Anonymous, that reminds me of some anecdote by Feynman where he has complex mathematical ideas described to him by young students. He wouldn't fully understand them, but he would imagine a shape, and for each new concept he'd add an extra bit, like a squiggly tail or other appendage. When something didn't fit in right, it would be instantly obvious to him, even if he couldn't explain exactly why.

Improvised sensory modality for maths?