A Venn Diagram of Eight Quadrilaterals

We construct eight quadrilaterals in the plane, with integer vertices, whose interiors and exteriors divide the plane into exactly 256 connected regions, one for each inside/outside pattern. The arrangement is a simple Venn diagram: the boundaries cross exactly 254 times and no point lies on three of them. Each quadrilateral has one reflex corner, and some nonconvexity is unavoidable, since a standard count shows that eight convex quadrilaterals cannot form a Venn diagram. Four sides are the fewest possible for any Venn diagram of eight polygons. The result answers, for quadrilaterals, a question recorded in Ruskey and Weston’s survey of Venn diagrams and attributed there to Grünbaum. Cutting one small corner off each quadrilateral gives eight simple nonconvex pentagons, which answers the pentagon part of the same question when the pentagons need not be convex. The construction is a product of two copies of a four-curve module whose interiors cut a line in a Gray-code pattern. It is fixed by a handful of integers, and a short exact argument proves it correct. Separately, we construct eight congruent strictly convex 125-gons forming a simple Venn diagram. This settles the case of eight sets of the survey’s question of whether convex Venn diagrams can be made from congruent curves. All three witnesses are exact and are confirmed by a standalone verifier.