Regular polyhedron
From Wikipedia, the free encyclopedia
A regular polyhedron is a polyhedron whose faces are congruent (all alike) regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-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.
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[edit] The nine regular polyhedra
There are five convex regular polyhedra, known as the Platonic solids:
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Tetrahedron {3, 3} Cube {4, 3} Octahedron {3, 4} Dodecahedron {5, 3} Icosahedron {3, 5}
and four regular star polyhedra, the Kepler-Poinsot polyhedra:
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Small stellated dodecahedron
{5/2, 5}Great stellated dodecahedron
{5/2, 3}Great dodecahedron
{5, 5/2}Great icosahedron
{3, 5/2}
[edit] 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 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. (Cromwell, 1997)
Concentric spheres
A regular polyhedron has all of three related spheres (other polyhedra lack at least one kind) 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 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.
[edit] 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 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 {5, 3} is {3, 5}.
For further information please see the individual articles or the general polyhedron article.
[edit] History
[edit] 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 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.
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[citation needed].
The earliest known written records of these shapes do come from Greek authors, who also gave the first known mathematical description of them.
[edit] 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.
[edit] 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, after Johannes Kepler and Louis Poinsot. Star polygons were first described in the 14th century by Thomas Bradwardine (Cromwell, 1997). Johannes Kepler realised that star polygons could be used to build star polyhedra, which have non-convex regular polygons, typically pentagrams 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 star vertex figures (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) the small stellated dodecahedron 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.
See also Regular polytope: History of discovery.
[edit] 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 (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 viruses 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 fullerenes (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 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 (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.
[edit] References
- Bertrand, J. (1858). Note sur la théorie des polyèdres réguliers, Comptes rendus des séances de l'Académie des Sciences, 46, pp. 79-82.
- Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press, p77. 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.
