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# Tessellations

For centuries, mathematicians and artists alike have been fascinated by tessellations, also called tilings. Tessellations of a plane can be found in the regular patterns of tiles in a bathroom floor, or flagstones in a patio. They are also widely used in the design of fabric or wallpaper. Tessellations of three-dimensional space play an important role in chemistry, where they govern the shapes of crystals.

A tessellation of a plane (or of space) is any subdivision of the plane (or space) into regions or "cells" that border each other exactly, with no gaps in between. The cells are usually assumed to come in a limited number of shapes; in many tessellations, all the cells are identical. Tessellations allow an artist to translate a small motif into a pattern that covers an arbitrarily large area. For mathematicians, tessellations provide one of the simplest examples of symmetry.

In everyday language, the word "symmetric" normally refers to an object with dihedral or mirror symmetry. That is, a mirror can be placed exactly in the middle of the object and the reflection of the mirrored half is the same as the half not mirrored. Such is the case with the leftmost tessellation in the figure. If an imaginary mirror is placed along the axis shown, then every seed-shaped cell, such as the one shown in color, has an identical mirror image on the other side of the axis. Likewise, every diamond-shaped cell has an identical diamond-shaped mirror image.

Mirror symmetry is not the only kind of symmetry present in tessellations. Other kinds include translational symmetry, in which the entire pattern can be shifted; rotational symmetry, in which the pattern can be rotated about a central point; and glide symmetry, in which the pattern can first be reflected and then shifted (translated) along the axis of reflection. Examples of these three kinds of symmetry are shown in the other three blocks of the figure. In each case, the tessellation is called symmetric under a transformation only if that transformation moves every cell to an exactly matching cell.

In the rightmost block of the figure, the tessellation has glide symmetry but does not have mirror symmetry because the mirror images of the shaded cells overlap other cells in the tessellation.

The collection of all the transformations that leave a tessellation unchanged is called its symmetry group. This is the tool that mathematicians traditionally use to classify different types of tilings. The classification of patterns can be further refined according to whether the symmetry group contains translations in one dimension only (a frieze group), in two dimensions (a wallpaper group), or three dimensions (a crystallographic group). Within these categories, different groups can be distinguished by the number and kind of rotations, reflections, and glides that they contain. In total, there are seven different frieze groups, seventeen wallpaper groups, and 230 crystallographic groups.

## Exploring Tessellations

To an artist, the design of a successful pattern involves more than mathematics. Nevertheless, the use of symmetry groups can open the artist's eyes to patterns that would have been hard to discover otherwise. A stunning variety of patterns with different kinds of symmetries can be found in the decorations of tiles at the Alhambra in Spain, built in the thirteenth and fourteenth centuries.

In modern times, the greatest explorer of tessellation art was M. C. Escher. This Dutch artist, who lived from 1898 to 1972, enlivened his woodcuts by turning the cells of the tessellations into whimsical human and animal figures. Playful as they appear, such images were based on a deep study of the seventeen (two-dimensional) wallpaper groups.

One of the most fundamental constraints on classical wallpaper patterns is that their rotational symmetries can only include half-turns, one-third turns, quarter-turns, or one-sixth turns. This constraint is related to the fact that regular triangles, squares, and hexagons fit together to cover the plane, whereas regular pentagons do not.

However, one of the most exciting developments of recent years has been the discovery of simple tessellations that do exhibit five-fold rotational symmetry. The most famous are the Penrose tilings, discovered by English mathematician Roger Penrose in 1974, which use only two shapes, called a "kite" and a "dart." Three-dimensional versions of Penrose tilings* have been found in certain metallic alloys. These "non-periodic tilings," or "quasicrystals," are not traditional wallpaper or crystal patterns because they have no translational symmetries. That is, if the pattern is shifted in any direction and any distance, discrepancies between the original and the shifted patterns appear.

*A Penrose tiling was used to design a mural in the science building of Denison University in Granville, Ohio.

Nevertheless, Penrose tilings have a great deal of long-range structure to them, just as ordinary crystals do. For example, in any Penrose tiling the ratio of the number of kites to the number of darts equals the "golden ratio," 1.618. Mathematicians are still looking for new ways to explain suchpatterns, as well as ways to construct new non-periodic tilings.

Dana Mackenzie

## Bibliography

Beyer, Jinny. Designing Tessellations. Chicago: Contemporary Books, 1999.

Gardner, Martin. Penrose Tiles to Trapdoor Ciphers. New York: W. H. Freeman, 1989.

Grunbaum, Branko, and G. C. Shephard. Tilings and Patterns. New York: W. H. Freeman, 1987.