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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 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