Question

1) A regular hexagon is mapped onto itself every time it is rotated

Answer 1

A regular hexagon has angles that measure 360/6=60°, between 0° and 360°, the multiples of this angle is 120°, 180°, 240°, 300°. Thus when the hexagon is rotated through this angles, it will be mapped onto itself. When rotated through any other angle, the same result won't be possible.
Thus we conclude that the hexagon it's mapped into itself every time it is rotated.