TILINGS BY REGULAR POLYGONS III: DODECAGON-DENSE TILINGS

In Tilings and Patterns, Grunbaum and Shephard claim that there are only four k- uniform tilings by regular polygons (for some k) that have a dodecagon incident at every vertex. In fact, there are many others. We show that the tilings that satisfy this requirement are either the uniform 4.6.12 tiling, or else fall into one of two infinite classes of such tilings. One of these infinite classes can be fully characterized, while the other can be shown to be equivalent to the class of all tilings by squares and equilateral triangles; i.e. a largely unconstrained infinite class. This characterization is, however, sufficiently powerful to determine all such k-uniform tilings for k ≤ 14.