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An AI-assisted search has produced simple, rotationally symmetric Venn diagrams with 17 curves and 19 curves. The constructions extend the previously known simple symmetric examples from 13 curves, but they do not establish 19 as a maximum. The result, reported by software engineer Chris Dzoba in a paper posted to arXiv on September 22, 2026, is about finding and checking intricate mathematical drawings—not asking an image generator to sketch a Venn diagram.
What makes these Venn diagrams extraordinary?
A Venn diagram for n sets must represent every possible combination of membership. Each combination must appear as its own connected region, so an n-set diagram has 2n regions. The 17-curve construction therefore represents 131,072 combinations; the 19-curve version represents 524,288. These counts follow from the definition of a Venn diagram; they are not counts of curves or crossings.
The reported drawings meet additional conditions: the curves map to one another when the diagram is rotated by one equal step, and every crossing is shared by exactly two curves. Dzoba describes them as Jordan curves—simple closed curves—not as ordinary circles. Circles cannot show every possible intersection for four or more sets, as the American Mathematical Society’s background on Venn diagrams explains.
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In this mathematical setting, a simple Venn diagram has no crossing where three or more curves meet. That restriction makes the new examples more specific than a drawing that merely contains many loops. Every set-membership combination must still form a present, connected region.
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Rotational symmetry describes how the curves relate
For a diagram with n curves, the reported symmetry means a rotation of 2π/n carries the curves around cyclically. It is not enough for the overall picture to look balanced: the rotation must map the curves as a mathematical structure.
What did the paper report?
Dzoba’s paper, Simple symmetric Venn diagrams with 17 and 19 curves, reports four diagrams with 17 curves and nine with 19. It says the earlier known simple rotationally symmetric examples went only as far as 13 curves. That is a statement about the examples known in the literature described by the paper, not proof that no larger example had ever been found or can be found.
The paper reports that each diagram has a machine-checkable certificate and that one certificate at each size was formally verified in Lean 4. Those checks support the constructions’ stated properties; they are not a theorem that 19 is the largest possible size.
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The paper describes a Metropolis walk over rotation-invariant quadrangulations of the sphere. The search began with a construction that had multiple crossings, which were resolved; during the search, regions could temporarily be duplicated. The reported final diagrams satisfy the Venn-region condition, even though intermediate search states did not necessarily do so.
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The method matters because the examples are non-monotone, according to the paper. That property meant searches based on crossing sequences—the approach used to find earlier 11- and 13-curve examples—could not find these diagrams. The result is therefore best understood as a computational search guided by mathematical structure, followed by certificate-based checking, rather than an AI system inventing a picture and declaring it correct.
Does this prove 19 is the largest symmetric Venn diagram?
No. The paper presents constructions with 17 and 19 curves, not a maximum-size theorem. It also discusses a distinct result: rotationally symmetric Venn diagrams exist for every prime number of curves. That prime-number condition concerns the number of sets in a rotationally symmetric diagram; it does not mean every such diagram is simple, or settle how large a simple one can be.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Why the result matters
The achievement pushes the known range of simple rotationally symmetric examples from 13 to 19 curves, while illustrating how computation can explore highly constrained mathematical objects. The key distinction is between producing examples and proving a limit: the paper reports rigorously checkable constructions at two new sizes, but leaves open whether larger simple symmetric Venn diagrams exist.
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