Growth Rates in the Quaquaversal Tiling

TL;DR

This paper analyzes how anisotropy in finite quaquaversal tilings diminishes with size, revealing specific scaling laws for tile orientations and their distribution, which has implications for identifying similar physical quasicrystals.

Contribution

It provides the first detailed quantitative analysis of the anisotropy scaling rates in finite quaquaversal tilings, including orientation distribution and approach to isotropy.

Findings

Tiles in a volume N appear in O(N^{1/6}) orientations.

A small subset of orientations accounts for most tiles.

Sample averages approach ergodic limits as N^{-1/336}.

Abstract

Conway and Radin's "quaquaversal" tiling of R^3 is known to exhibit statistical rotational symmetry in the infinite volume limit. A finite patch, however, cannot be perfectly isotropic, and we compute the rates at which the anisotropy scales with size. In a sample of volume N, tiles appear in O(N^{1/6}) distinct orientations. However, the orientations are not uniformly populated. A small (O(N^{1/84})) set of these orientations account for the majority of the tiles. Furthermore, these orientations are not uniformly distributed on SO(3). Sample averages of functions on SO(3) seem to approach their ergodic limits as N^{-1/336}. Since even macroscopic patches of a quaquaversal tiling maintain noticable anisotropy, a hypothetical physical quasicrystal whose structure was similar to the quaquaversal tiling could be identified by anisotropic features of its electron diffraction pattern.