1. **Problem:** Write the permutation cycle of vertices for a cube rotated $90^\circ$ around an axis through opposite faces. 2. **Step 1:** Identify the four corners around the axis and label them as $1, 2, 3, 4$. 3. **Step 2:** The $90^\circ$ rotation moves each corner to the next, forming the cycle $$Q = (1\ 2\ 3\ 4).$$ 4. **Step 3:** Powers of $Q$ describe multiples of $90^\circ$ rotations: - $$Q^2 = (1\ 3)(2\ 4)$$ swaps opposite corners, - $$Q^3 = (1\ 4\ 3\ 2)$$ cycles corners in the opposite direction, - $$Q^4 = \mathrm{id}$$ is the identity (no change). 5. **Step 4:** For a cube rotated $120^\circ$ around an axis through opposite vertices, label neighbors $a,b,c$ around the vertex. - The rotation cycle is $$P = (a\ b\ c),$$ - Powers: $$P^2 = (a\ c\ b),$$ $$P^3 = \mathrm{id}.$$ 6. **Step 5:** For a $180^\circ$ rotation around axis through opposite edges, the permutation is the product of two disjoint transpositions: $$ (a\ c)(b\ d).$$ 7. **Step 6:** For a tetrahedron with axis through vertex $1$ and opposite face vertices $2,3,4$: - $120^\circ$ rotation cycle: $$R = (2\ 4\ 3),$$ - $240^\circ$ rotation: $$R^2 = (2\ 3\ 4),$$ - $360^\circ$ rotation: $$R^3 = \mathrm{id}.$$ 8. **Step 7:** To find permutation for any "shape + axis + angle": - Label moving vertices/faces as $1,2,3...$ - Trace their movement around the axis to form a cycle - Raise the cycle to the power corresponding to the angle - Analyze order and parity if needed. **Final Answer:** The rotation of a cube or tetrahedron about specified axes corresponds to permutation cycles that can be expressed algebraically, with powers representing multiple rotations and cycles encoding the motion of vertices or faces.