The mathematical description of patterns adumbrated above is not the only approach that has been taken.
Another kind of mathematical symmetry is important in art: symmetry of similarity. In this kind of symmetry, studied by a mathematician named A.V. Shubnikov, a figure is translated, rotated, or subjected to something like glide reflection while being magnified or diminished in scale, while retaining its proportions. As noted in the previous section, these transformations do not relate to any of the traditional four isometries. Regrettably, Shubnikov named the three similarity transformations obscurely as K, L, and M.1 Certain circular decorations, if projected onto a plane (and perhaps even as is), would exhibit symmetry of similarity, and some spirals do, also. Kappraff's fig. 12.49, a pattern having the similarity symmetry 12.mL, is similar to the design of the interior of the dome of the Masjid-i Shaykh Luṭfallāh in Isfahan, which might perhaps also be described as a complex form of the scaling transformation described by Loeb (below). However, magnification or diminishment alone is not sufficient to a count as a similarity symmetry transformation: it must be done in a particular way.
A valuable discussion that treats symmetry as a subset of transformations of figures is to be found in Arthur Loeb's Color and Symmetry, cited by Chorbachi as particularly useful in relation to Islamic patterns.2
While it is written for the science student (but somewhat offhandedly) and not very accessible to others,3 Loeb's book is decently illustrated. Loeb approaches his subject by briefly considering transformations in general. “Two patterns can be transformed into each other if an operation exists that turns each point P0 in one pattern into a P1 in the other one,” and vice versa. Such transformations can clearly include mathematically definable stretchings and other deformations. Loeb instead considers a transformation he calls scaling: “all the linear dimensions are changed by the same factor, and all angles remain the same.” The two patterns are “similar.” Finally Loeb arrives at symmetry by observing that when the scaling factor is unity (1), the transformation is a “coincidence transformation” (or “coincidence operation”), and the two patterns are the same size (“homometric”). In such a case the two figures taken together will be symmetrical, although Loeb does not say so directly.4
It seems to me that this approach to similarity overlaps that of Shubnikov.
Loeb immediately goes on to consider what he calls a “direct coincidence operation that leaves a single point unaffected”, which is his way of describing a rotation; the point unaffected (that is, not displaced) is the center of rotation, or in Loeb's terminology, a rotocenter.5
Loeb goes on to describe the isometries discussed above in terms of rotations and glide reflections. Translation for him is the rotation of a figure around a rotocenter lying infinitely far from the original figure (so that a line from the rotocenter to some point in the original figure lies effectively parallel to a line from the rotocenter to the corresponding point in the figure after rotation).6 Similarly, reflection for Loeb is a special case of glide reflection.
A rotocenter has an integral “symmetry number” which is simply the number of rotational symmetries of the pattern of which it is the rotocenter. Loeb shows how patterns can be developed from the interaction of multiple rotocenters with different symmetry numbers. He also shows that, given certain constraints, there can be only seventeen combinations of rotocenters in a given tiling pattern, and this set of combinations is equivalent to the seventeen distinct symmetry groups mentioned above.
Loeb also considers rotational symmetries in which the components of a pattern have colors (which may be “reversed” or preserved during some transformation; cf. Washburn and Crowe, above). Color here is a stand-in for any variation in some property, for example the difference between sodium and chorine atoms in a salt crystal; the idea has been explored well.7
The tile pattern illustrated below, from the Hall of the Ambassadors in the Alhambra, is what Washburn and Crowe call a two-color pattern: a 90° rotation around the center of any of the larger white tiles replaces blue with black, and vice versa. The white tiles are considered to be “background”.8 Note that the same pattern has other symmetries also: symmetry of reflection and symmetry of glide reflection. Using Washburn and Crowe's methodology (assisted by two printed copies of a detail of the pattern, a light table, and a very close reading of the text), one can determine that this pattern is classified as p4'g'm.9
Figure 1. A two-color tile pattern.
Another tile pattern from the Hall of the Ambassadors can be considered as a two-color pattern in three different ways.
Figure 2. Another two-color tile pattern.
Considering the ochre and blue tiles only, and counting the black and white tiles as background (think of the black tiles as being white for this purpose, realizing that their borders then disappear so that the ochre and blue tiles float on a seamless sea of white), the pattern is c'mm.10
Considering the black and white tiles only, and counting the ochre and blue tiles as background (think of them both as gray for this purpose, and noting that no borders between tiles disappear in this case), the pattern is pmm.11
Finally, considering the blue, ochre, and black tiles as equivalent (think of them all as black), the pattern is p4'g'm, just as the first two-color example, and (necessarily) by the same reasoning.
Entertaining as such exercises as those with the Alhambra patterns may be, especially on a rainy day and once one can get the same result twice in a row, my reason for introducing them here is to point out that Washburn and Crowe's method for determining the color symmetry of a pattern (if it has color symmetry) turns out to rely on identifying rotocenters and axes of reflection rather than on the four transformations of patterns discussed earlier (their method does involve identifying translations and glide reflections for patterns with no color symmetry).
That is, the value of Loeb's approach is that by concentrating on the rotocenters and, more generally (and abandoning Loeb's terminology), points in a pattern (particularly a tiling pattern) that bear the same relationships to the rest of the pattern, he draws attention to aspects of symmetry particularly relevant to Islamic art.
Furthermore, Chorbachi believes the anonymous manuscript in the Bibliotheque Nationale, Paris, Persan 169, shows how to construct patterns by much the same method as Loeb employs, and that this was the method commonly used to generate such patterns.12
That said, the importance of any mathematical approach to symmetry is that it provides terminology to use, methods of analysis to employ instead of or alongside connoisseurship, and forces an awareness of certain characteristics of the work of art.13 While it may not be interesting to know that a pattern is of type p4'g'm, it can certainly be useful to know what aspects of the pattern contribute to its classification and to recognize that certain patterns are constructed (deliberately or not) with the same color symmetry.14
1. Kappraff, op. cit., pp. 446–49.
2. Arthur L. Loeb, Color and Symmetry, N.Y., 1971.
3. My copy, obtained used, has yellow highlighting through page 16 only, and seems not to have been opened at all after that point.
4. Ibid., pp. 1–3.
5. Loc. cit.
6. Although I do not see how this is what his illustration, p. 4, shows.
7. Washburn and Crowe, op. cit., pp. 5–6.
8. For the definition of background see ibid., p. 65; it is a technical concept, and background in this sense may not correspond to background considered artistically.
9. I do not understand from Washburn and Crowe's explanation of the notation (ibid., p. 70) why this is the correct notation (a prime is supposed to distinguish a movement that reverses colors, and g stands for glide reflection, yet no glide reflection reverses colors here). I assume that this is one of the exceptions to the general rules they allude to and one of the reasons they remark “a complete statement of the crystallographic rules which determine the notation is too complicated to be practically useful” (p. 72). Note also that this field of polychromatic-symmetry theory involves some arbitrary restrictions. (For color rotational symmetry, see Loeb, op. cit., p. 67; for the amusing confusions that would arise were gray to be an allowed color—it “reverses” to itself—and a horrified glance at the problem of mixing primary colors, pp. 99–100.) My path through Washburn and Crowe's flow chart, with its questions and my answers, is as follows:
P. 129, “What is the smallest rotation consistent with color?” 90°: the rotocenter is in the center of a large white tile, and “consistent with color” means that the motion either reverses colors or preserves them (not counting the background).
Pp. 154–55, “Is there a reflection consistent with color?” Yes: reflections can be made along any axis running through the middle of the smaller, hexagonal white tiles, the long way, and all these reflections preserve color.
“Are there reflections consistent with color in four directions?” No: the meaning of “four directions” is explained on p. 157 as “horizontal, vertical and diagonal in two directions”, and there are only horizontal and vertical reflections here.
“Is there a 90° turn which preserves colors?” No: all such rotations reverse colors.
“Is there a reflection which preserves colors?” Yes: all reflections preserve colors.
Note that one must identify all possible reflections and rotations, not just some of them.
10. My path through the flow chart:
P. 129, “What is the smallest rotation consistent with color?” 180°: the rotocenter is in the center of an ochre or blue tile, and no 90° rotations are symmetrical under the assumptions made about the significant colors and the background.
Pp. 140–41, “Is there a reflection consistent with color?” Yes: All reflections along an axis through the vertical lines of colored tiles or along an axis perpendicular to those lines and running through the center of a colored tile preserve color; a reflection along an axis parallel to those lines and centered between colored tiles, or perpendicular to those lines and running midway between colored tiles, reverses color.
“Is there a half-turn which reverses colors?” Yes: the rotocenter is halfway between the colored tiles, centered on either their short or long axes (that is to say, in the center of the actual white tiles, which have gone away as shapes of their own for the purpose of this exercise), or at a point equidistant from any four colored tiles that are closest to each other (that is to say, in the center of the actual black tiles, which have gone away as shapes of their own for the purpose of this exercise).
“Are there reflections, consistent with color, in two directions?” Yes: as described above.
“Are all rotation centers on reflection axes?” Yes: the only rotocenters in the patterns as construed are those described above, and all lie on reflection axes.
“Do all reflections in one (of the two) directions(s) preserve colors?” No: as described above, color is reversed or preserved depending on whether the axis of reflection runs through the colored tiles or midway between them; this is true of both directions.
“Do all reflections in one (of the two) direction(s) reverse colors?” No: as described immediately above.
11. My path through the flow chart:
P. 129, “What is the smallest rotation consistent with color?” 180°: In the pattern construed this way, there appear to be rotocenters at the tips of the 90°-angled sides of the black and white tiles (the points at which two white tiles, one black tile, and one gray tile meet). A 90° rotation around such a rotocenter carries all black tiles onto white tiles, but also carries grey tiles onto white tiles, so it is not consistent with color, and in fact not a symmetrical transformation. The same is true of a 180° rotation, which carries white onto white but black onto gray and gray onto black. The other rotocenters are in the center of every tile, and all 180° rotations around them preserve color.
Pp. 140–41, “Is there a reflection consistent with color?” Yes: as described immediately above.
“Is there a half-turn which reverses colors?” No: as described above.
“Are there reflections, consistent with color, in two directions?” Yes: as described above.
“Are all rotation centers on reflection axes?” Yes: the rotocenters at the corners of four tiles, described above, do not count for this question (I believe) because they do not produce a symmetrical transformation except in the degenerate case of a 360° rotation, which the authors implicitly rule out as relevant in their discussion of symmetry of rotation, pp. 48–49; were any point to count as a rotation center if only a 360° rotation around it produced a symmetrical transformation, the question would always have to be answered in the negative. All the other rotocenters, as described above, lie on reflection axes.
“Is there a reflection which reverses colors?” No: as described above, all reflections preserve colors.
12. The manuscript, given the title Fī tadākhul al-ashkāl al-mutashābiha aw al-mutawāfiqa by Chorbachi, is also discussed under “its shorter title”, Aʿmāl wa ashkāl, by Gülru Necipoğlu, The Topkapı Scroll—Geometry and Ornament in Islamic Architecture: Topkapı Palace Museum Library MS H. 1956, Santa Monica, 1995, pp. 166ff.
13. There is nothing wrong with connoisseurship, but art historians generally do not look at their subject matter harder than they need to in order to make progress, and a method of analysis that forces one to identify all possible symmetries necessarily forces that harder looking.
14. Those interested in advanced tiling designs may be interested in several articles in Fivefold Symmetry, ed. István Hargittai, Singapore, 1992: “800-Year-Old Pentagonal Tiling from Marāgha, Iran and the New Varieties of Aperiodic Tiling It Inspired”, by Emil Makovicky, pp. 67–86, which seems to suggest that the pattern on the lower part of the Gunbad-i Kabūd is or is related to the class of not strictly symmetrical tilings called Penrose tilings; “Pentagon and Decagon Designs in Islamic Art”, by Gilbert M. Fleurent, pp. 263–281, which shows how symmetric tilings of more than one kind of shape can be analyzed, and “An Islamic Pentagonal Seal (from scientific manuscripts of the geometry of design)”, by Wasma'a K. Chorbachi and Arthur L. Loeb, pp. 283–305.
For another discussion of color symmetry in relation to tilings see Grünbaum and Shephard, op. cit., ch. 8.