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
group 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
of colors and a given underlying symmetry group?
Color Symmetry Groups
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