1. The problem is understanding how to write cube and tetrahedron rotations as permutations of vertices or faces given an axis and an angle.\n\n2. For a cube with an axis through opposite faces and a 90° rotation: the cycle is $Q=(1\ 2\ 3\ 4)$ representing the four corners moving around the axis.\n\n3. Powers of $Q$ represent multiples of 90°:\n$Q^2=(1\ 3)(2\ 4)$ swaps opposite corners,\n$Q^3=(1\ 4\ 3\ 2)$ cycles in the opposite direction,\n$Q^4=\mathrm{id}$ is the identity.\n\n4. For a cube with axis through opposite vertices and 120° rotation: the cycle is a 3-cycle $(a\ b\ c)$ representing three neighbors turning around a vertex.\nPowers:\n$(a\ b\ c)^2 = (a\ c\ b)$\n$(a\ b\ c)^3 = \mathrm{id}$.\n\n5. For a cube with axis through opposite edges and 180° rotation: the permutation is two disjoint transpositions $(a\ c)(b\ d)$ swapping pairs. The square is the identity.\n\n6. For a tetrahedron with axis through vertex 1 and opposite face 2,3,4:\n120° rotation is $R=(2\ 4\ 3)$,\n240° rotation is $R^2=(2\ 3\ 4)$,\n$R^3=\mathrm{id}$.\n\n7. To use these diagrams for any "shape + axis + angle":\n- Label moving items as 1,2,3...\n- Trace items around the axis in the rotation direction to form a cycle\n- Raise the cycle to the appropriate power matching the angle\n- Optionally, calculate order and parity.\n\nThis method turns geometric rotations into algebraic permutations allowing easy computation of powers and understanding cycle structure.