Quaquaversal tilings and rotations (Q1127764)

Considered are tilings of the three-dimensional Euclidean space which are generated from triangular prisms by inflation rules. An example of a tiling is given in which the orientations of the tiles are uniformly distributed in \(SO(3)\). The number of such orientations which occur in a sphere of volume \(k\) grows polynomially in \(k\). (Only logarithmic growth is possible in analogous two-dimensional tilings). Moreover, local properties of the tilings concerned with the neighborhoods of tiles, are discussed.