Published Online: 15 DEC 2011
Copyright © 2011 John Wiley & Sons, Ltd. All rights reserved.
Encyclopedia of Inorganic and Bioinorganic Chemistry
How to Cite
King, R. B. 2011. Polyhedra. Encyclopedia of Inorganic and Bioinorganic Chemistry. .
- Published Online: 15 DEC 2011
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.
- Euler's theorem;
- Schlegel diagrams