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!


Steve USMA '85 said...

Wow, same thing happened to me 34 years ago with Calculus. Couldn't make sense of it for weeks and then all of a sudden something dropped into place and it all became clear. Went from doing C-F work to doing A-B work in a space of 24 hours. Almost scary.

socalmike said...

Happened to me, too. I took some calculus in college - kicked my butt - had no idea what the heck was going on. Brutal. Got to grad school, took a hydrology class, and presto! I had it. Is it a developmental thing? Not sure, but it's an odd thing.

PeggyU said...

Same thing happened to me with a discrete structures class - only it happened literally decades after the fact! I actually made it through the class with an A. This was a very uncomfortable feeling, since I knew I didn't really know what I was doing, but somehow managed to mimic the motions well enough to skate through unscathed. So, anyhow, one night my husband and I got into a conversation about things that make you say "WTF?" and I told him about my WTF quarter in that class. Upon which he, a software engineer, explained how useful that course would have been to him in his line of work - and it all suddenly made sense in light of the applications I had been unaware of. I felt like a complete dolt, but also was very relieved to have finally found some closure.