Technical Report CIS-2007-06

Title: Isoperimetrically Optimal Polyforms
Authors: Daniel Vainsencher and Alfred M. Bruckstein
Abstract: 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 redeļ¬ne 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.

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