Regular polyhedron

A regular polyhedron is a polyhedron with regular and congruent polygons as faces. Its 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, 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, and four regular star 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

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Tetrahedron	Cube	Octahedron	Dodecahedron	Icosahedron
χ = 2	χ = 2	χ = 2	χ = 2	χ = 2

Kepler–Poinsot polyhedra

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Small stellated dodecahedron	Great dodecahedron	Great stellated dodecahedron	Great icosahedron
χ = −6	χ = −6	χ = 2	χ = 2

Regular compounds

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Two tetrahedra 2	Five tetrahedra 5	Ten tetrahedra 10	Five cubes 5	Five octahedra 5
χ = 4	χ = 10	χ = 0	χ = −10	χ = 10

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 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
Icosahedral

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 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 is.

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. 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 the late 19th century of a dodecahedron made of soapstone, and dating back more than 2,500 years.

Greeks

The earliest known written records of the regular convex solids originated from Classical Greece. When these solids were all discovered and by whom is not known, but Theaetetus was the first to give a mathematical description of all five,. H.S.M. Coxeter credits Plato with having made models of them, and mentions that one of the earlier Pythagoreans, Timaeus of Locri, used all five in a correspondence between the polyhedra and the nature of the universe as it was then perceived – this correspondence is recorded in Plato's dialogue Timaeus. Euclid's reference to Plato led to their common description as the Platonic solids.
One might characterise the Greek definition as follows:

A regular polygon is a planar figure with all edges equal and all corners equal.
A regular polyhedron is a solid figure with all faces being congruent regular polygons, the same number arranged all alike around each vertex.

This definition rules out, for example, the square pyramid, or the shape formed by joining two tetrahedra together.
This concept of a regular polyhedron would remain unchallenged for almost 2000 years.

Regular star polyhedra

Regular star polygons such as the pentagram were also known to the ancient Greeks – the pentagram was used by the Pythagoreans as their secret sign, but they did not use them to construct polyhedra. It was not until the early 17th century that Johannes Kepler realised that pentagrams could be used as the faces of regular star polyhedra. Some of these star polyhedra may have been discovered by others before Kepler's time, but Kepler was the first to recognise that they could be considered "regular" if one removed the restriction that regular polyhedra be convex. Two hundred years later Louis Poinsot also allowed star vertex figures, enabling him to discover two new regular star polyhedra along with rediscovering Kepler's. These four are the only regular star polyhedra, and have come to be known as the Kepler–Poinsot polyhedra. It was not until the mid-19th century, several decades after Poinsot published, that Cayley gave them their modern English names: small stellated dodecahedron and great stellated dodecahedron, and great icosahedron and great dodecahedron.
The Kepler–Poinsot polyhedra may be constructed from the Platonic solids by a process called stellation. The reciprocal process to stellation is called facetting. Every stellation of one polyhedron is dual, or reciprocal, to some facetting of the dual polyhedron. The regular star polyhedra can also be obtained by facetting the Platonic solids. This was first done by Bertrand around the same time that Cayley named them.
By the end of the 19th century there were therefore nine regular polyhedra – five convex and four star.

Regular polyhedra in nature

Each of the Platonic solids occurs naturally in one form or another.
The tetrahedron, cube, and octahedron all occur as crystals. These by no means exhaust the numbers of possible forms of crystals, of which there are 48. Neither the regular icosahedron nor the regular dodecahedron are amongst them, but crystals can have the shape of a pyritohedron, which is visually almost indistinguishable from a regular dodecahedron. Truly icosahedral crystals may be formed by quasicrystalline materials which are very rare in nature but can be produced in a laboratory.
A more recent discovery is of a series of new types of carbon molecule, known as the fullerenes. Although C₆₀, the most easily produced fullerene, looks more or less spherical, some of the larger varieties are hypothesised to take on the form of slightly rounded icosahedra, a few nanometres across.
Regular polyhedra appear in biology as well. The coccolithophore Braarudosphaera bigelowii has a regular dodecahedral structure, about 10 micrometres across. In the early 20th century, Ernst Haeckel described a number of species of radiolarians, some of whose shells are shaped like various regular polyhedra. Examples include Circoporus octahedrus, Circogonia icosahedra, Lithocubus geometricus and Circorrhegma dodecahedra; the shapes of these creatures are indicated by their names. The outer protein shells of many viruses form regular polyhedra. For example, HIV is enclosed in a regular icosahedron, as is the head of a typical myovirus.
In ancient times the Pythagoreans believed that there was a harmony between the regular polyhedra and the orbits of the planets. In the 17th century, Johannes Kepler studied data on planetary motion compiled by Tycho Brahe and for a decade tried to establish the Pythagorean ideal by finding a match between the sizes of the polyhedra and the sizes of the planets' orbits. His search failed in its original objective, but out of this research came Kepler's discoveries of the Kepler solids as regular polytopes, the realisation that the orbits of planets are not circles, and the laws of planetary motion for which he is now famous. In Kepler's time only five planets were known, nicely matching the number of Platonic solids. Kepler's work, and the discovery since that time of Uranus and Neptune, have invalidated the Pythagorean idea.
Around the same time as the Pythagoreans, Plato described a theory of matter in which the five elements 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.