Labyrinth Tiling from Quasiperiodic Octonacci Chains

This Demonstration shows a two-dimensional labyrinth tiling [1]. Such objects have been widely investigated in order to understand the interplay between quasiperiodicity and electronic structure in quasicrystals. The labyrinth can be obtained from the grid (tensorial) product of two identical octonacci chains.

An octonacci 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 ().

In this Demonstration, the two letters and in the octonacci word sequence are graphically represented by a long and short spacing on the grid axes ( and ). The "nesting index " control lets you generate octonacci word sequences of different lengths, and you can see product grids of different complexity with the "add grid product lines" checkbox control. By changing the "long to short ratio " slider you can also investigate the transition between periodic and quasiperiodic order .

Snapshot 2: aperiodic order for tiling with and maximum available ratio ()

Snapshot 3: aperiodic order for tiling with and intermediate value ratio ()

Reference

[1] 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.