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
number

of colors and a given underlying symmetry group?

Color Symmetry Groups

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