Simple symmetric Venn diagrams found with 17 and 19 curves
Simple symmetric Venn diagrams with 17 and 19 curves
Computational GeometryDiscrete Mathematics
Summary
Drawing Venn diagrams with many overlapping circles is tricky, especially when every shape is connected and the drawing looks the same after spinning it around. The authors found new ways to create such perfectly balanced Venn diagrams with 17 and 19 circles, something that wasn’t done before for simple diagrams. They used a special search method starting from older complex diagrams and verified their results using computer proofs. These new diagrams don’t follow the usual layering rules, making them unique compared to smaller examples.
What this means in practice
- •For mathematicians and educators: Provide new examples of highly symmetrical Venn diagrams for teaching and exploring combinatorial geometry with prime numbers of sets.
- •For graphic designers: Create complex symmetric patterns based on verified Venn diagram structures for artistic visualizations and data representations.
Authors
Chris Dzoba
Abstract
We exhibit simple, rotationally symmetric Venn diagrams with 17 curves and with 19 curves: n Jordan curves carried to one another by rotation through 2π/n, with every one of the 2^n regions present and connected and, since the diagrams are simple, every crossing on exactly two curves. Symmetric Venn diagrams exist for every prime number of curves (Griggs, Killian and Savage, 2004), but those diagrams have many curves through a point; simple ones were known only up to 13 curves (Mamakani and Ruskey, 2014). Four 17-curve and nine 19-curve diagrams were found by a Metropolis walk on rotation-invariant quadrangulations of the sphere in which regions may temporarily be duplicated, started from the Griggs-Killian-Savage diagram with its multiple crossings resolved. Every diagram is given by a machine-checkable certificate; one certificate of each size has been verified by a formal proof in Lean 4. All of the diagrams are non-monotone, which is why the crossing-sequence searches that found the 11- and 13-curve diagrams could not have found them.