Henneberg surface

{{Short description|Non-orientable minimal surface}}

File:Henneberg surface.jpg

In differential geometry, the Henneberg surface is a non-orientable minimal surfaceL. Henneberg, Über salche minimalfläche, welche eine vorgeschriebene ebene curve sur geodätishen line haben, Doctoral Dissertation, Eidgenössisches Polythechikum, Zürich, 1875 named after Lebrecht Henneberg.

It has parametric equation

:$\backslash begin\{align\}$

x(u,v) &= 2\cos(v)\sinh(u) - (2/3)\cos(3v)\sinh(3u)\\

y(u,v) &= 2\sin(v)\sinh(u) + (2/3)\sin(3v)\sinh(3u)\\

z(u,v) &= 2\cos(2v)\cosh(2u)

\end{align}

and can be expressed as an order-15 algebraic surface.Weisstein, Eric W. "Henneberg's Minimal Surface." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/HennebergsMinimalSurface.html It can be viewed as an immersion of a punctured projective plane.Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny, Minimal Surfaces, Volume 1. Springer 2010 Up until 1981 it was the only known non-orientable minimal surface.M. Elisa G. G. de Oliveira, Some New Examples of Nonorientable Minimal Surfaces, Proceedings of the American Mathematical Society, Vol. 98, No. 4, Dec., 1986

The surface contains a semicubical parabola ("Neile's parabola") and can be derived from solving the corresponding Björling problem.L. Henneberg, Über diejenige minimalfläche, welche die Neil'sche Paralee zur ebenen geodätischen line hat, Vierteljschr Natuforsch, Ges. Zürich 21 (1876), 66–70.Kai-Wing Fung, Minimal Surfaces as Isotropic Curves in C3: Associated minimal surfaces and the Björling's problem. MIT BA Thesis. 2004 http://ocw.mit.edu/courses/mathematics/18-994-seminar-in-geometry-fall-2004/projects/main1.pdf