Face-balanced, Venn and polyVenn diagrams




Bultena, Bette

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A \emph{simple} $n$-\emph{Venn diagram} is a collection of $n$ simple intersecting closed curves in the plane where exactly two curves meet at any intersection point; the curves divide the plane into $2^n$ distinct open regions, each defined by its intersection of the interior or exterior of each of the curves. A Venn diagram is \emph{reducible} if there is a curve that, when removed, leaves a Venn diagram with one less curve and \emph{irreducible} if no such curve exists. A Venn diagram is \emph{extendible} if another curve can be added, producing a Venn diagram with one more curve. Currently it is not known whether every simple Venn diagram is extendible by the addition of another curve. We show that all simple Venn diagrams with $5$ curves or less are extendible to another simple Venn diagram. We also show that for certain Venn diagrams, a new extending curve is relatively easy to produce. We define a new type of diagram of simple closed curves where each curve divides the plane into an equal number of regions; we call such a diagram a \emph{face-balanced} diagram. We generate and exhibit all face-balanced diagrams up to and including those with $32$ regions; these include all the Venn diagrams. Venn diagrams exist where the curves are the perimeters of polyominoes drawn on the integer lattice. When each of the $2^n$ intersection regions is a single unit square, we call these \emph{minimum area polyomino Venn diagrams}, or \emph{polyVenns}. We show that polyVenns can be constructed and confined in bounding rectangles of size $2^r \times 2^c$ whenever $r, c \ge 2$ and $n=r+c$. We show this using two constructive proofs that extend existing diagrams. Finally, for even $n$, we construct polyVenns with $n$ polyominoes in $(2^{n/2} - 1) \times (2^{n/2} + 1)$ bounding rectangles in which the empty set is not represented as a unit square.



Venn diagram, graph theory, computational geometry, minimum area Venn diagram, Winkler's conjecture