Classify all regular polyhedra How can we classify all regular polyhedra? I know that there are five regular polyhedra as a hint. Thanks.

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. 
