Regular polyhedron

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.

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.

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.

[3] 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).

When these solids were all discovered and by whom is not known, but Theaetetus (an Athenian) was the first to give a mathematical description of all five (Van der Waerden, 1954), (Euclid, book XIII).

Coxeter (Coxeter, 1948, Section 1.9) credits Plato (400 BC) 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.

One might characterise the Greek definition as follows: This definition rules out, for example, the square pyramid (since although all the faces are regular, the square base is not congruent to the triangular sides), or the shape formed by joining two tetrahedra together (since although all faces of that triangular bipyramid would be equilateral triangles, that is, congruent and regular, some vertices have 3 triangles and others have 4).

It was not until the early 17th century that Johannes Kepler realised that pentagrams could be used as the faces of regular star polyhedra.

Although C60, the most easily produced fullerene, looks more or less spherical, some of the larger varieties (such as C240, C480 and C960) are hypothesised to take on the form of slightly rounded icosahedra, a few nanometres across.

[4] In the early 20th century, Ernst Haeckel described a number of species of radiolarians, some of whose shells are shaped like various regular polyhedra.

[5] Examples include Circoporus octahedrus, Circogonia icosahedra, Lithocubus geometricus and Circorrhegma dodecahedra; the shapes of these creatures are indicated by their names.

In Kepler's time only five planets (excluding the earth) were known, nicely matching the number of Platonic solids.

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.

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.

In the first decades, Coxeter and Petrie allowed "saddle" vertices with alternating ridges and valleys, enabling them to construct three infinite folded surfaces which they called regular skew polyhedra.

In H3 hyperbolic space, paracompact regular honeycombs have Euclidean tiling facets and vertex figures that act like finite polyhedra.

If the tiling is properly scaled, it will close as an asymptotic limit at a single ideal point.

On a spherical surface, the regular polyhedron {2, n} is represented as n abutting lunes, with interior angles of 2π/n.

[13] A regular dihedron, {n, 2}[13] (2-hedron) in three-dimensional Euclidean space can be considered a degenerate prism consisting of two (planar) n-sided polygons connected "back-to-back", so that the resulting object has no depth, analogously to how a digon can be constructed with two line segments.