I've recently read up on 2D and 3D finite rotation groups. But I'm struggling to find resources on the analogous results for 4D and I'd appreciate some help finding answers/resources.
For context: In 2D all finite rotation groups are cyclic, $C_n$, and are the rotations of an n-gon.
In 3D, what I found interesting/surprising is that there aren't really that many more finite rotational symmetry groups, in fact there are only 3 more than the circular ones you got in 2D. These "circular" ones inherited from 2D are $C_n$ again (for the cone of an n-gon) and also the dihedral group $D_n$ (for the prism of an n-gon), which is like $C_n$ but you can also flip over (if you lived in 2D this would look like a reflection, but it's a rotation in 3D).
In the above, orbits of single points live on one or two circles, and so they aren't really spherical rotation groups. There are only three spherical ones (where orbits of points live on 3D spheres) are the rotational symmetries of the platonic solids: the tetrahedron, cube (or its dual the octahedron), and the dodecahedron (or its dual the icosahedron).
I'm struggling for resources, online or in books, for what this looks like in 4D? Is it also the case that there aren't that many more over 3D, or does it explode into lots more? Are the only new finite rotation groups in 4D those of the 4D regular polytopes, or does other new and interesting things creep in? Where might I read about it? Any thoughts on dimensions 5 and up? Thank you!
In $4D$ you can use quaternions:
$q=a+bi+cj+dk$ where $i^2=j^2=k^2=ijk=-1$
In quaternions the product in non-commutative $(q_1q_2\neq q_2q_1)$
The product of the quaternion $q_1=x_1+y_1i+z_1j+t_1k$ for the quaternion $q_2=x_2+y_2i+z_2j+t_2k$ represent the rotation. You can find a good explanation here:
https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation#Using_quaternions_as_rotations