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COSTA'S SURFACE
Surface studied by da Costa in 1982.
Celso Jose da Costa (1949....): Brazilian mathematician. https://en.wikipedia.org/wiki/Costa%27s_minimal_surface http://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) nonperiodic minimal surfaces without selfintersection 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 halfturn 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 halfturn, the two faces are swapped. Opposite, two half surfaces. Check that each one is the image by a halfturn 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 halfturn 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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© Robert FERRÉOL 2017