The Reye configuration is a configuration of 12 planes and 12 points such that six points lie in every plane and six planes pass
through every point. Alternatively, the configuration consists of 16 lines and the
same 12 points such that four lines pass through every point and three points lie
on every line.

The points consist of the eight vertices of a cube together with its center and the three points at infinity
where parallel edges of the cube meet. The 12 planes are
the six faces of the cube and the six planes passing through diagonally opposite
edges. The 16 lines consist of the 12 edges and four space diagonals of the cube.

The Reye configuration can be realized without any points at infinity by squashing the cube and bringing the points at infinity to finite positions, as illustrated above.