A comment on my last YouTube video pointed out that the SL(2,3) group expressed by the Quaternion Cube is isomorphic to the group formed by multiplication of the unit Hurwitz quaternions.
Simulator
‘Order’
There are five different orders of unit Hurwitz quaternions: 1, 2, 3, 4, and 6; On the puzzle, the order represents how many times a sequence must be repeated to return to where you started.
Square roots
An unusual property of the unit Hurwitz quaternions is that every order 3 element has a unique square root of order 6. For the single left turn:
L→½(-1+i+j+k)(order 3)
Its square root is:
R̅LR̅→½(+1+i+j+k)(order 6).
Repeating R̅LR̅ twice has the same effect as L.
Selected sequences
| Sequence | Quaternion | Order |
|---|---|---|
| Identity | 1 | 1 |
| L | ½(-1+i+j+k) | 3 |
| R | ½(-1-i-j+k) | 3 |
| R̅LR̅ = sqrt(L) | ½(+1+i+j+k) | 6 |
| L̅RL̅ = sqrt(R) | ½(+1-i-j+k) | 6 |
| RL̅ | i | 4 |
| L̅R | j | 4 |
| RLR or LRL | k | 4 |
| RL̅RL̅ or L̅RL̅R | -1 | 2 |
