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KUMMER SURFACE
Surface studied by Kummer in 1864.
Ernst Kümmer (1810  1893): German mathematician. 
Cartesian equation:
where
so that pqrs = 0 is the equation of the union of the extended faces of a regular tetrahedron centered on O and the edges of which are at distance a from O (cf. figure on the right). Quartic surface. REMARK: the Kummer surface passes through the intersections of these 4 planes and the sphere . Compare to the Klein quartic. 
The Kummer surface is the surface with the above equation;
when ,
it has 16 ordinary singular points (i.e. non degenerate), the maximum number
for a quartic surface; the 16 points are real for .
Opposite, view of the case ,
so ; the
surface is composed of a central "tetrahedron", extended by 4 small "tetrahedra",
themselves linked by 6 infinite sheets, in parallel with the 6 "edges"
of the central tetrahedron.
The 16 singular points are the "vertices" of the 4 small "tetrahedra". 
The case , so , gives the Roman surface; the 6 infinite sheets have disappeared, and so has the central tetrahedron, and the 4 small tetrahedra are joined, which makes 3 double segment lines appear. 

REMARK: the equation of the Kummer surface in a frame
turned by 45° around Oz is
If the coefficients of the homogeneous terms above are
replaced by any coefficients, then we get the general equation of the surfaces
of degree
4 that have the symmetries of the tetrahedron or the cube (see
Goursat
surface).
Furthermore, the terms "Kummer surface" can refer to any quartic surface with 16 ordinary singular points.
See also the Barth sextic, which is the degree 6 equivalent of the Kummer surface.
Compare to the Klein
quartic (curve).
Kummer surface by Patrice Jeener 
Kummer surface by Alain Esculier,

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© Robert FERRÉOL 2017