The Art of Elegant Tiling
MATHEMATICAL RECREATIONS by Ian Stewart
Scientific American, July 1999
Although the basic mathematics of symmetry and tilings was worked out long ago, new discoveries continue to be made, often by artists. Rosemary Grazebrook, a contemporary British artist, has invented a remarkably simple tiling system that is eminently practical and different enough from the usual rectangular tiles to be interesting. It is also ingenious and, in the right hands, beautiful.
The mathematical definition of symmetry is simple but subtle. A symmetry of a design is a transformation that leaves the design unchanged. For example, the transformation “rotate by 90 degrees” leaves a square unchanged; the transformation “reflect from left to right” leaves the human form (superficially) unchanged. A design may have many different symmetries: together they constitute its symmetry group.
There are also many kinds of tilings. The type that has traditionally attracted the most interest from mathematicians is based on a two-dimensional lattice—in effect, a planar crystal. Ironically, the math here was first worked out in the hugely difficult case of three dimensions and only much later carried through in two dimensions. In 1891 Russian crystallographer E. S. Fedorov proved that lattices in the plane fall into 17 distinct symmetry types. The same goes for wallpaper designs and textile patterns. It may seem strange to say this when any home improvement store can show you dozens of thick books of wallpaper samples and rack after rack of tiles. In most cases, however, the differences lie in such features as color, texture and the nature of the underlying design elements. Important as these are to the customer, they do not affect the symmetry of the pattern, except that they may be constrained by it. For instance, square bathroom tiles bearing an image of a duck will have the same symmetry as similar tiles with the image of a length of seaweed—unless extra symmetry occurs in the images themselves.
Some patterns do not possess any great degree of symmetry, and these I shall ignore here. Among them are important modern discoveries such as the famous Penrose tiles, which completely cover the plane but never repeat exactly the same arrangement. The patterns of concern here are based around one “fundamental region”—a design that is repeated indefinitely in two independent directions. For example, imagine an array of standard square tiles, as seen in so many bathrooms. Our imaginary bathroom, however, has infinitely large walls, so the pattern never stops. Pick some tile. The pattern of that tile repeats in both the horizontal and vertical directions and in combinations of those. In fact, if you displace the tile by any whole number of tile widths horizontally, to the left or the right, and then by any whole number of tile widths vertically, up or down, you’ll find an identical tile. So the pattern repeats in two distinct directions. Here those directions happen to be at right angles to each other, but this is not a general requirement.
The existence of two such directions is what we mean by a lattice. Lattice symmetry is natural for wallpaper and textiles because they are usually made by forming a long roll of material along which the same pattern repeats over and over again—perhaps printed by a revolving drum or woven by a machine that repeats a fixed loop. When the paper is stuck to a wall or if the material is sewn together to cover a wider region, it is usual to match the pattern along the join. But this matching may involve what interior decorators call a “drop”: you slide the paper sideways and then up or down by some amount. If there is a drop, then the lattice repeats along two directions that are not at right angles.
The lattice condition is less natural for tiles, which are made individually, but it is an easy scheme for an artist to follow when placing them on a wall or a floor. The square bathroom-tile lattice, for example, has rotational symmetries through 90 degrees. It also has reflectional symmetries about vertical, horizontal and diagonal lines that pass through the center or vertex of each tile or through the midpoint of each tile edge. A “honeycomb” tiling by regular hexagons is also a lattice, but it has different symmetries, notably rotations through 60 degrees. For a more detailed discussion of lattice patterns, see Symmetry in Chaos, by Michael Field and Martin Golubitsky (Oxford University Press, 1992).
Grazebrook discovered that a particular pentagonal tile can be the building block of a multitude of lattice patterns. A key feature of the tile is that it has two angles of 90 degrees and three of 120 degrees, allowing the tiles to be arranged in both square and hexagonal lattices. A square tile, in contrast, has only 90-degree angles, so it can form just a few distinct lattices. Four of Grazebrook’s pentagonal tiles can be fitted together to make a wide, short hexagon, which tiles the plane like bricks in a wall. When the pentagonal tiles are augmented with regular hexagons, they can form all but one of the 17 symmetry types of lattice patterns. (I leave readers the pleasure of discovering which is the missing symmetry type and how to obtain the other 16.)
Grazebrook first got the idea for these tiles from this very column—or, more accurately, from its predecessor, Martin Gardner’s inimitable Mathematical Games column. She was studying for a Ph.D. at London’s Royal College of Art, focusing on the Islamic art at the Alhambra. She started a dissertation entitled “From Islam to Escher and Onwards ...”. (Readers are probably familiar with the remarkable drawings of M. C. Escher, many of which use animal shapes as tiles, arranged in mathematical patterns.) Grazebrook sensed a connection between Islamic art and Escher’s characteristic tiling patterns, but only after reading Gardner’s column did she realize that the link is the theory of the 17 lattice symmetry types. From that point on, she began to explore ways to make Islamic patterns using various lattice-based grids.
Grazebrook introduced two distinct schemes for coloring her pentagonal tiles. One scheme divides the tile into three triangles: this is called the “Pentland” set. The other coloring scheme divides the pentagon into four regions: two squares, one kite-shaped quadrilateral and a smaller pentagon. This is the “Penthouse” set. Of course, it is possible to divide and color the tiles in many other ways, but these sets alone can form an amazing variety of designs.