I began studying 3-manifolds after coming up with a novel way I preferred to draw their presentations. All approaches are formally equivalent, but they impose different cognitive loads in practice. My approach was trivially equivalent to triangulations, or spines, or Heegaard splittings, or ... but I found myself far more nimbly able to "see" 3-manifolds my way.

I showed various colleagues. Each one would ask me to demonstrate the equivalence to their preferred presentation, then assure me "nothing to see here, move along!" that I should instead stick to their convention.

Then I met with Bill Thurston, the most influential topologist of our lifetimes. He had me quickly describe the equivalence between my form and every other known form, effectively adding my node to a complete graph of equivalences he had in his muscle memory. He then suggested some generalizations, and proposed that circle packings would prove to be important to me.

Some mathematicians are smart enough to see no distinction between any of the ways to describe the essential structure of a mathematical object. They see the object.