I have recently started to read a bit about geometry and topology. Hopf fibration, Lense spaces, CW complexes, stuff that are discussed in Hatcher's Algebraic Topology and other things that require good visualization. What is apparent to me is that the further I go, the less I understand what is going on. I have searched on YouTube and found some really nice animations for some of these topics but good animations are rare like gems.

Advanced stuff in mathematics are less discussed and available on the internet. I have realized that if I want to understand math one day, at some point I should be able to create my own animations. Now, my question is rather directed at people with experience in teaching advanced mathematics or currently doing research in mathematics in areas where geometric intuition is absolutely necessary. What kind of tools do you use? Do you develop them on your own in your research team/group? Can an independent person have access to them? Is it possible for an independent person to develop this kind of tools on their own?

Can you think of a situation where you couldn't understand a geometric concept visually but you created an animation that demystified it for you?

What is apparent to me is that the further I go, the less I understand what is going on.ObVonNeumann: "Young [one], in mathematics you don't understand things. You just get used to them." And while visualizations are useful, you might find that mathematicians are less reliant on them than you might guess. Another not-really-joke that's relevant here:Student:How do you visualize 6-dimensional space?Teacher:Oh, it's easy, you just visualize $n$-dimensional space and then set $n=6$. $\endgroup$ – Nate Eldredge Aug 8 '18 at 18:101more comments