How many Faces can the Polycubes of Lattice Tilings by Translation of ${\mathbb R}^3$ have?

Abstract

We 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.