Double Enneper

Double k fold Enneper 001
Double k fold Enneper
double2fold enneper 001
double 2-fold enneper 001
double 4 fold enneper
double 4 fold enneper
double 6 fold enneper
double 6 fold enneper
double enneper sw
double enneper sw
double enneper st
double enneper st

The surfaces Wavy Enneper, Catenoid Enneper, Planar Enneper, and Double Enneper are finite total curvature minimal immersions of the once or twice punctured sphere — shown with standard polar coordinates. These surfaces illustrate how the different types of ends can be combined in a simple way.

The pure Enneper surfaces (Gauss(z)=z^k, k=ee+1) and the Planar Enneper have been re-discovered many times, because the members of the associate family are CONGRUENT surfaces (as can be seen in an associate family morphing) and the Weierstrass integrals integrate to polynomial (respectively) rational immersions.

Double Enneper was one of the early examples in which I joined two classical surfaces by a handle.

Formulas are taken from:

For a discussion of techniques for creating minimal surfaces with various qualitative features by appropriate choices of Weierstrass data, see:

Related Surface of Enneper

  1. Enneper Surface
  2. Double Enneper
  3. Planar Enneper
  4. Wavy Enneper
  5. Catenoid-Enneper

Double_k-fold_Enneper.pdf