Henneberg surface

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

It has parametric equation

${\displaystyle {\begin{aligned}x(u,v)&=2\cos(v)\sinh(u)-(2/3)\cos(3v)\sinh(3u)\\y(u,v)&=2\sin(v)\sinh(u)+(2/3)\sin(3v)\sinh(3u)\\z(u,v)&=2\cos(2v)\cosh(2u)\end{aligned}}}$

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]