About how many lines can one rotate a regular hexagon through some angle x, O<x<306 deg. so that the hexagon again occupies its original position
Hi,
I've some difficulties to understand your problem:
In IR² you can only rotate something around a point. So I presume that you would like to reflect the hexagon over a straight line.
There are 6 such lines: Three lines passing through opposite vertices, three lines passing through midpoints of sides.
I've attached an image to show you what I mean.
EB
If you are rotating about lines and you are allowing your rotations to be in 3D, then any of Earboth's lines of symmetry will do, just rotate by 180 degrees. And, of course, you've got an 6-fold axis of symmetry up through the center of the hexagon, where you can rotate by 60, 120, 180, and 240 degrees. So there would be a total of 7 such lines.
-Dan