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A regular polyhedron is a Polyhedron whose faces are Regular Polygons all alike ( Congruent ) and are assembled in the same way around each vertex. A regular polyhedron is Edge-transitive , Vertex-transitive and Face-transitive - i.e. it is '''transitive on its Flag s'''. 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:
Concentric spheres A regular polyhedron has all of three related spheres (other polyhedra lack at least one kind) which share its centre:
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:
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):
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 at Oxford University . It is impossible to know why these objects were made, or how the sculptor gained the inspiration for them. 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 1800s of a Dodecahedron made of Soapstone , and dating back more than 2,500 years (Lindemann, 1987). Pyritohedric Crystals are found in northern Italy. 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) 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'' . 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:
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 and Great Stellated Dodecahedron , and (Poinsot's) the 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 , 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 (Ref needed). 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 the Regular Icosahedron nor the Regular Dodecahedron are amongst them, although one of the forms, called the Pyritohedron , 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 of Radiolaria , 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 many Virus 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 the Fullerene s (see (Curl, 1991) for an exposition of this discovery). 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. In ancient times the Pythagorean s believed that there was a harmony between the regular polyhedra and the orbits of the Planet s. 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 (excluding the earth) were known, nicely matching the number of Platonic solids. Kepler's work, and the discovery since that time of Uranus , Neptune and Pluto , have invalidated the Pythagorean idea. REFERENCES |
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