>>5021010Platonic solids are convex polytopes in R^3. If we generalize to R^n there are many more than five. Just in R^2 we have infinitely many, namely every regular n-gon.
It is fairly easy to prove that there can only be five. This can be seen by noting that we need to have the sum of the angles in every vertex smaller than 360°. Since every face is a congruent n-gon we have only n = 3,4,5 left.
From there you try all possible configurations of vertices. We can arrange triangles in pairs of three, four or five without breaking the 360° rule. There is only one arrangement of squares, and only one arrangement of pentagons.
That's five, and that's all.
