Henneberg Surface
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)
The Henneberg surface, plotted with increasingly larger domain.
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.
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
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.
The Moebius band on the surface demonstrates that the surface is not orientable
(it has only one side).
Henneberg in wireframe, rotating.
Henneberg moebius cross-eyed stereogram.
CatalanHennebergScherk.pdf

3DXM Minimal Surfaces