Solid Geometry Notebook
A Platonic solid is a convex polyhedron built from one kind of regular polygon, arranged identically at every vertex — and there are exactly five of them.
The definition is short but demanding. Each face must be a regular polygon — equal sides, equal angles — all faces must be congruent, and the same number of faces must meet at every vertex. Convexity is assumed throughout: no dents, no spikes.
The five that qualify are the tetrahedron (four triangles), the cube (six squares), the octahedron (eight triangles), the dodecahedron (twelve pentagons), and the icosahedron (twenty triangles). Their face counts run 4, 6, 8, 12, 20, and every one satisfies V − E + F = 2.
Why no sixth exists is a one-line argument: the face angles at each vertex must sum to less than 360°. Triangles admit three, four, or five at a vertex; squares admit three; pentagons admit three; hexagons already sum to exactly 360°, and larger polygons fail faster. The five solids are the complete solution set.
On narrow screens, swipe or scroll the plate sideways.
The solids pair up by duality. Swap faces for vertices and the cube becomes the octahedron, the dodecahedron becomes the icosahedron, and the tetrahedron becomes itself. Dual pairs always share an edge count — twelve for the first pair, thirty for the second.
Their history is long. The solids were studied before Plato, but his dialogue Timaeus attached four of them to the classical elements and the fifth to the heavens, which fixed their name. Euclid's Elements, compiled around 300 BCE, devotes its final book to constructing all five and closing the list.
They still work for a living: dice and game pieces, the tetrahedral bond angles of carbon chemistry, virus capsids approximating icosahedra, and every textbook that needs a clean example of perfect symmetry.
Further reading