Quaquaversal tiling

The quaquaversal tiling is a nonperiodic tiling of Euclidean 3-space introduced by John Conway and Charles Radin.

The basic solid tiles are 30-60-90 triangular prisms arranged in a pattern such that some copies are rotated by π/3, and some are rotated by π/2 in a perpendicular direction.

[1] They construct the group G(p,q) given by a rotation of 2π/p and a perpendicular rotation by 2π/q; the orientations in the quaquaversal tiling are given by G(6,4).

[1] Radin and Lorenzo Sadun constructed similar honeycombs based on a tiling related to the Penrose tilings and the pinwheel tiling; the former has orientations in G(10,4), and the latter has orientations in G(p,4) with the irrational rotation 2π/p = arctan(1/2).

They show that G(p,4) is dense in SO(3) for the aforementioned value of p, and whenever cos(2π/p) is transcendental.