Pentagonal dodecahedron, symmetry groups

Look at Uie drawings accompanying Problem 3.6. Is it possible to superimpose the cube on the dodecahedron Castieman and coworfcers have recently detected a cation with mje = 28, identified as TigC. It b bebeved that the titanium atoms form a cube with the addition of twelve carbon atoms to complete a pentagonal dodecahedron. Draw the proposed structure. What e its point group symmetry ... 

The Pentagonal Dodecahedron and the Icosahedron. These bodies (Fig. A5-8) are related to each other in the same way as are the octahedron and the cube, the vertices of one defining the face centers of the other, and vice versa. Both have the same symmetry operations, a total of 120 We shall not list them in detail but merely mention the basic symmetry elements six C5 axes ten C3 axes, fifteen C2 axes, and fifteen planes of symmetry. The group of 120 operations is designated Ih and is often called the icosahedral group. 

The symmetry of many molecules and especially of crystals is immediately obvious. Benzene has a six-fold symmetry axis and is planar, buckminsterfullerene (or just fullerene or footballene) contains 60 carbon atoms, regularly arranged in six- and five-membered rings with the same symmetry (point group //,) as that of the Platonic bodies pentagon dodecahedron and icosahedron (Fig. 2.7-1). Most crystals exhibit macroscopically visible symmetry axes and planes. In order to utilize the symmetry of molecules and crystals for vibrational spectroscopy, the symmetry properties have to be defined conveniently. 

Figure 2. Two conformers proposed for met-cars (a) the structure of the pentagonal dodecahedron with eight equivalent metal sites (Th symmetry), proposed by Castleman s group (b) the structure of the tetracapped tetrahedron, with two sets of metal sites each occurring four times (Td symmetry), first proposed by Dance from DFT calculations ...

The spherical viruses are by far the largest group. Cubic symmetry includes the tetrahedron (4 triangles), the dodecahedron (12 pentagons), and the icosahedron (20 triangles). The icosahedron is shown in the example of the Tipula irridescens virus (an insect virus). Higher deltahedra are all derivatives of these three basic forms. [For a detailed dis-... 

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