mathe
matical Philosophy and occasionally lecturing at physics and mathematics conferences.
The book under review does not contain a list of Kahler's former PhD-students. Here is one, maybe incomplete: Walter Thimm (Konigsberg 1939), Gtinter 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
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 numerous contributions to our science which have come to bear so much fruit inside and outside mathematics. With his collected works, the editors have given him a worthy monument.
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
The 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 science of knots?, V. Decorative knots and other aspects. The editors, Turner and van de Griend, have chosen a wide variety of experts as authors for the individual chapters; more information about them may be found in "About the Authors" on page 419. The construction and use of knots, links, and braids from string-like objects predates known human history. They occur 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 polymer 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 applications. 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 mathematics. 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 chronology 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 required the use of knots. Chapter 2 is devoted to some speculations regarding the earliest forms of knots and their origin. Chapters 3 and 4 are written by archaeologists who examine the knotted structures found at many excavation sites in Europe and Egypt. The earliest 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-European civilizations. The use of knotted cords or qui pus by the Incas is detailed in Chapter 5. There is an interesting discussion here of the construction of knotted cords and their use in representing 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 description of the decorative use of knots which began in ancient China and continues 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 readily 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 subject in just a few pages. Chapter 9 discusses what the author calls life-support knots. It covers the use of knots in rock climbing, rescue work etc., with particular attention to their properties in life-support tasks.
Part N is the longest, with five chapters. Behaviour, under load, of single stranded knots tied in fiber rope is studied in Chapter 10. Chapter 12 is devoted 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 altering one tuck at a time) is dealt with in Chapter 13. The last chapter is devoted to the history and techniques of crochet 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 interest to the readers of this magazine. To a mathematician, a knot is an embedding of a circle in the three-dimensional 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 systematic study of knots was begun in the second half of the 19th century by Tait and his followers. They were motivated by Kelvin's theory of atoms modeled 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 invariants.
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 orthogonal projection of the knot such that at any crossing in the projection exactly two strands intersect transversely. Tait made a number of observations about some general properties of knots which have come to be known as the "Tait conjectures." In its simplest form the classification problem for knots can be stated as follows: Given a projection of a knot, is it possible to decide in infinitely many steps if it is equivalent to an unknot. This question was answered affirmatively by W. Haken in 1961 , 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 computer 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 special 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 projection 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 number 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 published posthumously as part of his estate. They deal mostly with his attempts to classify Tracifiguren or
plane closed curves with a finite number of transverse self-intersections. Such figures arise as regular plane projections 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 analytic formula for the linking number of a pair of knots. This number is a combinatorial 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 Electrodynamik" ("On Electrodynamics") and his continuing work with Weber on the properties of electric and magnetic fields leads us to guess that it originated in the study of the magnetic field generated by an electric current flowing in a curved wire. Maxwell knew Gauss's formula for the linking number and its topological significance and its origin in electromagnetic theory. In obtaining a topological invariant by using a physical field theory, Gauss had anticipated Topological Field Theory by almost 150 years. Even the term topol
ogy was not used in his era. It was introduced in 184 7 by J. B. Listing, a student 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 considered as an example of an index theorem. Starting with this, a far-reaching generalization of the Gauss integral to higher self-linking integrals can be obtained. 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 topological field theories and Feynman diagrams 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 polynomial 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 generalizing both the Alexander polynomial and
the Jones polynomial. This polynomial is called the HOMFLY polynomial (a name formed from the initials of authors 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 developments 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 longstanding 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-example to the chirality conjecture is given in [2]. Jones did not provide a geometrical or topological interpretation of his polynomial invariants. Such an interpretation was provided by Witten [6], who applied ideas from Quantum Field Theory (QFT) to the ChemSimons Lagrangian. In fact, Witten's model allows us to consider the knot and link invariants in any compact 3-manifold M. Witten's ideas have led to the creation of a new area called Topological Quantum Field Theory (TQFT) which, at least formally, allows us to express topological invariants of manifolds by considering a QFT with a suitable 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 structure of DNA. These early results have led molecular biologists to believe that knot theory may play an increasingly significant role in understanding the geometric and topological properties of DNA and that these in turn may help in resolving some of the riddles encoded in these basic building blocks of life. Understanding 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 decorative 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 describes 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 sophisticated 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
Codebreakers: Arne Beurling and the Swedish Crypto Program during World War I I by Bengt Beckman
PROVIDENCE, Rl, AMERICAN MATHEMATICAL SOCIETY,
2003. 259 PP , US $39, ISBN 0·821 8-2889-4
REVIEWED BY HAKAN HEDENMALM
The author of this book, Bengt Beckman, is one of early members of
the Swedish cipher bureau FRA of Forsvarsstaben (Defense Staff Headquarters), which was operational in 1941 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 breaking 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 during the same war. In England as well as in Sweden, mathematicians played a vital role for the intelligence deciphering part of the war effort; perhaps the most famous mathematician working 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 successful; indeed, the importance of Beurling's contribution to Sweden's ability to keep out of the war was, until recently, known only to narrow circles 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), produced by Beckman and Olle Hager. In the book, Beckman is able to tell a
much more detailed story, of course. With the translation to English, commissioned by the American Mathematical Society, the material is now available to a considerably wider audience. The translator, Kjell-Ove Widman, who himself has been involved in cryptology at the Swiss company Krypto AG, has done a very thorough job and produced a very readable text. However, the style differs a little from the original: Beckman tells a story by the campfire, 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 following 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 little bit further southeast. It is also unfortunate that the suggested explanation of Beurling's analysis is marred by cipher typos (on pp. 80, 82).
The book takes a historical perspective on ciphers and begins with an exercise to decipher a coded message from the 18th century. This is quite enjoyable; the cipher is of simple substitution 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 little, 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 particularly dire on April 9, 1940, when Germany invaded neighboring Denmark and Norway. As Sweden could not afford a massive military build-up, it became imperative to be able to secondguess the German intentions regarding Sweden. Then Sweden's foreign policy could be modified to be more palatable for the Germans, and hence avoid actual invasion. This was done quite successfully-German shipments of materials 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 transmitted 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 automatically. One day in 1940 Beurling, who had already been involved with some simpler cryptanalysis tasks for the Swedish Defense Department, collected 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 Germans. If we think of the Geheim
schreiber encryption as a kind of substitution cipher, then the cipher key apparently was changing with each new letter of the message. Also, the initial key settings were altered every few days. The way the Geheimschreiber
was made, it would not begin cyclically repeating its cipher on any given message, for the number of possible encodings was much much bigger than the total quantity of information exchanged over the entire war.
Beurling never revealed how he performed his feat; he would say that a magician never reveals his tricks. Nevertheless, 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 allowed Beurling to get a foot in the
door, and using some sound hypotheses regarding the nature of possible codes on teleprinters (where each letter corresponds to a sequence of five Os and Is )-essentially combinations of permutations and transpositionshe 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 Geheimschreiber 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, machinescalled Apps-were doing the job. The value of having cracked the Geheimschreiber depreciated toward the end of the war. The Germans sensed that their transmissions were being read, and reacted to it. By that time, however, the risk that Sweden would get dragged into the war was much reduced.
Beurling was a deep mathematician equipped with a difficult temperament. The stories about his disagreements, rows, or even outright fights with colleagues are widely known in mathematical circles in Scandinavia. Some of these stories are retold in this book Beurling was apparently quite charming to the ladies, and this aspect of his life, based on interviews with AnneMarie 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 theory, harmonic analysis, and complex
From Newspeak to Cyberspeak. A History of Soviet Cybernetics by Slava Gerovitch
CAMBRIDGE, MA, THE MIT PRESS, 2002. xiv + 369 PP
US $42 ISBN 0-262-07232-7
REVIEWED BY PAUL JOSEPHSON
Most readers know of the impact
of ideological interference in the
practice of Soviet scientists. In the case
of biology, the peasant agronomist
Trofun Lysenko rose to the top of the
scientific establishment with Stalin's
personal endorsement. He advanced
Lamarckian notions of the influence
of acquired characteristics that sup-
planted genetics. After a 1948 national
conference that proclaimed Lysen
koism the only true Soviet biology,
dozens of geneticists lost their jobs
and were exiled from the laboratories
to sheep-breeding farms at the end of
the empire; and genetics was purged
from university curricula. In physics,
too, several scientists joined ideo
logues in an attack on quantum me
chanics and relativity theory. The at
tack nearly had disastrous results for
physicists, many of whom were forced
to abandon research. In both of these
cases, questions of epistemology, class
struggle, and other issues of impor
tance in the official Soviet philosophy
of science, dialectical materialism,
played a role, as did Cold War pres
sures to promote a new, Soviet science
that was different from, and better than
that in the West, particularly in the
United States.
In a superb contribution to the his
tory and philosophy of science, Slava
Gerovitch considers the place of cy
bernetics in Soviet philosophical dis
putes, and the development of what
he calls "cyberspeak" in the postwar
USSR. Gerovitch, a research associate
at the Dibner Institute, has studied the
history of Soviet computing and cy
bernetics for some time, and beyond
From Newspeak to Cyberspeak, is now
focusing on human-machine interaction
in the Soviet space program. Cyberspeak he defines as a universal language of man-machine metaphors described in
such terms as information and feedback
and control, e.g., the organism as an en
tropy-reducing machine, the computer
as a brain, the brain as a computer.
Gerovitch joins several other scholars in
rejecting the notion that cybernetics was
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,
and scientists learned how to manage it.
Yet the fact that attacks were short
lived and ignorant should not lessen
our appreciation of the way in which
they reflected the dangers of doing sci
ence in the USSR generally. Those who
carried on the anti-cybernetics campaign
employed accepted ways of discourse
and dispute to rescue their careers and
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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 cybernetics was needed for radar and rocketry, the Cold War saved it from further attacks-in the way that research on the atomic and hydrogen bombs saved relativity theory and quantum mechanics from ideological interference.
Gerovitch provides a detailed discussion of what made attacks on cybernetics 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 establishment-the government bureaucracy and Party apparatus-in their rhetorical styles, modes of thought, and arguments 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 developed along many of the same lines along which cybernetics developed in the West. Such scientists as Andrei Kolmogorov 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 Norbert 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 computing 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 disappeared.
An important facet of the history of computer science and the first Soviet computing machines was the connection between cyberneticians and their powerful military and Communist Party patrons. One such example is Mikhail Lavrent'ev, who, as director of the Institute of Computer Technology, received 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, Akademgorodok, with its own new Computer Center in the late 1950s. When it came to military purposes, computers were a technology without ideological deviations.
After the death of Stalin, in the mid-1950s Soviet computers were deified and entered the public realm, no longer to be held under wraps of military secrecy. Cyberneticians touted computers as paragons of objectivity based on quantitative knowledge, precise language, and precise concepts. They also commenced an attack against past ideological interferences of the philosophers and their allies among scientists and Party officials. Gerovitch's discussion of the battle against dogmatism and calcification of philosophical discourse under Stalin at this time is engaging. Ultimately, Gerovitch writes, cyberneticians "overturned earlier ideological criticisms of mathematical methods in various disciplines and put forward the goal of the 'cybernetization' of the entire science enterprise" in search of objectivity in the life sciences and social sciences alike. (p. 199) This conclusion reinforces the sense that an excessive scientism developed in the USSR.
In the 1960s cybernetics became a full-fledged science in the Soviet establishment. 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 influential such publications in the world. Its promoters claimed that cybernetics had become a universal, objective language, and as such would break interdisciplinary barriers and legitimize the use of mathematical methods in other sciences. Its power was evident in the fact that even before Lysenko was deposed, cyberneticians promoted genetic research in physics and chemistry institutes, speaking about genes as units of hereditary information. Supporters thought cybernetics might be a panacea for reforming the Soviet economy through the creation of "optimal" models for planning and management. 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, administration, direction, and government purposes. The goal was to control the entire national economy, technological processes, and so on, to ensure optimal 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 efforts 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 suggests that scientism or not, cybernetics, like other sciences in other countries, could not avoid reflecting the social, political, 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 "thinking machines" that occurred from 1950
to 1965. He does not consider the decision to build computers based on reverse engineering after so many decades of success in building indigenous machines. In addition, because his focus is on philosophy and intellectual history, some readers will need to seek other sources to gather the impact of the broader context of Soviet history and politics on cybernetics.
Gerovitch's study is based on a thorough use of archival materials and unpublished memoirs and interviews. This book will be of interest to advanced undergraduate students, graduate students, and teachers, as well as to computer scientists, historians, and philosophers. I recommend it highly.
Traditiona l 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
The 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 mathematicians developed their own "traditional
mathematics," which, in the 1850s, began 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 mathematical tablets (SANGAKU) which were hung under the roofs of shrines or temples in Japan during two centuries 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 universities 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 displaying their discoveries to other geometers . . . with the implicit challenge: "See if you can prove this!" The 2002 book continues the col-
lection 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, solutions, 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 published in German, 1913).
Mikani places sangaku in the perspective 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 development of the Japanese mathematics 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 problems in the 1989 and 2002 books involve circles.
One of the foremost mathematicians represented in sangaku was Ajima Chokuen (1732-1798). (The family 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 inscribed 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, Example 2.3 and Problems 2.3. 1 to 2.3. 7 are presented under the heading "Three Circles and Triangles," followed by Examples 2.4(1) and 2.4(2) and Problems 2.4.1 to 2.4. 7 under "Four Circles and Triangles." The presentation is in two-column format with figures in the right column. Each problem proposal is labeled "Example" or "Problem." Here are three items from the 1989 book:
Example 2.3: The three circles, 01Cr1), Oz(rz), and 03(r3) have external contact with each other. The triangle ABC is formed by the common 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) respectively 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
also I(r) externally, and 02(r2), and 03(r3) are defined similarly. Show that
1 1 1 1 - = -- + -- + --
r v;v; y,;;; � . 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 construction 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 considerable 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 improvement over "center of perspective") is a point known as the AjimaMalfatti point. For a discussion, visit the Encyclopedia of Triangle Cen
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 Theorem" at the beginning of Chapter 4 and "Ajima's second theorem" at the beginning of Chapter 5. The second theorem is elsewhere cited in the book on several pages, as is a third theorem attributed to Ajima, labeled "Boushajutu".
Another traditional mathematician represented in both books was Shoto Kenmotu (1790-1871). His configuration in Problem 3.2. 1 of the 2002 book is the earliest known (1840) construction of a point now known as the Kenmotu point. The configuration, involving three congruent isosceles right triangles, extends easily to the one depicted 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) Ellipses (and One Hyperbola), (6) Ellipses and Circles, (7) Ellipses and Polygons, (8) Ellipses, Circles and Quadrilaterals, and (9) Spheres. The total number of problems is 249.
The 2002 book chapter headings are (1) Number theory, (2) Numerical Analysis, (3) Geometry ofPolygons, (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 originated in various prefectures at various times: Fukusima in 1891, Iwate undated, Iwate in 1842, Miyagi in 1857, and Fukusima in 1901. Regarding such places and dates, Dr. Hiroshi Kotera offers a very attractive website:
http://www.wasan.jp/english/
(There is also a Japanese version.) There you will find a Clickable SANGAKU Map of Japan, showing 24 labeled 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 particular, you can click Tokyo, then select 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 recommended: Tony Rothman's (with the cooperation of Hidetoshi Fukagawa) Japanese Temple Geometry,
Both books have extensive bibliographies. 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 ninepoint 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 geometric circles. For example, Ross Honsberger'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 Math World:
http://mathworld. wolfram. com/ RigbyPoints.html.
Physicist Freeman Dyson's Foreword (or "Forward", as it not inappropriately 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 important scientific enterprises of the twentieth century is the search for extraterrestrial intelligent species." Once
contact is made, there will be the problem of communication. Dyson writes, "Eighteenth-century Japan is stranger to me, in language and in historical tradition, 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 directly from The Charles Babbage Research Centre, P. 0. Box 272, St. Norbert 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).
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
This is a very short review of Mathematics: A Very Short Introduc
tion, by Timothy Gowers, a very smart mathematician and an excellent communicator 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 mathematics [ 1 ] , an argument for the second. He has done a wonderlul job in producing this rich little book on the essentials 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 arguments showing its power and suggesting 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, examples, and well-illustrated concepts such as limits, estimates, dimension, infinity . . . . Gowers has chosen a list of simple concepts of great mathematical importance, which he presents to the curious reader. Finally, the chapter "Some frequently asked questions" responds to what people generally ask about mathematicians (or so mathematicians imagine). Among the questions, "Why do so many people positively dislike mathematics?"
When people asked Henri Poincare why they never understood mathematics at school, he answered them, "What I don't understand is that people don't understand mathematics!". Gowers suggests using the abstract approach. We wish him all the best!
I would strongly recommend this book to first-year undergraduates, although professional mathematicians will also find it a useful introduction to many beautiful examples on Gowers's Web site, such as the astonishing results that won him the Fields Medal [2]. The general audience too will fmd much of interest on the Web site, including an article about what is definable in mathematics, and the text version [3] of the Clay Lecture, in which Gowers shows how mathematics is a subject where importance and beauty are connected.
Basebal l's Al l-Time Best H itters by Michael Schell
PRINCETON, NJ: PRINCETON UNIVERSITY PRESS, 1 999.
xxi + 295 PP. US $1 7.95 ISBN 0691- 1 2343·8 (PAPER)
REVIEWED BY JIM ALBERT
Baseball 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 American and National Leagues, and millions of people watch games in ballparks 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 offense, and in the bottom half, the teams reverse roles. The team that is batting during a particular half-inning, the offensive 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 advances 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
confrontation is called the batter's atbat. 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 enter the strike zone in flight.
After a number of thrown pitches, the batter will either be put out or become 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 fielders, (2) he hits a ball in "fair" territory, and first base is tagged before the batter 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-onballs 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 outfield 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
player's ability to hit is the batting average (AVG), defined as the proportion 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 attempt 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 beginning 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 comparing 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 "batters," but rather the best "hitters"-the ones who have the best chance of getting a hit. Given the current popularity
of the AVG as a measure of hitting per-formance, I believe that this is a rea-sonable aim. The definition of bat-ting 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 hit-ting performance.)
What Adjustments Should
Be Made?
To illustrate some of the issues discussed 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 superficial comparison ignores some relevant concerns. First, we question whether 0.366 and 0.344 are representative measures 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 typical 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 retirement. 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 during the middle years of his career.
A second general concern in comparing 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 playing conditions that affect the effectiveness of the pitchers have changed, and these changes have had a significant effect on the batting averages of players. It is difficult to say that Cobb was a better hitter than Williams, because they played against different pitchers and the game was different in the two eras. For these reasons, a reported batting
Chapter 7 examines the social conditions in which sampling techniques originated. Chapter 8 considers problems associated with choosing categories into which to classify people and things. Moving into the twentieth century, chapter 9 traces the history of modem econometrics.
The book has a number of interesting 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 exploring 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, particularly in the context of efforts to identify the causes of the cholera epidemic 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 debates about the productivity of Northem versus Southern states, of disagreements about immigration policy, and of Franklin D. Roosevelt's New Deal.
Readers not familiar with the historical or mathematical details of the topics treated by Desrosieres may find his sociological analysis more accessible after first reading for themselves the sources from which the author draws his material (for example, [ 1 , 2, 3] discuss the history of probability before the twentieth century). In these works, they will find the writings of the historical figures analyzed carefully; they will learn something of the biographies of people who play important roles in Desrosieres's discussion and of the professional and scientific contexts in which they worked. Mathematicians will need no introduction to Blaise Pascal or Simeon Poisson. But what about Wesley Mitchell? The author 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 occurs thirty pages later. About Tjalling Koopmans, whose ideas figure prominently 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 diverse 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' diverse 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 exposition relies on specialized vocabulary and rather complicated sentences. Setting 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 development-through a costly investment
of technical and social forms. This enables us to make disparate things hold together, thus generating things of another order" (p. 9, italics in the original). Desrosieres thus seeks to understand the development of these "forms," as well as how social phenomena like unemployment, and technical phenomena 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 conditions the solidity, durability, and space of validity of objects thus constructed: this idea is interesting precisely in that it connects the two dimensions, economic and cognitive, of the construction of a system of equivalences. The stability and permanence of the cognitive forms are in relation to the breadth of investment (in a general sense) that produced 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 translation.
Desrosieres's exploration of the relationship 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 answers.
[ I ] Lorraine Daston, Classical Probability in the
Enlightenment, Princeton University Press,
1 988.
[2] Theodore Porter, The Rise of Statistical
Thinking: 1820- 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, I l l by Howard Eves
WASHINGTON, DC, THE MATHEMATICAL ASSOCIATION
OF AMERICA, 2003. US $98.00. ISBN 0-88385-542-9, 0-
88385-543-7, 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 exceptions are figures like Gauss, Riemann, and Cauchy.) While only a handful 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 sixvolume Mathematical Circles collection.
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 credibility, 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 history 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 Circles volumes constitute one of our collective treasures. The original Prindle, Weber, and Schmidt editions are now
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
out of print, and we are fortunate that the Mathematical Association of America 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 Andrew Wiles, about the Riemann hypothesis, 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 understanding. It should be noted that Eves was one of the pioneers of mathematical popularization and of mathematical 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 enterprise, 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 valuable. We are fortunate to have these collections of stories.
Among the more charming anecdotes 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 Marcellus'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 possible 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 Johann Bernoulli.
• A determination of who were the second and third most prolific mathematicians in history (after Leonhard Euler).
• The story of how the Indiana legislature passed a law to declare it possible 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 Pennsylvania 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 sequels: 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 lifetime" that you set forth in Volume I.
A CALCULUS BOOK WORTH READING • Clear narrative style • Thorough explanations and accurate proofs • Physical interpretations and appl ications
"Unlike any other calculus book I have seen . . . Meticulously written for the intelligent person who wants to understand the subject. . . Not only more intuitive in 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
beginner . . . A course based on it would in my opinion definitely have a much greater chance of producing 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
Mathematics, University 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
Encyclopedia of Genera l Topology edited by K. P. Hart,
Jun-iti Nagata, and J. E. Vaughan ----- · --------- -- ---- . -----
AMSTERDAM, ELSEVIER, 2004. 526 PP € 1 45, ISBN 0-444-50355-2.
REVIEWED BY HANS-PETER A. KUNZI
As a student you may have been
taught that normality is neither
hereditary nor finitely productive.
Later on in your career you may have
encountered intricate problems in var
ious contexts that could be reduced to
seemingly plain topological questions
which, nevertheless, you could not im
mediately solve. For instance, in my
case, I could not find answers to some
questions about the behaviour of nor
mality in box products, or to the prob
lem of whether each compact topology
is coarser than a compact topology in
which all compact subsets are closed.
It is difficult to be a mathematician
and not use some basic concepts and
methods from General Topology every
now and then. Over the past decades a
few polished techniques and some
fundamental basic terminology of this
field have become so well known and
efficient that they now belong to the
folklore of today's university mathe
matics. Less known to the mathemati-
cal community are many modem-and
often much more sophisticated and
complex-developments that thus far
have not had many applications be
yond their area of origin.
The encyclopedia under review tries
to cover the basic as well as many of
the more specialized and new develop
ments in the field of topology. It ad
dresses both established mathemati
cians and students, independent of their
area of specialization, and it is designed
to lead them quickly to the terminology
and results that might be useful to their
own investigations in other areas.
It explains terms like "Hedgehog",
"Weak P-Point", "Cliquish Function",
"Talagrand Compactness", "Thick Cov
ers", and "Resolutions". Similarly, it
discusses concepts like "the Sous
lin Hypothesis", "Hit-and-Miss Topolo
gies", "the Normal Moore Space Con
jecture", "the Blumberg Property", "the
Class MOBI", or "Bing's Example G". It
also deals with difficult open problems:
Is the class of meta-Lindelof spaces
preserved under perfect maps, or is
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?
When is the completion of a topologi
cal ring a field? Does the Sorgenfrey
line have a connected Hausdorff com
pactification? How can one prove with
the method of elementary submodels
that the cardinality of a first-countable
compact Hausdorff space is at most
that of the continuum?
The encyclopedia includes about 120 articles contributed by a similar number
of topologists from all over the world.
Mostly written by experts in the spe
cialized fields, these articles outline, in
a short sequence of definitions and re
sults, many basic aspects of the treated
topics. The average length of a contri
bution is about four pages. Very few
proofs are given. The articles are col
lected under ten headings that imitate
Section 54 of the 2000 Mathematics Sub
ject Classification as used by Mathe
matical Reviews and Zentralblatt MATH.
The key words listed below will give
the reader an idea about the contents of
the work. They roughly follow the titles
of the articles as they are listed in the
table of contents of the book. Some
readers might want to skip the list below. 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 theorems and techniques, as they are discussed in the present review.
GENERALITIES: Topological Spaces, Modified Open and Closed Sets, Cardinal Functions, Convergence, Several Topologies on One Set (Minimal and Maximal Topologies).
BASIC CONSTRUCTIONS: Subspaces, Relative Properties, Product (Quotient, Adjunction, and Cleavable) Spaces, Hyperspaces, Inverse and Direct Systems, Covering Properties, 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, Absolute Retracts, Extensors, Generalized Continuities, Spaces of Functions in Pointwise Convergence, Radon-Nikod:Ym (Corson, Rosenthal, and Eberlein) Compacta, Topological Entropy, and Function Spaces.
FAIRLY GENERAL PROPERTIES: Separation Axioms, Frechet (Sequential, and Pseudoradial) Spaces, Compactness (Local Compactness, Sigma-Compactness, Countable Compactness, and Pseudocompactness ), The LindelOf Property, Realcompactness, k-Spaces, Dyadic Compacta, Paracompact Spaces (Generalizations and Countable Variants), Extensions of Topological Spaces, Remainders, The Cech-Stone Compactification (in Particular of N and �), Wallman-Shanin Compactification, H-Closed Spaces, Connectedness, Connectifications, and Special Constructions.
SPACES WITH RICHER STRUCTURES: Metric Spaces, Metrization, Special Metrics, Completeness,
Baire Spaces, Uniform and QuasiUniform Spaces, Proximity Spaces, Generalized Metric Spaces, Monotone Normality, Probabilistic Metric Spaces, and Approach Spaces.
SPECIAL PROPERTIES: Continuum Theory, Dimension Theory (General, and of Metrizable Spaces), Infinite Dimension, Dimension Zero, Linearly Ordered and Generalized Ordered Spaces, Unicoherence and Multicoherence, Topological Characterizations of Spaces, and Higher-Dimensional Local Connectedness.
SPECIAL SPACES: Extremally Disconnected (Scattered, and Dowker) Spaces.
CONNECTIONS WITH OTHER STRUCTURES: Topological Groups (Rings, Division Rings, Fields, and Lattices), Free Topological Groups, Homogeneous Spaces, Transformation Groups and Semigroups, Topological Discrete Dynamical Systems, Fixed Point Theorems, and Topological Representations of Algebraic Systems.
INFLUENCES OF 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 Theory, Manifolds, and Infinite-Dimensional Topology. Throughout the book it is assumed
that the reader has some basic knowledge 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 insufficiently. 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 theory 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 uneven in many respects, even after some unifying work was done by the editors. In particular the articles about the more specialized topics often and unavoidably have to assume a certain familiarity of the reader with basic concepts. This familiarity cannot be gained by reading a few introductory articles in the encyclopedia. Thus while the encyclopedia will certainly look impressive on the shelves of a library or on the desk of a mathematician, the question remains whether it is of practical use.
As can be guessed from the short description 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 replace the usual text books, monographs, or original research papers. In some sense, the value of encyclopedias in mathematics is quite limited: Essentially they can only provide quick orientation for informed readers. But they can hardly be relied on to teach complete novices, and they are generally too superficial for the working specialist.
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 article, and from a general list of basic references that is used throughout the book Persons who are simply interested 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 dealing 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 various areas closely related to their own field of interest.
Certainly some discussions in topology seminars could be based on parts
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 duplications and subsequent disappointments. Without such books, complex theories cannot develop properly, and even excellent mathematical concepts and results are soon forgotten and do not survive their discoverers or inventors.
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 Publishers, 1997-2001.
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The Ph i lamath' s Alphabet-L
Lagrange: Joseph-Louis Lagrange (1736-1813) wrote the first 'theory of functions', using the idea of a power series to make the calculus more rigorous, 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
Lagrange
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'Laplace transform' of a function. His monumental five-volume work on celestial mechanics, Traite de mechanique celeste, earned him the title of 'the Newton of France'. Leibniz: Although Newton could claim priority for the calculus, Gottfried Wilhelm Leibniz (1646-1716), who developed 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 calculation of areas and volumes, the evaluation of roots, and the systematic solution of simultaneous equations. Around 260 AD, while revising the Jiuzhang suanshu, Liu Hui calculated the
Laplace
areas of regular polygons with 96 and 192 sides and deduced that 7T lies between 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 mathematician 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; Lyapunov shot himself the same day and died shortly after.