But if this is beautiful to behold, and a mathematician uses a formula to fully describe their movements with exact precision, wouldn't he be no different than a painter or an artist who each use their own tools to describe the movements? A mathematician who uses formulas to describe reality, then, is an artist as well.

I see what you're saying. I wonder, are you familiar with the term, "ekphrasis"?

For me, I would have to say that art is not about describing reality, but rather interpreting, re-interpreting, contextualizing and playing with reality where as math is much more about what things are, are not, could, or could not be. Certainly, I think that math, science, art, language, music and religion all overlap often and often quite freely, but partly because the people performing those actions, creating those works and ultimately digesting them, are human and imperfect in broadly similar ways.

Another thing is intent. Should a mathematician derive certain formulas and use them to convey something more than the factual, something interpretive, something that reveals an element of the human experience, then yeah, I think that would be art. But, if the mathematician sets out simply to describe reality, then personally, I don't think I would consider that art.