I’m gonna tell you right now: The clockwise rotation of a square is isomorphic to the group {1, i, -1, -i} multiplied by -i.
So basically, let’s say we had a table of rotations for a square with counterclockwise vertices A, B, C, and D. But let’s assign the values 1, i, -1, and -1 (the powers of i) to these four vertices, respectively. It’s clear that each position can be determined by multiplying the original group by -i in order to rotate the square clockwise (left-to-right):
| position | 0 degree | 90 degrees | 180 degrees | 270 degrees |
| 1 | A (1) | D (-i) | C (-1) | B (i) |
| 2 | B (i) | A (1) | D (-i) | C (-1) |
| 3 | C (-1) | B (i) | A (1) | D (-i) |
| 4 | D (-i) | C (-1) | B (i) | A (1) |
I think this is amazing! A connection between geometry and complex numbers! The tools math has given us are wonderful and amazing. So beautiful.
Mathematics to Blow Your Mind 