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© Copyright 2003, Jim Loy
Here are the
three regular tesselations (tilings) of the plane (the patterns shown can be
repeated to fill the entire plane). See Regular
Solids. Here, regular means that the polygons are regular and identical,
and that each vertex joins up with vertices (not edges) identically. As you can
see, the regular pentagon is missing, as it cannot tile the plane. Similarly,
regular polygons of 7, 8, 9, . . . sides cannot tile the plane.
The arrangement of bricks in a wall would be more or less regular.
Here is a tesselation that is even less regular. Each triangle being a
different size. This one is not drawn to any systematic scheme. Such a tiling
would be similar to a fractal. I assume that some such scheme could be devised.
Of course, a tiling can be much more irregular than that.
There are eight "regular tilings of the plane by two or more convex regular polygons such that the same polygons in the same order surround each vertex" (quoted from MathWorld). See below. Such a tiling is called a semi-regular tiling. To clarify the definition, we have vertices next to vertices and edges next to edges. No vertex is adjacent to another polygon's edge.
Of
course, any triangle tiles the plane. So does any quadrilateral (left),
including concave quadrilaterals. Although a regular pentagon cannot tile the
plane, there are irregular pentagons which can tile the plane (one is shown on
the right).
The above diagrams were drawn with the program Cinderella.