Chromatic Number of the Plane

How many colors are needed to paint the plane so that no two points a unit distance apart are painted the same color? If the same question is asked of the line, the answer is $2$: Coloring $[0,1)$ red, $[1,2)$ blue, etc., ensures that no two unit-separated points have the same color. One can view the question as asking for the chromatic number $\chi(\mathbb{E}^2)$ of the infinite unit-distance graph $G$, with every point in the plane a vertex, and an edge between two vertices if they are separated by a unit distance.