Fang Fang, Garrett Sadler, Julio Kovacs and Klee Irwin (2012)

Since antiquity, the packing of convex shapes has been of great interest to many scientists and mathematicians. Recently, particular interest has been given to packings of three-dimensional tetrahedra. Dense packings of both crystalline and semi-quasicrystalline have been reported. It is interesting that a semi-quasicrystalline packing of tetrahedra can emerge naturally within a thermodynamic simulation approach. However, this packing is not perfectly quasicrystalline and the packing density, while dense, is not maximal. Here we suggest that a “golden rotation” between tetrahedral facial junctions can arrange tetrahedra into a perfect quasicrystalline packing. Using this golden rotation, tetrahedra can be organized into “triangular”, “pentagonal”, and “spherical” locally dense aggregates. Additionally, the aperiodic Boerdijk-Coxeter helix (tetrahelix) is transformed into a structure of 3-or 5-fold periodicity—depending on the relative chiralities of the helix and rotation—herein referred to as the “philix”. Further, using this same rotation, we build (1) a shell structure which resembles a Penrose tiling upon projection into two dimensions, and (2) a “tetragrid” structure assembled of golden rhombohedral unit cells. Our results indicate that this rotation is closely associated with Fuller’s “jitterbug transformation” and that the total number of face-plane classes (defined below) is significantly reduced in comparison with general tetrahedral aggregations, suggesting a quasicrystalline packing of tetrahedra which is both dynamic and dense. The golden rotation that we report presents a novel tool for arranging tetrahedra into perfect quasicrystalline, dense packings.