ABSTRACT: In 1917, a Japanese mathematician S. Kakeya asked the following question:
What is the smallest area of a planar region within which a line segment of length 1—a.k.a. a “needle” - can be turned around through 360 degrees, always remaining within this region?
A first example of such a figure that comes to mind is a circle of diameter 1: place the midpoint of the “needle” at the center and rotate it 360⁰ about the center. Thus, the area is π/4. A non-convex example could be a three-cornered hypercycloid (pictured) whose area is π/8.
But it turns out that such sets may actually have an arbitrarily small area. The proof relies on the construction of what is known as a Besicovitchset, a set of measure zero, which contains a unit line segment in every direction. In particular, one of the constructions that we will consider involves creating a “monster” with long “arms” and a tiny “heart” (M. Furtner).