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What makes a Venn diagram a Venn diagram?
A Venn diagram with n sets must represent every possible combination of membership in those sets. Each combination must appear as its own connected region. That means a diagram for 17 sets has 217 possible membership combinations, while one for 19 sets has 219.
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The curves are not necessarily ordinary circles. In mathematics they are treated as simple closed curves, or Jordan curves. Circles cannot realize every possible intersection pattern for four or more sets, so more general shapes are needed. The American Mathematical Society’s background on Venn diagrams explains the definition and construction history.
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What is special about the 17- and 19-curve examples?
Dzoba’s paper, Simple symmetric Venn diagrams with 17 and 19 curves, reports four diagrams with 17 curves and nine with 19. In each, every possible region is present and connected, the curves map to one another under a rotation by 2π/n, and every crossing lies on exactly two curves.
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“Simple” means no triple crossings
In this context, a diagram is simple when no crossing is shared by three or more curves. Each crossing involves precisely two curves. This is a stricter condition than merely drawing many curves that intersect.
Rotational symmetry means equal turns cycle through the curves
For an n-curve rotationally symmetric diagram, turning the drawing by one step—an angle of 2π/n—maps the curves around cyclically. The paper’s constructions satisfy this symmetry as well as the requirement to show all 2n connected regions.
How did the search find them?
The paper says the diagrams were found using a Metropolis walk on rotation-invariant quadrangulations of the sphere. The search began with a construction containing multiple crossings, resolved those crossings, and allowed regions to be temporarily duplicated while exploring possibilities.
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How does this compare with earlier symmetric Venn diagrams?
Dzoba’s paper describes the previous known simple rotationally symmetric examples as reaching 13 curves, citing earlier work by Mamakani and Ruskey. There is also an older result that rotationally symmetric Venn diagrams exist for every prime number of curves. The prime-number result concerns symmetry and does not guarantee that a diagram is simple; simplicity is an additional condition.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Does 19 curves set a record that cannot be beaten?
No. The paper presents constructions with 17 and 19 curves; it does not prove that larger simple rotationally symmetric Venn diagrams are impossible, or that 19 is the largest possible. The result raises the known example count, rather than establishing a maximum-size theorem.
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Sources
- Chris Dzoba, “Simple symmetric Venn diagrams with 17 and 19 curves,” arXiv, posted September 22, 2026.
- American Mathematical Society, background on Venn diagrams.
- ScienceX, news summary and headline framing.
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