# Platonic Solids

31 30 the PoInSot Polyhedra the great dodecahedron great icosahedron Louis Poinsot 17771859 investigated polyhedra independently of Kepler, rediscovering Keplers two icosahedral hedgehogs and also discovering the two polyhedra shown here, the great dodecahedron top and the great icosahedron lower. Both of these polyhedra have five faces to a vertex, intersecting each other to give pentagram vertex figures. The great dodecahedron has twelve pentagonal faces and is the third stellation of the dodecahedron. The great icosahedron has twenty triangular faces and is one of an incredible fiftynine possible stellations of the icosahedron, which also include the compounds of five octahedra and of five and ten tetrahedra. A nonconvex regular polyhedron must have vertices arranged like one of the Platonic Solids. Joining a polyhedrons vertices to form new types of polygon within it is known as faceting. The possibilities of faceting the Platonic Solids produce the compounds of two and ten tetrahedra, the compound of five cubes, the two Poinsot polyhedra below left and the two Kepler star polyhedra below right. The four KeplerPoinsot polyhedra are therefore the only nonconvex regular polyhedra.