Originally posted by geepamoogle
If someone wishes to submit a description of the discovered methods thus far, it might help in the search for some others.
Describing the shape can be difficult, but perhaps numbering the dots clockwise from the starting dot could help.
For instance, a regular hexagon would run from [b]1 to 2 to 3 to 4 to 5 to 6 to 1[/b]
Or a bit more dense notation (123456).
This is also what is called a cycle, there is a mathematics of cycles and I believe that it's connection with Group Theory would give an answer.
Untill I actually find that book I have no idea if it helps or what the answer is.
A very crued ansver is There is 6! permutations of (123456) but for each there are 6 that are equivalent up to rotation, further there are 6 mirror axis that transfer one immage in to an other, and finaly direction does not matter.
This would give 6! / 6 * 6*2 * 2 = 5
Since some rotations give the same as a mirroring we are obvious double counting something. So only thing gained from this is that there are more than 5 ways to do it (And you can all easyli construct 6 examples to prove that), so not much gained.