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]

FrettlĂ¶h, D and Sing, B

**Computing modular coincidences**

*Preprint*