Regular Patterns, Tiling the Plane

Introduction

A regular tiling places a fixed number of a specific n-gon at every vertex. For instance, an infinite checkerboard tiles the plane, and since four squares meet at every vertex, the tiling is regular. Other surfaces can be tiled as well. Draw 3 orthogonal circles around the sphere and produce a regular tiling, where four equilateral triangles meet at every vertex. Now for the formal definition.

In a regular tiling of the plane, a rigid rotation or reflection, followed by a translation, maps any vertex p onto any vertex q, and maps all other vertices and edges onto vertices and edges. In other words, all vertices look alike. In addition, any face can be mapped onto any other face; all faces look alike. And any edge can be mapped onto any other edge; all edges look alike. Finally, we will stipulate that corners meet corners, and do not run into edges.

Start with a face f and map one of its edges onto any of the others, hence they all have the same length, which we will call one. Walk around the vertices of f, and they all have the same interior angle. Thus f is a regular n-gon. Since all faces look like f, they are all the same n-gon. Since all vertices look alike, there are k copies of f at each vertex, for some fixed k.

How many such tilings are possible?

Let p be a vertex with k regular n-gons around it. Thus k times the interior angle of the n-gon yields 360°. Turn this around, and 360 divided by the interior angle must be an integer. Recall that the interior angle is 180(n-2)/n, hence 2n/(n-2) must be an integer. When n = 3, six triangles meet at p. When n = 4, four squares meet at p. When n = 6, three hexagons meet at p. For higher values of n, 2n/(n-2) is trapped between 2 and 3.

There are three regular tilings of the plane. You are probably most familiar with the squares, building a checkerboard that goes on forever. Triangles also group themselves into rows, where each row consists of alternating triangles, half pointing up and half pointing down. Rows of triangles stack up to cover the plane. Hexagons also form rows, standing next to each other like soldiers. The rows above and below are shifted by half a hexagon, so that they interlock perfectly.

Additional patterns appear if we allow a vertex to run into a wall. Start with the checkerboard and shift the first row of squares by x. Shift the next row by 2x, the next row by 3x, and so on. Once again there is no way to distinguish one vertex from another, or one face from another. Rows of triangles can be shifted similarly. This presents an infinite continuum of different tilings, so we usually force corners to meet corners.