TITLE: $\mathbb{Z}$ is the symmetry group of what?
QUESTION [11 upvotes]: It is often said that the notion of group has a kinship with that of symmetry: many groups appear as the symmetry group of some object -- take,for example, dihedral groups.
What is the object related to $\mathbb{Z}$?

REPLY [6 votes]: $\mathbb{Z}$ is a $1$-dimensional Euclidean crystallographic group. In dimension $2$, they are called wallpaper groups, and there are $17$ non-isomorphic ones.
See also One-dimensional symmetry groups.