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