The Clebsch Diagonal Surface

The Clebsch Diagonal Surface is one of the most famous cubic surfaces because of its symmetry and the fact that it's the only one with ten Eckardt Points.

From its equation x^3+y^3+z^3+w^3 - (x+y+z+w)^3, it is easy to see, that this cubic is invariant under permutations on 5 letters (by setting t:=-(x+y+z+w)). This is the reason for the symmetry of the set of the corresponding six points in the plane. From the picture below the surface (where those points and the lines and conics connecting them are drawn) one sees immediately, that there are ten Eckardt Points (points, where three lines meet in a point).

A bibliography of Alfred Clebsch.

The Cayley Cubic

The Cayley Cubic is very famous, too, because of its symmetry and the fact that it contains four double points (which is the maximum number for any cubic surface).

Its equation is given by 4(x^3+y^3+z^3+w^3) - (x+y+z+w)^3. From the picture below the surface, one sees immediately, that there are four double points on the surface (each one corresponds to a set of three points on a line in the plane).

The fact, that the 15 lines between the six points in the plane and the six conics (through five of the six points) degenerate into seven lines corresponds to the fact, that on the surface, lots of lines coincide, too.

A bibliography of Arthur Cayley.