A Venn Diagram of Six Congruent Triangles
John Seamons · Zenodo (CERN European Organization for Nuclear Research) · 2026
We construct six congruent triangles in the plane whose interiors and exteriors divide it into exactly 64 connected regions, one for each inside/outside pattern. The arrangement is a simple Venn diagram: the boundaries cross exactly 62 times and no point lies on three of them. The triangle can be the right triangle with vertices (0, 0), (20, 0) and (0, 10), half of a 2 × 1 domino, or an isosceles triangle; the six copies differ only by rotations and translations. Cutting corners gives six congruent convex k-gons forming a Venn diagram for every k ≥ 3. This answers, for six sets, the question in Ruskey and Weston’s survey of Venn diagrams of whether a convex Venn diagram can be made from congruent curves; the survey records it as settled only for prime n and for n = 4, so six sets is its smallest open case. The witnesses were found by simulated annealing over one shared shape and six rigid placements, followed by collapsing a short side of a quadrilateral and snapping the triangle to an exact shape. Each diagram is fixed by a few short rational numbers and is confirmed by an exact rational verifier.