Quaternion Cube
The Quaternion Cube is a rather easy-to-solve puzzle that I designed to visually demonstrate the Quaternion Group Q₈.
The puzzle turns over the top 4 vertices of a cube, with 1:2 gearing connecting the opposing corners. There are linear slides under the edge pieces that prevent the static corners from blocking the overall movement.
The puzzle has 24 possible positions, which express the group SL(2,3). SL(2,3) contains the Quaternion Group (Q₈) as a subgroup, and these Q₈ states are expressed over the 8 edges whenever the corners are solved.
Every 3 turns in the same direction is a corner-preserving sequence, and these sequences can be assigned to the elements i, j, and k of quaternion algebra to construct a graph. Notable is a common ‘inverted state’ that is reachable by repeating any of the 6 intermediate sequences. This also happens to be the most distant state from solved - requiring 4 moves to solve.
By playing with the puzzle you can observe the parallels between quaternion algebra rules and the positions of the edge pieces.
