I think there is a lot of value in working on visual/reactive representation of algebraic structures. I have stopped counting the number of times I spent toilin

I think there is a lot of value in working on visual/reactive representation of algebraic structures. I have stopped counting the number of times I spent toiling on a result to understand its "depth" only to realize that much of that so-called depth comes from the narrow perspective from which I am contemplating it. A good amount of undergraduate mathematics is exactly that: working on fundamental results long enough and deeply enough that you get a coherent mental model that lets you manipulate, compose, and spot them. It seem to me that most of the attempts at visual mathematics fall short of not being general enough to capture different sorts of objects or ideas, or not in a way that is visually helpful anyway. And I'm thinking maybe this has more to do with the expressiveness of the visual medium rather anything else. Maybe images/visuals are just not powerful enough to map with these objects, maybe you need something at least as powerful as the "physical world" (albeit with different law of physics - dependent on the system you are studying). Example: the same way you cannot match the language of matched parenthesis with anything less powerful than a CFG. Now, this is speculative and a conjecture: suppose that you have a perfectly immersive VR world, then could you find a way to perfectly match the axiomatic system+semantics/syntax of a mathematical object (e.g a Field) to a visual representation along with some physical rules (e.g 1+1=0) such that your actions in that world are always sound and correct with respect to the mathematical object it maps to.