The Limits of Analysis: OrnamentationNovember 14, 2009
Color photo of a handbag (top) created by a Tonga weaver in Mozambique, illustrating the transformation of a weave structure that can be achieved simply through the arrangement of colors.
Philip J. Davis
Sipatsi: Basketry and Geometry in the Tonga Culture of Inhambane and Sipatsi Images in Colour. By Paulus Gerdes, Universidade Pedagogica, Maputo, Mozambique/Lulu, Morrisville, North Carolina; main book: 422 pages, $29.95 (print) and $5.95 (download); colour supplement: 56 pages, $27.95 (print) and $4.95 (download).
"Painting and sculpture cannot be translated into words. One art cannot evoke another."--André Malraux
Many readers will find a few situating definitions helpful in approaching this review. Inhambane is a province on the southern coast of Mozambique. Tonga is the name of an ethnic group and its culture. Sipatsi refers to the baskets, hats, purses, bags, and other items woven in distinctive patterns by Tonga women. Maputo is the capital and largest city of Mozambique.
What we have in Sipatsi and its supplement is a part of both ethno- and applied mathematics. As I surfed the pages, my eyes were drawn initially to the figures, especially the many instances in which steps in the sipatsi production process appear side by side with the final products. I was intrigued by the intricate geometric patterns, all of which share a family resemblance that is distinct from, say, the patterns of Navajo blankets or Celtic jewelry. One sipatsi hat was woven in a spiral pattern (like the arrangement of sunflower seeds); I was able to produce such a pattern by some simple but nonlinear iterations.
Gerdes describes in some detail how the sipatsi patterns are woven and what rules or well-established procedures the women employ. He also provides a count of the frequency with which particular patterns occur. His book can be recommended to all who are interested in the art of weaving and the global omni-presence of weavers and their products.
Mathematics has often been described as consisting in part of the creation and study of generalized patterns. It should then come as no surprise that the text of Sipatsi provides an analysis of what is going on in the weavings in mathematical terms: translations, reflections, rotations, and so forth---in short, in terms of symmetry groups.
Looking at the many pictures in Sipatsi put me in mind of the general subject of the mathematical analyses of ornamentation, and it is to this very well plowed (and still being plowed) field that I want now to turn.
Color photo of a handbag (at top of article) created by a Tonga weaver in Mozambique, illustrating the transformation of a weave structure that can be achieved simply through the arrangement of colors. The black and white illustrations show the original weave structure (a), its woven appearance (b), an alternative coloring of the weave structure (c), and its woven appearance (d). From Sipatsi and Sipatsi Images in Colour.
Within a rigid interpretation of the word "evoke," Malraux's admonition in my epigraph is undoubtedly correct. The accumulation of adjectives on the label of a bottle of fine wine may delight a Madison Avenue wordsmith, but to the oenophile these words are no substitute for the experience of swishing the wine in a goblet, sniffing, and tasting.
In the name of purity, Aristotle railed against the "error of metabasis" (i.e., the mixing of disciplines or crafts). Presumably, he would have proscribed The Mathematical Mechanic, a recent book.* We do, however, allow or even encourage one art to mix with, to suggest, to evoke another. Such evocations might be incomplete, inadequate, exploratory, fruitful, and can sometimes emerge as ridiculous (as in Kepler's planetary model via the regular polyhedra). Yet, as Amos Funkenstein pointed out,† it was by means of metabasis that Galileo, Descartes, Newton, and others were able to render nature understandable, accessible, and manipulable. Nonetheless, one must be aware of the "paradox or the fallacy of Pygmalion," wherein the model, the representation, or the algorithm takes precedence over what is being modeled, e.g., the thought-experiment known as Laplace's demon (1814), in which the whole cosmos is reduced to a system of ordinary differential equations.
The mathematization of ornaments is a minor subfield of group theory that I first came across as a student, in Andreas Speiser's Die Theorie der Gruppen von Endlicher Ordnung. Speiser's book appeared in 1922, but he pointed to the strong relation of ornament to crystallography and gave references to studies of this connection that go back to the late 1800s. The 22 pages of Speiser's chapter 6, "Die Symmetrien der Ornamente," contain about fifty illustrations, some of them stick figures that detail the enumeration of the 17 possible symmetry groups. Other figures show ornamental designs well known in the world of art. Speiser embedded these figures in group-theoretic analyses involving translations, reflections, rotations, and other transformations. Such a mix of art and mathematics provides an analysis and classification along group lines of what it is that we see. We might now question whether this analysis of ornamental objects is a mere taxonomic goal, a terminus ad quem, or whether something within it lies deeper. In crystallographic physics, the group-theoretic analyses go very deep.
Geometric patterns are culturally universal. The works of ornamental art pictured in Sipatsi are among the myriad specimens that abound in books as well as on the Internet. With the electronic versions has come the potential for interactive design. As a result, many questions, beyond the popular reduction of ornamental pattern to group theory, now come to mind:
- What varieties of representations have been used to describe or create ornamental patterns? Surely mathematical formulas or algorithms are involved. In the design of coats-of-arms, a specialized heraldic vocabulary is employed: gules, vair, saltire, etc. A recipe for knitting a pair of socks contains a mix of words, abbreviations, and numbers:
Ndl. 1: k to 3 last, ktog k1
Ndl. 2 – 3: in pattern
Ndl. 4: k1 ssk. K rest
Next round: knit ndl. 1-4. Ndl. 2-3 in pattern
Repeat until 14 st. pr ndl.
- How do artists or designers produce ornaments? Freehand? What rules, stencils, templates, aids, mechanical devices, if any, have been employed? Embedded in the first Jacquard looms (1801) was a punch-card-like "computer." The lace-making machine (1808) of John Heathcoat translated the dynamic geometric arrangement of the threads into a mechanical process. Thanks to computer graphics and simple iterations, fractals, an absolutely new type of design (love 'em or leave 'em), has emerged.
- What have been the social implications of the mechanization of pattern creation? The industrial revolution took off from weaving, and Luddism was its backlash. What are the artistic or economic implications of the hand-made vs. the machine-made? Does not the hand-made enter into the machine-made process through the design or coding phase?
- How widespread is a particular ornament, and what semantic or psychological content does it carry? As a few examples, consider the heart shape, the peace symbol, and the swastika. To what extent is such content culture-dependent? Mathematicians of Pythagorean bent used to (and still do) interpret numbers and visual patterns in mystic ways.
Enough questioning---on to comparing and contrasting. In the villages of Mozambique, the Tonga women, with their sense of beautiful design and their craft traditions, weave and sell useful objects. Thousands of miles away, in Australia‡ (and perhaps elsewhere), mathematicians with their own aesthetic criteria have reduced design to group theory and combinatorics. Such reductions cannot capture the fullness of the aesthetic experience. Yet the visual and the symbolic together create more than either can alone. The full grasp and appreciation of ornamental patterns constitute a multi-semantic and semiotic enterprise.
Metabasis is as old as the hills and has many faces. In academic circles it is called interdisciplinary studies and is now a hot topic. But the question of what to mix and what not to mix abides and is not easily answered.
Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at firstname.lastname@example.org.
*Mark Levi, Princeton University Press, 2009.
†Theology and the Scientific Imagination, Princeton University Press, 1986.
‡Cheryl Praeger, Mathematics and Weaving, 1987.