Jet Wimp*

The Emperor's N e w Mind: Concerning Computers, Minds and the Laws of Physics by Roger Penrose New York: Oxford University Press, 1989 xiv + 466 pp. US$24.95 paperback edition Penguin, 1991, US$12.95

Visions of Symmetry: Notebooks, Periodic Drawings, and Related Work of M. C. Escher by Doris Schattschneider New York: Freeman, 1990 xiv + 354 pp. US$39.95

Reviewed by Marjorie Senechal

theme of his book and local-global problems in aperiodic tiling theory. This prompted me to try to discuss both books in a single essay. It is unfair to both authors since each book is much richer than I will even begin to indicate. But on the other hand--9 The reviews of the central argument of Penrose's book already amount to an entire body of literahare; indeed, they are worthy of a review themselves, since they constitute a learned (sometimes not so learned) debate on the strengths and weaknesses of Penrose's attack on "strong AI." However, none of the reviewers seems to have been struck by Penrose's interesting suggestion that there may be analogies between our thought processes and quasicrystal growth; my aim here is to highlight these remarks.

"'How did he do it? The work of M. C. Escher (Figure 1) provokes that irrepressible question," writes Doris Schattschneider in her preface to the beautifully illustrated and encyclopedic Visions of Symmetry: Notebooks, Periodic Drawings, and Related Work of M. C. Escher. How did he do it? Everyone who has studied the Penrose filings, in any of their guises (Figure 2 and Figure 6), eventually asks this question. And w h y does it seem to be so difficult to establish a theory of aperiodic files? How do they do it ? Seven years after their discovery, solid state scientists all over the world are still trying to understand h o w and w h y atoms organize themselves into crystalline-like alloys with aperiodic atomic struchares known as quasicrystals. These three questions may or may not be related to a fourth: How do we do it? That is, how do we think? This is the central question of Penrose's recent book, The Emperor's New Mind. A deep question underlying all these questions is the classic problem of understanding the relation between local configurations and global structure. Escher's work has been used for many years by crystaUographers to illustrate just this relation for periodic patterns. In several very thought-provoking passages, Penrose discusses a possible analogy between the * Column Editor's address: Department of Mathematics, Drexel Figure 1. From the notebooks of M. C. Escher (Visions of University, Philadelphia, PA 19104USA. Symmetry). 72

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Schattschneider's book is also being reviewed elsewhere; in any case, you should look at it instead of reading about it. No review can begin to c o n v e y its f a s c i n a t i n g v i s u a l c o n t e n t . Schattschneider's discussion of Escher's classification system is invaluable, but his ideas become comprehensible only by examining the 150 notebook drawings themselves. (The publisher should be thanked for reproducing all of the drawings in beautiful color, and for keeping the price of the book within reasonable bounds.) Tilings are often compared to jigsaw puzzles. Like the pieces in a puzzle, tiles must be fitted together to fill a given region or space without gaps or overlaps. But jigsaw puzzles are mathematically trivial, while things decidedly are not. The puzzle maker starts with a global solution--a picture--and then cuts it up into shapes in any way she likes. Usually, the shapes are fairly complicated but no two are alike. Thus the puzzle solver, who works locally by matching the outlines and colors of the shapes, cannot make a mistake. Sooner or later the picture is completely reassembled. How does one create a tiling? Any resemblance to jigsaw puzzle-making is superficial: things are to jigsaw puzzles as sonnets are to free verse. There are three principal methods that may be used in combination. In Method A, you start with a simple thing, such Figure 2. Three versions of Penrose's aperiodic tiles (The as the tiling of the plane by squares or hexagons, and Emperor's New Mind). then modify it by altering their boundaries (Figure 3),

Figure 3. Method A: modifying simple filings. From the notebooks of M. C. Escher (Visions of Symmetry). THE M A T H E M A T I C A L [NTELLIGENCER VOL. 14, NO. 2, 1992 73

Figure 4. Method B: trial and error. From Johann Kepler's Harmonice MuncH.

or by dissecting the tiles and regrouping the pieces into new shapes. The mathematical challenge is to do this in such a way that the modified tiles are copies of one or two prototiles. In Method B, you begin with a few prototiles and try to fit copies of them together around a point. If this can be done, then you try to extend the pattern to the entire plane (Figure 4). Method C can only be used if your prototiles can be dissected into smaller copies of themselves. After dissection, you inflate the small tiles to the sizes of the original ones, and dissect again. Continuing ad infinitum, you get a tiling of the plane. Escher's notebooks confirm what we already knew from earlier studies of his work: Escher was a master of Method A. All, or almost all, of his tilings are modifications of simple periodic tilings by polygons. Consequently they all have "'crystallographic" symmetry, that is, rotational symmetries of order two, three, four, or six. Escher's interest in "regular division of the 74

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plane" was first stimulated by Moorish ornament at the Alhambra palace. After reading a paper by P61ya in which the seventeen plane crystallographic groups were enumerated and illustrated, he began to develop his own pattern theory. He did not use group theory; in any case, group theory provides only a coarse classification because it ignores most of the geometrical and combinatorial relations among the tiles. (See [3] for a detailed discussion of various classification schemes.) In the course of his work, Escher also developed rules for coloring patterns symmetrically and for effecting transitions between classes (Figure 5). Later, guided by diagrams of Coxeter, he created several tilings of the hyperbolic plane, and adapting a puzzle of Penrose, made one tiling with congruent but symmetrically inequivalent tiles. However, despite the enthusiasm of many mathematicians for Escher's art, only his work on color symmetry seems to have prodded us to expand our own theories.

Figure 5. Transitions among tilings. From the notebooks of M. C. Escher (Visions of Symmetry).

Figure 6. Penrose's first tiling. From The Emperor's New

Mind.

Schattschneider argues that Escher's viewpoint was that restrict the ways in which the pentagons and the local: "The crystallographers' and mathematicians' gap tiles can be juxtaposed or matched. He then incorquest is for a logical analysis of a given structure; porated the rules into the tiles themselves by modifyEscher's quest was to discover the various ways in ing their boundaries (Figure 2a). Finally, by applying which to create original periodic patterns in the plane. Method A to his six tiles, Penrose reduced the number They always begin with a pattern; he always began of tiles to two, in two different ways: kites and darts with a blank sheet of paper. Their point of view is a (Figure 2b), and thick and thin rhombs (Figure 2c). All global one--what is the structure of the whole molec- Penrose tilings reflect their pentagonal origins: they all ular array and what are the symmetries of the whole have five-fold rotational symmetry over arbitrarily pattern. Escher's view was a local o n e - - h o w can a sin- large regions. gle motif be surrounded by copies of itself?" But as we One of the most fascinating questions in tiling thehave seen, the piece of paper was not entirely blank ory is: what is the relation between Methods A, B, and because Escher created his motifs by modifying simple C? Robert Ammann, whose aperiodic tiles are distilings in ingenious ways. Method A is only quasilocal. cussed in detail in [3], says he began his investigations Method B---trial and error--is really local. If Escher by studying intersection patterns of superimposed ever used it in its pure form, he must have abandoned grids (families of parallel lines). Independently, N. G. it by the time he began his notebooks. Trial and error deBruijn showed, in 1981, that Penrose's tilings by is not a recommended method for discovering tilings, since it is known that no algorithm exists for determining whether or not an arbitrary shape will tile the plane. The nonexistence of such an algorithm implies, and is implied by, the existence of aperiodic tiles (tiles which fill the plane only nonperiodically). On the other hand, if we want to create a genuinely new tiling we may need to use Method B. Kepler's "notebook"--that is, his drawings in Harmonice Mundi--show that he did use trial and error and began to push beyond the boundaries of periodicity (see Aa of Figure 4). Penrose's discovery of his nonperiodic tilings began with the observation that a regular pentagon can be dissected into six regular pentagons and five triangles. Also, w h e n the pentagons are again divided, only a few more gap shapes appear and thus the gaps can be regarded as tiles (Figure 6). Iteration of the subdivision leads to a tiling of the plane via Method C. Penrose then showed that his tiling could be reconstructed by Figure 7. A tiling with local seven-fold rotational symmejuxtaposing regular pentagons edge-to-edge, as Kepler try. No matching rules are known. Drawing courtesy of A. did in constructing his Aa. To do this, he devised rules Katz. THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 2, 1992 7 5

rhombs are the topological duals of grid intersection patterns [1]. Consequently, we can construct nontrivial nonperiodic tilings with a small number of rhombic prototiles which have, locally, seven-fold, twelve-fold, or any other desired rotational symmetry (Figure 7). These tilings are completely determined by the orientation and relative positions of the grids. Could we reconstruct them by Method B, starting with loose tiles and fitting them together? Since any rhombus (indeed, any quadrilateral) can tile the plane periodically, we would need some sort of rules to preclude periodicity. Whether or not such rules exist and what forms they can take is a major unsolved problem. So is the relation between matching rules and hierarchical structures. As far as I am aware, all of the nonperiodic filings with matching rules also have hierarchical structures since the matching rules force us to group the tiles into larger tiles, similar to the original ones, over and over again. In other words, the tiling could also have been produced by Method C. It is striking and perhaps significant that the plane filings for which both matching rules and hierarchical structures are known to exist have precisely the same local rotational symmetries as the quasicrystals that have been found so far: eight-fold, ten-fold (including fivefold here), and twelve-fold. No matching rules are known for the tiling of Figure 7, and no quasicrystal with that symmetry has been discovered either. The art historian E. H. Gombrich has written, "aesthetic delight lies somewhere between boredom and confusion" [2]. Escher's tilings are delightful to look at because they are both boring and confusing. The shapes are so surprising that at first one is confused. It does not seem possible that such wiggly creatures can tile. But if you cut out copies of an Escher tile, and play the jigsaw game of reassembling them, you will find that there is usually only one way to put them together, and you will quickly become bored by the tiling's periodicity. Penrose's tilings, on the other hand, are aesthetically delightful in the opposite way: the shapes are not themselves exciting but matching them is fascinating and instructive. At each stage there are choices to be made, even within the constraints imposed by the matching rules. You are unlikely to construct the same tiling twice. The choices imply an uncountable infinity of Penrose tilings. (Penrose has lamented the fact that Escher died shortly before Penrose discovered his tiles; he is sure that Escher could and would have applied Method A to create some startling realizations of his simple shapes. But what would have happened to the delicate balance between boredom and confusion?) One of the most interesting features of the Penrose matching rules is not their power, but their weakness. If you have ever tried to assemble them, you have probably found it difficult to avoid creating regions that cannot be tiled according to the rules. Indeed, Con76

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way has shown [3] that there are 61 different kinds of "essential holes," for which there is no remedy except to remove some of the tiles you have already placed, and try again. The holes raise the question of whether Penrose's rules can be improved by imposing additional conditions. Interest in this problem has been stimulated by the apparently parallel problem of understanding how quasicrystals grow. Early specimens were riddled with defects, suggesting that defects might be a necessary feature of quasicrystal structure. But more recently, quasicrystals have been created that are almost defectfree. Evidently, the atoms "know" how to assemble themselves. Can we discover how they do it and mimic the process with tiling rules? A few years ago a group of physicists proposed an additional "local" rule which, w h e n added to Penrose's, eliminates the defect problem. Before adding any new tiles, you must search the entire boundary of the configuration for unfilled positions which are forced by the existing configuration. After filling these positions, you can place the next unforced tile as you like (consistent, of course, with Penrose's rules). This procedure works, but is it local? As the configuration grows, so does the region that must be searched. Can a rule be called local if the range to which it must be applied increases without bound? Penrose has shown elsewhere [5] that rules which apply only in a uniformly bounded region cannot preclude defects, because defects can be created at any level of the hierarchy. Thus, they cannot be avoided by the application of truly local rules. Then how do the atoms do it? Penrose suspects that, in contrast with the classical model of crystal growth, quasicrystal growth is nonlocal. Recognizing that the puzzle of quasicrystal growth still appears to be far from being resolved (indeed, the statistical mechanics of ordinary crystal growth is not well understood), he writes, "Nevertheless, one may speculate; and I shall venture my own opinion. First, I believe that some of these quasicrystalline substances are indeed highly organized, and their atomic arrangements are rather close in structure to the tiling patterns that I have been considering. Second, I am of the (more tentative) opinion that this implies t h a t . . , there must be a non-local essentially quantum-mechanical ingredient to their assembly." How does it work? "Many alternative arrangements must coexist in complex linear superposition. A few of these superposed alternatives will grow to very much bigger conglomerations and, at a certain p o i n t . . , one of the alternative arrangements--or, more likely, still a superposition, but a somewhat reduced superposition--will become singled out as the 'actual' arrangement . . . . we have a global problem to solve. It must be a cooperative effort among a large number of atoms all at once." Notice that the global solution eventually

arrived at is not selected a priori at the beginning of growth, but it emerges during the growth process itself. It is not simply the concatenation of independent local decisions, but the result of a sequence of collective reconfigurings by the atoms of the growing crystallite. Is it possible that these decisions are made along a hierarchical chain of command? Whether or not Penrose's arguments turn out to be correct, they are thought-provoking. What are matching rules, anyway? What do we want them to do, and how can they be formulated? Perhaps the search for local, infallible matching rules is too narrowly construed. In fact, more general formulations are being explored (see, for example, [4]). And what does this have to do with the physics of the mind? To quote Penrose once more, "Whichever atomic arrangements finally get resolved (or reduced) as the actuality of the quasicrystal involve the solution of an energy-minimizing problem. In a similar way, so I am speculating, the actual thought that surfaces in the brain is again the solution of some problem, but now not just an energy-minimizing problem. It would generally involve a goal of a much more complicated nature, involving desires and intentions that themsel'ves are related to the computational aspects and capabilities of the brain. I am speculating that the action of conscious thinking is very much tied up with the resolving out of alternatives that were previously in linear superposition." We are still far from a reasonable theory of aperiodic tiles, far from a reasonable theory of quasicrystal growth, and farther still from a reasonable theory of how we think. Still, aperiodic tiles do exist, quasicrystals do exist, and (arguably) we do think. Penrose has given us a lot to think about. In particular, we might pay more attention to transitions between, and hierarchies among, structures. Perhaps we still have things to learn from Escher's notebooks.

References 1. Bruijn, N. G. de, "Algebraic theory of Penrose's nonperiodic filings of the plane, I, II," Nederl. Akad. Wetensch. Indag. Math. 43 (1981) 39-52, 53-66. 2. Gombrich, E. H., The Sense of Order, Ithaca: Comell University Press (1979). 3. Gr/imbaum, B. and Shephard, G. S., Tilings and Patterns, New York: W. H. Freeman, 1987 (the 1989 paperback edition does not include all of the material relevant to this review). 4. Levitov, L. S., "Local rules for quasicrystals," Communications in Mathematical Physics 119 (1988) 627-666. 5. Penrose, R., "Tilings and quasicrystals: a nonlocal growth problem?" in Introduction to the Mathematics of Quasicrystals, edited by M. Jaric, San Diego: Academic Press (1989). Department of Mathematics Smith College Northampton, MA 01063 USA

Mathematical Visions: The Pursuit of G e o m e t r y in Victorian England by Joan Richards San Diego: Academic Press, 1988 266 pp. US$24.95 Reviewed by Thomas Drucker Joan Richards's Mathematical Visions: The Pursuit of Geometry in Victorian England makes enjoyable reading as it chronicles the history of the usefulness of mathematics in a certain society. It also raises questions, if only because of its success, about the usefulness of the history of mathematics. When the history of mathematics is badly done, it is a waste of time to look for ways to justify it as a practice. It is more challenging to ask what the benefits of history of mathematics as a discipline are when it is done well. The history of usefulness and the usefulness of history can be used to cast light on one another. Mathematicians can study the lessons of history, and bask in a sense of the usefulness of their discipline. Instances abound of mathematics done for the sake of beauty and having unexpected applications to other branches of mathematics and science. The contributions of number theory to cryptography and of non-euclidean geometry to the study of astronomy do not prove that everything has an application outside itself, but do indicate the difficulty of dismissing work in any field as inapplicable. Much greater than the time devoted to doing mathematics, however, is the time devoted to teaching it in schools at various levels, with the assumption that most of those engaged in learning will not go on to become professional mathematicians. There are two general arguments for the vast amount of time devoted to the subject: mathematics serves to train the mind, regardless of ultimate career; and the tools of mathematics are skills that are relevant to specific tasks outside the scholastic routine. Both of these arguments are set in the context of the Victorian age in Richards's book. The contrast with chess is instructive (although not a subject discussed by the Victorians). Chess has its grandmasters, the equivalent of professional mathematicians. The mental discipline afforded by the study of chess resembles that inculcated by mathematics. Where chess falls short is in the applicability of its tools to problems outside the game. As Borel once noted, without the element of applicability it would be hard to justify the universal preference for mathematics over chess in the schools. If one asks about the usefulness of the history of mathematics, one group of clients that has had a good deal to say has been the mathematical community. There is no single use to which the history of their discipline has been put by mathematicians. In their role as practitioners of mathematics, they build on both the human beings, and the mathematics of the past. THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 2, 1992 7 7

As far as personalities are concerned, a good deal of the history of mathematics as used by mathematicians sounds like an Imitation of Newton. The trials and difficulties encountered by those who have left the most lasting marks on the field serve as a warning to later generations not to expect an easy victory. Even if the picturesque details of Galois's life in Eric Temple Bell's Men of Mathematics go beyond the evidence of the sources, Galois cannot be said to have pursued any royal road into mathematics.

ety-century students. This approach has received endorsement from the historical school in the analysis of mathematics presented by Imre Lakatos and Philip Kitcher, among others. At the least, it shakes the student's conviction of the immutability of mathematics. Historians of mathematics live in two worlds. Because the community of historians of mathematics is not large, members of the community spend most of their time in a setting either where the study of mathematics prevails or where the study of history dominates. After having made all the efforts to find an accommodation with mathematicians, the historians of There is always the hope of finding the right mathematics are judged by historical canons as well. Part of the value of Joan Richards's book is its ability to pair of shoulders to stand o n . say something to both communities. The task of understanding history involves seeing The same sort of biographical consolation comes the past through the eyes of its inhabitants. Just as even to those w h o do not see themselves cast in the social sciences, like anthropology, try to approach Newtonian mold. "If I have seen further it is by stand- other cultures on their terms, so the historian uses the ing on ye shoulders of Giants." Robert K. Merton's On evidence of the past to think back into the past. As the Shouldersof Giants traces the history of the aphorism proponents and opponents of relativism have obbest known in its Newtonian form. There is a sort of served, it is impossible to rid oneself of all the intellecfellowship in being part of the mathematical enterprise tual baggage of the present, but it is essential to remain in succession to Newton and Gauss and there is al- as mobile as possible under the load. ways the hope of finding the right pair of shoulders to One fundamental problem afflicting any historian is stand on. the survival of sources from the past. First, just finding Those looking at the mathematics, rather than the the sources can be an infuriatingly slow process. Then mathematicians, of the past are frequently searching there is the problem, once the surviving sources have after nuggets that may have been ignored or misun- been assembled, of ascertaining how typical they are of derstood in the subsequent development of the sub- the age that produced them. The best the historian can ject. Richard Askey practices and advocates this ap- usually do is speculate on the reasons why the survivproach, most accessibly in his contribution to the As- ing sources may not be typical and eschew dogmatism pray and Kitcher collection, History and Philosophy of as a result. Modern Mathematics (University of Minnesota Press, 1988). From this point of view, the historian's obligation is to avoid getting in the way of the mathematics It is impossible to rid oneself of all the inin texts of previous generations. This mining approach to the history of mathematics requires less historical tellectual baggage of the present, but it is essophistication from the mathematician seeking to prac- sential to remain as mobile as possible under the load. tice it. In addition to the use of history by mathematicians outside the classroom, there is some enthusiasm for history's playing a role in the classroom as well. The The practicing historian tends to be wary of drawing simplest way of introducing history is in the form of morals from the past, although popular historians (and anecdotes to spice up the tedium of a lecture or to catch the scholar writing for a general audience) are not rethe interest of the guaranteed nonmathematician. luctant to take up the slack. The best-selling history Views differ on the pedagogical effectiveness of histor- will point out the analogies between previous ages and ical seasoning, but it is easy to see that historical accu- the present. What the historical article will do, in a racy has very little to do with maintaining student in- much more painful, plodding fashion, is reconstruct terest. Stories of dubious authenticity are frequently the past on its own terms and speculate on the relathe most effective---part of the reason for the contin- tionship between people, institutions, and events, ued success of Bell's book. with echoes in later times. A more serious argument for the presence of history The era to which Richard's book is devoted offers a in the classroom is the advantage of following the his- good deal of material to the historian. She uses newstorical line of development. Harold Edwards has de- papers, periodicals, books, and manuscripts of the pevoted several books to reconstructing the work of nine- riod to describe the characters on her stage. In her teenth-century mathematicians for the sake of twenti- search for typicality, she calls up figures little remem78

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bered now, but who propounded certain ideas in their time. The reader has the feeling of being in the midst of a lively discussion on pedagogy and the status of mathematics in society. Richards traces a change in attitude, from the justification of mathematics (and geometry, in particular) in the schools on the grounds of mental discipline, to its serving the training of engineers and businessmen. The changing role of the universities in the Victorian era is seen against the changes in society. The fear of German technological superiority in the wake of the Franco-Prussian war goes far towards explaining curricular alterations. Even within the c o m m u n i t y of mathematicians one can contrast the conservatism of Cayley in his later years with the radicalism of Clifford. Perhaps more unexpected is Richards's way of finding events within mathematics that serve to explain new attitudes toward geometry and the role it should serve. Non-euclidean geometry, she argues on the basis of contemporary quotations, struck a raw nerve, and just could not obtain a fair hearing within the educational (or, indeed, the mathematical) community. What served as a sort of sugar coating for noneuclidean geometry was projective geometry in a form capable of including non-euclidean as well as euclidean geometries. The abuse that non-euclidean geometry had received dwindled into a sort of respectful admiration for the generality that projective geometry achieved. The book ends with the abandonment of the centrality of euclidean geometry and the early work of Bertrand Russell. In so complicated a story, there are some threads that were neglected. Developments in algebra by mid-century receive some attention, but later developments must also have influenced the discussion. The author mentions a number of the unsuccessful proposals for reforming Euclid, but leaves out what ultimately took his place. Even if her focus was on the change in the nature of the geometric curriculum, discussing the terminus would have rounded out the story. Albert Lewis, in a review in Historia Mathematica (August 1990), indicates a number of errors that crept into Richards's text (as well as some cautions about her interpretations). Let me just note that the name of the historian of mathematics Ivor Grattan-Guinness is consistently misspelled and that Grace Chisholm was ranked as a Wrangler at Cambridge in the days before women were admitted to degrees there. None of the other errors on this level got in the way of the text. Richards's book uses the Escher " H a n d with Reflecting Globe" both on the cover and in the text in front of the chapter on projective geometry. The imagery is applicable to the historical enterprise in general, as it seeks to piece together a world that cannot be seen directly. The historian of mathematics uses both internal factors (those of most interest to mathematicians)

M. C. Escher, "Hand with Reflecting Globe" 9 1935 M. C. Escher/Cordon Art, Baarn, Holland

and external (those more accessible to other historians) to map out the geometry of the past. The resulting picture will be a faithful reproduction of neither past nor present, but the task of putting it together is involved in making sense of both. Department of Mathematics Dickinson College Carlisle, PA 17013-2896 USA

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