- Regular polyhedron
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**regular polyhedron**is apolyhedron whose faces are congruent (all alike)regular polygons which are assembled in the same way around eachvertex . A regular polyhedron is highly symmetrical, being all ofedge-transitive ,vertex-transitive andface-transitive - i.e. it is transitive on its flags. This last alone is a sufficient definition.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.**The nine regular polyhedra**There are five convex regular polyhedra, known as the

:Platonic solid s:

and four regular star polyhedra, the

:Kepler-Poinsot polyhedra :

**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 the polyhedron all lie on a

sphere .

*All thedihedral angle s of the polyhedron are equal.

*All thevertex figure s of the polyhedron areregular polygon s.

*All thesolid angle s of the polyhedron are congruent. (Cromwell, 1997)**Concentric spheres**A regular polyhedron has all of three related spheres (other polyhedra lack at least one kind) which share its centre:

* Aninsphere , tangent to all faces.

* An intersphere ormidsphere , tangent to all edges.

* Acircumsphere , tangent to all vertices.**Symmetry**The regular polyhedra are the most symmetrical of all the polyhedra. They lie in just three

symmetry group s, which are named after them:

*Tetrahedral

*Octahedral (or cubic)

*Icosahedral (or dodecahedral)**Euler characteristic**The five Platonic solids have an

Euler characteristic of 2. Some of the regular stars have a different value.**Duality of the regular polyhedra**The regular polyhedra come in natural pairs, with each twin being dual to the other (i.e. the vertices of one polyhedron correspond to the faces of the other, and vice versa):

* Thetetrahedron is self dual, i.e. it pairs with itself.

* Thecube andoctahedron are dual to each other.

* Theicosahedron anddodecahedron are dual to each other.

* Thesmall stellated dodecahedron andgreat dodecahedron are dual to each other.

* Thegreat stellated dodecahedron andgreat 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}.

For further information please see the individual articles or the general

polyhedron article.**History****Prehistory**Stones carved in shapes showing the symmetry of all five of the Platonic solids have been found in

Scotland and may be as much as 4,000 years old. These stones show not only the form of each of the five Platonic solids, but also the relations of duality amongst them (that is, that the centres of the faces of the cube gives the vertices of an octahedron, and so on). Examples of these stones are on display in the [*http://www.ashmol.ox.ac.uk/ash/guide/t-text/room29.html John Evans room*] of theAshmolean Museum atOxford University . Why these objects were made, or how their creators gained the inspiration for them, is a mystery.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 NorthernItaly ) in the late 1800s of adodecahedron made ofsoapstone , and dating back more than 2,500 years (Lindemann, 1987). Pyritohedric crystals are found in northern ItalyFact|date=February 2007.The earliest known "written" records of these shapes do come from Greek authors, who also gave the first known mathematical description of them.

**Greeks**The Greeks were the first to make "written" records of the regular Platonic solids. Some authors (Sanford, 1930) credit

Pythagoras (550 BC) with being familiar with them all, whereas others indicate that he may only have been familiar with the tetrahedron, cube, and dodecahedron, crediting the discovery of the other two to Theaetetus (an Athenian), who in any case gave a mathematical description of all five (Van der Waerden, 1954), (Euclid, book XIII).H.S.M. Coxeter (Coxeter, 1948, Section 1.9) creditsPlato (400 BC) with having made models of them, and mentions that one of the earlierPythagoreans ,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". It is from Plato's name that the term "Platonic solids" is derived.**Regular star polyhedra**For almost 2000 years, the concept of a regular polyhedron remained as developed by the ancient Greek mathematicians. One might characterise the Greek definition as follows:

*A regular polygon is a (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 would be equilateral triangles, that is, congruent and regular, some vertices have 3 triangles and others have 4).However, in addition to the Platonic solids, the modern definition of regular polyhedra also includes the regular star polyhedra, otherwise known as the

Kepler-Poinsot polyhedra , afterJohannes Kepler andLouis Poinsot . Star polygons were first described in the 14th century byThomas Bradwardine (Cromwell, 1997). Johannes Kepler realised that star polygons could be used to build star polyhedra, which have non-convex regular polygons, typicallypentagram s as faces. Some of these star polyhedra may have been discovered by others before Kepler's time, but he was the first to recognise that they could be considered "regular" if one removed the restriction that regular polyhedra be convex. Later, Poinsot realised that starvertex figure s (circuits around each corner) can also be used, and discovered the remaining two star polyhedra. Cayley gave them English names which have become accepted. They are: (Kepler's) thesmall stellated dodecahedron andgreat stellated dodecahedron , and (Poinsot's) thegreat icosahedron andgreat dodecahedron .The Kepler-Poinsot polyhedra may be constructed from the Platonic solids by a process called

stellation . The reciprocal process to stellation is calledfacetting (or faceting). 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.**See also**.**Regular polyhedra in nature**Each of the Platonic solids occurs naturally in one form or another.

The tetrahedron, cube, and octahedron all occur as

crystal s. These by no means exhaust the numbers of possible forms of crystals (Smith, 1982, p212), of which there are 48. Neither theregular icosahedron nor theregular dodecahedron are amongst them, although one of the forms, called thepyritohedron , has twelve pentagonal faces arranged in the same pattern as the faces of the regular dodecahedron. The faces of the pyritohedron are, however, not regular, so the pyritohedron is also not regular.Polyhedra appear in biology as well. In the early 20th century,

Ernst Haeckel described a number of species ofRadiolaria , some of whose skeletons are shaped like various regular polyhedra. (Haeckel, 1904) 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 manyvirus es form regular polyhedra. For example,HIV is enclosed in a regular icosahedron.A more recent discovery is of a series of new types of

carbon molecule, known as thefullerene s (see (Curl, 1991) for an exposition of this discovery). Although C_{60}, the most easily produced fullerene, looks more or less spherical, some of the larger varieties (such as C_{240}, C_{480}and C_{960}) are hypothesised to take on the form of slightly rounded icosahedra, a few nanometres across.In ancient times the

Pythagorean s believed that there was a harmony between the regular polyhedra and the orbits of theplanet s. In the 17th century,Johannes Kepler studied data on planetary motion compiled byTycho 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 (excluding the earth) 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.**References***Bertrand, J. (1858). Note sur la théorie des polyèdres réguliers, "Comptes rendus des séances de l'Académie des Sciences",

**46**, pp. 79-82.

*cite book

last = Cromwell | first = Peter R. | title = Polyhedra | publisher = Cambridge University Press | date = 1997 | pages = p77 | id = ISBN 0-521-66405-5

*Haeckel, E. (1904). "Kunstformen der Natur ". Available as Haeckel, E. "Art forms in nature", Prestel USA (1998), ISBN 3-7913-1990-6, or online at http://caliban.mpiz-koeln.mpg.de/~stueber/haeckel/kunstformen/natur.html

*Smith, J. V. (1982). "Geometrical And Structural Crystallography". John Wiley and Sons.

* Sommerville, D. M. Y. (1930). "An Introduction to the Geometry of n Dimensions." E. P. Dutton, New York. (Dover Publications edition, 1958). Chapter X: The Regular Polytopes.**ee also***

Polytope s

**List of regular polytopes

**Regular polytope

* Polyhedra

**Hosohedron

**Semiregular polyhedron

**Uniform polyhedron **External References*** [

*http://www.slyman.org/right_projects_math.php Angle between surfaces of a regular polyhedron*]

* [*http://mathworld.wolfram.com/RegularPolyhedron.html Regular Polyhedron*]

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