Henneberg surface

Henneberg surface.

In differential geometry, the Henneberg surface is a non-orientable minimal surface^[1] named after Lebrecht Henneberg.^[2]

It has parametric equation

{\begin{aligned}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{aligned}}

and can be expressed as an order-15 algebraic surface.^[3] It can be viewed as an immersion of a punctured projective plane.^[4] Up until 1981 it was the only known non-orientable minimal surface.^[5]

The surface contains a semicubical parabola ("Neile's parabola") and can be derived from solving the corresponding Björling problem.^[6]^[7]

References[]

^ L. Henneberg, Über salche minimalfläche, welche eine vorgeschriebene ebene curve sur geodätishen line haben, Doctoral Dissertation, Eidgenössisches Polythechikum, Zürich, 1875
^ Lebrecht Henneberg from the German-language Wikipedia. Retrieved on September 25, 2012.
^ Weisstein, Eric W. "Henneberg's Minimal Surface." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/HennebergsMinimalSurface.html
^ Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny, Minimal Surfaces, Volume 1. Springer 2010
^ 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
^ 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

[1] L. Henneberg, Über salche minimalfläche, welche eine vorgeschriebene ebene curve sur geodätishen line haben, Doctoral Dissertation, Eidgenössisches Polythechikum, Zürich, 1875

[2] Lebrecht Henneberg from the German-language Wikipedia. Retrieved on September 25, 2012.

[3] Weisstein, Eric W. "Henneberg's Minimal Surface." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/HennebergsMinimalSurface.html

[4] Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny, Minimal Surfaces, Volume 1. Springer 2010

[5] 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

[6] L. Henneberg, Über diejenige minimalfläche, welche die Neil'sche Paralee zur ebenen geodätischen line hat, Vierteljschr Natuforsch, Ges. Zürich 21 (1876), 66–70.

[7] 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

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