There are only five regular convex polyhedra that can exist in our three dimensional Euclidean space. These polyhedra were known since the ancient times. Plato (427-348 BC) wrote about them in his work Timaeus (written in about 340 BC) to identify five principles upon which everything is modelled: he identified these principles as fire, earth, air, water, and cosmos (or divine force). He identified each with the elementary principle

Tetrahedron - fire

Cube - earth

Octahedron - air

Icosahedron - water

Dodecahedron - cosmos or divine force

Euclid's Elements (c. 300 BC) list all these solids in mathematical way, giving their properties in the last book, book XIII.

Around the same time Archimedes was working on Platonic solids and discovered new polyhedra which are now called Archimedean polyhedra. The difference between platonic and Archimedean polyhedra is that, while the former are regular polyhedra containing only regular and same faces, the latter contain two or more different types of faces (which are also regular polygons).

Have a look at the Platonic polyhedra below.

cube

tetrahedron

octahedron

icosahedron

dodecahedron