TITLE: Platonic Solids
QUESTION [11 upvotes]: It´s a theorem that there exist only five platonic solids ( up to similarity). I was searching some proofs of this, but I could not. I want to see some proof of this, specially one that uses principally group theory. 
Here´s the definition of Platonic solid Wikipedia Platonic solids

REPLY [2 votes]: This is another proof which is not based on group theory. You may see it as an expanded version of Ross' answer, so read this if you need more details on one of the steps there.
Suppose you have a patonic solid where each face is a regular $n$-gon and at each corner $m$ faces meet. Then you have several constraints:

$m,n\in\mathbb N$ (since they obviously have to be non-gnegative integers)
$n>2$ (to have at least triangles not line segments)
$m>2$ (since for $m=2$ the “solid” would be a pair of polygons glued back to back)
$m\cdot\frac{n-2}{n}\cdot180°<360°$, which means the inner angles at a corner add up to less than one whole turn. You need this to ensure that if you place the faces next to one another, they form a convex cone instead of a flat or even overlapping (or hyperbolic) surface. Laying out the faces in the plane leaves a gap, and closing that gap produces the cone.
This condition can be rewritten to $m<\frac{2n}{n-2}$.

For $n=6$ this last formula would result in $m<\frac{12}4=3$ which together with $m>2$ leaves no valid integer solutions. For $n>6$ this only gets worse. So you can conclude that $n<6$. Then you can start enumerating things:

$n=3\implies2<m<6\implies m\in\{3,4,5\}$ (tetrahedron, octahedron, icosahedron)
$n=4\implies2<m<4\implies m=3$ (cube)
$n=5\implies2<m<\tfrac{10}3\implies m=3$ (dodecahedron)

So there are only very few possible combinations of $m$ and $n$ which satisfy all of the given constraints. Each of them can indeed be realized as a platonic solid, demonstrating that these neccessary conditions are also sufficient. Furthermore, each solid is unique up to similarity, since $m$ and $n$ tell you exactly how to combine the faces according to the rules (in terms of convexity and symmetry) of a platonic solid.