Non lattice periodic tilings of R3 by single polycubes

International audienceIn 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 has anisohedral number. That is k copies of Tk are assembled by translation in order to form a metatile. We prove that this metatile is lattice periodic while Tk is not a lattice periodic tile

Non lattice periodic tilings of R3 by single polycubes

International audienceIn 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 has anisohedral number. That is k copies of Tk are assembled by translation in order to form a metatile. We prove that this metatile is lattice periodic while Tk is not a lattice periodic tile

How many faces can polycubes of lattice tilings by translation of R3 have?

International audienceWe construct a class of polycubes that tile the space by translation in a lattice- periodic way and show that for this class the number of surrounding tiles cannot be bounded. The first construction is based on polycubes with an L-shape but with many distinct tilings of the space. Nevertheless, we are able to construct a class of more complicated polycubes such that each polycube tiles the space in a unique way and such that the number of faces is 4k + 8 where 2k + 1 is the volume of the polycube. This shows that the number of tiles that surround the surface of a space-filler cannot be bounded