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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 )
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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