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 $\mathbb{Z}$?
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1You can think of elements of $\mathbb Z$ as the characters of the circle $S^1$ or the deck transformations of the covering map $\mathbb R \to S^1$, in other words, the fundamental group of the circle. – Arkady Aug 15 '17 at 17:49
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1I'd say cyclic groups ($\Bbb Z/n\Bbb Z$ and $\Bbb Z$) and connected one-dimensional Lie groups ($S^1=\Bbb R/\Bbb Z$ and $\Bbb R$) are symmetry groups of themselves under translation. – anon Aug 15 '17 at 17:49
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$\mathbb Z$ 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!)
lhf
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See also https://www.maa.org/sites/default/files/images/upload_library/4/vol1/architecture/Math/seven.html. – lhf Aug 15 '17 at 18:11
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+1 for the great choice of symbol. Also worth noting that $\mathbb{Z}$ itself would be a possible choice, making it at least somewhat self-referential. – Tobias Kildetoft Aug 15 '17 at 18:23
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2@TobiasKildetoft, the symbol is by John Conway. Also, $\sf ZZZ$ has $180^\circ$ rotational symmetry. – lhf Aug 15 '17 at 18:25
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Maybe it goes without saying that different types of symmetry may correspond to the same group, but I think this is important to emphasize. Even in the realm of friezes, there are two distinct frieze groups isomorphic to $\mathbb{Z}$. One of these is the group of a frieze having only translation symmetry (like the hop above). The second is the group of a frieze having only translation and glide reflection symmetries (like the walk). Since performing two glide reflections in succession is the same as translation, this frieze group is an infinite cyclic group with glide reflection as generator. – Will Orrick Aug 27 '17 at 17:54
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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.
Tobias Kildetoft
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2Your string also has reflection symmetries, so that the full symmetry group is infinite dihedral. – Moishe Kohan Aug 15 '17 at 17:56
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@MoisheCohen Ohh, right, good point. We need the triangles to be tilted slightly to one side. – Tobias Kildetoft Aug 15 '17 at 17:58
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Isn't a symmetry group of a geomtric figure, by definition, a collection of rotations and reflections? (I know that the group of symmetries of a set $S$ is the group of all bijections of $S$ onto $S$, but as far as I know the definition is different in geometry) – Cauchy Aug 15 '17 at 18:06
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@Francisco I am not sure what you mean by ad-hoc here. Sure, it might not be some naturally occurring figure, but the ideas behind how to create it were not random. – Tobias Kildetoft Aug 15 '17 at 18:07
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2@Francisco what do you mean? You asked for an example of an object for which $Z$ is the symmetry group. By definition, any answer to your questions is adhoc! – Cauchy Aug 15 '17 at 18:08
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@Cauchy , symmetries don't have to just be rotations and reflections. A symmetry group is a group of symmetries, i.e., transformations that map the original object to itself. We get used to objects whose only symmetries are rotations and reflections, but the frieze groups are a well known formalization of symmetries that involve translations as well. – G Tony Jacobs Aug 15 '17 at 18:12
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$\mathbb{Z}$ 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.
Dietrich Burde
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