The Five Platonic Solids

Plato believed the universe was built from five perfect solids — the tetrahedron (fire), cube (earth), octahedron (air), icosahedron (water), and dodecahedron (cosmos). Euclid proved, in the final proposition of the Elements, that only five such solids can exist: polyhedra where every face is the same regular polygon and every vertex is identical. The proof is one of the most beautiful in mathematics — an infinite space contains exactly five perfect objects, and we can prove there are no others.