Kusner (Dihedral Symmetric)

Differential Equations, Mechanics, and Computation
minimal surface
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Pseudo-Code for “Kusner's Dihedral Symmetry Surfaces”:

Let p = Max[Round[ee], 2] 
and let 

  z.re = u * Cos[v]
	z.im = u * Sin[v]

Then P[z] = Re[ a[z] * V[z] ] + (0, 0, aa)

where a[z] is the complex number

a[z] = 1/(z^p - z^(-p) + (2/(p-1))*Sqrt[2p - 1])

and V[z] is the complex vector

V[z] = { i ( z^(p-1) - z^(1-p) ) ,  z^(p-1) + z^(1-p) , (i(p-1)/p) ( z^p + z^(-p) ) }

This is a minimal surface with dihedral symmetry of order 2p if p is
odd and 4p if p is even.

The default value of ee is 4.  This gives the inversion in the unit
sphere of the “Morin Sphere Eversion Midpoint” Willmore surface (see
the surface surface menu).  On the other hand, when ee = 3 this gives
the Inverted Boy's Surface (on this menu).

   For full details, see:

R.  Kusner, Conformal Geometry and Complete Minimal Surfaces,
   Bulletin of the AMS, v.17, Number 2, October 1987, pp291—295.