Elliptic Clebsch Cubic

This represents the 27 lines upon Clebsch's diagonal cubic surface.  This realization is unusual because it represents the "elliptic" model, which is a double cover of a real section of the surface.  Thus, every line is represented by a great circle in the 3-dimensional sphere, and intersecting lines intersect in pairs of antipodal points.  Since this represents a stereographic projection from the 3-sphere, all the angles between lines are preserved.  For example, at each Eckardt point one sees 3 circles (i.e lines) mutually intersecting at 60 degrees.