Symmetry pentagonal dodecahedron

Finally, we turn to the pentagonal dodecahedron and the icosahedron. These two polyhedra have the same symmetry. They are related to each other as the cube and octahedron are related. The symmetry elements and operations are as follows. 

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 ... 

Figure B.2 shows polyhedra commonly encountered. The five Platonic (or regular) solids are shown at the top. Beside the octahedron and cube, the octahedron is shown inside a cube, oriented so the symmetry elements in common coincide. These solids are conjugates one formed by connecting the face centers of the other. The tetrahedron is its own conjugate, because connecting the face centers gives another tetrahedron. The icosahedron and pentagonal dodecahedron are conjugates. The square antiprism and trigonal...

Describe the symmetry elements of a pentagonal dodecahedron. It has 12 faces and 30 edges. See Figure 2.16. 

Many primitive organisms have the shape of the pentagonal dodecahedron. As will be seen later, pentagonal symmetry used to be considered forbidden in the world of crystal structures. Belov [82] suggested that the pentagonal symmetry of primitive organisms represents their... 

The pentagonal dodecahedron and the icosahedron are in the same symmetry class. The fivefold, threefold and twofold rotation axes intersect the midpoints of faces, the vertices and the midpoints of edges of the dodecahedron, respectively (Figure 2-50). On the other hand, the corresponding axes intersect the vertices and the midpoints of faces and edges of the icosahedron (Figure 2-50). 

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. 

FIG. A5-8. The two regular polyhedra having / symmetry, (a) The pentagonal dodecahedron and (b) the icosahedron. 

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. 

While the pentagon does not tile the plane, it does indeed tile the sphere. Twelve pentagons make up ti e pentagonal dodecahedron whose vertices lie on the sphere. Similarly, the icosahedron is a triangular tiling of the sphere with fivefold rotational symmetry. 

The pentagonal dodecahedron is the fifth regular polyhedron. It has thirty edges, twenty corners, and twelve faces, which are regular pentagons. It is closely related to the icosahedron the relation involves interchanging corners and faces. Its symmetry elements are the same as those for the icosahedron. 

The most common polyhedron is the pentagonal dodecahedron. It is formed by 20 water molecules and has 12 pentagonal faces, 20 vertices and 30 edges. With a hydrogen bond distance of 280 pm, its volume is 170 x 10 pm (Fig. la). Because of its fivefold symmetry, the close packing of such polyhedra cannot fill space the pentagonal dodecahedron is therefore associated with slightly different polyhedra ... 

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 ...

Figure 16.2 The icosahedron (top) and dodecahedron (bottom) have identical symmetries but different shapes. Protein subunits of spherical viruses form a coat around the nucleic acid with the same symmetry arrangement as these geometrical objects. Electron micrographs of these viruses have shown that their shapes are often well represented by icosahedra. One each of the twofold, threefold, and fivefold symmetry axes is indicated by an ellipse, triangle, and pentagon, respectively.

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