|VW - CVM 1.1|
A natural way to generalize the equation governing negating symmetries , is to ask that a function satisfy:
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
One simple homomorphism from the group p3 is:
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).|