On isoperimetrically optimal polyforms

In the plane, the way to enclose the most area with a given perimeter and to use the shortest perimeter to enclose a given area, is always to use a circle. If we replace the plane by a regular tiling of it, and construct polyforms i.e. shapes as sets of tiles, things become more complicated. We need to redefine the area and perimeter measures, and study the consequences carefully. A spiral construction often provides, for every integer number of tiles (area), a shape that is most compact in terms of the perimeter or boundary measure; however it may not exhibit all optimal shapes. We characterize in this paper all shapes that have both shortest boundaries and maximal areas for three common planar discrete spaces.