For any tiling T, the translation module is the $\mathbb{Z}$-span of all translations $t$, such that there is a tile $T$ in $T$, and $T+t$ is also in $T$.
If $T$ is periodic, the translation module is a lattice, and the set of periods is a subset of the translation module. Even if $T$ is nonperiodic, the translation module can be a lattice (for instance, see the Chair tilings). Whenever the translation module of a tiling is a lattice, this is a hint that the tiling may be a model set with $p$-adic internal space [LMS03]
, [FS]
.
[LMS03]
Lee, J E S and Moody, R V and Solomyak, B
Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems
Discrete and Computational Geometry
2003,
29,
pp. 525-560,
MR1702375
[FS]
Frettlöh, D and Sing, B
Computing modular coincidences
Preprint