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Isoperimetrically Pareto-optimal shapes on the hexagonal grid

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 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. In this paper we characterize all shapes that have both shortest boundaries and maximal areas for one particular boundary measure on the hexagon tiling. We show this set of Pareto optimal shapes is the same as that induced by a different boundary measure that was studied in the context of theoretical chemistry.