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Polyhedra

  1. R. Bruce King

Published Online: 15 DEC 2011

DOI: 10.1002/9781119951438.eibc0183

Encyclopedia of Inorganic and Bioinorganic Chemistry

Encyclopedia of Inorganic and Bioinorganic Chemistry

How to Cite

King, R. B. 2011. Polyhedra. Encyclopedia of Inorganic and Bioinorganic Chemistry. .

Author Information

  1. University of Georgia, Athens, GA, USA

Publication History

  1. Published Online: 15 DEC 2011

Abstract

The concept of a polyhedron is a useful way of describing diverse chemical structures, including those of coordination compounds and metal clusters. In this context, a polyhedron may be regarded as a set consisting of vertices, edges, and faces. Familiar examples of polyhedra are the five regular (Platonic) polyhedra, that is, the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. The relationship between the numbers of vertices (v), edges (e), and faces (f) in a polyhedron can be described by Euler's theorem, that is, v − e + f = 2. A given polyhedron can be converted to larger polyhedra by capping one or more of its faces. Interchanging the vertices and faces of a polyhedron leads to its dual, which has the same symmetry as the original polyhedron. A projection of a polyhedron onto a plane, known as a Schlegel diagram, is often a useful way of visualizing polyhedra, which supplements the usual perspective drawings. There are no formulas, direct or recursive, for which the number of distinct polyhedra having a given number of vertices, edges, or faces can be calculated.

Keywords:

  • polyhedra;
  • deltahedra;
  • topology;
  • Euler's theorem;
  • Schlegel diagrams