Minimum Area Polyomino Venn Diagrams
Bette Bultena,
Department of Computer Science,
University of Victoria, Canada.
Matthew Klimesh,
Jet Propulsion Laboratory,
Pasadena, California, USA.
Frank Ruskey,
Department of Computer Science,
University of Victoria, Canada.
Abstract:
Polyomino Venn (or polyVenn) diagrams are Venn diagrams whose curves are
the perimeters of orthogonal polyominoes drawn on the integer lattice.
Minimum area polyVenn diagrams are those in which each of the
2^{n} regions, in a diagram of n polyominoes,
consists of exactly one unit square.
We construct minimum area polyVenn diagrams in bounding rectangles of size
2^{r} X 2^{c} whenever
r, c > 2.
Our construction is inductive, and depends on two "expansion" results.
First, a minimum area polyVenn diagram in a
2^{r} X 2^{c} rectangle can be
expanded to produce another that fits into a
2^{r+1} X 2^{c+1} rectangle.
Secondly, when r = 2, it can also be expanded to produce a polyVenn
diagram in a
2^{r} X 2^{c+3} bounding rectangle.
Finally, we construct polyVenn diagrams in bounding rectangles of size
2^{n/2 1} X 2^{n/2 +1} if n is even,
but where the empty set is not represented as a unit square.

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Appears in Journal of Computational Geometry,
(3)1 (2012) 154167.

Mark Thompson's page on
Venn
Polyominos is what got us started on this research.
Unfortunately, his page seems to no longer be active, but we
have reproduced it
here using the
"Wayback Machine".

For more on Venn diagrams check out the Survey
of Venn Diagrams, Dynamic Survey #5 at the
Electronic Journal of
Combinatorics.
Selected papers that refer to this one:
Six and Seven Set PolyVenn Diagrams
Back to list of publications.