Tessellations, Making

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Tessellations, Making

The Dutch artist Maurits Cornelius Escher was, more than anyone else, responsible for bringing the art of tessellations to the public eye. True, tessellations have been used for decorative purposes for centuries, but the cells were usually very bland: squares, equilateral triangles, regular hexagons, or rhombuses . The visual interest of such a tessellation lies in the pattern as a whole, not in the individual cells. But Escher discovered that the cells themselves could become an interesting part of the pattern. His tessellations feature cells in the shape of identifiable figures: a bird, a lizard, or a rider on horseback.

To create an Escher-style tessellation, start with one of the polygons mentioned above. (This is not essential, but it is easiest.) Next, choose two sides and replace them with any curved or polygonal path. The only rule is that anything that is added to the polygon on one side has to be taken away from the other. If a protruding knob has been added to one side, the other side must have an identically shaped dimple.

This procedure can be repeated with other pairs of sides. Usually, the result will vaguely resemble something, perhaps a lizard. The outline can then be revised so that the result becomes more recognizable. But every time one side is modified, it is imperative that its corresponding side is modified in precisely the same way. This makes it nearly impossible to draw a convincing lizard, but by adding a few embellishments—eyes, scales and paws—a reasonable caricature can be produced. Escher, thanks to his years of experience, was a master at this last step.

The figure illustrates the process of making tessellations. Starting with a hexagon, the top and bottom sides are replaced with identical curves that are related by a translation . The upper curve suggests a head, and the bottom curve suggests two heels. Next the top left and top right sides are replaced with two identical curves that are related by a glide reflection . (Note: A translation will only work if the two sides are parallel.) One curve bulges outward and the other inward, in accordance with the rule that anything removed from one side must be given back to the other.

Then the bottom left and bottom right sides are replaced with S-shaped curves that represent the feet and legs. Again, these two curves are identical and related by a glide reflection. Finally, the figure is decorated so that it looks a little bit like a ghost wearing boots. (The complete tessellation is shown here, but obviously Escher did it better.)

Another way of creating interesting tessellations has been called the kaleidoscope method. In this method, one begins with a triangular "primary cell," which may have angles of 60°, −60°, −60°; 30°, −60°, −90°; or 45°, −45°, −90°. After it is decorated in any way, the triangle should be reflected through one side after another until a hexagon or a square is completed. Finally, this figure can be used as a tile to cover the plane.

The kaleidoscope method can be modified by using rotations about the midpoint of a side or about a vertex, instead of reflections , to generate the additional copies of the decorated primary cell. Different patterns of rotation or reflection can yield a dazzling variety of different tessellations, all arising from the same primary cell.