Schläfli double six

In geometry, the Schläfli double six is a configuration of 30 points and 12 lines in three-dimensional Euclidean space, introduced by Ludwig Schläfli in 1858.

Each of the 12 lines of the configuration contains five intersection points, and each of these 30 intersection points belongs to exactly two lines, one from each subset, so in the notation of configurations the Schläfli double six is written 302125.

[3] An alternative construction, shown in the illustration, is to place twelve lines through the six face centers of a cube, each in the plane of its face and all making the same angles with respect to the cube's edges.

[4] Once constructed in either of these ways, the double six can be projected into the plane, forming a two-dimensional system of points and lines with the same incidence pattern.

A generic cubic surface contains 27 lines, among which can be found 36 Schläfli double six configurations.