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

$\mathbb Z$ 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!)

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.

$\mathbb{Z}$ 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.


Group Theory