A Kakeya set in R^n is a set of points that contains a unit line segment in every direction. We study the structure of Kakeya sets in R^3 and show that  for any Kakeya set K, there exists well-separated scales 0<\delta<\rho\leq 1 so that the \delta-neighborhood of K is almost as large as the \rho-neighborhood of K.  As a consequence, every Kakeya set in R^3 has Assouad dimension 3.  This is joint work with Josh Zahl.