Henneberg surface

Henneberg surface.

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

It has parametric equation

x ( u , v ) = 2 cos ⁡ ( v ) sinh ⁡ ( u ) − ( 2 / 3 ) cos ⁡ ( 3 v ) sinh ⁡ ( 3 u ) y ( u , v ) = 2 sin ⁡ ( v ) sinh ⁡ ( u ) + ( 2 / 3 ) sin ⁡ ( 3 v ) sinh ⁡ ( 3 u ) z ( u , v ) = 2 cos ⁡ ( 2 v ) cosh ⁡ ( 2 u ) {\displaystyle {\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.[2] It can be viewed as an immersion of a punctured projective plane.[3] Up until 1981 it was the only known non-orientable minimal surface.[4]

The surface contains a semicubical parabola ("Neile's parabola") and can be derived from solving the corresponding Björling problem.[5][6]

## References[edit]

1. ^ L. Henneberg, Über salche minimalfläche, welche eine vorgeschriebene ebene curve sur geodätishen line haben, Doctoral Dissertation, Eidgenössisches Polythechikum, Zürich, 1875
2. ^ Weisstein, Eric W. "Henneberg's Minimal Surface." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/HennebergsMinimalSurface.html
3. ^ Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny, Minimal Surfaces, Volume 1. Springer 2010
4. ^ 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
5. ^ L. Henneberg, Über diejenige minimalfläche, welche die Neil'sche Paralee zur ebenen geodätischen line hat, Vierteljschr Natuforsch, Ges. Zürich 21 (1876), 66–70.
6. ^ 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

## Further reading[edit]

* E. Güler; Ö. Kişi; C. Konaxis, Implicit equations of the Henneberg-type minimal surface in the four-dimensional Euclidean space. Mathematics 6(12), (2018) 279. doi:10.3390/math6120279.
* E. Güler; V. Zambak, Henneberg's algebraic surfaces in Minkowski 3-space. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 68(2), (2019) 1761–1773. doi:10.31801/cfsuasmas.444554.

* v
* t
* e

Minimal surfaces

* Associate family
* Bour's
* Catalan's
* Catenoid
* Chen–Gackstatter
* Costa's
* Enneper
* Gyroid
* Helicoid
* Henneberg
* k-noid
* Lidinoid
* Neovius
* Richmond
* Riemann's
* Saddle tower
* Scherk
* Schwarz
* Triply periodic

*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template