
Visual Group Theory - erwan
http://web.bentley.edu/empl/c/ncarter/vgt/index.html
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erwan
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.

