Statistics In Repeats

Here is a fragment of a tiling made of triangles and squares that will cover the plane. I have indicated two possible repeats of this tiling. You might want to spend a few moments convincing yourself that the two regions really are repeats.

The repeat on the right shows that it consists of two triangles and two squares. So the mean number of edges of a tile in this tiling is (3+3+4+4)/4 = 14/4 = 7/2 sides.

The repeat on the left shows that it contains three vertices, two 5-valent and one 4-valent. So the mean valence of a vertex is (4+5+5)/3 = 14/3.

Summing the reciprocals of these two means shows that

2/7 + 3/14 = 7/14 = 1/2

This is an instance, not a proof, of the assertion that in any periodic tiling

1/(mean valence of each tile) + 1/(mean valence of each vertex) = 1/2