# This is not a Venn 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?**

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.*

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.

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