2 days ago
Given that a circle is perimeter minimizing for given area in the Euclidean plane (i.e., R2), prove that a semicircle is perimeter minimizing in the Euclidean halfplane (i.e., just the part above the x-axis), and that a quarter circle is perimeter minimizing in the first quadrant.I have proven it. It is a very short proof.
Gaussian space is endowed with variable density, so that points near the origin are denser than points far away from the origin. The distribution follows a normal distribution (bell-shaped curve). So area is worth more near the origin, but perimeter also costs more near the origin. What is the perimeter-minimizing shape to enclose a given percentage (say, 10%, or 30%, etc.) of the Gaussian plane?Hint: it's a shape you've heard of.
We think it may be possible to organize geometry courses around a set of problems. This approach seems unusual. We have not collected enough problems for a complete course but we have started such a collection. We will share examples of problems we think are interesting, challenging and instructive.So before the conference I e-mailed them and told them that I had learned not only geometry, but also everything else after algebra and all the way up through BC calculus via a problem-oriented approach. I gave them the URL for the Exeter teaching materials and they wrote back and said something like "that's very interesting, thanks, I'll check it out sometime." They did not sound very interested. But when I went to talk to them on Saturday, they were very excited about the curriculum, said it was the best thing ever (an exaggerated paraphrase) and said that they were going to print out the whole thing (no joke, all 400 pdf pages). I told them it was most definitely the best math curriculum ever because people actually learn math, and they don't forget it, even over the summer, even now that I haven't done that math for two years I still remember the problems from ninth grade and the strategies to solve them. It is the best math curriculum ever. Don't waste your time with a textbook. Go straight to the heart of real mathematics and actually learn something. Please.