The Mathologer attacks the hundred-year-old Kakeya needle problem with his trusty squeegee: What is the smallest amount of area required to continuously rotate a (mathematical) needle in the plane by 180 degrees? The surprising answer is the starting point for a huge amount of very deep mathematics. For the really intrepid amongst you here is a survey by Australian Fields Medalist Terry Tao: http://www.ams.org/notices/200103/fea...
And here is the link to the Numberphile video mentioned in our video:    • Kakeya's Needle Problem  - Numberphile  

Enjoy :)