Platonic solids
Five shapes. The only convex regular polyhedra that can exist in three dimensions — proven by Euclid in Book XIII of the Elements. Drag to rotate, scroll to zoom, and watch one solid morph into another.
V − E + F = 2 → Euler’s polyhedron formula
Why only five?
A Platonic solid requires identical regular polygonal faces meeting at identical vertices. The angle constraint at each vertex — at least three faces, total angle less than 360° — limits the possibilities to exactly five: tetrahedron, cube, octahedron, dodecahedron, and icosahedron. This was known to the ancient Greeks and proved rigorously by Euclid.
Duality
Every Platonic solid has a dual: swap vertices and face-centers. The cube and octahedron are duals of each other. The dodecahedron and icosahedron are duals. The tetrahedron is self-dual. Duality swaps V and F while preserving E.
Euler’s formula
For any convex polyhedron, V − E + F = 2. This topological invariant holds for all five Platonic solids and is one of the most beautiful results in mathematics. It was discovered by Euler in 1752 and is the foundation of algebraic topology.