I have a hard enough time with Euclidean geometry. Especially in my current class, wherein we're proving a number of Euclid's propositions, I can't keep track of them in order, which means I don't know "what I'm allowed to know" and/or use for each proposition; for example, I can't use Proposition 32 to prove Proposition 29. It's driving me nuts.
But at least Euclidean geometry makes some sense to me. It "exists" in the same world I do. Non-Euclidean geometry is just insane. I can't make heads or tails of it.
Last night I was working on a non-Euclidean proof--I had to show that the summit angles of saccheri quadrilaterals are equal. Can I use the "fact" that diagonals or rectangles are equal, or not? Am I "allowed" to know that for this proof, or not? If so, the proof is trivial. If not... I punted, gave it up for the night.
This morning I was thinking about someone I used to know who died yesterday--coincidentally, a former geometry teacher. Out of nowhere the proof jumped into my mind, I'll give it a shot when I get home this afternoon. I think I can do it without assuming the diagonals are equal (I should be able to conclude they are using congruent triangles).
I'm reminded of a time 33 years ago, when I was first taking calculus. I beat my head against the wall and did my homework by following the example problems in the book without understanding what I was doing. This went on for a few weeks, I'd never not understood a subject before. It was a most disconcerting feeling. One night, though, I went to bed, not even thinking about calculus, and when I woke up it all just made sense to me. I have no idea how.
The subconscious mind is a scary and powerful thing indeed!