# Kepler-Poinsot solids

The four Kepler-Poinsot solids. Wenniger, Magnus J. *Polyhedron Models for the Classroom*. NCTM
1966. p. 11.

The Kepler-Poinsot solids are four regular non-convex polyhedra that exist in addition to the five regular convex polyhedra known as the Platonic solids. As with the Platonic solids, the Kepler-Poinsot solids have identical regular polygons for all their faces, and the same number of faces meet at each vertex. What is new is that we allow for a notion of "going around twice," which results in faces that intersect each other.

In the **great stellated dodecahedron** and the **small
stellated dodecahedron**, the faces are pentagrams (five-pointed
stars). The center of each pentagram is hidden inside the polyhedron. These
two polyhedra were described by Johannes Kepler in 1619, and he deserves credit for first understanding them mathematically,
though a 16th century drawing by the Nuremberg goldsmith Wentzel Jamnitzer
(1508–1585) is very similar to the former and a 15th century mosaic
attributed to the Florentine artist Paolo Uccello (1397–1475) illustrates
the latter.

The **great icosahedron** and **great dodecahedron** were described by Louis Poinsot in 1809,
though Jamnitzer made a picture of the great dodecahedron in 1568. In these
the faces (20 triangles and 12 pentagons, respectively) which meet at each
vertex "go around twice" and intersect each other, in a manner that is the
three-dimensional analog to what happens in two-dimensions with a pentagram.
Together, the Platonic solids and these Kepler-Poinsot polyhedra form the
set of nine regular polyhedra. Augustin Cauchy first proved that no other polyhedra can exist with identical regular faces
and identical regular vertices.