Today Vojislav and I were working in the math library on topology, and Professor Morgan was talking with us as he was about to leave. Then it apparently occurred to him that Vojislav is from last year's Geometry Group, I am going to be in this year's Geometry Group, and he was going to an open house hosted by a member of the 1990 Geometry Group (the one that published the paper proving that the double bubble is perimeter-minimizing in R2 in Euclidean space). So he invited us to come along. So half an hour later, Vojislav and I were on our way to Pownal, VT where we got to see this guy's new house. It was a nice house, with a big field going down to a marsh. And Professor Morgan took us to get ice cream afterwards. I had graham cracker flavored ice cream (on his recommendation). It was yummy.
We went out to look at their field, which slopes down towards the marsh, and I found myself trying to figure out what had happened in the past to make it look like this, just as we did at the Mountain School. I determined that the part by the house had been a field for a while, because it was flat and mown. There were about six trees dispersed through this field, and since they all looked different, I decided that they were different species of trees that had been planted, rather than growing naturally from seeds. On the edge of the field at the bottom there was a peculiar swath leading down to the marsh with no trees on it, which really perplexed me. To the left was a big tree on the corner, which means that it was on the edge of the field, and behind the big tree to the left were trees about 10 inches in radius, which were all about the same size. By contrast, on the right of the cleared swath was an area full of little trees, pretty tall (20-30 feet) but all under 4 inches in diameter. So I figured that area had been pasture much longer than the part on the left, and had at some point been left to grow into trees, at which time all the little trees grew in.
For anyone who kept reading, try this, which is our Geometry Group homework:
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.
If you finish that, try this:
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.
Next hint: It's not a circle. But that was a good guess.