In geometry, 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 . The basic solid tiles are 30-60-90 triangular prisms arranged in a pattern such that some copies are rotated by , and some are rotated by in a perpendicular direction.
They construct the group given by a rotation of and a perpendicular rotation by ; the orientations in the quaquaversal tiling are given by . are cyclic groups, are dihedral groups, is the octahedral group, and all other are infinite and dense in ; if and are odd and ≥3, then 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 , and the latter has orientations in with the irrational rotation . They show that is dense in for the aforementioned value of , and whenever is transcendental.
References .
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External links A picture of a quaquaversal tiling Charles Radin page at the University of Texas