Tilings with regular polygons are said to be k-uniform if there are precisely k different kinds of vertices in the tiling.
There are 20 tilings with regular polygons that are 2-uniform, that is, have two kinds of vertices. Each is shown twice: once to illustrate the different vertices and again as a colored pattern.
Once we go beyond k=2 the problem of discovering tilings gets more complex. For example, consider the tiling where a row of triangles alternates with three rows of squares. There are two types of vertex: 33344 and 4444. Now consider the tiling with
Mathematician Otto Krotenheerdt enumerated the tilings where there are k distinct types of vertices. That rules out the example above where we have two kinds of 4444 vertices. He found that for each k the number of tilings was;
If we allow tilings where vertices can be the same type as long as they are in topologically different settings, there is no limit to k. Even for k=4 a complete enumeration has not been done, nor is there any way of estimating the number of tilings for large k.
Branko Grunbaum and G.C. Shephard, 1989; Tilings and Patterns, an Introduction, Freeman, 446p. sections 2.1 and 2.2
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Created 2 July. 1999, Last Update 2 July. 1999