# Platonic Solids

more exPloSIonS and unseen dimensions 49 48 Exploding the rhombic dodecahedron, or its dual the cubocta hedron, results in an equaledged convex polyhedron of fifty faces top right opposite. The exploded rhombic triacontahedron, or exploded icosidodecahedron, has one hundred and twentytwo faces opposite lower right Ludwig Schlfi 18141895 proved that there are six regular four dimensional polytopes generalisations of polyhedra the 5cell made of tetrahedra, the 8cell or tesseract made of cubes, the 16cell made of tetrahedra, the 24cell made of octahedra, the 120cell made of dodecahedra, and the 600cell made of tetrahedra. The rhombic dodecahedron is a threedimensional shadow of the fourdimensional tesseract analogous to the hexagon as a two dimensional shadow of the cube. In a cube two squares meet at every edge. In a tesseract three squares meet at every edge. Squares through the same edge define three cubes, shaded below with an alternative tesseract projection. Schlfi also proved that in five or more dimensions the only regular polytopes are the simplex, or generalised tetrahedron, the hypercube, or generalised cube, and the orthoplex, or generalised octahedron.