Simple symmetric Venn diagrams with 17, 19 and 23 curves
Abstract
We exhibit simple, rotationally symmetric Venn diagrams with 17 curves, with 19 curves and with 23 curves: n Jordan curves carried to one another by rotation through 2π/n, with every one of the 2n 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, twelve 19-curve and six 23-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. At 23 curves the walk was held for two weeks by duplicated regions near the poles; the two lineages that finished were the first whose E=92 states had none. Every diagram is given by a machine-checkable certificate; one certificate each of the 17-, 19- and 23-curve sizes 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.
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BibTeX
@misc{dzoba2026simple,
title = {{Simple symmetric Venn diagrams with 17, 19 and 23 curves}},
author = {Chris Dzoba},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.26546},
eprint = {2609.26546},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26546},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Simple symmetric Venn diagrams with 17, 19 and 23 curves
AU - Chris Dzoba
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2609.26546
UR - https://arxiv.org/abs/2609.26546
AB - We exhibit simple, rotationally symmetric Venn diagrams with 17 curves, with 19 curves and with 23 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, twelve 19-curve and six 23-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. At 23 curves the walk was held for two weeks by duplicated regions near the poles; the two lineages that finished were the first whose E=92 states had none. Every diagram is given by a machine-checkable certificate; one certificate each of the 17-, 19- and 23-curve sizes 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.
ER -