Growth Rates in the Quaquaversal Tiling

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.