On Topological Form in Structures

M.J. BUCKNUM; E.A. CASTRO

J.Math.Chem.

Año: 2006 vol. 39 p. 33 - 46

            This paper begins with a review of the Euler relation for the polyhedra and presents the corresponding Schläfli relation in n, the polygonality, and p, the connectivity of the polyhedra. The use of ordered pairs is given by (n,p), the Schläfli symbols, to organize the mapping of the polyhedra and its extension into the two dimensional (2D) and three dimensional (3D) networks is described. The topological form index, represented by l, is introduced and is defined as the ratio of the polygonality, n, to the connectivity, p, in the structure, it is given by l = n/p. Next a discussion is given of establishing a conventional metric of length in order to compare toplogical properties of the polyhedra and networks in 2D and 3D. A fundamental structural metric is assumed for the polyhedra. The metric for the polyhedra is, in turn, used to establish a metric for tilings in the Euclidean plane. The metrics for the polyhedra and 2D plane are used to establish a metric for networks in 3D. Once the metrics have been established, a conjecture is introduced, based upon the metrics assumed, that the area of the elementary polygonal circuit in the polyhedra and 2D and 3D networks is proportional to a function of the toplogical form index, l, for these structures. Data of the form indexes and the corresponding elementary polygonal circuit areas for a selection of polyhedra and 2D and 3D networks is tabulated, and the results of a least squares regression analysis of the data plotted in a Cartesian space are reported. From the regression analysis it is seen that a quadratic in l, the form index, successfully correlates with the corresponding elementary polygonal circuit area data of the polyehdra and 2D and 3D networks. A brief discussion of the evident rigorousness of the Schläfli indexes (n,p) over all the polyhedra and 2D and 3D netweorks, based upon the correlation of the topological form index with elementary polygonal circuit area in these structures, and the suggestion that an Euler-Schläfli relation for the 2D and 3D networks, is possible, in terms of the Schläfli indexes, concludes the paper.