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A prototypical example of a system with local symmetry is a grid of completely black balls.
TODO IMAGE GRID OF BILLARD BALLS.
We can rotate each ball individually and no observer could tell the difference. Hence this system has a local rotational symmetry.
The symmetry vanishes as soon as we put a little red dot onto each ball. Now it's easy to keep track of the orientation of each ball and there is no longer any local symmetry.