Tilings and Patterns

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UW mathematician Branko Grunbaum has taken his inspiration from decorative patterns used in arts and crafts to advance basic theories of geometry. Since the early 1980s, Grunbaum and colleague G. C. Shephard of the University of East Anglia, Norwich, England, have pioneered new ways of analyzing the intricate patterns found in tilings and textiles and have elaborated what they call a "theory of patterns."

The results of this work are far-ranging. They have important ramifications for divisions of mathematics including group theory, combinatorics, geometry and topology; they may benefit other technical fields such as crystallography and engineering; and they may find use in the fields of design, art, and anthropology.

Grunbaum and Shephard observe that the beginnings of geometry were, in ancient times, stimulated by practical problems of building, surveying, and decorating. "From the beginnings of civilization, peoples of every culture have manufactured and used objects decorated with repeating geometric patterns. This is still true today and everywhere patterns of many kinds can be seen all around us," they write. "Moreover, repeating patterns frequently arise naturally in many areas of science and engineering, and their investigation has proved a useful tool in such diverse areas as crystallography and the ethnological classification of primitive artifacts." footnote 1

What is surprising, they note, is that there had been relatively little work on these patterns from a theoretical standpoint. The practice of weaving is a typical example. "Weaving is one of the oldest activities of mankind and so it is hardly surprising that there exists a vast literature on the subject. But this literature is almost entirely concerned with the practical aspects of weaving; any treatment of the theoretical problem of designing fabrics with prescribed mathematical properties is conspicuously absent," note the researchers.

In 1980, Grunbaum and Shephard published "Satins and Twills: An Introduction to the Geometry of Fabrics,"footnote 2 which revealed subtle problems in combinatorics and geometry. Traversing uncharted territory, they had to establish new concepts with an entirely new vocabulary to describe textile patterns.

Much the same situation was encountered with repetitive ornamentation. Previous studies had been restricted to analyzing the patterns with the well-established tool of symmetry groups. But Grunbaum found them to be "very coarse tools for the characterization and description of repeating patterns." Much finer classifications were developed by the team; the concepts are elaborated in Grunbaum and Shephard's detailed text, Tilings and Patterns.footnote 3

The researchers were able to elucidate "certain mysterious aspects" of interlace patterns frequently found in Islamic and Moorish art. They verified that, despite the complexity of the designs, most of the interlaces are formed by strands of a small number of shapes, often just a single shape stretching over many repeats of the design. A plausible explanation is that the early artisans used stencils to draw the patterns; for practical reasons, the stencils were made as small as possible for a given pattern, and they may have consisted of just one translational repeat unit.footnote 4

Other work by Grunbaum contributed to the mathematical understanding of aperiodic tilings, in which the constituent units repeat but the pattern lacks symmetry over the long range. This work subsequently became of interest to solid matter physics because of the discovery of actual substances with this type of aperiodic symmetry, called quasicrystals; and it was applied to an analysis of decorations of ancient Peruvian fabrics that are not covered by the usual symmetry groups.

  1. "Patterns," B. Grunbaum and G. C. Shephard, Handbook of Applicable Mathematics, Vol V: Combinatorics and Geometry, Wiley, Chichester, 1985, p. 677.
  2. B. Grunbaum, G. C. Shephard, Mathematics Magazine, 53 (3), 139 (1980).
  3. Branko Grunbaum and G. C. Shephard, Tilings and Patterns, W. H. Freeman and Co., New York 1987.
  4. "Interlace Patterns in Islamic and Moorish Art," B. Grunbaum and G. C. Shephard, Leonardo, 25 (3/4), 331 (1992).

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