# Besicovitch set

A Besicovitch set (also called Kakeya set) is a set in the plane which contains at least one unit line segment in every direction[1].

The right triangle with unit base and height makes one quarter of the set

The right triangle (n=0 in the applet) forms one quarter of a Besicovitch set, if its base and height are exactly one unit long. It is possible to place a unit line segment in any direction between N-S and NW-SE, that forms a quarter of all possible orientations, simply by placing one end at the top corner. Another example of Besicovitch set with rotating needle is described in [2]. Besicovitch set can also be defined for higher dimensions.

Around the turn of the century, it was thought that there was a certain minimum size to Besicovitch sets. This minumum was expected to be somewhere abount / 8 and surely above / 32. However A.S. Besicovitch has disproved this, showing that you could have a Besicovitch set of arbitrarily small (even zero!) area.

Building small Besicovitch set by cutting the triangle. n controls how many times you cut the triangle up, alpha controls how much you shove things closer together. By default, triangle is cut once (n=1), set n=0 to see it intact. We suggest setting alpha to 0.8 and incrementing n from 0 to 9 to understand an idea of the construction. An approximate upper bound for the area of the set is also given at the bottom of the applet.

## Possible way of building a set with arbitrary small area

One of the possible ways to construct a small Besicovitch set is to cut up the triangle and shove the pieces together so that there is a lot of overlap. From the applet[3] demo it is seen that by choosing the two parameters carefully, it is possible to make a quarter-Besicovitch set whose area is as small as you like. By putting four such quarter-Besicovitch sets together, you can manufacture a complete Besicovitch set with arbitrarily small area.

## References

1. 1 E.M. Stein, "Harmonic Analysis", Princeton University Press, 1993, Chapter X.
2. 2 Kakeya set article in Wikipedia.
3. 3 Besicovich set demo in Terence Tao project