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Regular Polyhedra Cube, Tetrahedron, Octahedron, Icosahedron & Dodecahedron from Curious and Interesting Geometry by David Wells

A polyhedra is regular if it has as its faces just one type of regular polygon, and all its vertices are congruent. There are only five: the cube, regular tetrahedron, regular octahedron, regular icosahedron and regular dodecahedron.

The regular polyhedra are called 'Platonic' by tradition, though the last book of Euclid's Elements states, 'In this book, the thirteenth, are constructed the five figures called Platonic, which however do not belong to Plato. Three of these five figures, the cube, pyramid and dodecahedron, belong to the Pythagoreans, while the octahedron and icosahedron belong to Theætetus.'

That the dodecahedron was discovered early is not surprising, since iron pyrites crystals often occur as almost regular dodecahedra and fine examples are found in southern Italy. Artificial dodecahedra have been found in Italy dating from before 500 BC.

As can be seen from the table, the Platonic solids all satisfy Euler's relationship, that

Vertices + Faces = Edges + 2

	Vertices	Faces	Edges
Tetrahedron	4	4	6
Cube	8	6	12
Octahedron	6	8	12
Dodecahedron	20	12	30
Icosahedron	12	20	30