Pictures of Platonic Solids Why only five Platonic Solids?

DodecahedronNumber of faces: 12Number of edges: 30Number of vertices: 20

TetrahedronNumber of faces: 4Number of edges: 6Number of vertices: 4

CubeNumber of faces: 6Number of edges: 12Number of vertices: 8

OctahedronNumber of faces: 8Number of edges: 12Number of vertices: 6

IcosahedronNumber of faces: 20Number of edges: 30Number of vertices: 12

A Platonic solid is a polyhedron all of whose faces are congruent regular convex polygons*, and where the same number of faces meet at every vertex.

The Greeks recognized that there are only five platonic solids. But why is this so? The key observation is that the interior angles of the polygons meeting at a vertex of a polyhedron add to less than 360 degrees. Tetrahedron: Three triangels at a vertex: 3*60 = 180 degreesOctahedron: Four triangles at a vertex: 4*60 = 240 degreesIcosahedron:Five triangles at a vertex: 5*60 = 300 degrees Cube: Three squares at a vertex: 3*90 = 270 degreesDodecahedron: Three pentagons at a vertex: 3*108 = 324 degrees Note:   Six triangles: 6*60 = 360 degrees  Four squares: 4*90 = 360 degrees  Four pentagons: 4*108 = 432 degrees  Three hexagons: 3*120 = 360 degrees So there are only five Platonic Solids!  *) Regular means that the sides of the polygon are all the same length. Congruent means that the polygons are all the same size

Pictures of Archimedean Solids CuboctahedronNumber of faces: 14Number of edges: 24Number of vertices: 12

IcosidodecahedronNumber of faces: 32Number of edges: 60Number of vertices: 30

Truncated TetrahedronNumber of faces: 8Number of edges: 18Number of vertices: 12

Truncated OctahedronNumber of faces: 14Number of edges: 36Number of vertices: 24

Truncated CubeNumber of faces: 14Number of edges: 36Number of vertices: 24

RhombicuboctahedronNumber of faces: 26Number of edges: 48Number of vertices: 24

Truncated CuboctahedronNumber of faces: 26Number of edges: 72Number of vertices: 48

Truncated IcosidodecahedronNumber of faces: 62Number of edges: 180Number of vertices: 120

Snub CubeNumber of faces: 38Number of edges: 60Number of vertices: 24

Snub DodecahedronNumber of faces: 92Number of edges: 150Number of vertices: 60

Truncated Icosahedron(football)Number of faces: 32Number of edges: 90Number of vertices: 60

Truncated DodecahedronNumber of faces: 32Number of edges: 90Number of vertices: 60

RhombicosidodecahedronNumber of faces: 62Number of edges: 120Number of vertices

Pictures of Kepler-Poinsot Polyhedra Great IcosahedronNumber of faces: 20Number of edges: 30Number of vertices: 12

Great DodecahedronNumber of faces: 12Number of edges: 30Number of vertices: 12

Small Stellated DodecahedronNumber of faces: 12Number of edges: 30Number of vertices: 12

Great Stellated DodecahedronNumber of faces: 12Number of edges: 30Number of

kepler

Archimedes

Rhombicosidodecahedron Truncated Dodecahedron Truncated Icosahedron

Truncated Octahedron Truncated Cube Rhombicuboctahedron

Snub Cube Snub Cube Truncated Cuboctahedron

Truncated Tetrahedron Icosidodecahedron Cuboctahedron

Platonik

Dodecahedron Tetrahedron Octahedron

Icosahedron cube