Regular polyhedron

The property of having a similar arrangement of faces around each vertex can be replaced by any of the following equivalent conditions in the definition:

A convex regular polyhedron has all of three related spheres (other polyhedra lack at least one kind) which share its centre:

Any shapes with icosahedral or octahedral symmetry will also contain tetrahedral symmetry.

The five Platonic solids have an Euler characteristic of 2. This simply reflects that the surface is a topological 2-sphere, and so is also true, for example, of any polyhedron which is star-shaped with respect to some interior point.

In a dual pair of polyhedra, the vertices of one polyhedron correspond to the faces of the other, and vice versa.

This concept of a regular polyhedron would remain unchallenged for almost 2000 years.

By the end of the 19th century there were therefore nine regular polyhedra – five convex and four star.

Around the same time as the Pythagoreans, Plato described a theory of matter in which the five elements (earth, air, fire, water and spirit) each comprised tiny copies of one of the five regular solids. Matter was built up from a mixture of these polyhedra, with each substance having different proportions in the mix. Two thousand years later Dalton's atomic theory would show this idea to be along the right lines, though not related directly to the regular solids.

The 20th century saw a succession of generalisations of the idea of a regular polyhedron, leading to several new classes.

These occur as dual pairs in the same way as the original Platonic solids do. Their Euler characteristics are all 1.

The usual nine regular polyhedra can also be represented as spherical tilings (tilings of the sphere):