Triangular faces, equilateral

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... 

The mineral fluorite, CaF2, in Figure 13-5 has a cubic crystal structure and often cleaves to form nearly perfect octahedra (eight-sided solids with equilateral triangular faces). Depending on impurities, the mineral takes on a variety of colors and may fluoresce when irradiated with an ultraviolet lamp. 

FIGURE 19.10 One equilateral triangular face of a tetrahedral P4 molecule showing the 60° bond angles and the 90° angles between the p orbitals. The relatively poor orbital overlap in the bent bonds accounts for the high reactivity of white phosphorus. 

Figure 2.13. The dodecahedron and the icosahedron are two of the five Platonic solids (regular polyhedra), the others being the tetrahedron, the cube, and the octahedron, (a) The dodecahedron has twelve regular pentagonal faces with three pentagonal faces meeting at a point, (b) The icosahedron has twenty equilateral triangular faces, with five of these meeting at a point.

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... 

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

From the fact that the smallest kind of interstice between spheres in contact is a tetrahedral hole it follows that we should expect to find coordination polyhedra with only triangular faces, in contrast to those in, for example, cubic closest packing which have square in addition to triangular faces. Moreover, it seems likely that the preferred coordination polyhedra will be those in which five or six triangular faces (and hence five or six edges) meet at each vertex, since the faces are then most nearly equilateral. It follows from Euler s relation (p. 61) that for such a polyhedron, 1)5 -h Oug = 12, where 1)5 and are the numbers of vertices at which five or six edges meet, so that starting from the icosahedron (vg = 12) we may add 6-fold vertices to form polyhedra with more than twelve vertices. 

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

The octahedron is one of the regular polyhedra. It has eight equilateral triangular faces, twelve edges, and six corners. 

The regular icosahedron, the fourth of the regular polyhedra, has twenty equilateral triangular faces, thirty edges, and twelve corners. Its name is derived from the Greek eikosi, twenty, and hedra, seat or base. It has many symmetry elements, including six fivefold axes of rotational symmetry, ten threefold axes, and fifteen twofold axes. 

Again, as in (LiMe)4 (34), the hypercoordinate carbon atom forms three normal two-center bonds within the alkyl group and one multicenter bond to the bridged metal atoms. The molecules of benzene of crystallization are located over the equilateral triangular faces of the Lig antiprism. 

Fig. 3.8 The tetrahedron T symmetry), octahedron (Oh S5mmetry) and icosahedron (7h symmetry) possess four, six and twelve vertices respectively, and four, eight and twenty equilateral-triangular faces respectively.

The names of many solid figures are based on the numbers of plane faces they have. A regular tetrahedron is a three-dimensional figure with four equal-sized equilateral triangular faces (the prefix tetra- means four ). 

A (derived from halving the square face of the cube) and a right-triangle B (derived from halving the equilateral triangular face of the tetrahedron, octahedron, or icosahedron). Earth was composed of triangle A. Air, fire, and water were composed of triangle B and could therefore be interconverted. ... 

The simplest host chat may be constructed using chemical subunits that approximate a sphere is a shell based on =4 subunits with a structure that conforms to a tetrahedron. " Such a shell consists of four identical subunits in the form of equilateral triangles, where edge-sharing by the triangular faces provides the curvature along the surface of the shell. The polygonal subunits of the tetrahedron are related by combinations of twofold and threefold rotation axes such that the framework is of cubic symmetry (i.e., 32 symmetry). 

Reciprocal salt pair solutions may be represented on an isothermal space model, in the form of either a square-based pyramid or a square prism. Figure 4.30a indicates the pyramidal model the four equilateral triangular faces stand for the four ternary systems AX-AY-W, AY-BY-W, BY-BX-W and AX-BX-W (W = water) for the salt pair represented by the equation... 

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