If a discrete Group of displacements in the plane has more than one center of rotation, then the only rotations
that can occur are by 2, 3, 4, and 6. This can be shown as follows. It must be true that the sum of the interior angles
divided by the number of sides is a divisor of 360°.

where is an Integer. Therefore, symmetry will be possible only for

where is an Integer. This will hold for 1-, 2-, 3-, 4-, and 6-fold symmetry. That it does not hold for is seen by noting that corresponds to . The case requires that (impossible), and the case requires that (also impossible).

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1999-05-25