Geometry
Platonic solids
There are exactly five — tetrahedron, cube, octahedron, dodecahedron, icosahedron.
A Platonic solid is a 3D shape where every face is the same regular polygon and the same number of faces meet at every corner. There are exactly five — and that's it. No more are possible.
Spin the solid
Tetrahedron
faces
4
4
edges
6
6
vertices
4
4
V − E + F = 4 − 6 + 4 = 2
drag to rotate
Spin the solid
Cube
faces
6
6
edges
12
12
vertices
8
8
V − E + F = 8 − 12 + 6 = 2
drag to rotate
Spin the solid
Octahedron
faces
8
8
edges
12
12
vertices
6
6
V − E + F = 6 − 12 + 8 = 2
drag to rotate
Spin the solid
Icosahedron
faces
20
20
edges
30
30
vertices
12
12
V − E + F = 12 − 30 + 20 = 2
drag to rotate
The famous five
- Tetrahedron — 4 triangular faces. The fire element to the ancients.
- Cube (hexahedron) — 6 square faces. The earth element.
- Octahedron — 8 triangular faces. The air element.
- Dodecahedron — 12 pentagonal faces. The cosmos itself.
- Icosahedron — 20 triangular faces. The water element.
Why exactly five? At each corner, the angles around the corner have to add to less than 360°. With triangles you can fit 3, 4, or 5 (giving tetra, octa, icosa). With squares only 3 (giving the cube). With pentagons only 3 (the dodecahedron). Hexagons hit 360° flat — no corner possible.