  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.
