Henneberg

Henneberg is a minimal surface, given by a parametrisation, 1 end, self-intersecting.

Open Henneberg in the interactive viewer →

aperiodic def-parametric implemented minimal non-orientable noncompact self-intersecting tradition-classical

About

A minimal surface that is not orientable -- it contains a Möbius band -- which for a long time was thought impossible for a minimal surface. It is also the only classical example that is algebraic, and it contains a pair of straight lines meeting at right angles.

Formula

x(u, v)

2 \sinh\left(u\right) \cos\left(v\right) - \frac{2}{3} \sinh\left(3 u\right) \cos\left(3 v\right)

y(u, v)

2 \sinh\left(u\right) \sin\left(v\right) + \frac{2}{3} \sinh\left(3 u\right) \sin\left(3 v\right)

z(u, v)

2 \cosh\left(2 u\right) \cos\left(2 v\right)

Properties

Family: minimal
Given by: a parametrisation
Curvature: zero mean curvature
Periodicity: not periodic
Ends: 1
Embedding: self-intersecting
Orientable: no
Fidelity: exact
Exactness: elementary

Sources

Chart as curated in tools/surfdb/charts.py, verified numerically against the curvature condition the record claims (measured over the chart by the validator and the charts self-test).

L. Henneberg, 'Uber salche Minimalflachen, welche eine vorgeschriebene ebene Curve zur geodatischen Linie haben' (1875).
E. L. Henneberg, 'Ueber diejenige Minimalflaeche, welche die Neil'sche Parabel zur ebenen geodaetischen Linie hat' (1876); M. Weber (mirror: minsurf posts ch384): parametrizes a projective plane, 'the puncture at 0 represents an Enneper end'.