Polygons, planar

Generally, minimum adhesion forces occur for particles of isometric shape, that is, particles that look more like spheres or regular polygons. Planar particles have higher adhesion forces than isometric shapes. One also finds the adhesion forces of needles and fibers to be larger than both isometric and planar particles. 

Baeyer strain theory (Section 3 4) Incorrect nineteenth century theory that considered the nngs of cycloalkanes to be planar and assessed their stabilities according to how much the angles of a corresponding regular polygon devi ated from the tetrahedral value of 109 5°... 

Coordination polyhedron. It is standard practice to think of the lignnd atoms that are directly attached to the central atom as defining, i coordination polyhedron (or polygon) about the central atom. Thus Co(NH,)h]J+ is an octahedral ion and [PtCIJ2- is a square planar ion. In this way the coordination number may equal the number of vertices in (he coordination polyhedron. Exceptions are common among orgunonietnlhtf compounds. 

Different geometrical arrangements of the atoms attached to the central atom are possible for all coordination numbers greater than one. The coordination polyhedron (or polygon in planar molecules) may be denoted in the name by an affix called the polyhedral symbol. This descriptor clearly distinguishes isomers differing in the geometries of their coordination polyhedra. 

Figure 9-16. Planar networks of regular polygons with up to eight-fold symmetry.

The diagram illustrates a general mode of puckering that occurs for all even-membered rings. The resulting conformation is known as the crown form and derives from a regular distortion of the planar D h polygon. 

A further set of semi-regular polyhedra, many of which are also important in Chemistry, follows on relaxation of the requirement for equivalent polygonal faces. The thirteen Archimedean polyhedra have equivalent vertex figures on all vertices and all faces remain planar and equilateral, but are of two or three distinct kinds. In orbit terms, the Platonic and Archimedean polyhedra all have single orbits of vertices, but, whereas, the face centres of Platonic solids also fall into single orbits, those of Archimedean solids span either two or three. Again, the Archimedean solids fall into cubic and icosahedral families. 

Problem 9.13 The energies of the -k m.o.s of monocyclic hydrocarbons of n atoms can be directly obtainedfrom the vertical coordinates of the corners ofan n-sided regular polygon inscribed in a circle of radius 2(3, with one corner at the lowest point see Fig. 9.9) ref. 121). Apply this construct to a square planar cyclobutadiene molecule the real geometry is rectangular) and conclude that this is not an aromatic hydrocarbon. 

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