MATH325: Lecture 2, Exercise 1.9

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Sliding the paper along your desk moves all the points on it, but that move is probably a translation. That answer was too easy; the author won't accept it. Flipping the paper fixes an entire line -- that's no good either. Rotating it leaves the center of rotation unmoved.

What if you rotate it while translating it? That moves every point on the paper, but we're talking about the whole plane. Can you be sure that no point, on or off the paper, stays still while you do that? Can you think of other ways of combining rotations, reflections and translations that might work? (Yes, combining two isometries makes a new isometry -- there's a homework question about this.)