CLEBSCH (DIAGONAL CUBIC) SURFACE

In yellow the 15 diagonals with the Eckardt points, in red and blue the principal double six.

 Surface studied by Clebsch in 1871. Alfred Clebsch (1833-1872): German mathematician.

 Pentahedral equation in : . Smooth cubic surface with hyperbolic points.

The Clebsch surface is the surface given by the above equation.
It is, up to homography, the only cubic surface for which the group of homography under the action of which it is invariant is the group S5 of permutations of five objects (cf. the invariance under the 120 permutations of the coordinates  ).

It has the distinctive feature that its 27 lines (that all smooth cubic surfaces have) are all real.
These 27 lines are divided into two groups:

 1) the 15 so-called diagonal or hyperbolic lines  with ; the reason for this name being that these lines are the diagonals of the pentahedron the faces of which are the 5 tritangent Sylvester planes:  and the vertices (points common to 3 faces) of which are the 10 so-called Eckardt points; the line , joining two Eckardt points  and , is indeed one of the 3 diagonals of the complete quadrilateral cut on the face Fm by the 4 other faces. The three diagonals ,  and  passing by  are coplanar (they join  to the points  aligned on the edge ); their plane  is one of the 10 tritangent Eckardt planes.  Every Eckardt point  is common to 4 Eckardt planes . Face Fm of the Sylvester pentahedron with the 6 Eckardt points and the three diagonals included in the cubic. General view of the pentahedron.

The Clebsch surface is, up to homography, the only cubic surface to have 10 triple Eckardt points.

 2)  The 12 lines of the principal double six   with  , {k, l} = {1,2,3,4} \ {i, j}, where  is the golden ratio, positive solution of . In the complex space, these lines have a very simple definition: they are the lines joining the point with homogeneous coordinates  to its conjugate , where  is the set of the 5 fifth roots of 1 (with the normalization u1 = 1, we indeed find the 12 possibilities); for example, the line  corresponds to  where .

It can be proved that these twelve lines form the so-called principal "double six"

(there are 35 others). Two lines are on the same line of the double six if the permutation changing the indices of one to the indices of the other is even.

Here are various affine representations of this surface:

 The change of coordinates  gives the  Homogeneous equation: , hence the affine equation: . This representation has the inconvenient that the plane  and the three lines included in it are at infinity. here, the diagonals in blue, and the double six in red.

Affine representation chosen by Clebsch and Klein in 1872 for the model created by the sculptor Adolf Weiler:

 The change of coordinates defined by , i.e.  gives for k = 4, l= 3 (chosen for aesthetic reasons) the Homogeneous equation:  Associated affine Cartesian equation:  i.e. . This transform was chosen so that the Euclidian surface obtained has an order 3 rotation symmetry around Oz; the first 4 Sylvester planes form a isosceles tetrahedron with base F4, and F5 is parallel to F4, so that the Eckardt points S12, S23, and S31, are at infinity.

Here are representations using this equation: notice the parallel lines intersecting at an Eckardt point at infinity.

 In yellow, the 15 diagonals with the Eckardt points, in red and blue, the principal double six; notice the tetrahedron formed by the first 4 Sylvester planes. Sculpture at the cafeteria of the university Heinrich Heine of Düsseldorf.

Here is a rational transform such that the first 4 Sylvester planes form a regular tetrahedron:

 The change of coordinates defined by , i.e. gives the Homogeneous equation:  Associated affine Cartesian equation:  i.e. . The first 4 Sylvester planes form the regular tetrahedron with vertices the Eckardt points , the other 3 Eckardt points at finite distance being the middle points of the edges passing by A45: . The points  are at infinity. here, the diagonals in blue, and the double six in red.

Here is a series of views of a representation for which the 10 Eckardt points are at finite distance: