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.<p>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.<p>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.<p>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.