Bour's minimal surface

In mathematics, Bour's minimal surface is a two-dimensional minimal surface, embedded with self-crossings into three-dimensional Euclidean space.

It is named after Edmond Bour, whose work on minimal surfaces won him the 1861 mathematics prize of the French Academy of Sciences.

[1] Bour's surface crosses itself on three coplanar rays, meeting at equal angles at the origin of the space.

The points on the surface may be parameterized in polar coordinates by a pair of numbers (r, θ).

Each such pair corresponds to a point in three dimensions according to the parametric equations[2]

The surface can also be expressed as the solution to a polynomial equation of order 16 in the Cartesian coordinates of the three-dimensional space.

The Weierstrass–Enneper parameterization, a method for turning certain pairs of functions over the complex numbers into minimal surfaces, produces this surface for the two functions