Regular polyhedron

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A regular polyhedron is a polyhedron whose faces are identical (or, technically, congruent) regular polygons and which has the same number of faces around each vertex. Regular polyhedra are edge-transitive, vertex-transitive and face-transitive.

The second condition in the definition (same number of faces around each vertex) is necessary in order to distinguish the regular polyhedra from other non-regular deltahedra. It can be replaced by any of the following equivalent conditions:

• 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)

Contents 1 The nine regular polyhedra 2 Duality of the regular polyhedra 3 Regular polyhedra in nature 4 References 5 See also

[edit] The nine regular polyhedra

There are five convex regular polyhedra, known as the Platonic solids:

Main article: Platonic solid

and four regular star polyhedra, the Kepler-Poinsot polyhedra:

Main article: Kepler-Poinsot polyhedra

[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 Kepler-Poinsot polyhedra also come in dual pairs.

For further information please see the individual articles or the general polyhedron article.

[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, Neptune and Pluto, have invalidated the Pythagorean idea.

[edit] References

• Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press, p77. ISBN 0-521-66405-5. 
• (Curl, 1991) Curl, R. F.; Smalley, R. E.; Fullerenes, Scientific American 265 4 (1991) pp32–41.
• (Haeckel, 1904) Haeckel, E.; Kunstformen der Natur (1904). 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, 1982) Smith, J. V.; Geometrical And Structural Crystallography, (John Wiley and Sons, 1982).

[edit] See also

• Polytopes
  • List of regular polytopes
  • Regular polytope
• Polyhedra
  • Hosohedron
  • Semiregular polyhedron
  • Uniform polyhedron