In the mid 90s Conway and Radin introduced the Quaquaversal Tiling. It is a hierarchical tiling of three-dimensional space that exhibits statistical rotational symmetry, in the sense that the distribution of tiles chosen uniformly at random from a large sphere has a nearly uniform distribution of orientations. Any hierarchical tiling has an associated operator whose spectrum can be analyzed to study the distribution of orientations in a large sample. Radin and Conway showed that 1 has multiplicity 1 in the spectrum of this operator to show that the operator exhibited statistical rotational symmetry. By numerically analyzing the spectrum of this operator Draco, Sadun, and Wieren found eigenvalues very close to 1 and concluded that the rate with which the distribution approaches uniformity is fairly slow, mentioning that a galactic scale sample of a material with this crystal structure at the molecular level would exhibit noticeable anisotropy. Bourgain and Gamburd proved, on the other hand, that a certain class of operators including this one has a nonzero gap between 1 and the second largest eigenvalue, concluding that the distribution must approach uniformity at an exponential rate.
In this talk I will introduce hierarchical tilings, discuss results like those above, and prove that the spectrum of this operator is real. Answering a question of Draco, Sadun, and Wieren.
Josiah Sugarman