the symmetry of things.pdf

The Symmetries of Things by John H. Conway; Heidi Burgiel; Chaim Goodman-StraussReview by: Branko GrünbaumThe American Mathematical Monthly, Vol. 116, No. 6 (Jun. - Jul., 2009), pp. 555-562Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/40391162 .

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REVIEWS Edited by Jeffrey Nunemacher

Mathematics and Computer Science, Ohio Wesleyan University, Delaware, OH 43015

The Symmetries of Things. By John H. Conway, Heidi Burgiel, and Chaim Goodman-Strauss. A K Peters, Wellesley, MA, 2008, xviii + 426 pp., ISBN-13: 978-1-56881-220-5, ISBN-10: 1-56881-220-5, $69.

Reviewed by Branko Grünbaum

The word "symmetry" has many different meanings, so it seems appropriate to discuss the notion before reporting on The Symmetries of Things. In my view, any symmetry is an orderly or systematic disposition of parts in a whole, and vice versa, any such disposition is a symmetry. The orderliness or system can be of many kinds - leading to very different situations and developments. Some of the earliest cultural artifacts exhibit symmetries. The ancient Egyptians have long been presented as being unsurpassed masters of symmetry in ornamentation, but other cultures have had their own contributions. As examples, one may mention the well-known Islamic ornaments, which are totally independent of any Egyptian influence, or the stunning decorations of textiles made by pre-conquest Peruvians. In Figure 1 we show in a schematic way the patterns of two of the still preserved ancient Peruvian fabrics; as explained, illustrated, and ref- erenced in detail in [5], many of these textiles show a great creativity in combining shapes and colors to generate very orderly but attractive patterns. (The journal Symme- try was discontinued by the publisher, VCH Publishers, after a single issue. The paper [5] was reprinted as [7], but without the dedication to Heinrich Heesch, and with no color illustrations.)

A A A A A A A A A V V A A V A /" A A A A A A V V A A V

A /" A /" A /" A A/^vvaav

A A A A A A A A V V A A V

A A A A A A A AAVVAAV A A A A A A A A v v A A v

A A A A A A A AAVVAAV Figure 1. Schematic representations of two Peruvian textiles; see [6, page 46] or [7, page 23]. The first contains two copies of the motif in each translational fundamental region, the second eight copies.

The mathematical side of symmetry was very slow to develop. While we tend to think of the regular polygons, polyhedra, and tessellations as the most symmetric objects of their kinds, one has to bear in mind that they were singled out - ever since Euclid and Archimedes - by various local properties. This included the requirement of

June-July 2009] reviews 555



equal sides and angles for polygons, and the requirements of regular polygons (of the same kind) as faces and congruent vertices, or similar requirements, for regular polyhedra. Only in the nineteenth century, with the development of group theory, did mathematicians develop the insight that the symmetry of these objects may be explained as a global attribute, resulting from transitivity under groups of isometric self-maps. Crystallography contributed greatly to the understanding of objects with various levels of symmetry (such as polyhedra, or discrete sets of points), which led to isohedral and to isogonal polyhedra, and to other particular types of symmetric objects. Coxeter and others made great contributions to these investigations, and the research was expanded into other kinds of spaces and objects. Earlier, Klein's "Erlanger program" declared that each geometry is the study of properties invariant under some group acting on a set. While there is no doubt that this program had a positive influence on geometry in general, the spirit that came with it was (and is) stifling the study of ornaments, polyhedra, and similar objects. It is obviously much easier to investigate objects that have a group of isometries acting transitively on their elements than to study objects of the same kind that do not admit such isometries.

The case of polyhedra presents a clear example. Regular polyhedra, polytopes, and related objects have been studied for ages from every point of view, and generalizations to other spaces continue to attract a lot of attention. But it took linear programming and other optimization techniques to bring (around 1950) attention to polyhedra (and polytopes) not necessarily endowed with any particular symmetry or regularity.

In the study of tilings of the plane we encounter a similar situation. Tilings that have vertices (or edges, or tiles) in a single orbit under isometries have been inten- sively studied, and there are extensive and detailed accounts of the results obtained; for example, see [9]. However, even the slight generalization from isohedral tilings to monohedral ones (that is, tilings such that all tiles are congruent) leads to a wealth of simple problems, which are still open despite long and intensive efforts. Among them is the question of what pentagons are monohedral tilers, open even for convex pentagons. Another question concerns the possible symmetries of tiles in monohedral tilings: can each tile have, say, five-fold rotational symmetry? Is there a tile that can be used to construct monohedral tilings with each of the 17 symmetry groups?

In Figure 2 we show schematic drawings of tilings discovered by Peter Raed- schelders. In several of the artist's original works the tiles are zoomorphic, in the style of Escher, with no individual symmetry. However, the tiles are arranged in such a way

Figure 2. Schematic representation of some of the zoomorphic tilings of P. Raedschelders (from [13] and private communications).

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that a minimal translational region of the tiling has 64 tiles, and the arrangement has the particular property that each row and each column of any 8 by 8 patch contains tiles of all eight aspects (translationally inequivalent positions) - hence represents an 8 by 8 Latin square. Which Latin squares are representable in a similar manner -

possibly using more than one basic tile, or using colors to distinguish among tiles of the same aspect? This question, and many others (see [5]), have still to find appropriate answers.

Much of The Symmetries of Things is devoted to symmetry groups of tilings (of the plane and of other spaces). While this is quite interesting from a purely mathematical point of view, it is a severe restriction for the applicability to ornaments and other topics. Classifying "things" by symmetry groups is somewhat analogous to classifying animals by the number of their legs (0, 2, 4, 6, 8, 100, 1000,. . . ?). In both cases some information is conveyed, but usually not enough information comes through to be really useful. From ancient Egypt to modern quilt makers, from crystallography to art, there are symmetries - systematic dispositions - that are trivialized by describing them in terms of groups of symmetry.

Granting the above general considerations about symmetry, it is time to discuss the book on its own terms. That's what we shall do in the rest of this review. There is a lot to be enthusiastic about, but also some serious shortcomings.

The book is divided into three parts. The first is entitled "Symmetries of Finite Ob- jects and Plane Repeating Patterns," and it gives an introduction to the symmetries of the figures considered, to the notation, and much more. One of the central points is the introduction of symbols, called signatures, that directly express the various symmetries and the symmetry group of a pattern. The presentation in this part is quite leisurely and achieves a whole lot besides the introduction of the Conway signatures. The assigning of "costs" to the components of the signature enables one to easily enumerate the possible groups of symmetry by using the "Magic Theorem," which is eventually shown to be equivalent to Euler's theorem. The symmetry groups of rosettes, friezes, wallpa- pers, and patterns on the sphere are all determined. This part of the book concludes with an introduction to "orbifolds," which are used, together with appropriate signatures, throughout the book. In all, this is a pedagogically excellent presentation of the material; it is the best introduction to orbifolds I have seen. One can only hope that it will attain the goal of spreading the use of the signatures and the orbifold tools. The ease with which one enumerates the groups for different manifolds, and the analogy between the collections of symmetry groups for these, are here presented in a really valuable way.

Nevertheless, there are self-inflicted injuries to the authors' aims. The type of isometries called glide-reflections or glides by everybody else is here called "miracles" (page 24), and instead of "translation" the authors use "wonder." Do they really expect that these cute terms will be generally accepted? The signatures are said to have been developed recently from "Murray MacBeath's mathematical language for discussing symmetry" - and that's all we get concerning MacBeath and the origins of the signatures (except that on page 1 19 his name is given as McBeath).

In some cases poor wording causes problems. For example:

• On page 31, glides ("miracles") are supposed to be found if "you can walk from some point to a copy of itself without ever touching a mirror line." But what happens if there are no mirror lines?

• On pages 38-39 (and others), is it accidental or intentional that the order of listing sets of types does not coincide with the order of the illustrations? It can confuse the beginner.




• On page 57 we read that the group denoted here by *532 "is generated by reflections in a triangle of angles n/5, n/3, n/2, and a spherical pattern with this symmetry exists because there is a spherical triangle with these angles." What about angles 7T/5.01, 7T/3.02, 7T/2.05 ? There is a spherical triangle with these angles.

• There is no indication anywhere that only patterns with discrete symmetry group are considered. Like many other writers, the authors leave the reader wondering: What is the signature of a circle, or a circular disk, or of one straight line or a family of parallel lines? If the signatures are to be applied to the study of ornaments, this is a big drawback that could be eliminated with a few words.

Even more serious is the omission - in all of the first part - of any other work about symmetries of the patterns discussed. How is a student supposed to get acquainted with the relevant literature needed for any serious study? It is only on page 1 19 in the "Introduction to Part II" that we are told that the tables in the Appendix contain "dic- tionaries" between signatures and the other systems used in the literature. However, even here only two sources are specified, and the others only referred to by name of an author - with no references.

The second part of the book deals mainly with color symmetry groups, after introducing material regarding the connections between signatures and generators of the symmetry groups. It also covers several aspects of classification of tilings of the plane and the sphere. The symmetry groups of 2-colored patterns in the plane are classified giving the well-known 46 types. There is no indication that the authors are aware of any previous determination of this classification, which in fact goes back to the 1930s.

The groups of 3-colored patterns are enumerated in a separate chapter, and another chapter covers the only slightly more complicated enumeration of /7-color patterns for prime p. On page 120 (as well as in the Preface) it is claimed that enumeration of /7-color symmetry groups is carried out for the first time in this book. Again, this is not the case, as the groups of «-colored patterns have been determined in [8] for n = 3 in 1979, by Jarratt and Schwarzenberger [11] in 1979 for n < 15, by Wieting [18] in 1982 for n < 60, and for many (but not all) n by Senechal [15] in 1979; however, Senechal's results cover all prime n. The disregard of the existing literature results in an error in the enumeration for both threefold and primefold colorings. In both cases, one group is missed. On page 156, the entry for 22* = pmg in Table 12.1 asserts that the only 3-coloring for this group is 22*//** = pmg[3]2 (in the notation of [9]); however, this is wrong. There is also the 3-coloring 22*//o = pmg[3]i ; see Figure 3. The analogous error is repeated for p-colorings in Table 13.1 on page 164.

Figure 3. The two distinct 3-colorings of a pattern with symmetry group 22* = pmg.


22*//o = pmg[3] j 22*//** = pmg[3]2



There are several other glitches in this part, but one is serious enough to be explicitly mentioned. On page 188 the authors say: "There may be symmetries of the tile itself that do not extend to the tiling." This is wrong: they do extend, but the extended symmetries would force additional symmetries of the tile itself. Although they say "We indicate this possibility,. . . ," it is neither stated nor is it clear how this is indicated. The situation can be best understood by distinguishing between tiles (endowed with a specific shape), and marked tiles (that carry an imbedded subset as a mark); this is described in detail in Section 6.2 of [9].

I found the explanations of enumeration of isohedral types of tilings of the plane (Chapter 15) hard to follow, and the results incomplete. On the one hand, there is no indication of the fact that some of the tilings require specifically marked tiles since they cannot be realized by tiles of appropriate shape. On the other hand, the list of isohedral types for 2*22 = cmm (page 195) misses one type and has a wrong drawing for another type. I cannot vouch for the other isohedral types. Here too comparison with easily accessible literature would have prevented the errors.

The third part deals with "Repeating Patterns in Other Spaces." It is far longer than the earlier two, and the authors state (page 217): "We expect that Part III will be completely understood only by a few professional mathematicians." The first 80 or so pages of this part deal with hyperbolic groups, Archimedean tilings and polyhedra, and tilings of 3-space. The rest is devoted to an enumeration of the crystallographic groups in 3-space, and to infinite Archimedean and pseudo-Platonic polyhedra. These polyhedra appear to be just a sampling of the possibilities. The authors say (page 336) "there are some subtleties on which we shall not elaborate." No explanations or references are given.

I have to admit that I am not among the few that completely understand the third part. Hence I will comment only on the chapters I am acquainted with. It seems to me that Chapter 19 entitled "Archimedean Tilings" attempts to cover too much material in too little space, with too short explanations and too few illustrations. While this may be due to my own shortcomings, there seem to be some intrinsic contradictions and probable errors. The discussion is meant to apply to the Euclidean and hyperbolic planes as well as to the sphere. The working definition requires regular polygons as faces (tiles), and vertices in a single orbit under symmetries of the tiling. The tra- ditional Archimedean tilings are termed "absolute," meaning that all the symmetries of the tiling are considered. In addition, if a subgroup H of the symmetry group G acts transitively on the vertices, then the tiling is said to be Archimedean relative to H . The authors state (page 251): "The complete classification of all Archimedean tilings, both relative and absolute, appears for the first time in this book." On page 250 are shown "The thirty-five relative Archimedean tilings of the Euclidean plane by squares." While no additional explanations are given, for the tilings shown (some have colored tiles, some have markings decreasing the symmetry of the squares, some have both) it is clear that the subgroup in each case is meant to preserve the markings and/or colors while acting transitively on the vertices. Although I would prefer that the relative tilings that are determined by colors alone be distinguished from the ones that need markings, I realize that this is not inherent in the algebraic-topological way of deriving the tilings. However, a comparison of the tilings shown on page 250 with the uniform colorings of tilings in Figure 2.9.2 of [9] shows that at least three "relative" tilings are missing on page 250. But even more confusing is Table 19.1 (on pages 262-263): "The Archimedean polyhedra and tessellations. The spherical and Euclidean Archimedean tilings are shown. Each absolute tiling is shown The relative tilings are lightened. . . ." The use of the definite article would seem to imply that all relative tilings are shown, but this is not the case. Of the four relative tilings of




the square tiling shown on page 250 by colors, only two appear in Table 19.1. Several relative tilings of the regular tiling by triangles are missing, as are a relative tiling of (3.6.3.6) and one of the relative tilings of (63).

The next chapter, "Generalized Schiaffi Symbols," is very interesting and innova- tive, although based on earlier related work by Andreas Dress (no reference given). Especially attractive is the graphical representation of the adjacencies of the cells in the barycentric subdivision of a given polygon, polyhedron, or tiling. The applications provided include classifications of hexagons and octagons into types determined by their symmetry properties, and the lattices of subgroups of the octahedral and icosahe- dral symmetry groups.

Chapter 21 deals with Archimedean and Catalan polyhedra and plane tilings. These are presented in a manner different from the usual, but with a plethora of new names and terms that may discourage some readers. It also contains a list of 13 vertex- transitive tilings of 3-space by Archimedean polyhedra. It is reasonably well known (Andreini [1] in 1905, with later corrections by various authors) that there are 28 types of such tilings. The authors state: "The most interesting ones are those whose symmetry group is one of the 'prime' space groups of Chapter 22, and we shall restrict ourselves to these." In some sense the authors may be correct - but it is regrettable that they did not add here a single short sentence that would be sufficient to explain the term "prime." On the other hand, the illustrations and descriptions provided for these 13 tilings are the best I have ever seen.

Introducing Chapter 22 the authors say ". . . we discuss the 35 most interesting crystallographic space groups, namely the 'prime' ones that don't fix any family of parallel lines." This is followed by an algebraic description, and by comments on generators and relations of these groups. The next chapter describes a variety of objects that illus- trate the "prime" groups. Among them are the three Coxeter-Petrie infinite polyhedra and some of their Archimedean relatives. The authors assert: "We believe that no- body has yet enumerated the hundreds of 'Archimedean' polyhedra in 3-space. The only further ones we'll discuss here are pseudo-Platonic, meaning that all their faces are the same shape." They describe such infinite polyhedra that have 7, 8, 9, or 12 equilateral triangles incident with each vertex, or 5 squares (two kinds of these last). They do not claim completeness of their list; in fact, Hughes Jones [10] has shown in 1995 that there are many other such polyhedra with triangular faces, and his list is far from complete. There is also the catalog of Wachman et al. [17], that lists close to a hundred such infinite polyhedra. Surprisingly, there is also no mention of the Goodman-Strauss and Sullivan [4] paper on polyhedra with six squares incident with each vertex.

Among the remaining chapters the one likely to attract the most interest is Chap- ter 26, entitled "Higher Still." It contains (among other material) a description of the enumeration of the 4-dimensional Archimedean polytopes carried out more than forty years ago by Conway and M. J. T. Guy, which was briefly announced by Conway [2]. Also given is the list of the 4-dimensional star-polytopes. Together with the two convex regular polytopes with pentagonal symmetries, they are placed at the 12 vertices of a cuboctahedron, on which their relationships are indicated in a visually attractive way.

Several typos - especially of names of people - are annoying. There is Murray MacBeath or McBeath; then there is the "Kline bottle" on page 217, the notation of the 17 wallpaper groups by "Spieser" on page 415, and even a modification of my first name on page 420. A more serious error on page 415 is the claim that Niggli "inad- vertently interchanged" the notation of "Spieser." In fact, it is the other way around: Niggli [12] in 1924 has things correct, and Speiser [16] in 1927 messed them up. No




reference is given for either work. Speiser's mix-up was repeated by many mathematicians, until the publication of Schattschneider [14]. See also Cundy [3].

The reason for the great length of this review is that I admire the book for what it achieves, but I do not like many parts of it. The authors are very gifted mathematicians with many startling results among their achievements, and I expected more from the book. While a quick glance seemed to confirm that this is an exceptional book, with fascinating illustrations, a close reading of the second and third parts found many shortcomings, only some of which are mentioned above.

The disdain with which previous work on symmetry (even in the restricted, group- based meaning adopted here) is completely ignored is damaging in at least two different ways. First, it renders it very difficult for the reader of this book to establish any connection with the knowledge available in the literature. The idiosyncratic terminol- ogy ("miracle," "wonder," and many other terms) adds to this difficulty. Second, many of their ideas and results do appear in works of earlier writers, and the customary ap- proach is to acknowledge such priority. But beyond courtesy, they might have profited from comparing their results with those in the literature. For example, the rather ob- vious discrepancy regarding the number of prime-fold color types should have alerted them to the need to find out whether the accepted enumerations deal with concepts different from theirs, or whether one of the parties made an error. Whatever the answer had turned out to be, both the authors of the book and the mathematical public would have benefited.

Most of us will profit by reading the book, or at least parts of it - but bearing in mind that one should not take all that is written at face value.

ACKNOWLEDGMENT. The hospitality in Summer 2008 and resources of the Helen Riaboff Whiteley Cen- ter at the Friday Harbor Laboratories of the University of Washington are gratefully acknowledged.

REFERENCES

1. A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative, Mem. Società Italiana delle Scienze 14 (1905) 75-129.

2. J. H. Conway, Four-dimensional Archimedean polytopes, in Proceedings of the Colloquium on Convexity, Copenhagen 1965, K0benhavns Universitets Mathernatiske Institut, Copenhagen, 1967, 38-39.

3. H. M. Cundy, p3ml or p31m?, Math. Gaz. 63 (1979) 192. 4. C. Goodman-Strauss and J. M. Sullivan, Cubic polyhedra, in Discrete Geometry: In Honor ofW. Kuper-

berg's 60th Birthday, Monographs and Textbooks in Pure and Applied Mathematics, 253, A. Bezdek, ed., Marcel Dekker, New York, 2003, 305-330.

5. B. Grünbaum, Periodic ornamentation of the fabric plane: Lessons from Peruvian fabrics, Symmetry: An Interdisciplinary and International Journal 1 (1990) 45-68.

6. , Levels of orderliness: Global and local symmetry, in Symmetry 2000, Proc. of a Symposium at the Wenner-Gren Centre, Stockholm, vol. I, I. Hargitai and T. C. Laurent, eds., Portland Press, London, 2002, pp. 51-61.

7. , Periodic ornamentation of the fabric plane: Lessons from Peruvian fabrics, in Symmetry Comes of Age: The Role of Pattern in Culture, D. K. Washburn and D. W. Crowe, eds., University of Washington Press, Seattle, 2004, 18-64.

8. B. Grünbaum and G. C. Shephard, Incidence symbols and their applications, in Relations Between Com- binatorics and Other Parts of Mathematics, Proc. Sympos. Pure Math., vol. 34, American Mathematical Society, Providence, RI, 1979, 199-244.

9. , Tilings and Patterns, Freeman, New York, 1986. 10. R. Hughes Jones, Enumerating uniform polyhedral surfaces with triangular faces, Discrete Math. 138

(1995)281-292. 11. J. D. Jarratt and R. L. E. Schwarzenberger, Coloured plane groups, Acta Cry st. A36 (1980) 884-888. 12. P. Niggli, Die Flächensymmetrien homogener Diskontinuen, Z. Kristallographie 60 (1924) 283-298. 13. P. Raedschelders, Semimagic tiling based on an asymmetrical tile, Geombinatorics 10 (2000) 45-50.




14. D. Schattschneider, Symmetry groups: Their recognition and notation, this Monthly 85 (1978) 439- 450.

15. M. Senechal, Color groups, Discrete Appl. Math. 1 (1979) 5 1-73. 16. A. Speiser, Die Theorie der Gruppen von endlicher Ordnung, Springer, Berlin, 1927. 17. A. Wachman, M. Burt, and M. Kleinmann, Infinite Polyhedra, Technion-Israel Institute of Technology,

Faculty of Architecture and Town Planning, Haifa, 1974. 18. T. Wieting, The Mathematical Theory of Chromatic Plane Ornaments, Marcel Dekker, New York, 1982.

University of Washington, Seattle, WA 98195-4350 grunbaum@math. Washington, edu

Mathematics Is ...

"Mathematics is, on the one side, the qualitative study of the structure of beauty, and on the other side is the creator of new artistic forms of beauty."

James B. Shaw, Mathematics - The subtle fine art, in Mathematics: Our Great Heritage, William L. Schaaf, ed.,

Harper & Brothers, New York, 1948, p. 50.

-Submitted by Carl C. Gaither, Killeen, TX




the symmetry of things.pdf

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