A recent Radiolab episode on Numbers had some interesting content, including an introduction to Benford's Law, unfortunately presented as something utterly mysterious and only comprehensible to mathematicians. I don't think it would be too terribly oversimplifying to explain it with dice. If you throw a normal die, the first digit of the results will tend to be evenly distributed among numbers from 1 to 6. A ten-sided or twelve-sided die will favor the initial digit 1. This effect will be even more pronounced with a 20-sided die, which will also show a bit of favoritism toward an initial 2.
There was also a short profile of Paul Erdös, with the first recording I've heard of his voice and that of his biographer, Paul Hoffman.
Somewhere in the story an elliptical swimming pool was mentioned. I thought it was going to be something similar to the elliptical billiard table, with waves instead of balls converging on a focus, but it turned out to be a clever problem. If the elliptical swimming pool has a uniformly wide border around the edge, is the outer edge of the border also an ellipse?
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My intuition was that outer edge obviously and trivially is an ellipse. It just seems like it has to be.
But if my calculations are correct then my intuition is a dupe.
We can let the pool be an ellipse centered at the origin. Then its equation is:
x^2/a^2 + y^2/b^2 = 1.
If the outer edge is an ellipse, then its equation must be:
x^2/(a + k)^2 + y^2/(b + k)^2 = 1,
where k is the width of the border (consider the points of intersection withe the x and y axes).
Then for the outer edge to be an ellipse, we need that for each (x1, y1) on the outer ellipse and (x, y) on the inner ellipse, sqrt(x1^2 + y1^2) = sqrt(x^2 + y^2) + k.
I don't feel like reproducing the calculations here, but it turns out those two things aren't equal everywhere. Hence the outer edge can't be an ellipse.
Here's an intuitive approach, and therefore not rigorous but pretty convincing to me.
If the elliptical swimming pool is circular, the outer edge of the border is also a circle and therefore an ellipse.
So imagine a pool with an extremely eccentric elliptical shape. It looks like a toothpick -- very long and narrow. The outer edge of the border looks like a frankfurter, long and almost uniformly wide, with round, almost semicircular ends. Exxagerating the eccentricity makes the two "almosts" more nearly true.
The frankfurter shape already seems unelliptical. Recall the principle that the sum of the distances from any point on an ellipse to the foci is constant (and the trick of drawing an ellipse with two thumbtacks, a loop of string and a pencil). Imagine the pencil point at one end of the frankfurter, the loop of string wrapped around the far focus. Now trace around the rounded end until the pencil point is at a right angle to the long axis. If the end is semicircular, there's enough string for two strands to go to the near focus and both to go across to the far focus. If the frankfurter shape were an ellipse one side of the string loop would go straight to the far focus.
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