Date: Fri, 08 Oct 2000 10:27:06 -0400
From: Alex Bogomolny
It was a conjecture by Kakeya that deltoid has the smallest area needed for a line segment to turn 360 degrees.
Besicovitch showed that there is no least area. In other words, for any given number, however small, there exists a figure of that area such that a given segment may be moved without crossing the figure's border and at the end to get rotated by 360 degrees.
All the best,