Imagine you have a plane, flat surface with a square grid drawn on it. You have a standard cubic die which is placed flat on the surface. Its length is the same as the length of side of each grid square. The only way to move the die to an adjacent square is by tipping the die into the square. So obviously the die cannot move diagonally.
Since this grid is infinitely large, there are infinite round trips that can be made with the die. Some of these round-trips can cause a rotation of the die. The set of all round trips can be categorized into equivalence sets, where each class represents a particular rotation following a round-trip. These rotations naturally constitute a group.
Can anyone list out all possible rotations in this configuration? Also, when moving the die around, I noticed that it was impossible for a round-trip to produce a 90 degree rotation about the the axis perpendicular to the surface. Can anyone give me a simple proof of this? It seems to be an easy proof, but I am unable to show it.
Also, if we generalize this to a n-sided die, what are the impossible rotations? If anyone could provide a related paper or reference as well, I would really appreciate it.