Playful as they appear, such images were based on a deep study of the seventeen (two-dimensional) wallpaper groups. 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. In modern times, the greatest explorer of tessellation art was M. 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. Nevertheless, the use of symmetry groups can open the artist's eyes to patterns that would have been hard to discover otherwise. To an artist, the design of a successful pattern involves more than mathematics. In total, there are seven different frieze groups, seventeen wallpaper groups, and 230 crystallographic groups. Within these categories, different groups can be distinguished by the number and kind of rotations, reflections, and glides that they contain. 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). This is the tool that mathematicians traditionally use to classify different types of tilings. The collection of all the transformations that leave a tessellation unchanged is called its symmetry group. 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. In each case, the tessellation is called symmetric under a transformation only if that transformation moves every cell to an exactly matching cell. Examples of these three kinds of symmetry are shown in the other three blocks of the figure. 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. Mirror symmetry is not the only kind of symmetry present in tessellations. Likewise, every diamond-shaped cell has an identical diamond-shaped mirror image. 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. Such is the case with the leftmost tessellation in the figure. 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. In everyday language, the word "symmetric" normally refers to an object with dihedral or mirror symmetry. For mathematicians, tessellations provide one of the simplest examples of symmetry. Tessellations allow an artist to translate a small motif into a pattern that covers an arbitrarily large area. The cells are usually assumed to come in a limited number of shapes in many tessellations, all the cells are identical. Tessellations of three-dimensional space play an important role in chemistry, where they govern the shapes of crystals.Ī 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. They are also widely used in the design of fabric or wallpaper. Tessellations of a plane can be found in the regular patterns of tiles in a bathroom floor, or flagstones in a patio. For centuries, mathematicians and artists alike have been fascinated by tessellations, also called tilings.
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