We consider a variation on the problem of determining the chromatic number of the Euclidean plane and define the -unit distance graph to be the graph whose vertices are the points of E², in which two points are adjacent whenever their distance is within of 1. For certain values of we are able to show that the chromatic number is exactly 7. For some smaller values we show the chromatic number is at least 5. We offer a conjecture, with some supporting evidence, that for any > 0 the chromatic number is 7.