Regular pentagon

In this section, all of the symmetries of a pentagon (regular 5-gon) were discovered. This was part of a general discussion of symmetries for planar or two-dimensional objects. However, symmetries of three-dimensional objects are also interesting. Consider the two three-dimensional objects (a cube and a tetrahedron) presented below. By using reflections and rotations, all of the symmetries of each figure can be obtained. 

At one time all cycloalkanes were believed to be planar It was expected that cyclopentane would be the least strained cycloalkane because the angles of a regular pentagon (108°) are closest to the tetrahedral angle of 109 5° Heats of combustion established that this is not so With the exception of cyclopropane the rings of all cycloalkanes are nonplanar... 

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. 

FIGURE 5.36 A plane surface cannot be covered by regular pentagons without leaving gaps. The same is true ot regular heptagons. 

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.

The existence of a stable Cgg molecule, Buckminsterfullerene(BF) was recently proposed(14). In the suggested structure, twelve regular pentagons and twenty hexagons are connected to form an almost perfectly spherical pattern of icosahedrai symmetry as in Figure 10. 

The given conditions do not always allow for equivalent positions for all atoms. Take as an example the following conditions composition MX5, covalent M-X bonds, all X atoms bonded to M atoms. In this case all X atoms can only be equivalent if each set of five of them form a regular pentagon around an M atom (as for example in the XeFj ion). If this is not possible for some reason, then there must be at least two non-equivalent positions for the X atoms. According to the symmetry principle the number of these non-equivalent positions will be as small as possible. 

Five points can be arranged on the surface of a sphere such that they are all equivalent, only in a planar pentagonal arrangement, which does not maximize the distance between the points. In other words, there is no regular polyhedron with five equivalent vertices. There are two... 

What is the sum of the measures of the exterior angles of a regular pentagon ... 

The molecular structure of the closo 12-vertex 1-SBnHn cluster (17) has been studied by electron diffraction methods augmented by ab initio calculations. Substantial distortions away from the regular icosahedron occurred by expansion of the pentagonal belt adjacent to sulfur.73 The UV-PES spectra of 17 has also been reported,71 and its microwave spectrum has been investigated and demonstrated that the molecule had C5v symmetry.74... 

Cyclopentane has a smaller amount of strain than does cyclohexane when forced to have all its carbons in a plane (flat). Cyclopentane is nearly flat since the bond angles in a regular pentagon (108°) are near the tetrahedral angle (109.5°). The bond angles in a flat cyclohexane are 120°, which are not near the 109.5° bond angle. To avoid this strain, the cyclohexane ring buckles and becomes nonplanar. 

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