Towards an Empirical Relation between the Polygonal Circuit Area and the Topological Form Index,

Science Direct Working Paper No S1574-0331(04)70809-7

19 Pages Posted: 24 May 2017 Last revised: 23 Dec 2017

See all articles by Michael J. Bucknum

Michael J. Bucknum

Georgia College & State University - Department of Chemistry and Physics; Cornell University - Department of Chemistry and Chemical Biology

Date Written: January 2002

Abstract

This paper begins with a review of the Euler relation for the polyhedra and derives the corresponding Schläfli relation in n, the polygonality, and p, the connectivity of the polyhedra. The use of the Schläfli symbols to organize the mapping of the polyhedra and its extension into the 2D and 3D networks is described. The topological form index, represented by , is introduced and is defined as the ratio of the polygonality, n, to the connectivity, p, in a structure. Next a discussion is given of establishing a conventional metric of length in order to compare topological 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 that the area of the elementary polygonal circuit in the polyhedra and 2D and 3D networks is proportional to the topological form index, , 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 successfully correlates the topological form indexes with the corresponding elementary polygonal circuit area data of the polyhedra and 2D and 3D networks. A brief discussion of the evident rigorousness of the Schläfli indexes over all the polyhedra and 2D and 3D networks, based upon the correlation of the topological form index with elementary polygonal circuit area, and the suggestion that an Euler-Schläfli relation for the 2D and 3D networks in terms of the Schläfli indexes is possible, concludes the paper.

Keywords: Physical Chemistry > Solid State Chemistry and Materials, physchem/0201001

Suggested Citation

Bucknum, Michael J., Towards an Empirical Relation between the Polygonal Circuit Area and the Topological Form Index, (January 2002). Science Direct Working Paper No S1574-0331(04)70809-7, Available at SSRN: https://ssrn.com/abstract=2969599

Michael J. Bucknum (Contact Author)

Georgia College & State University - Department of Chemistry and Physics ( email )

CBX 82
Milledgeville, GA 31061
United States

Cornell University - Department of Chemistry and Chemical Biology ( email )

Ithaca, NY 14853
United States

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