This is not a Venn diagram

Fig. 1: Four-set Euler diagram

Do you love both circles and diagrams, but struggle to find the overlap between them? You’re in luck! Introduced in 1880 by British mathematician John Venn, the Venn diagram shows all possible logical relationships between a collection of finite sets.

The classic Venn diagram, featuring two interlinking circles, has seen widespread use as a graphic organizer, yet it is but one member of an infinite family. An n-Venn diagram shows all relationships between n sets. Although Fig. 1 appears at first glance to be a 4-Venn diagram, it leaves out a few intersections, namely red ⋂ green and blue ⋂ yellow. Instead, this is an example of an Euler diagram, which typically only shows the relevant relationships.

Pointing this out may seem annoyingly pedantic (it is), but it leads to a simple question with an interesting answer: If that’s an Euler Diagram, can we make a proper 4-Venn diagram?

Fig. 2: 4-Venn diagram with circles and elbow shape

Fig. 3: 4-Venn diagram with ellipses

As it turns out, you can, just not with circles alone. The diagram featuring ellipses [Fig. 3] is visually pleasing, but lacks a certain symmetry. An 𝑛-Venn diagram is symmetric if it is left fixed by a rotation of the plane by 2𝜋/n radians. This survey article summarizes the problem of finding n-fold symmetric diagrams, and references a condition first proved by David W. Henderson.

Theorem. A necessary condition for the existence of a symmetric n-Venn diagram is that n be a prime number.

Alas, 4-Venn diagrams are doomed to be forever askew. I have included below a few drawings for small values of n.

Fig. 4: Symmetric 5-Venn diagram [Gr75]

Fig. 5: Hamilton, a symmetric 7-Venn diagram

Fig. 6: Newroz, the first simple symmetric 11-Venn diagram.

The diagrams get complicated quickly, with the number of faces growing exponentially. Mamakani and Ruskey, searching for diagrams with “crosscut symmetry,” discovered the first simple symmetric 11-Venn diagram, which they called Newroz [Fig. 6]. The same authors later found a 13-Venn diagram [Fig. 7], but their efforts to find a diagram for n > 13 have proved unsuccessful.

Fig. 7: Simple symmetric 13-Venn diagram

Will we ever find a 17-Venn diagram? The sequences of crossings blow up with n, making searches for these elusive graphs computationally expensive, but not impossible.

Sources

• Ruskey, F., Weston, M. A Survey of Venn Diagrams. The Electronic Journal of Combinitorics (ed. June 2005), DS #5.
• Mamakani, K., Ruskey, F. New Roses: Simple Symmetric Venn Diagrams with 11 and 13 Curves. Discrete Comput Geom 52, 71–87 (2014). https://doi.org/10.1007/s00454-014-9605-6