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it is known that the symmetry groups on the 2d Euclidean plane are given by the point-groups (n-fold and dihedral symmetries) and the wallpaper groups.

However we can create more symmetries on the plane than just those.

For example we can stereographically project the 2d plane onto the unit sphere, and consider all the spherical symmetry groups (that are much more than those on the plane), and stereographically re-project the sphere onto the plane to obtain new symmetries.

Has this idea been explored already? I bet it was, but I can't find information on this.

And ultimately, why do people say that the symmetries of the plane are just the point-groups and the wallpaper groups?

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# Symmetry groups on the plane

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