A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.
A regular polyhedron is identified by its Schläfli symbol of the form { n, m}, where n is the number of sides of each face and m the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra (the Platonic solids), and four regular star polyhedra (the Kepler–Poinsot polyhedra), making nine regular polyhedra in all. In addition, there are five regular compounds of the regular polyhedra.
The regular polyhedra
There are five convex regular polyhedra, known as the Platonic solids; four regular star polyhedra, the Kepler–Poinsot polyhedra; and five regular compounds of regular polyhedra:
Platonic solids
Kepler–Poinsot polyhedra
Regular compounds
Characteristics
Equivalent properties
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:
The vertices of a convex regular polyhedron all lie on a sphere. All the dihedral angles of the polyhedron are equal All the vertex figures of the polyhedron are regular polygons. All the solid angles of the polyhedron are congruent.
Concentric spheres
A convex regular polyhedron has all of three related spheres (other polyhedra lack at least one kind) which share its centre: An insphere, tangent to all faces. An intersphere or midsphere, tangent to all edges. A circumsphere, tangent to all vertices.
Symmetry
The regular polyhedra are the most symmetrical of all the polyhedra. They lie in just three symmetry groups, which are named after the Platonic solids: Tetrahedral Octahedral (or cubic) Icosahedral (or dodecahedral)
Any shapes with icosahedral or octahedral symmetry will also contain tetrahedral symmetry.
Euler characteristic
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.
Interior points
The sum of the distances from any point in the interior of a regular polyhedron to the sides is independent of the location of the point (this is an extension of Viviani's theorem.) However, the converse does not hold, not even for tetrahedra.
Duality of the regular polyhedra
In a dual pair of polyhedra, the vertices of one polyhedron correspond to the faces of the other, and vice versa.
The regular polyhedra show this duality as follows: The tetrahedron is self-dual, i.e. it pairs with itself. The cube and octahedron are dual to each other. The icosahedron and dodecahedron are dual to each other. The small stellated dodecahedron and great dodecahedron are dual to each other. The great stellated dodecahedron and great icosahedron are dual to each other.
The Schläfli symbol of the dual is just the original written backwards, for example the dual of { 5, 3} is { 3, 5}.
History
Prehistory
Stones carved in shapes resembling clusters of spheres or knobs have been found in Scotland and may be as much as 4,000 years old. Some of these stones show not only the symmetries of the five Platonic solids, but also some of the relations of duality amongst them (that is, that the centres of the faces of the cube gives the vertices of an octahedron). Examples of these stones are on display in the John Evans room of the Ashmolean Museum at Oxford University. Why these objects were made, or how their creators gained the inspiration for them, is a mystery. There is doubt regarding the mathematical interpretation of these objects, as many have non-platonic forms, and perhaps only one has been found to be a true icosahedron, as opposed to a reinterpretation of the icosahedron dual, the dodecahedron.
It is also possible that the Etruscans preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery near Padua (in Northern Italy) in the late 19th century of a dodecahedron made of soapstone, and dating back more than 2,500 years (Lindemann, 1987).