Henneberg surface - Alchetron, The Free Social Encyclopedia

In differential geometry, the Henneberg surface is a non-orientable minimal surface 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 )

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

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

References

Henneberg surface Wikipedia

(Text) CC BY-SA