COMBINATORIAL SELF-SIMILARITY

Combinatorial (or numerical) self-similarity is an apparently new conc ept, introduced here in an attempt to explain the similarity of proper ties of the members of a homologous series that are not (geometrically ) self-similar and whence are not (deterministic) fractals. The term i s defined in the following steps: a) Select a numerical invariant, phi , characteristic of the member of the series b) Partition this propert y, phi, into a finite number of parts through a prescribed algorithm c ) Members are described so as to be combinatorially self-similar (or t o represent a ''numerics'' fractal) if the limits of the ratios of phi of two successive members at infinite stages of homologation are equa l for all parts, and equal the corresponding limit for the total prope rty. In the present work, phi is taken to be the Kekule count, K, when dealing with benzenoid systems and the topological index, Z, (H. Hoso ya, Bull. Chem. Sec. Japan 44 (1971) 2332) when dealing with saturated hydrocarbons. The previously described equivalence relation, I, [S. E I-Basil, J. Chem. Sec. Faraday Trans. 89 (1993) 909; J. Mel. Struct. ( Theochem) 288 (1993) 67; J. Math. Chem. 14 (1993) 305; J. Mel. Struct. (Theochem) 313 (1994) 237; J. Chem. Sec. Faraday Trans. 90 (1994) 220 1], is used to partition K when the number of terminal hexagons remain s constant throughout the series; otherwise the method of Klein and Se itz [D. J. Klein and W. A. Seitz, J. Mel. Struct. (Theochem) 169 (1988 ) 167] is used. For alkanes, an appropriate recurrence relation is use d to partition the Z values. it was found that phi for any homologous series of unbranched benzenoids, alkanes, Clar graphs, rook broads and King polyominos are all scaled by the golden mean, tau = 1,618033989, while homologous series of other types of benzenoids also represent ' 'numerical'' fractals, but the characteristic scaling factors depend o n the closed form expressions of their K values. in all cases, self-si milarities were manifested by expressing the ratios of adjacent phi's in the form of continued fractions, which in some cases led to exact s elf-similarity but in most cases self-similarity was only approximate.