## Wednesday, September 23, 2015

### Saccheri Quadrilaterals, A Calculus Coincidence

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!