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February 12th, 2017, 01:49 PM   #1
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Ideas for exploration in geometry

I'm taking a course called "College Geometry" right now at my university. It's a laid back course that essentially builds up Euclidean geometry from Euclid's axioms and then proceeds on to non-euclidean geometry and if I'm not mistaken we'll review a chapter at the end on something called "Hyperbollic Geometry". From what I've gathered, it's mostly a course to give insight into an early subject of mathematics and give math teachers the chops to deal with pesky questions from students.
We have an assignment called "exploration", in which we are given an unnaturally large amount of freedom. We are to simply explore the depths of geometry and write our thoughts on what we have found. Very vague, I know. My professor tells us that he is more concerned with us actually taking the time out of our day to explore concepts of geometry rather than actually coming up with concrete results. So here I am, not sure what to look into. What makes this seemingly easy task even more of a problem is that geometry (to my understanding) is a dead topic of math (like trigonometry). Maybe this is because I am only considering Euclidean Geometry. Can anyone help me on ideas for topics to look into? The only topic of geometry I've come to be interested in is differential geometry, but from my understanding, it is very different from the euclidean geometry that I'm studying right now. It still bears the title "geometry", so maybe it can still be a fruitful topic to explore. Any thoughts?

Last edited by skipjack; March 25th, 2017 at 08:09 PM.
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March 25th, 2017, 07:53 PM   #2
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Wasn't sure whether you're still looking for a topic to explore? You could look into the geometry of the snowflake; why are they all six-sided? That's just the first thing that came to mind, could be interesting...

Last edited by skipjack; March 25th, 2017 at 08:24 PM.
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March 25th, 2017, 09:09 PM   #3
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Geometry is extremely active in higher math. Not relevant to your course, but algebraic geometry, arithmetic geometry, and nonabelian geometry are three areas that come to mind.
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March 25th, 2017, 09:23 PM   #4
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the merging of geometry and quantum physics is a pretty fascinating topic if maybe a bit too advanced.

https://www.quantamagazine.org/20130...antum-physics/

and you can also google amplituhedron
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