If the number of vertices of a convex polyhedron is v, the number of edges is e, and the number of faces is f, then they always have the relationship: f+ve=2.
From this, it can be concluded that there are only 5 kinds of regular polyhedra: regular tetrahedron, cube, regular octahedron, regular dodecahedron, and regular icosahedron.
For an understanding of a non-rigorous proof, see reference link 1:
Link1: How many kinds of regular polyhedra are there? Actually it can be simpler « Half a bottle of ink – Ren Zhongfang http://www.2maomao.com/blog/zheng-duo-mian-ti/