A **Besicovitch set** is a set in the plane (or in higher dimensions) which
contains at least one unit line segment in every direction. The triangle
pictured below forms one quarter of a Besicovitch set, if its base and height
are exactly one unit long: you can
place a unit line segment in any direction between
N-S and NW-SE (which forms a quarter of all possible orietnations), simply by placing one end at the
top corner.

Around the turn of the century, it was thought that there was a certain minimum
size to Besicovitch sets; that you could never have a Besicovitch set which
had an area smaller than `pi/8`. (In particular, you could never have
a quarter-Besicovitch set whose area was smaller than `pi/32`).
However, this conjecture was **disproved** by A.S. Besicovitch, who showed
that you could have a Besicovitch set of arbitrarily small area (or even zero
area)!

The applet below gives one version of Besicovitch's construction. Basically
the idea is to cut up the triangle you see below and shove the pieces together
so that there is a lot of overlap. There are two parameters: `n`,
which controls how many times you cut the triangle up, and `alpha`,
which controls how much you shove things closer together. By choosing
the two parameters carefully, you can make a quarter-Besicovitch set
whose area is as small as you please. By putting four of these
quarter-Besicovitch sets together, you can manufacture a genuine
Besicovitch set with arbitrarily small area.

In this demo, `n` can be set to any integer from 0 to 9, and
`alpha` can be any number from 0.5 to 1. You have to click
the "Redraw" button to actually redraw the Besicovitch set.
A crude upper bound
for the area of the set is also given at the bottom of the applet;
it is off by a factor of two or so, but I'm too lazy to find the exact
area. I suggest setting `alpha` to 0.8 and incrementing
`n` from 0 to 9 to get an idea of the construction.

Details of the construction (together with applications to summation of multiple Fourier series) can be found in

- E.M. Stein, "Harmonic Analysis", Princeton University Press, 1993, Chapter X.