Were does the notation T^2 for oriented real projective space come from? That's just bad, because it is not a torus but a sphere, and the two are topologically very different!

Very nice "visual" introduction a topic that's usually treated very abstractly in math textbooks! If you'd like more of such a visual perspective on differential geometry, I recommend Tristan Needham's book [1].

[1]: https://press.princeton.edu/books/paperback/9780691203706/vi...

I'm not sure if this is the main point of the article (but it does at least mention it), a good intuitive visualization of how the sphere can be thought of as asymmetric is the Hairy Ball Theorem[0], which basically states: "you can't comb a hairy ball flat without creating a cowlick"

[0]: https://en.wikipedia.org/wiki/Hairy_ball_theorem

No mention of quaternions and SLERP?

Something that may be of interest to CS people and reveals the complexity of the unit sphere is the following problem:

Find an efficient (class of) algorithm(s) to select a large number N of uniformly distributed points on S^2, where "uniformly distributed" is given in a more flexible sense than the usual one. For instance, you may want to minimize the Weyl discrepancy between average and integral, or you may want to focus more on minimizing the number of ε-clusters of distances.

One of the most elegant approaches to this problem is the classic work of Lubotzky, Phillips and Sarnak: Hecke Operators and Distributing Points on the Sphere I and II. They translate the problem to one of generating good sequences of elements of SO(3), which they attack with a combination of harmonic analysis on the semisimple groups, homogeneous dynamics and number theory with Hecke operators as their central tool.