$\mathbb{Z}$ 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 $\mathbb{Z}$?

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

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