The Number of Circles Covering a Set

Abstract

It is well known that the best way to cover a given area with circles of a given radius E, or to pack such circles within a given region is to place the centers of the circles on an equilateral triangle network, i. e., to circumscribe (inscribe) the circles about the hexagons of a regular hexagon network or honeycomb. This, of course, is not a precise statement, and, in fact, it is difficult to make a precise statement in this direction that is true. Roughly, the statement becomes more true as E is taken smaller in relation to the area of the given region. The most usual 1 precise statement of this fact is that the densest plane Punktgitter is that of the equilateral triangle. This statement avoids the difficulties caused by the boundedness of the bounded region but is less general than might be desired in that permissible packings or coverings are limited to those in which the centers of the circles form a Punktgitter. The object of the present paper is to give, a new aiid elementary proof of a precise statement in this direction; a statement involving no restriction on the nature of permissible coverings or on the nature of the given region. Specifically the statement to be proved is the following: