Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a Kakeya set, or Besicovitch set, is any set of points in Euclidean space which contains a unit line segment in every direction. While many types of objects satisfy this property, several interesting results and questions are motivated by considering how small such sets can be. Besicovitch showed that there are Besicovitch sets of measure zero. A Kakeya needle set (sometimes also known as a Kakeya set) is a (Besicovitch) set within which a unit line segment can be rotated continuously through 180 degrees, returning to its original position with reversed orientation. Besicovitch showed that there are Kakeya needle sets of arbitrarily small positive measure.