Given a three dimensional tensor T, how can I figure out:
- How many 'fundamental' states T has? By 'fundamental' I mean states needed to derive all other possible states of the T by rotating it arbitrarily along x, y and z axis by 90° increments.
- Given a particular state of T, how can I figure the number of different states that can be derived by rotating it?
The above is my best try at generalizing the question. Now I'm going to give a specific example.
Consider a 2x2x2 tensor holding boolean [0, 1] values. Such a tensor has $2^8$ possible states. Imagine it as a 3d cube composed of 8 cells. This cube can be rotated in euclidean space by 90° steps along any axis. Some examples:
- A uniform tensor with all values equal either 0 or 1 can not be used to derive other states by rotating it.
- The below tensor can be used to derive 7 other states $$ \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix} \begin{bmatrix} 0 & 0 \\ 0 & 0 \\ \end{bmatrix} $$
- The below tensor can be used to derive 11 other states $$ \begin{bmatrix} 1 & 0 \\ 1 & 0 \\ \end{bmatrix} \begin{bmatrix} 0 & 0 \\ 0 & 0 \\ \end{bmatrix} $$
- The below tensor can be used to derive 3 other states $$ \begin{bmatrix} 0 & 0 \\ 1 & 0 \\ \end{bmatrix} \begin{bmatrix} 0 & 1 \\ 0 & 0 \\ \end{bmatrix} $$
PS: I would like to, in advance, apologize to any mathematicians offended by my uninformed, crude and imprecise definitions above. I'm a mere engineer, have mercy.