In a unit circle, flip the four quarter-circles inside the circle. Draw $n$ line segments from the centre to the flipped quarter-circles, so that the angles between neighboring line segments are equal. (Any collective orientation of the line segments will do.)
It can be shown algebraically that the product of the lengths of the line segments aproaches $2^{-n}$ as $n\to\infty$.
Is there an intuitive explanation for this elegant property? The numbers are so simple ($1$ and $2$) that I wonder if there is some hidden, intuitive reason for this result. I kind of doubt that there is, but I don't want to dismiss the possibility too quickly.
To give you an idea of what I mean by an intuitive explanation, here is an example. On the graph of $y=\tan{x}$, $0<x<\pi/2$, draw $2n$ zigzag line segments that, with the x-axis, form equal-width isosceles triangles whose top vertices lie on the curve. Here is an example with $n=6$.
As $n\to\infty$, the product of the lengths of the line segments converges to some positive number. Amazing? Not really: there is an intuitive explanation why it should converge. Almost all of the lengths of the zigzag line segments are very close to the value of $\tan{x}$ at odd multiples of $\frac{\pi}{2n}$ between $0$ and $\frac{\pi}{2}$. These values can be paired together, taking them from the ends and working towards the middle. Each pair's product is $1$, because they are reciprocal ratios of side lengths in a right triangle. So it is reasonable to expect (but not with certainty) that the product of the lengths of the zigzag line segments converges. (As to what it converges to, that is certainly not obvious, but it can be found.)
Anyway, going back to the circle: is there an intuitive explanation for why the product of lengths converges to reciprocal powers of $2$?
Or equivalently, is there an intuitive explanation for why, in a circle of radius $2$, the product of lengths converge to $1$? (It takes its time getting there: with $n=4, 40, 400, 4000, 40000$, the products are $16, 2.52, 1.34, 1.10, 1.03$ to 2 decimal places, respectively.)
