ABSTRACT
Summary ................................................................................................. 58 3.1 A Chemical Topology Scheme ....................................................... 59 3.2 Mapping of Tessellations and Networks ........................................ 62 3.3 Survey of Mapped Patterns and Carbons Allotropes ..................... 64 3.4 Rational Approximations to Φ, E and π ......................................... 78 3.5 Conclusions .................................................................................... 84 Keywords ................................................................................................ 86 References ............................................................................................... 86
SUMMARY
In this chapter, we describe the tenets of a chemical topology of crystalline matter and certain associated rational approximations to the transcendental mathematical constants φ, e and π, that arise out of considerations of both: (1) the Euler relation for the division of the sphere into vertices, V, faces, F, and edges, E, and: (2) its simple algebraic transformation into the so-called Schl�fli relation, which is an equivalent mathematical statement for the polyhedra, in terms of parameters known as the polygonality, defined as n = 2E/F, and the connectivivty, defined as p = 2E/V. It is thus the transformation to the Schl�fli relation from the Euler relation, in particular, that enables one to move from a simple heuristic mapping of the polyhedra in the space of V, F and E, into a corresponding heuristic mapping into Schl�fli-space, the space circumscribed by the parameters of n and p. It is also true, that this latter transformation equation, the Schl�fli relation, applies only directly to the polyhedra, again, with their corresponding Schl�fli symbols (n, p), but as a bonus, there is a direct 1-to-1 mapping result for the polyhedra, that can be seen to also be extendable to the tessellations in 2-dimensions, and the networks in 3-dimensions, in terms of coordinates in a 2-dimensional Cartesian grid, represented as the Schl�fli symbols (n, p), as discussed earlier, which do not involve rigorous solutions to the Schl�fli relation. For while one could never identify the triplet set of integers (V, F, E) for the tessellations and networks, that would fit as a rational solution within the Euler relation, it is in fact possible for one to identify the corresponding values of the ordered pair (n, p) for any tessellation or network. The identification of the Schl�fli symbol (n, p) for the tessellations and networks emerges from the formulation of its so-called Well’s point symbol, through the proper translation of that Well’s point symbol into an equivalent and unambiguous Schl�fli symbol (n, p) for a given tessellation or network, as has been shown by Bucknum et al. previously. What we report in this communication, are the computations of some, certain Schl�fli symbols (n, p) for the so-called Waserite (also called platinate, Pt3O4, a 3-,4-connected cubic pattern), Moravia (A3B8, a 3-,8-connected cubic pattern) and Kentuckia (ABC2, a 4-,6-,8-connected tetragonal pattern) networks, and some topological descriptors of other relevant structures. It is thus seen, that the computations of
the polygonality and connectivity indexes, n and p, that are found as a consequence of identifying the Schl�fli symbols for these relatively simple networks, lead to simple and direct connections to certain rational approximations to the transcendental mathematical constants φ, e and π, that, to the author’s knowledge, have not been identified previously. Such rational approximations lead to elementary and straightforward methods to estimate these mathematical constants to an accuracy of better than 99 parts in 100.
