Classify all regular polyhedra How can we classify all regular polyhedra? I know that there are five regular polyhedra as a hint. Thanks.
 A: Since your polyhedron is regular, all faces are regular $n$-gons with some $n\in\mathbb{N}$, $3\leq n$. Look at the $k \geq 3$ faces meeting at an vertex (because of regularity, the choice of the vertex doesn't matter). Each of the $k$ faces has an internal angle of $(\pi - 2\pi/n)/2 = \pi(1/2 - 1/n) = \pi(n-2)/n$. Since your polyhedron is convex, you get $k \pi(n-2)/n < 2\pi$, or equivalently $k < 2n/(n-2)$. There are five solutions to these conditions on $n,k$: $$(n,k)\in\{(3,3),(3,4),(3,5),(4,3),(5,3)\}$$
Each solution corresponds to one of the $5$ Platonian solids. 
A: If you ask about "all" regular polyhedra, the answer must depend on what you mean by a "polyhedron". In Euclidean space there are five regular convex polyhedra, the Platonic solids, and four regular stars, the Kepler-Poinsot polyhedra. There are also a good many infinite polyhedra or apeirohedra known as the Grünbaum-Dress polyhedra.
All these regular polyhedra, bar one subsequently discovered by Dress, are described in Grünbaum, B.; "Regular polyhedra - old and new", Aequationes Math. Vol 16, 1977, pp.1-20.
For a more elegant treatment confirming Dress' list, see McMullen, P. and Schulte, E. "Regular Polytopes in Ordinary Space", Discrete & Computational Geometry, Vol 17, No 4, June 1997, pp 449–478.
In other kinds of space there are many more, including regular projective polyhedra (such as the hemicube) and regular hyperbolic polyhedra.
