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

When a regular hexagon maps into itself, it implies that vertices must map to vertices and edges to edges. A regular hexagon has 6 angles between the adjacent vertices, all of which are equal and they all sum up to 360 degrees. Each angle has a measure of 60 degrees. Each subsequent rotation by 60 degrees maps the hexagon onto itself. There are 5 such rotations in the interval 0°≤θ≤360°: 60, 120, 180, 240, 300. Thus we conclude that not all rotation maps the hexagon onto itself every time it is rotated.