Regular-faced polyhedron

The electron-counting scheme which allows to understand this situation and which also works for the simpler clusters was developed from the MO theory for boron hydride polyhedra 389-393). It rests on the fact that a regular triangular faced polyhedron of n boron atoms in the molecule requires n + 1 bonding... 

Generalized Archimedean polyhedron A polyhedron derived from an Archimedean solid by replacement of one set of regular faces by nonregular ones (e.g., squares by rectangles), but which has the fiiU symmetry of the parent Archimedean figure. 

Octahedron (regular) A polyhedron with eight equal-sized, equilateral triangular faces and six apices (comers). 

Proteins with cyclic or dihedral symmetry are particularly common. More complex rotational symmetries are possible, but only a few are regularly encountered. One example is icosahedral symmetry. An icosahedron is a regular 12-cornered polyhedron having 20 equilateral triangular faces (Fig. 4-24c). Each face can... 

Several different types of reaction patterns may form on the sphere as the reaction proceeds. In cases where the reaction products actually build up on the surface of the metal, as in oxidation, the oxide film will form more rapidly on one face than on another. When the films are in the range of 200 to 2,500 A. and the sphere is examined by placing a tube of white paper over the crystal, the different thicknesses of oxide on the different faces appear as a regular pattern of interference colors of great beauty. The symmetry of one of these highly colored patterns is shown in Fig. 1. In electrodeposition on a crystal sphere at a low current density the metal will deposit more rapidly on one face than another, and the sphere is converted into a polyhedron, or small facets are formed on the different faces which may be seen under the microscope. 

The B12 icosahedron is a regular polyhedron with 12 vertices, 30 edges, and 20 equilateral triangular faces, with B atoms located at the vertices, as shown in Fig. 13.2.2. The B12 icosahedron is a basic structural unit in all isomorphic forms of boron and in some polyhedral boranes such as h has 36... 

From the above diagram, we see why the bonds in SFe are called sp3d2 bonds. Mathematical treatment can predict the most stable way of orienting six bonds formed from hybridization of these orbitals the configuration with bond directions toward the comers of a regular octahedron (a polyhedron with eight faces and six corners) may be shown... 

A solid figure with 12 faces A regular dodecahedron is a regular polyhedron with 12 faces. Each face is a rgular pentagon. 

A polyhedron having 6 faces. The cube is a regular hexahedron, hexomino... 

On the basis of these data and the calculations of Schwartz, Krotov [73] made the conclusion that the number of faces, edges and vertexes of the average polyhedron in foam corresponds to a compact tetradecahedron with an accuracy of 0.5%. This is a hypothetical non-regular polyhedron with 24 vertexes and 36 edges. Krotov proposed a method for calculating the parameters of this body. 

Icosahedral symmetry. The symmetry displayed by a regular polyhedron that is composed of 20 equilateral triangular faces with 12 corners. 

This is a general form of the well-known Gibbs-Thomson (Lord Kelvin) equation applied to the case of electrochemical metal deposition. It gives the size of the critical nucleus and its equilibrium form in terms of the normal distances of the equilibrium form faces from the Wulff point, hi, as a function of the overvoltage. When this form is a regular polyhedron (cr,- = const.), the size of the nucleus can be given by the radius of the inscribed sphere, perit = h, so that... 

Recently quasicrystals having the shape of a triacontahedron have been discovered in specially prepared alloys of aluminum and other metals. A triacontahedron is a regular polyhedron with 30 identical, diamond-shaped faces (Fig. 3.35). Quasicrystals seemingly defy the rules of symmetry that do not allow a periodic structure having unit cells with five-fold symmetry. What is the symmetry of a triacontahedron Can you make a model of it similar to the polyhedra given in Appendix H ... 

---