Color Tiling

In most presentations of tiling, the colors of the tiles are merely decorative or are used to make it easier
to distinguish the tiles from each other.  Here we give a brief introduction to Color Tiling.  Specifically,
given a periodic tiling, how do the transformations of the group of symmetries of the tiling act on the
colors of the tiles? 

If T is a tiling with some finite number of colored tiles, let G be the group of symmetries of T (with colors ignored).
Let P(G) be the subgroup of G that maps every colored tile to a tile of the same color.  P(G) is known as the
color-preserving symmetry group of the colored tiling.

Call a color symmetry of T a symmetry that is compatible with the coloring of T.  That is, if  F is a color symmetry,
and if F maps one blue tile to a green tile, then F will map all blue tiles to green tiles.  Call C(G) the color symmetry
of a colored tiling.

So, for example, one color symmetry of the checkerboard at left is
the one which shifts the picture up by one square.  On the other hand,
one color-preserving symmetry would by a rotation by 180 degrees
about a point centered at any of the square tiles.  The symmetry group
of this (uncolored) tiling is p4m.  For this colored tiling, the color-
preserving symmetry group is also isomorphic to p4m, as is the
color symmetry group.

How many different color symmetry groups are there for a given number
of colors and a given underlying symmetry group?
Color Symmetry Groups




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