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!
3 comments:
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.
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.
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.
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