Bour's minimal surface

Bour's surface.

Bour's surface, leaving out the points with r < 0.5 to show the self-crossings more clearly.

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]

Description[]

Bour's surface crosses itself on three coplanar rays, meeting at equal angles at the origin of the space. The rays partition the surface into six sheets, topologically equivalent to half-planes; three sheets lie in the halfspace above the plane of the rays, and three below. Four of the sheets are mutually tangent along each ray.

Equation[]

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]

{\begin{aligned}x(r,\theta )&=r\cos(\theta )-{\tfrac {1}{2}}r^{2}\cos(2\theta )\\y(r,\theta )&=-r\sin(\theta )(r\cos(\theta )+1)\\z(r,\theta )&={\tfrac {4}{3}}r^{3/2}\cos \left({\tfrac {3}{2}}\theta \right).\end{aligned}}

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.

Properties[]

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 $f(z)=1,g(z)={\sqrt {z}}$ . It was proved by Bour that surfaces in this family are developable onto a surface of revolution.^[3]

References[]

^ O'Connor, John J.; Robertson, Edmund F., "Edmond Bour", MacTutor History of Mathematics archive, University of St Andrews.
^ Weisstein, Eric W. "Bour's Minimal Surface." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/BoursMinimalSurface.html
^ Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny, Minimal Surfaces, Volume 1. Springer 2010

[1] O'Connor, John J.; Robertson, Edmund F., "Edmond Bour", MacTutor History of Mathematics archive, University of St Andrews.

[2] Weisstein, Eric W. "Bour's Minimal Surface." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/BoursMinimalSurface.html

[3] Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny, Minimal Surfaces, Volume 1. Springer 2010

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