Polyhedra regular convex

There are only five regular convex polyhedra, a very small number indeed. The regular convex polyhedra are called Platonic solids because they constituted an important part of Plato s natural philosophy. They are the tetrahedron, cube (hexahedron), octahedron,... 

Regular convex polyhedra are called Platonic solids because they play an important role in Plato s philosophy. Plato (ca. 428-347), however, was not the first to write about regular polyhedra - the mathematician Theatetus (ca. 380 B.C.), a friend of Plato s and pupil of Socrates (ca. 470-399), discovered the octahedron and the icosahedron and was also the first to write about the five regular polyhedra.256... 

Explicit Euler equations for 3-, 4- and 5-connected polyhedra elegantly show the existence of the 5 regular convex polyhedra and the semi-regular Archimedean and Catalan polyhedra. 

A practical implementation of the above procedure can be obtained by replacing the sphere S by a quasi-regular convex polyhedron with a large number of vertices. The buckminsterfullerene (60 vertices) is a good initial candidate. The implementation of this procedure to analyze the dynamical changes in small enzymes is under development [25]. 

Polyhedra related to the pentagonal dodecahedron and icosahedron In equation (1) for 3-connected polyhedra (p. 62) the coefficient of is zero, suggesting that polyhedra might be formed from simpler 3-connected polyhedra by adding any arbitrary number of 6-gon faces. Although such polyhedra would be consistent with equation (1) it does not follow that it is possible to construct them. The fact that a set of faces is consistent with one of the equations derived from Euler s relation does not necessarily mean that the corresponding convex polyhedron can be made. Three of the Archimedean solids are related in this way to three of the regular solids ... 

As an example, in a cube one has v = 8,e = 12, f = 6, and hence 8 — 12-1-6 = 2. The 2 in the right-hand side of Eq. (6.128) is called the Euler invariant. It is a topological characteristic. Topology draws attention to properties of surfaces, which are not affected when surfaces are stretched or deformed, as one can do with objects made of rubber or clay. Topology is thus not concerned with regular shapes, and in this sense seems to be completely outside our subject of symmetry yet, as we intend to show in this section, there is in fact a deep connection, which also carries over to molecular properties. The surface to which the 2 in the theorem refers is the surface of a sphere. A convex polyhedron is indeed a polyhedron which can be embedded or mapped on the surface of a sphere. Group theory, and in particular the induction of representations, provides the tools to understand this invariant. To this end, each of the terms in the Euler equation is replaced by an induced representation, which is based on the particular nature of the corresponding structural element. In Fig. 6.8 we illustrate the results for the case of the tetrahedron. 

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