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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> 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 )

Are you sure this works? I would rather suggest
(here the reflection is at the xz-plane):

     U = Re( (x+iy)^n )
     V = Im( (x+iy)^n )^2

> 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!

I am not completely sure that I understand what you want.
If your solution    for rotation and reflection works, then so should:

     U = Re( (x+iyz)^n )
     V = Im( (x+iyz)^n )
     W = (y+z)^2

Otherwise how about:

     U = Re( (x+iy)^n )
     V = z*Im( (x+iy)^n )
     W = (y+z)^2


-- 
merge{#local i=-11;#while(i<11)#local
i=i+.1;sphere{<i*(i*i*(.05-i*i*(4e-7*i*i+3e-4))-3)10*sin(i)30>.5}#end
pigment{rgbt 1}interior{media{emission x}}hollow}//  Mark Weyer

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