Quaquaversal tiling
The quaquaversal tiling is a nonperiodic tiling of Euclidean 3-space introduced by John Conway and Charles Radin. It is analogous to the pinwheel tiling in 2 dimensions having tile orientations that are dense in SO(3). 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.
They construct the group G given by a rotation of 2π/p and a perpendicular rotation by 2π/q; the orientations in the quaquaversal tiling are given by G. G are cyclic groups, G are dihedral groups, G is the octahedral group, and all other G are infinite and dense in SO; if p and q are odd and ≥3, then G is a free group.
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, and the latter has orientations in G with the irrational rotation. They show that G is dense in SO for the aforementioned value of p, and whenever cos is transcendental.