This article is cited in 2 papers

The Fibonacci–Penrose semigroup formalism and morphogenetic synthesis of quasicrystal mosaics

V. V. Yudin, E. S. Startsev, I. G. Permyakova

Far-East State University, Vladivostok, Russia

Abstract: We construct the group bundle for the inverse Fibonacci semigroups in the axiomatic approach framework. The proper Fibonacci semigroup is the corresponding group bundle for the Penrose semigroups, which can be interpreted as the generating grammar of the morphogenetic synthesis of the pentasymmetric Penrose parquet in the numbers of tiling by golden rhombuses. This morphogenetic synthesis of the Penrose parquet satisfies the scaling principle. Parquet plates are not absolutely rigid, and the relations between their metric characteristics are governed by the golden section and other magic numbers. The characteristic form factors of three-level dual alphabets are the corresponding invariants. We realize the morphogenetic synthesis in the examples of square-octagonal and bihexagonal lattices. We consider cumulative properties of magic series and the evolutionary aspects of semigroup orbits in the entropy representation.

Keywords: group bundle of inverse semigroups, Fibonacci generator, Penrose generator, morphogenesis, square-octagonal lattice, bihexagonal lattice.

Received: 02.04.2010
Revised: 13.09.2010

DOI: 10.4213/tmf6630

 English version: Theoretical and Mathematical Physics, 2011, 167:1, 517–537

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