670 NOTICES OF THE AMS VOLUME 53, NUMBER 6

What Symmetry Groups ArePresent in the Alhambra?Branko Grünbaum

On the occasion of the approaching InternationalCongress of Mathematicians (ICM) (Madrid 2006)it is appropriate to renew the enjoyment of thearts—modern as well as historical—that grace manylocations in Spain. The cover of the February 2006issue, and the article by Allyn Jackson (starting onp. 218) are helpful, as is the note of Bill Casselman(on p. 213). Two sentences in the latter made mecurious. Casselman states that “The geometric na-ture of Islamic design, incorporating complex sym-metries, has been well-explored from a mathe-matical point of view. A fairly sophisticateddiscussion, referring specifically to the Alhambra,can be found in the book Classical Tessellations andThree-manifolds by José Maria Montesinos.” I hadvisited the Alhambra more than twenty years agoand had seen Montesinos’ book soon after it ap-peared; that’s a long time ago, and I had forgottenthe details. I was about to get the book from ourlibrary, but before that I checked the Math Reviews.There I found an assertion that ran counter to mymemories; so I eagerly started looking at the bookitself and recovering old papers and notes on thetopic.

The question which of the seventeen wallpapergroups1 are represented in the fabled ornamenta-tion of the Alhambra has been raised and discussed

quite often, with widely diverging answers. Thefirst to investigate it was Edith Müller in her 1944Ph.D. thesis at the Universität Zürich, written underthe guidance of Andreas Speiser.2 In her thesis [7]Müller documents the appearance of twelve wall-paper groups among the ornaments of the Al-hambra.3 (She also investigates other kinds ofgroups, but this is not relevant for our discussionat this time.) Due to a misunderstanding of Müller’scomment that minor changes would have yieldedtwo additional groups, several writers claimed thatshe found examples of fourteen groups. Some later

Branko Grünbaum is professor emeritus at the Universityof Washington in Seattle. He gratefully acknowledges the help given by Bill Casselman and John Jaworski to the genesis of this note. The author can be reached at grunbaum@math.washington.edu; he will be happy tosupply references for all statements in this note.1Classes of discrete groups of isometries, with two inde-pendent translations.

2Speiser’s 1922 text on the theory of groups deals exten-sively with the investigation of symmetry groups of orna-ments and illustrates the topic with several patterns fromancient Egypt. His rather biased opinion about Egyptiandecorations is best seen from his assertion that “…Egypt,which is the source of all later ornamentation” (“…Ägypten,denn hier is the Quelle aller späteren Ornamentik”). He alsoquotes approvingly (in the original English) the opinion ofFlinders Petrie that “Practically it is very difficult, or al-most impossible, to point out decoration which is provedto have originated independently, and not to have beencopied from the Egyptian stock.” More on this topic can befound in [2].3She also investigated other kinds of groups as well, in par-ticular the eighty groups of the two-sided Euclidean plane.These are the topic of a short note by Müller [8], which in-cludes also illustrations of mosaics from the Alhambra rep-resenting nine different wallpaper groups. Despite hermathematical beginnings, Müller became a well-known as-tronomer, and was for several years the General Secre-tary of the International Mathematical Union. In a letterfrom 1984 she mentioned that there is continuing inter-est in her thesis and that there is a plan to republish it.Regrettably, this seems not to have happened.

JUNE/JULY 2006 NOTICES OF THE AMS 671

writers gave examples they claimed show the pres-ence of one or the other of the missing groups,while others just stated that all seventeen are pre-sent in the Alhambra. This latter phenomenon canbe most readily explained as authors copying fromauthors who copied from others—all without anyactual investigation. An exception to this is part ofJosé Maria Montesinos’ book [6], in which he arguesthat the photos he presents show the appearanceof all seventeen groups in the Alhambra. This wasstressed in the review of [6] by Roger Fenn [1] thatstartled me: “…Incidentally, for the benefit of His-panophiles, this book produces photographic evi-dence once and for all that all seventeen planesymmetry patterns appear in the Alhambra.” Butdoes it really? My memory contradicts this.

Before justifying my standpoint, let us brieflyconsider the situation in which one is asked tocount the number of trees in a forest. Clearly, forthe effort to mean anything one needs to knowwhere to count them—in the whole forest, or acertain square mile, or some other part. But it alsohas to be decided1. What kinds of trees to count;2. What is a tree? Is a sapling a tree? Should a min-

imum of 3 inches diameter be required? If so,where is it to be measured (just above ground,3 feet above ground or some other way)?

3. What about dead but standing trees? What aboutfallen trees, possibly decomposed to the extentthat no visual examination can determine theirkind?In the light of this metaphor we may accept that

Montesinos considers all mosaics, paintings, andplasterworks present in the whole Alhambra com-plex. But then we encounter problems:

There is no explanation concerning what is beingconsidered in any particular ornament: Do we countthe symmetries of the underlying tiling, without tak-ing into account the colors of the tiles, or do weinsist on color-preserving symmetries? (Similarquestions about interlaces. The interlace inFigure 14 has 4-fold rotational symmetries with noreflections if the interlacing is considered, but hasmirrors if it is not.) In fact, Montesinos countswhatever he finds convenient. In one case he re-places all non-white colors by black in order to findan example with 3-fold rotational symmetry and noreflections (see Figure 2). But one could equally wellabstract color altogether and get an example with6-fold symmetry. In another case (not illustrated)he does disregard colors altogether.

There is no explanation as to what is the size orextent of an ornament that is sufficient to accept

it as a representative of a certain group. Inone case a single decorated tile is consid-ered, while in another case miniature copiesof the pattern shown in Figure 3 are claimedto represent the group with 3-fold rotationsand mirrors through all the rotation cen-ters—although the sets of four triangles arein a pattern with 4-fold symmetry, and theseare again arranged in a larger pattern with4-fold symmetry, the whole just part of adecoration on the back of a chair.

Several of the ornaments shown are deterio-rated to such an extent that it is impossible to seethe pattern. Montesinos states that for several ofthese ornaments better examples can be foundwithin the Alhambra, but does not show them.

As pointed out in a private communication fromJohn Jaworski, it is easy to verify that by assign-ing appropriate colors, just two of the Alhambramosaics could yield all seventeen groups. The

4All photos were taken by the author in 1983, during avisit to the Alhambra while on a sabbatical from the Uni-versity of Washington and with the support of a Guggen-heim fellowship.

Figure 1.

Figure 2.

Figure 3.

672 NOTICES OF THE AMS VOLUME 53, NUMBER 6

ornaments shown in Figures 4and 5 are suitable for that pur-pose; photos of the same orna-ments have been used by Ja-worski in his very interestingwork [5].

Due to these objections, andsimilar ones that could be madeconcerning some other publica-tions, Fenn’s enthusiasm seemspremature. Moreover, during sev-eral days in 1983 of examiningthe decorations in the Alhambra,I found representatives of thetwelve wallpaper groups listedby Müller and one she missed; itis illustrated in Figure 6 (part ofwhich is Montesinos’ #4). A moredetailed consideration of the dif-ficulties in consistently counting

the groups in the Alhambra, and what other kindsof groups (color symmetry, interlace symmetry, …)might be more appropriate for some of the orna-ments, appears in [4].

In view of the above discussion, one might won-der whether it is at all possible to arrive at a final,generally accepted count of the groups present inthe Alhambra. The answer must be affirmative,but only if the counting is based on actual exami-nation of the ornaments, presented in a consistentand well-explained manner, and following explicitcriteria. It is possible that the presence of thousandsof mathematicians at the International Congressmay lead some of them to visit the Alhambra andbe sufficiently taken by its splendor to invest theirtime and energy in such a count.

On the other hand, one may well ask why any-body would wish to do this, and what—if any-thing—would be the significance of the result. Itseems to me that there is no more meaning to thedetermination of that number than, say, to the par-ity of the number of attendees of the ICM. Groupsof symmetry had no relevance to artists and arti-sans who decorated the Alhambra. They certainlycould have produced equally attractive ornamen-tation in any of the symmetry groups had anybodywished them to do so. Naturally, nobody did, sincenobody knew about symmetry groups for the nextfive centuries. Thus it is only our infatuation withthe idea that any attractive ornamentation must beexplained in group-theoretic terms that leads us totry to find them there. It is probably worth men-tioning that the analogous infatuation of crystal-lographers with groups crashed with the discoveryof quasicrystals.

Does this mean that there is no role for mathe-matics in the study of ornaments, in the Alhambraor anywhere else? I feel very strongly that there is,provided we approach the task in a way consistentwith the culture we are trying to understand andinterpret. Thus, we have to think, or at least try tothink, in terms that people creating the artifactswould understand and follow.

As an example, there is a lot of what could becalled symmetry in the tiling in Figure 6. Given theshape of the tiles, they are arranged in the onlypossible way; it entails periodicity. On this, the de-signer imposed coloration rules: Half the tiles arewhite; of the other half, half are black and the re-mainder are equally divided between green, blue,and brown tiles. This is a way of looking that wouldhave been understood by the Moorish artisans andmay well have been their intention. We could saythat we find there an example of color symmetries(some horizontal mirrors preserve the white, black,and green tiles, while interchanging blue and brownones, while other mirrors and glide axes lead toother permutations of colors)—but this would havebeen totally extraneous to the thinking of people

Figure 4.

Figure 5.

Figure 6.

JUNE/JULY 2006 NOTICES OF THE AMS 673

500 years ago, hence it is entirely irrelevant. Thereare many additional examples of similar assign-ments of colors, in the Alhambra as well as in theornamentation of other cultures. For example, Fig-ure 7 shows an example in which one half of thetiles are white, one quarter black, and the last quar-ter evenly divided between tan and green. A math-ematical investigation of the possibilities wouldappear to be both interesting and doable and pos-sibly even useful to anthropologists. On the otherhand, in many cases there is no such orderlinessin the colors of the tiles; one has the feeling thatthe artists destroyed the symmetries to make thetilings less monotonous.

In decorations on pottery, as well as other sur-faces, patterns that are not discrete often appear.Circles around pots and vases, straight lines andstrips on flat surfaces, are examples of (admittedlyrather minimal) decorations. They are not ap-proachable through the study of discrete groups—but their historic development within a culturestill can be of interest.

In the study of the exquisite textiles from ancientPeru discrete patterns are common, and quite or-derly. The motifs in some of them form one orbitunder isometric symmetries, but in others this isnot the case; often the colors “spoil” any symme-try. Moreover, just as in the case of Moorish orna-mentation, investigation of the symmetry groupsof the patterns is totally irrelevant. On the otherhand it can be shown that taking into account thestructure of the “fabric plane” in which the patternsare imbedded, one can devise (see [3]) an explana-tion for the orderliness of the patterns that couldhave been understood and transmitted among theilliterate weavers of long ago. As it turns out, thereis only a finite number of possibilities.

There are probably many other situations inwhich a more flexible approach of mathematical in-terpretation would be not only more productive butalso more relevant. In particular, this applies to thebeautiful ornamentation in the Alhambra, but alsoto those in Sevilla and other locations.

References[1] R. FENN, Review of [6], Math. Reviews, MR 0915761

(89d:57016).[2] B. GRÜNBAUM, The Emperor’s clothes: Full regalia, G

string, or nothing? Math. Intelligencer 6, No 4, (1984),47–53.

[3] ——— , Periodic ornamentation of the fabric plane:Lessons from Peruvian fabrics, Symmetry5 1 (1990),45–68. Reprinted6 in Symmetry Comes of Age, The Roleof Pattern in Culture, (eds. D. K. WASHBURN and D. W.CROWE), Univ. of Washington Press, Seattle, 2004, pp.18–64.

[4] B. GRÜNBAUM, Z. GRÜNBAUM, and G. C. SHEPHARD, Symmetryin Moorish and other ornaments, Computers and Math-ematics with Applications 12B (1986), 641–653 = Sym-

metry, Unifying Human Understanding, (ed. I. Hargit-tai), Pergamon, New York, 1986, pp. 641–653.

[5] J. JAWORSKI, A Mathematician’s Guide to the Alhambra,2006 (unpublished).

[6] J. M. MONTESINOS, Classical Tessellations and Three-Manifolds, Springer, New York, 1987.

[7] E. MÜLLER, Gruppentheoretische und Strukturanalytis-che Untersuchungen der Maurischen Ornamente ausder Alhambra in Granada, Ph.D. thesis, UniversitätZürich, Baublatt, Rüschlikon, 1944.

[8] ——— , El estudio de ornamentos como aplicación dela teoría de los grupos de orden finito, Euclides (Madrid)6 (1946), 42–52.

5This interdisciplinary journal, with a blue-ribbon Advi-sory Board, was started after meticulous preparations,and its first issue contained contributions by A. L. Mackay,A. L. Loeb, M. J. Wenninger, C. A. Pickover, V. Vasarely,and others. The publication of the journal was cancelledafter the first issue, by its publisher VCH Publishers, dueto low rate of subscriptions. So much for investment in theinterdisciplinary approach.6Unfortunately, the dedication of the paper to HeinrichHeesch was omitted, and the colored illustrations have beenrendered in black-and-white.

Figure 7.