Non-lattice-periodic tilings of R3 by single polycubes

In this paper, we study a class of polycubes that tile the space by translation in a non-lattice-periodic way. More precisely, we construct a family of tiles indexed by integers with the property that Tk is a tile having k≥2 as anisohedral number. That is k copies of Tk are assembled by translation in order to form a metatile. We prove that this metatile is a lattice-periodic tile while Tk is not a lattice-periodic tile.