VW - CVM 1.1

# Color-turning wallpaper

A natural way to generalize the equation governing negating symmetries , is to ask that a function satisfy:

f(gx) = P(g) f(x)

for all g in G, where P is a homomorphism from a wallpaper group G to some group that acts on the range of f.

For real-valued f we used the set {1, -1} for the target of P. If f is complex-valued, we can use the group of nth roots of unity.

One simple homomorphism from the group p3 is:

 P(t1) = 1 P(R3) = e2pi/3

We constructed the following function using two simple wallpaper waves, along with the technique of group averaging.

A wallpaper with group p3 whose colors are interchanged by certain rotations of 120 degrees.

We call it "Fish" because when you rotate through 120 degrees about certain points, the yellow fish-shape turns into the blue fish, which turns into the magenta fish, and so on. Still, this is not the simplest color-turning wallpaper, because it has color-preserving half-turns in addition to the color-turning 3-centers. A more basic example was computed recently:

A wallpaper with color-interchanging 3-centers, but no color-preserving half-turns.

The black borders on this one occur because the values of the wallpaper function become very high and are thus colored black.

A wallpaper whose function values get very large (black).

Communications in Visual Mathematics, vol 1, no 1, August 1998.