$\mathbb{Z}$ is the symmetry group of what?

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 $\mathbb{Z}$?

Solution 1:

$\mathbb Z$ is one of the seven frieze groups. It is the symmetry group of the simplest frieze:

Such decorative friezes occur very frequently in architecture and art. (Perhaps not with feet!)

Solution 2:

One nice "geometric" object for which the integers form the symmetries is an infinite string of evenly spaced identical symbols which do not have reflection symmetries vertically or horizontally, such as $$\cdots-\Gamma-\Gamma-\cdots$$ In this way the symmetries can shift to the left or right, but nothing else.