Editing Regular polyhedron (section)

==History==<!-- This section is linked from [[Polyhedron]] -->

{{see also|Regular polytope#History of discovery}}

=== 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.<ref>[http://www.neverendingbooks.org/the-scottish-solids-hoax The Scottish Solids Hoax].</ref>

It is also possible that the [[Etruscan civilization|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).

=== 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 (mathematician)|Theaetetus]] (an [[Athens|Athenian]]) was the first to give a mathematical description of all five (Van der Waerden, 1954), (Euclid, book XIII). [[H.S.M. 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 (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 ([[Convex polygon|convex]]) planar figure with all edges equal and all corners equal.
*A regular polyhedron is a solid (convex) 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]] (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).

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

=== Regular star polyhedra ===

Regular star polygons such as the [[pentagram]] (star pentagon) 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 polyhedron|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 figure]]s (circuits around each corner), 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: (Kepler's) [[small stellated dodecahedron]] and [[great stellated dodecahedron]], and (Poinsot's) [[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]] (or faceting). Every stellation of one polyhedron is [[Dual polyhedron|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.