The Kepler–Poinsot Star Polyhedra

In addition to the five convex regular polyhedra (the Platonic Solids) there are four non-convex regular polyhedra. These four solids were discovered by Johannes Kepler (1571-1630) and Louis Poinsot (1777-1859), and were unknown to the ancient world. Three of the Kepler–Poinsot Star Polyhedra are the three stellations of the regular dodecahedron. Namely:

The fourth is one of the many stellations of the icosahedron.

These solids' faces intersect one-another due to their 'star' faces or vertices (hence they are known as star polyhedra). This shown by the fractional '5/2' values in their vertex symbols.