I got the idea to illustrate the three-dimensional point symmetry groups
as pigments --

	noise( U(x,y,z), V(x,y,z), W(x,y,z) )

where U,V,W are independent functions (preferably polynomial) with the
appropriate symmetries.

If the group consists of only n-fold rotation around the z axis (C_n),
no problem:
	U = Re( (x+iy)^n )
	V = Im( (x+iy)^n )
	W = z

If the z plane is reflective (C_nh), no problem:
	W = z^2

If a reflective plane passes through the axis of rotation (C_nv):
	U = Re( (x^2+iy)^n )
	V = Im( (x^2+iy)^n )

But I'm stuck on the dihedral groups without reflection (D_n):
	f(x,y,z) = f(cx-sy,sx+cy,z) = f(x,-y,-z)
where c=cos(2pi/n), s=sin(2pi/n).  Note that I do *not* want
	f(x,y,z) = f(x,-y,z) or f(x,y,-z).

Any ideas?  This is somewhat arcane but I can't believe no one
has faced it before!

The relevant groups:
http://www.uwgb.edu/dutchs/SYMMETRY/3dptgrp.htm

-- 
Anton Sherwood, http://www.ogre.nu

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