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Surface studied by Barth in 1994.
Wolf Barth: German mathematician.
For examples of surfaces having the maximum possible number of ordinary double points
and the state of knowledge on these surfaces:

Cartesian equation:  where  (golden ratio) and ;
is the equation of the reunion of the 6 planes containing the edges of an icosidodecahedron (cf. figures on the right).
Sextic surface.
Cartesian equation in a frame where one of the planes is the plane z = 0, and another one is the plane z = 2x:

The Barth sextic is the surface with the above equation; its essential characteristic is to have 65 real ordinary (i.e. non-degenerate) double points, which is the maximum possible number for a sextic surface; only 50 of them are at finite distance, the 15 others being at infinity.
The surface is composed of 20 small "tetrahedra" "placed" on the 20 triangular faces of an icosidodecahedron; 3 vertices of each tetrahedron are connected to another tetrahedron (which makes 20*3/2= 30 double points) and the other vertex is connected to an infinite surface (which makes another 20 double points).

Since it has the symmetries of the dodecahedron, it is a dodecahedric surface of Goursat.

It is to the degree 6 what the Kummer quartic is to the degree 4, or the Togliatti quintic is to the degree 5.

Barth sextic by Patrice Jeener

Barth surface by Alain Esculier.
The 50 double points at finite distance are represented along the four circles, intersections with the 6 planes pi=0, qi=0, ri=0, and traced on the sphere x²+y²+y² = a², spherical projection of the edges of an icosidodecahedron.

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© Robert FERRÉOL  2017