I was going to post this as a comment on Jeff Carr's blog in response to his recent post, but it's a little lengthy for that. Plus it may be fairly interesting to some of you. So here we go.
I took a class called "Modern Geometry" at Augustana, which was based on assumptions seemingly contradictory to some of the things we observe on a day-to-day basis. Example: given 2 points, there is one and only one line which intersects both points. This seems fairly obvious if you think about it. Euclidean Geometry (which is what most people take in high school) is based on a lot of those seemingly obvious assumptions. But try and create proofs based on the the assumption that, given 2 points, there is at least one line that intersects both points. This makes things much more subjective. There could be one line, there could be 10 lines, there could be an infinite number of lines through those two points. You can only imagine some of the crazy things we had to prove. The most interesting part is that, as you shrink the limits of the area you're working in, the "universe" becomes more and more like the one created by Euclidean Geometry. So imagine how large our universe is, and imagine how comparitively small we are. Maybe that's why Euclidean Geometry makes so much sense.
If you didn't understand any of that, than I apologize. But you didn't have to read it! Back to work.
I took a class called "Modern Geometry" at Augustana, which was based on assumptions seemingly contradictory to some of the things we observe on a day-to-day basis. Example: given 2 points, there is one and only one line which intersects both points. This seems fairly obvious if you think about it. Euclidean Geometry (which is what most people take in high school) is based on a lot of those seemingly obvious assumptions. But try and create proofs based on the the assumption that, given 2 points, there is at least one line that intersects both points. This makes things much more subjective. There could be one line, there could be 10 lines, there could be an infinite number of lines through those two points. You can only imagine some of the crazy things we had to prove. The most interesting part is that, as you shrink the limits of the area you're working in, the "universe" becomes more and more like the one created by Euclidean Geometry. So imagine how large our universe is, and imagine how comparitively small we are. Maybe that's why Euclidean Geometry makes so much sense.
If you didn't understand any of that, than I apologize. But you didn't have to read it! Back to work.
2 Comments:
you are the coolest.
i understood all of that. i just taught a lesson on spherical geometry.
you should watch the sophomores heads explode when you them that they can make triangles with three 90 degree angles. its awesome.
I assumed that you would understand, Tree! I think we called it Hyperbolic Geometry or something though. Same thing. Poincare Disc!
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