Henneberg Surface

henneberg patch rot 001
The Henneberg surface, rotating.

The Henneberg surface was studied a lot because of its branch points (derivative of the representation is zero). branch points make visualizations of the surface difficult. Until the 1980's it was the only non-orientable minimal surface.

A parametrization:

  x = 2 * sinh(u) * cos(v) - 2 * sinh(3 * u) * cos(3 * v) / 3
  y = 2 * sinh(u) * sin(v) - 2 * sinh(3 * u) * sin(3 * v) / 3
  z = 2 * cosh(2 * u) * cos(2 * v)
henneberg domain grow 001
The Henneberg surface, plotted with increasingly larger domain.
henneberg far 001
The Henneberg surface, computed far out and looked at from far away, looks like an Enneper Surface. View the surface as an image of the (Riemann) sphere, then: antipodal points have the same image so that any halfsphere is mapped to the full Henneberg surface. The standard equator is mapped to the selfintersection segment between the two branch points. The animation ends with cut-away views to see the selfintersection better.
henneberg openup 001
The Henneberg surface as image of the halfsphere without the equator is two-sided: we can color one side blue, the other orange. Along the selfintersection segment the blue side connects up with the orange side so that the complete Enneper surface is only one-sided. The Henneberg surface is an immage of the projective plane, it can be viewed as image of the Cross-Cap Surface .
henneberg surface hole
henneberg surface hole
henneberg conjugate
The conjugate of the Henneberg surface is twice as large, because the Henneberg surface maps opposite points of the (Riemann) sphere to the same point in R^3 and the conjugate surface does not do that.
henneberg moebius ana
The Moebius band on the surface demonstrates that the surface is not orientable (it has only one side).
henneberg wireframe rot 001
Henneberg in wireframe, rotating.
henneberg moebius cross
Henneberg moebius cross-eyed stereogram.

Related Surface

  1. Catalan Surface
  2. Henneberg Surface
  3. Scherk Surface
  4. Saddle Tower

CatalanHennebergScherk.pdf