Proof that \(\mathrm{SO}(3)\) is diffeomorphic to the projective space \(\mathbb{R}\mathrm{P}^3\), which has fundamental group \(\mathbb{Z}_2\). Proof that the universal cover of \(\mathrm{SO}(3)\) is \(\mathrm{SU}(2)\).
Picture proof that the fundamental group of \(\mathrm{SO}(3)\) is \(\mathbb{Z}_2\): this is a fancy version of the belt trick.
Note that after the cube rotates 720°, the belts are back to their original positions!
The root vectors of \(\mathrm{Spin}(8)\), the double cover of the rotation group in 8 dimensions.
© 2021 John Baez
baez@math.removethis.ucr.andthis.edu