Regular pentagonal dodecahedron

Fig. 4. Regular pentagonal dodecahedron (RPD) (a), Kelvin s minimal tetrakaidecahedron (Kelvin s cell) (b) and /J-tetrakaidecahedron (c)...

Figure 2-53. Two artistic representations of the regular pentagonal dodecahedron, (a) Pentagonal dodecahedron as part of the sculpture symbolizing Industry at the Commons in Boston (photograph by the authors) (b) Leonardo da Vinci s dodecahedron in a book of Luca Pacioli, De Divina Proportione, published in 1509.

The shape of foam films and border profiles in large interval of foam expansion ratio from 10 to 1500 has been experimentally studied in [83], A regular pentagonal dodecahedron made up of transparent organic glass with an elastic rubber balloon inside it which took the shape of a sphere at inflation (Fig. 1.10) was used as a model of foam cell. 

This compares to values of 1.0990 for the planar tetrakaidecahedron 1.1053 for the rhombic dodecahedron and 1.0984 for the regular pentagonal dodecahedron. The latter - though often considered as a unit cell in foam modeling - is not really a viable candidate either, as it not only violates Plateau s laws but is also not space filling.)... 

The size of a bubble is expressed in the model in terms of the bubble radius R, which is the radius of a sphere having the same volume as that of the bubble. All other parameters that depend on the bubble size, such as the area of the films (faces) and the length of the Plateau border channels, are expressed in terms of R as shown below. The volume of a regular pentagonal dodecahedron with edge length / is ... 

In one of the cages within which gas molecules are trapped in methane hydrate, water molecules form a pentagonal dodecahedron, a three-dimensional figure in which each of the 12 sides is a regular pentagon. 

Fig. 1.—The arrangement of 45 spheres in icosahedral closest packing. At the left there is shown a single sphere, which constitutes the inner core. Next there is shown the layer of 12 spheres, at the corners of a regular icosahedron. The third model shows the core of 13 spheres with 20 added in the outer layer, each in a triangular pocket corresponding to a face of the icosahedron these 20 spheres lie at the corners of a pentagonal dodecahedron. The third layer is completed, as shown in the model at the right, by adding 12 spheres at corners of a large icosahedron the 32 spheres of the third layer lie at the corners of a rhombic triaconta-hedron. The fourth layer (not shown) contains 72 spheres.

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.

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...

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. 

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