As we discussed a couple years ago, I think that Venn diagrams have no practical use beyond three circles, but we should all be able to agree that the above pictures are pretty.
Jack Murtagh supplies the background:
Have you ever seen a proper Venn diagram with four overlapping circles? No, because it’s impossible. . . . John Venn knew of the shortcoming with circles, so he proposed ellipses to represent four sets. . . . This overcomes the limitations with circles but only temporarily. Ellipses work for four and five sets before failing in the same way that circles did. As the number of sets grows, we need more and more exotic shapes to portray them.
After reading that, I assumed that, in six or more dimensions, a complete Venn diagram would require at least one nonconvex shape. But I was wrong–a quick google search led to this article from 1997-2005 by Frank Ruskey and Mark Weston with lots of details.
Murtagh continues:
One could reasonably argue that beyond four sets of elements, Venn diagrams lose their utility. The four-ellipse image is already pretty chaotic. Maybe for five-plus sets we should abandon visual representations. But utility does not animate the mathematician so much as beauty and curiosity.
Agreed! Except that I’d say that the four-ellipse image is already too complicated for me to imagine it being useful in any case. Even the three-circle Venn diagram doesn’t seem to me to serve any practical purposes other than pedagogical.
In any case, here’s the fascinating punchline:
Venn and his successors believed that ellipses couldn’t portray all 32 regions required for a five-set diagram. Not until 1975 did mathematician Branko Grünbaum prove them wrong by example [see image above] . . . Grünbaum’s diagram displays a pleasing rotational symmetry. . . . Typical two- and three-circle Venn diagrams share this property. . . . But the four-ellipse diagram doesn’t have rotational symmetry. Can that be fixed? What do two, three and five have in common that four doesn’t?
In 1960 a then undergraduate student at Swarthmore College, David W. Henderson, answered this question with a surprising discovery (Stan Wagon and Peter Webb filled in some gaps later): Rotationally symmetric Venn diagrams are possible only when the number of sets is a prime number—a number divisible only by 1 and itself, such as 2, 3 and 5 but not 4. Henderson only showed that a prime number of sets is necessary, not that you can always design a symmetric Venn diagram for every prime number. . . . Mathematicians at the University of South Carolina settled the question in 2004 by showing that rotationally symmetric Venn diagrams exist for every prime number of sets.
Cool! Just one thing. I have no idea why Murtagh named Henderson, Wagon, and Webb but left anonymous the “mathematicians at the University of South Carolina” who settled the question. He links to the paper, though, so I can share the authors’ names: they’re Jerrold Griggs, Charles E. Killian, and Carla D. Savage. Thanks, Jerrold, Charles, and Carla!
