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

Abstract

The problem of tiling or tessellating (i.e., completely filling) three-dimensional Euclidean space R3 with polyhedra has fascinated people for centuries and continues to intrigue mathematicians and scientists today. Tilings are of fundamental importance to the understanding of the underlying structures for a wide range of systems in the biological, chemical, and physical sciences. In this paper, we enumerate and investigate the most comprehensive set of tilings of R3 by any two regular polyhedra known to date. We find that among all of the Platonic solids, only the tetrahedron and octahedron can be combined to tile R3. For tilings composed of only congruent tetrahedra and congruent octahedra, there seem to be only four distinct ratios between the sides of the two polyhedra. These four canonical periodic tilings are, respectively, associated with certain packings of tetrahedra (octahedra) in which the holes are octahedra (tetrahedra). Moreover, we derive two families of an uncountably infinite number of periodic tilings of tetrahedra and octahedra that continuously connect the aforementioned four canonical tilings with one another, containing the previously reported Conway-Jiao-Torquato family of tilings [Conway et al., Proc. Natl. Acad. Sci. USA 108, 11009 (2011)] as a special case. For tilings containing infinite planar surfaces, nonperiodic arrangements can be easily generated by arbitrary relative sliding along these surfaces. We also find that there are three distinct canonical periodic tilings of R3 by congruent regular tetrahedra and congruent regular truncated tetrahedra associated with certain packings of tetrahedra (truncated tetrahedra) in which the holes are truncated tetrahedra (tetrahedra). Remarkably, we discover that most of the aforementioned periodic tilings can be obtained by “retessellating” the well-known tiling associated with the face-centered-cubic lattice, i.e., by combining the associated fundamental tiles (regular tetrahedra and octahedra) to form larger polyhedra.