Labyrinth Tiling from Quasiperiodic Octonacci Chains
This Demonstration shows two-dimensional square and labyrinth tilings based on Octonacci (Pell) and Fibonacci sequences, respectively [1, 2]. Such objects have been widely investigated in order to understand the interplay between quasiperiodicity and electronic structure in quasicrystals. Both tilings can be obtained from the grid (tensorial) product of two identical quasiperiodic chains.
An octonacci (Pell) chain is a quasiperiodic sequence obtained by applying the two-letter substitution rule , recursively to the initial word ; in the limit this gives a semi-infinite aperiodic sequence . The term octonacci comes from the similarity to the Fibonacci substitution rule , , for which the resulting self-similar semi-infinite aperiodic sequence is . For an infinitely long Fibonacci chain, the limit of the ratio of the frequencies of the two letters and is the golden ratio ; for the octonacci chain, the limit of the ratio is the silver ratio ().
In this Demonstration, the two letters and in the octonacci (Fibonacci) word sequence are graphically represented by a long and short spacing on the grid axes ( and ). The "nesting index " control lets you generate octonacci (Fibonacci) word sequences of different lengths, and you can see product grids of different complexity with the "add grid product lines" checkbox control (only for octonacci enabled, since for Fibonacci the superposition is complete). By changing the "long to short ratio " slider you can also investigate the transition between periodic and quasiperiodic order .
Snapshot 1: periodic order for Fibonacci sequence with square tiling and ()
Snapshot 2: aperiodic order for Octonacci-Pell sequence with labyrinth tiling, and maximum available ratio ()
Snapshot 3: aperiodic order for Fibonacci sequence with square tiling, and intermediate value ratio ()
 U. Grimm and M. Schreiber, "Energy Spectra and Eigenstates of Quasiperiodic Tight-Binding Hamiltonians," in Quasicrystals: Structure and Physical Properties (H.-R. Trebin, ed.), Weinheim, Germany: Wiley-VCH, 2003 pp. 210–235. arxiv.org/abs/cond-mat/0212140.
 W. Steurer and S. Deloudi, "Tilings and Coverings", in Cristallography of Quasicrystals: Concepts, Methods and Structures, Springer-Verlag Berlin Heidelberg, Germany 2009 pp. 7-47