In differential geometry, the Henneberg surface is a non-orientable minimal surfaceL. Henneberg, Über solche Minimalflächen, welche eine vorgeschriebene ebene Curve zur geodätischen Linie haben, Doctoral Dissertation, Eidgenössisches Polythechikum, Zürich, 1876 named after Lebrecht Henneberg.
It has parametric equation
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 Parabel zur ebenen geodätischen Linie hat, Vierteljahr. Naturforsch. 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
References
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. .