C o m m e n t s on Gri inbaum's Article

Peter Hilton and Jean Pedersen

Branko Gr~nbaum has performed a great service in drawing attention to the danger of accepting authority blindly�9 This danger is especially real when the au- thority is speaking outside his or her area of expertise. However, Gr~nbaum's article is concerned with many issues beyond that of Hermann Weyl's attribution to the ancient Egyptians of the discovery of the 17 classes of symmetry groups of the plane. Grfinbaum writes with passion of the unfortunate consequences of this error, and of o thers- - i t is natural for him to do so- - but perhaps we might be forgiven for viewing the matter a little more temperately (though not dispas- sionately) and looking at Gr/inbaum's four principal themes in sequence.

First, then, to the extent that it is possible to prove a negative, Gr/inbaum has demonstrated that, of the 17 patterns in question, the Egyptians did not invent any of the five which exhibit 3-fold symmetry. He points out that Weyl depended on the authority of Speiser and that Speiser himself, while recognizing the fundamental research of Owen Jones 1, did not appre- ciate that Jones himself never mentioned these five patterns in discussing Egyptian decorations.

We ourselves had the opportunity to talk with P61ya about this aspect of Gr/inbaum's historical research, and P61ya's response was that he believed that Weyl never really meant his remarks about Egyptian wall- paper patterns to be regarded as definitive. Weyl is certainly not the first great thinker to be ill-served by those who offer him uncritical adulation (compare Sig- mund Freud and Karl Marx).

Second, GrOnbaum has also performed an invalu- able service by his own research on Moorish art, im- proving in a particular respect on the work of Jones and Edith Muller. For, whereas they only discovered 11 of the 17 patterns in the Alhambra, Gr~nbaum, on a study visit to Granada in 1982, discovered 13. He also remarks that Ostwald in his definitive study of Islamic art seems to have overlooked the symmetry groups pg and pgg, which are, in fact, missing from the Alhambra, along with p2 and p3ml.

1 Gr / inbaum appea r s to describe Jones as Speiser ' s "Ph .D. s tuden t " , bu t p e r h a p s we d id no t correct ly u n d e r s t a n d G r / i n b a u m at tha t point.

We remark that between them the Egyptians and the Moors only missed the one symmetry group (p3ml). One of us (JP) had the opportunity to intro- duce Kazuko Yamamoto, a specialist in the very an- cient art of Temari, to P61ya. On that occasion P61ya gave Kazuko (as she asked us to call her) a Japanese book on pattern design that had been given to him by a botany professor at the ETH in ZUrich in the early 1920's, about the time when he was working on the paper 2 in which he proved that there were precisely 17 symmetry groups in the plane. Kazuko told us that the book is over a hundred years old (from the Meiji era), that it is the first of a two-volume set, and that it appears to have been written for the purpose of ex- plaining pattern design to artists and craftsmen�9 We were interested to note that the Japanese book con- tains examples of all the 3-fold symmetries except the elusive p3ml and also contains examples of the sym- metries pg and pgg missing from the Ostwald catalog and the Alhambra (see Figures 1, 2) 3.

We are indebted to our colleague Professor Doris Schattschneider for pointing out to us that the book by Dye [2], on Chinese lattice, appears to be the richest single published source of periodic designs and does, in fact, include an example of the symmetry group p3ml. The reader should consult Schattschneider's fundamenta l article [3], which contains a definitive treatment of methods of identification of the 17 pat- terns; indeed her recognition chart for identifying a pattern must surely have been the basis of Crowe's flow diagram [1]. We quote from a letter of hers:

�9 . . I found 14 symmetry types in this single source [2] and of course the origins of the lattice designs are probably quite old.

Perhaps an important point to make is that other (far Eastern) cultures have different approaches to the creation of decorative art, and here some of the symmetry types which seem to be missing in Islamic art can be found. (Almost all Islamic designs have reflection lines and most were developed from circle constructions--but the

2 Pub l i shed in 1924.

3 The J a p a n e s e book has , in fact , 14 of t he 17 s y m m e t r i e s , t h e m i s s i n g ones be ing ping, cram a n d p3ml. Of course, these m i g h t be in the s econd vo lume.

54 THE MATHEMATICAL INTELLIGENCER VOL. 6, NO. 4 �9 1984 Springer-Verlag New York

Figure 1. A page of the 19 th century Japanese design book. The second pattern exhibits 3-fold symmetry (p31m).

Figure 2. Patterns corresponding to the symmetry groups pg and pgg missing from Moorish art (taken from the same Japanese book).

Chinese and Japanese designs seem to be free of these constraints and show rotations and glide reflections much more strongly. As your Figure 1 makes clear, the under- lying lattice of squares, equilateral triangles, rhombuses, or parallelograms is the basis of each design--quite in agreement with the basis of modern crystallographic clas- sification. Half-turns and glides in fact seem to be espe- cially popular in Japanese stencil designs.) The difference may also have something to do with the difference in func- tional purpose of the designs.

Gr~inbaum's third point is that one should not iden-

tify the not ions of symmet ry available to ancient civi- lizations wi th the mode rn concepts of g roup theory. At one level this is obviously true; however , it seems too sweeping to say that symmet ry as we unders tand it was not a mot ivat ion of the Egyptians. To the extent that they were seeking repeat ing pat terns they surely had an i n tu i t i ve c o n c e p t of s y m m e t r y , t h o u g h the mathematical formulat ion and conceptual izat ion were certainly not available to them. Moreover , we can in- terpret their ideas in terms of a potential infinity even

THE MATHEMATICAL 1NTELLIGENCER VOL. 6, NO. 4, 1984 5 5

though neither they nor we are concerned with ac- tually papering an infinite wall. Similarly their designs show an intuitive awareness of the concept of rota- tional symmetry even though they did not propose to turn either their walls or their baths through a right angle. They probably did, however, have the notion of invariance of appearance from different vantage points. Further, Grfinbaum makes the very significant point that their view of symmetry appears to have been more local than global; it is interesting that the global view of symmetry seems to be a strong feature of the design of Temari balls, an art which goes back over a thousand years, having originated in China.

Grfinbaum's fourth point is of a different nature from the other three, being mathematical rather than historical. It is that group theory provides a too-re- stricted view of pattern design. He poses some fasci- nating problems and gives examples of very new con- cepts of pattern, among them one in which the plane is covered by a pattern admitting no infinite symmetry group, and another in which the symmetry group does not act transitively on the constituent convex penta- gons of the pattern. While entirely agreeing with both the significance and the validity of Grfinbaum's point of view, we do note that Grfinbaum has found group- theoretical language adequate for pointing out the in- sufficiency of the group-theoretical description of the features of these patterns.

Let us close on an appropriate note of congratula- t i o n - t o the editors of the Mathematical Intelligencer for stimulating this spirited discussion and, even more, to our friend Branko Grfinbaum for his painstaking his- torical research, for his splendid advocacy of the role of geometry in mathematics, in art, and in science, and for his provocative and enlightening article.

References

1. Donald W. Crowe (1981) The Geometry of African Art. III The smoking pipes of Begho. In The Geometric Vein: The Coxeter Festschrift. Springer-Verlag, New York, 177- 189

2. Daniel S. Dye (1937) A Grammar of Chinese Lattice, Har- vard-Yenching Institute Monograph Series, vol. VI, Har- vard University Press, Cambridge, Mass. (Reprinted as Chinese Lattice Designs, Dover, New York, 1974.)

3. Doris Schattschneider (1978) The Plane Symmetry Groups: Their recognition and notation. The American. Mathematical Monthly 85(6), 439-450.

Peter Hilton Department of Mathematical Sciences SUNY Binghamton Binghamton, NY 13901

Jean Pedersen Department of Mathematics University of Santa Clara Santa Clara, CA 95053

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