The Euclidean plane consists of ordered pairs of real numbers p=(x,y) equipped with the well-known distance function d(p_1,p_2)=((x_1-x_2)^2+(y_1-y_2)^2)^{1/2}. An isometry is a self-map of the plane that preserves all distances between pairs of points.
After briefly classifying the isometries of the plane, I'll prove the following surprising result: If a self-map of the plane carries pairs of points that are distance one apart to pairs of points that are distance one apart, then the map is an isometry. That is, maps that preserve distance one preserve all distances!
Time permitting, I'll discuss the relationship of this problem to the more combinatorial problem of determining the chromatic number of the plane and some open problems.