# Kusner Surface

On the one hand this minimal surface is so complicated that we can only show images of 4 different smaller parts of the surface. On the other hand, when the surface is described by a map, then the domain is very simple: a sphere with 8 points missing (4 of these so called 'punctures' are on the northern half sphere and 4 on the southern half.

Pseudo-Code for “Kusner’s Dihedral Symmetry Surfaces”:

Let p = max(Round(ee), 2) and let

```R(z) = u cos(v) I(z) = u sin(v)```

Then `P(z) = R(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 iseven. 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 Surface.

For full details, see: [R. Kusner, Conformal Geometry and Complete Minimal Surfaces, Bulletin of the AMS, v.17, Number 2, October 1987, pp291--295.]