In 1917, Sochi Kakeya formulated the space dominated by a samurai's sword during battle as: a subset of the plane in which a unit line segment may be continuously rotated all the way around. An n-dimensional Besicovitch set is the weaker notion of a subset of R^n containing a unit line segment in every direction. The Kakeya Conjecture predicts the "dimension" of the contents of Besicovitch sets.
While this conjecture remains far out of reach, an analogue in finite fields was proposed by Wolff in 1999, and independently came up in computer science in 2003. It was finally resolved by Dvir in 2008 using a short elegant technique. We will present this proof and construct the smallest known Kakeya sets in every finite field.