New family of tilings of three-dimensional Euclidean space by tetrahedra and octahedra

Abstract

It is well known that two regular tetrahedra can be combined with a single regular octahedron to tile (complete fill) three-dimensional Euclidean space R 3 . This structure was called the "octet truss" by Buckminster Fuller. It was believed that such a tiling, which is the Delaunay tessellation of the face-centered cubic (fcc) lattice, and its closely related stacking variants, are the only tessellations of R 3 that involve two different regular polyhedra. Here we identify and analyze a unique family comprised of a noncountably infinite number of periodic tilings of R 3 whose smallest repeat tiling unit consists of one regular octahedron and six smaller regular tetrahedra. We first derive an extreme member of this unique tiling family by showing that the "holes" in the optimal lattice packing of octahedra, obtained by Minkowski over a century ago, are congruent tetrahedra. This tiling has 694 distinct concave (i.e., nonconvex) repeat units, 24 of which possess central symmetry, and hence is distinctly different and combinatorically richer than the fcc tetrahedra-octahedra tiling, which only has two distinct tiling units. Then we construct a one-parameter family of octahedron packings that continuously spans from the fcc to the optimal lattice packing of octahedra. We show that the "holes" in these packings, except for the two extreme cases, are tetrahedra of two sizes, leading to a family of periodic tilings with units composed four small tetrahedra and two large tetrahedra that contact an octahedron. These tilings generally possess 2,068 distinct concave tiling units, 62 of which are centrally symmetric.space-filling | nonoverlapping solids | polytopes T ilings have intrigued artists, architects, scientists, and mathematicians for millenia (1). A "tiling" or "tessellation" is a partition of Euclidean space R d into closed regions whose interiors are disjoint. Tilings of space by polyhedra are of particular interest. Certain periodic polyhedral tilings are intimately connected to lattices (2-5) and crystal states of matter (6), and can provide efficient meshings of space for numerical applications (e.g., quadrature and discretizing partial differential equations) (7). Polyhedral tilings arise in the structure of foams and Kelvin's problem (3,8,9,10). Remarkably, crystalline forms of DNA can be generated by using specifically constructed mathematical tiling analogs (11). Some aperiodic tilings of Euclidean space underlie quasicrystals (12, 13), which possess forbidden crystallographic symmetries, and glassy states of matter (14). Tilings of high-dimensional Euclidean space also have important applications in communications, cryptography, information theory, and in the search for gravitational waves (2, 15).Tilings of two-dimensional (2D) Euclidean space R 2 by regular polygons have been widely used since antiquity. Kepler was the first to provide a systematic mathematical treatment of 2D tiling problems in his book entitled Harmonices Mundi (16). Kepler showed that there are just 11 uniform tilings of R 2 with re...

Families of tessellations of space by elementary polyhedra via retessellations of face-centered-cubic and related tilings

“…These include the FCC tiling, the tiling associated with the conjectured densest packing of octahedra (the DO tiling), the tiling associated with the dense uniform packing of regular tetrahedra (the UT tiling) [19] in which the holes are small congruent octahedra, and the tiling associated with a layered packing of octahedra (the LO tiling) in which the holes are small congruent tetrahedra. Although the FCC tiling is well known, the tiling associated with the conjectured densest octahedron packing was only discovered recently [23]. The latter two tilings are derived here for first time to the best of our knowledge.…”

“…In addition, we report a tiling by large regular tetrahedra and small regular truncated tetrahedra that heretofore has not been identified. We find that this tiling, the tilings by large regular truncated tetrahedra and small regular tetrahedra associated with recently discovered dense packings of regular truncated tetrahedra [19,23], and a tiling by irregular truncated tetrahedra associated with the β-tin structure [28] can also be obtained by retessellations of FCC tiling. Thus, the retessellation process provides a unified approach to determine the possible tilings of R 3 by elementary polyhedra including regular octahedra, regular tetrahedra, as well as regular and irregular truncated tetrahedra.…”

“…This new tiling consists of fundamental repeat units composed of one octahedron and six tetrahedra whose edge length is one third of that of the octahedra. Moreover, these authors [23] also discovered an a one-parameter family of periodic tilings containing an uncountable number (i.e. a continuous spectrum) of tilings by congruent octahedra and small tetrahedra with two different sizes, connecting the tiling associated with the conjectured densest octahedron packing [25] at one extreme and the FCC tiling at the other extreme.…”

“…Very recently, Conway, Jiao and Torquato [23] have shown that one such tiling by congruent octahedra and small congruent tetrahedra can be derived from the densest known packing of regular octahedra [24], which has been conjectured to be the densest among all octahedron packings [25]. This new tiling consists of fundamental repeat units composed of one octahedron and six tetrahedra whose edge length is one third of that of the octahedra.…”

“…This family contains the Conway-Jiao-Torquato family of tilings [23] as a special case. We also obtain a one-parameter family of tilings by large tetrahedra and small octahedra, connecting the UT tiling to the FCC tiling.…”