Henneberg Surface

The Henneberg surface was the first known non-orientable minimal surface, first discovered by 1875, by Lebrecht Henneberg.

Until the 1981 it was the only known non-orientable minimal surface.

The Henneberg surface is topologically equivalent to the Cross-Cap Surface, thus is a representation of projective plane.

henneberg patch rot 001
The Henneberg surface, rotating.

The Henneberg surface was often studied due to its branch points (i.e. derivative of the parametrization is zero). Branch points make visualizations of the surface difficult.

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 crosscut 001
The henneberg surface, cut across the selfintersection segment, shows, how the surface behaves near the selfintersection segment. The angle of selfintersection is 90 degrees in the middle and decreases to zero towards the endpoints of this segment, i.e. towards the branch points. One can also see that, without this segment, the surface has two sides, colored in blue and orange. But along the selfintersection segment the blue side and the orange side meet: the henneberg surface is onesided or non-orientable. Note that the points (u,v) and (-u, v + π) in the parameter domain have the same image point on the surface. The points (0,v), 0 ≤ v ≤ 2 π, are mapped 2-1 onto the selfintersection segment.
henneberg far 001
Far away view of the Henneberg surface. When looked far-away, it looks like 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 equator is mapped to the self-intersection segment between the two branch points. The animation ends with cut-away views to see the self-intersection better.
henneberg openup 001
The Henneberg surface as image of the half sphere without the equator is two-sided: we can color one side blue, the other orange. Along the self-intersection 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 image of the projective plane, so it is topologically the same as Cross-Cap Surface .
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 wf 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

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