Today I watched another Numberphile video and read more of Infinite Powers.
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Having had a good day on Monday, I didn’t manage to do any study at all on Tuesday. My mind was on another project and time got away from me.
Today I tried to continue working on the proposition about trapezoids, but found I didn’t have enough mental energy. I relistened to more of A Brief History of Mathematics instead, hearing about Fourier and Galois. (Galois is one of math’s romantic heroes. Political radical, briefly imprisoned for his republican activities. Killed in a duel at 21, but leaving highly original work behind.) Made curious about Galois’s work, I left off listening and went to look him up in a lovely reference book I have called The Princeton Companion to Mathematics. I read the article on Galois himself and part of the article covering the branch of math called Galois theory. I did lose the thread of that article after a while.
In other news, I’m currently working through the involved application process for a volunteer position I hope to get. There are many applicants, so there is a good chance I won’t be selected. If I am, though, I’m not sure what it will mean for my math review project and this blog. I’m not sure what the future of those things should be, in any case. Despite my good day on Monday, I still feel as if I’ve lost something I need to sustain them.
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Today I had fun working on a geometric proposition about trapezoids and reading some mathematical correspondence. I may do more math activities later, as well.
The proposition I worked on, which comes from Elementary College Geometry, is that the base angles of an isosceles trapezoid are equal. (That book defines a trapezoid as a quadrilateral with exactly two parallel sides. An isosceles trapezoid is a trapezoid that has legs of equal length. The legs are the non-parallel opposite sides.)
I drew a number of pictures trying to find a good way to prove this proposition.
The first one shows the approach I came up with when I first looked at the problem some time ago: construct isosceles triangles on either end of the trapezoid, show those are congruent (likely requiring some lemmas), use alternate interior angles. I think this will probably work, but it will need some details ironed out. For instance, what if $\angle ABC$ is acute? (It can’t actually be with Elementary College Geometry’s definition of a trapezoid and $AD$ equal to $BC$, but I may need to prove that.)
This method is not very elegant, even ignoring the niggly details, so before working those out, I decided to see if I could find something better. In the process, I came up with the figures below. None of these approaches has born fruit so far, though.
In the first drawing, I tried the most basic quadrilateral proof move, but just went around in circles. The second drawing didn’t really go anywhere at all. Something resembling the third drawing will be useful for proving the converse of the proposition, I think, but didn’t yield a proof in this direction.
I’m feeling much encouraged and less inclined to simply drop this project, which I had been resisting the urge to do.
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Today I watched this fun video from Numberphile. I watched it earlier in the month—it’s the one I stopped so I could investigate its topic a bit myself—but I had since forgotten the details. I’d recommend it to all of my readers. It is very accessible but has connections to more advanced topics. When I am feeling better, I would like to work on proving the theorem presented at the end of the video.
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Today was a bumpy day, and I didn’t manage to do much math. I did read for a while at the Math Help Forum and in some of the Reddit math communities. I was looking to see if answering questions in one of those venues might be a good way to share my math knowledge with others. I’m not feeling sharp enough to pursue online tutoring at the moment.
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Hello, faithful readers. The circumstances that caused me to put the blog on hold last week have eased, so here I am again. I’m still having a crisis of confidence in my ability to continue this project, but hopefully that can be resolved, as well.
Today I worked for a short time on some past American Mathematics Competition problems. I also thought for a while about how to prove Notion 3 from my post Fun With Logarithms, since I’d used it while solving one of the AMC problems.
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Today was bad. I read for a few minutes, but I didn’t do any other math.
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Today I worked on questions from the video that I watched part of on Thursday. Having answered those to my satisfaction, I watched the rest of the video, which raised further questions. I hope to share some of that next week.
I think a recent change in my medication may be make it harder for me to work on math, which is very frustrating. I hope to have it changed again soon with better results.
