Elliptic Reye Configuration

This is a 3-dimensional model of Reye's configuration of 12 points and 16 lines.  This model is "elliptic" in the sense that, while the Reye configuration is projective, and thus lives in real projective 3-space, this is a double cover from the 3-dimensional sphere S^3.  Thus, each point is represented by a pair of antipodally-symmetric points and each line is represented by a great circle in S^3.  More properly, therefore, this represents a stereographic projection of this configuration.

This configuration also serves as a model of the edge skeleton of the regular 24-cell.  That is, this represents the stereographic projection of the image of the central projection of the edges of the 24-cell onto the unit sphere S^3.  The ball represent the vertices of the 24-cell and the 16 circles each represent a regular hexagon of edges in the 24-cell.