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

Description

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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

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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.

Properties

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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  . It was proved by Bour that surfaces in this family are developable onto a surface of revolution.[3]

References

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  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