Suppose you have a circle of radius ε. Now you want to inscribe a rectangle of arbitrary proportion into it. We'll have it be well behaved; it's aligned with the x and y axes and necessarily concentric with the circle. It touches the circle at 4 places, at angles we call ±θ and π±θ. Without loss of generality we let the longer sides be horizontal, so θ is between 0 and π/4.
Now draw a second circle, concentric with both previous shapes and within the original circle. We want to maximize the length of the circle's edge that falls inside the rectangle. What should the radius of the circle be?
I mean, of course it's εsinθ but how do you prove it?
Enters Nathan: "But you don't really need a proof. You can just multiply that by four and it's obviously an upper bound."
Duh.
That step had been hanging me up for a while today. I'm working at the moment on something called "concentric circle density." It has to do with taking a set of points on the plane and finding places you can draw lots of concentric circles, using only points in your set. That's all I'll say about it since a rigorous definition in text would be awful.
In fact, even Wilczynski's explanation on the board wasn't entirely clear. However, I came up with a definition based on what he said that may be what he meant. Really the only important thing is that I need to have worthwhile findings to show him when he returns.
The concentric circle density operator has sorta become my pet project while the other guys work on other problems Wilczynski posed. It's fun. I get to draw a lot of pictures and use old math like trigonometry and geometry along with the set theory. Some of the other things I find to be interesting, some are boring and some I do not understand. I like what we are doing, but I am also sure that I do not want to do measure theory for the rest of my life.
Relatedly, I did some more grad school researching today. It is overwhelming.
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