classification of Platonic solids


The regular tetrahedronMathworldPlanetmathPlanetmathPlanetmath, regular octahedron, regular icosahedron, cube, and regular dodecahedron are the only Platonic solids.


Each vertex of a Platonic solid is incidentMathworldPlanetmathPlanetmath with at least three faces. The interior anglesMathworldPlanetmath incident with that vertex must sum to less than 2π, for otherwise the solid would be flat at that vertex. Since all faces of the solid have the same number of sides, this implies bounds on the number of faces which could meet at a vertex.

The interior angle of an equilateral triangleMathworldPlanetmath has measure π3, so a Platonic solid could only have three, four, or five triangles meeting at each vertex. By similarMathworldPlanetmath reasoning, a Platonic solid could only have three squares or three pentagonsMathworldPlanetmath meeting at each vertex. But the interior angle of a regular hexagon has measure 2π3. To avoid flatness a solid with hexagonsMathworldPlanetmath as faces would thus have to have only two faces meeting at each vertex, which is impossible. For polygonsMathworldPlanetmathPlanetmath with more sides it only gets worse.

Since a Platonic solid is uniquely determined by the number and kind of faces meeting at each vertex, there are at most five Platonic solids, with the numbers and kinds of faces listed above. But these correspond to the five known Platonic solids. Hence there are exactly five Platonic solids. ∎

Title classification of Platonic solids
Canonical name ClassificationOfPlatonicSolids
Date of creation 2013-03-22 15:53:07
Last modified on 2013-03-22 15:53:07
Owner mps (409)
Last modified by mps (409)
Numerical id 5
Author mps (409)
Entry type Result
Classification msc 51-00