When: 
Thursday, November 6, 2014 - 12:15pm - 1:15pm
Where: 
Pardee 217
Presenter: 
Olena Kozhushkina
Price: 
Free & there will be pizza!

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 360 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).

 

Sponsored by: 
Department of Mathematics

Contact information

Name: 
c. jayne trent
Phone: 
610-330-5267
Email: 
trentj@lafayette.edu