Planar Enneper

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Planar Enneper surface plotted with increasing range

Planar Enneper surface is made by fitting plane with Enneper Surface.

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Planar Enneper, associate family.
planar enneper assofsc 001
Planar Enneper, associate family.

Planar Enneper with Higher Order Dihedral Symmetry

planar enneper2
Planar Enneper with 2-fold symmetry
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planar enneper associat
The associate family of Planar Enneper consists of congruent surfaces. Observe that the enneper tongues in the middle during their "rotation" also deform the color strips. This shows that they not really rotate but that they keep growing from different parts of the outside planar ring.

Dihedral Symmetry D3

planar enneper3
Planar Enneper with 3-fold symmetry

Dihedral Symmetry D5

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Planar Enneper with 5-fold symmetry

Dihedral Symmetry D8

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Planar Enneper with 8-fold symmetry

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planar enneper st
planar enneper sw
planar enneper sw

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