Minimum Area Polyomino Venn Diagrams

Polyomino Venn (or polyVenn) diagrams are Venn diagrams whose curves are the perimeters of polyominoes drawn on the integer lattice. Minimum area polyVenn diagrams are those in which each of the 2n intersection regions, in a diagram of n polyominoes, consists of exactly one unit square. We construct minimum area polyVenn diagrams in bounding rectangles of size 2r×2c whenever r, c ≥ 2. Our construction is inductive, and depends on two “expansion” results. First, a minimum area polyVenn in a 22×2c rectangle can be expanded to produce another one that fits into a 22×2c+3 bounding rectangle. Second, a minimum area polyVenn diagram in a 2r × 2c rectangle can be expanded to produce another one that fits into a 2r+1 × 2c+1 rectangle. Finally, for even n we construct n-set polyVenn diagrams in bounding rectangles of size (2n/2 − 1)× (2n/2 + 1) in which the empty set is not represented as a unit square.