Five Platonic Solids

Theorem
There exist exactly five platonic solids:
 * $\paren 1: \quad$ the regular tetrahedron
 * $\paren 2: \quad$ the cube
 * $\paren 3: \quad$ the regular octahedron
 * $\paren 4: \quad$ the regular dodecahedron
 * $\paren 5: \quad$ the regular icosahedron.