Henneberg surface explained

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

It has parametric equation

\begin{align} 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) \end{align} 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]}

Further reading

• 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. .
• 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. .

Notes and References

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