Costa's surface

COSTA'S SURFACE

Surface studied by da Costa in 1982.
Celso Jose da Costa (1949-....): Brazilian mathematician.
en.wikipedia.org/wiki/Costa%27s_minimal_surface
mathworld.wolfram.com/CostaMinimalSurface.html

Gray parametrization: , ,
where refers to the Weierstrass zeta function associated to the pair (c,0) (WeierstrassZeta[z, {c, 0}] in Mathematica),
refers to the Weierstrass P function also associated to the pair (c,0)(WeierstrassP[z, {c, 0}] in Mathematica)
c and are two constants associated to the Weierstrass functions equal respectively to 189.7272... and 6.87519....

Until 1982, it was conjectured that the only complete (i.e. without boundary) non-periodic minimal surfaces without self-intersection were: the plane, the catenoid and its associated surfaces. Costa's surface, and other ones afterwards, refuted this conjecture.

It can be obtained by taking in the Weierstrass parametrization of a minimal surface: .

Costa's surface is invariant under the action of a half-turn around the axis x = y, z = 0 (in the above parametrization, y(u,v)=x(v,u) and z(v,u)=-z(u,v)), and under the action of this half-turn, the two faces are swapped.
Opposite, two half surfaces. Check that each one is the image by a half-turn of the other one

Animation of the surface the two faces of which have different colors.
Imagine water poured inside the upper funnel; it will come out by the external face of the bottom funnel...

Costa's surface is topologically equivalent to a torus minus 3 points (it is therefore of genus 1); see an animation of it here.

It is also topologically very close to the cubic algebraic surface with the very simple Cartesian equation: , which is also invariant under the action of a half-turn that swaps the two faces.
Note that the horizontal sections of the latter surface are ellipses or hyperbolas.
Note also that this surface is asymptotic for large x and y to Plücker's conoid .

Costa's surface can be generalized to a surface with order n rotational symmetry, see here.

Compare Costa's surface to the finite Riemann minimal surface, which also has a plane sheet and two flared sheets, but that intersect with each other.

Costa's surface, by Alain Esculier

Costa's surface by Patrice Jeener, with his kind authorization

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