More often than I care to admit, I find myself sitting in the audience of a maths lecture or seminar completely and utterly lost as to what the speaker is going on about. What are they talking about? How does this relate to stuff I know about? Where does this fit within the sphere of mathematics as a whole? In fact, most of the time I am lost beyond the first slide of a presentation. In an endeavour to minimise the possibility that audience members would experience this feeling that I know all too well, I recently introduced myself at the start of a talk with this slide:
You could be forgiven for remarking “My, what a beautiful Venn diagram you have there!” Indeed, I too was under the impression that what I had created was in fact a Venn diagram.
You almost certainly will have come across Venn diagrams before. Though in case you have not, allow me to briefly introduce them. Venn diagrams are widely-used visual representation tools that show logical relationships between sets. Popularised by John Venn in the 1880s, they illustrate simple relations between sets, and are nowadays used across a multitude of scenarios and contexts. And no one can be blamed for their wide usage—they truly are things of beauty. But who was John Venn? And how did he come to create such an iconic tool that’s used so broadly today?John Venn (1834–1923) was an English mathematician, logician and philosopher during the Victorian era. In 1866, he published The Logic of Chance, a groundbreaking book which advocated the frequency theory of probability—the theory that probability should be determined by how often something is forecast to occur, as opposed to ‘educated’ assumptions.
The Victorian era in general saw significant shifts in the way that science and experimental measurements were thought about. It was at this point in history that science began to shift towards a new paradigm of statistical models, rather than exact descriptions of reality. Previously, scientists had believed that mathematical formulas could be used to describe reality exactly. But nowadays, we talk about probabilities and distributions of values, and not about certainties. We now interpret individual experimental measurements knowing that, no matter how precise or controlled an experiment is, some degree of randomness always exists. Using statistical models of distributions is what enables us to describe the mathematical nature of that randomness.
But I digress—back to Venn diagrams. When Venn actually first created his namesake diagrams, he referred to them as Eulerian circles, after everyone’s favourite Swiss mathematician Leonard Euler, who created similar diagrams in the 1700s. And this is where, I regret to inform you, the thing is—and I’m very sorry to be the one to tell you—that Venn diagram up there? Not actually a Venn diagram.
By definition, in a Venn diagram, the curves of all the sets shown must overlap in every possible way, showing all possible relations between the sets, regardless of whether or not the intersection of the sets is empty: $\emptyset$. Venn diagrams are actually a special case of a larger group of visual representations of sets: Euler diagrams. Euler diagrams are like Venn diagrams, except they do not necessarily show all relations.
When thinking about Venn diagrams, we normally picture something like my beautiful introductory slide above, right? Namely, there are circles, and they overlap. The interior of each of the circles represents all of the elements of that set, while the exterior represents things that are not in that set. For example, in a two-set Venn diagram, one circle may represent the group of all Chalkdust readers, while the other circle may represent the set of tea drinkers. The overlapping region, or intersection, would then represent the set of all Chalkdust readers that drink tea (a verifiably non-empty set). It is common for the size of the circles to represent relative size of the set that circle is representing (eg one’s undergraduate maths education being much (much) smaller compared with the sphere of mathematics as a whole).
We also commonly see nested Venn diagrams, where one set is completely situated within another set (again, one’s undergraduate maths education being entirely nested in the realm of mathematics as a whole [though this is debatable—I’ll save this for the next Chalkdust article]). But in a traditional, true-to-definition Venn diagram, every single possible combination of intersections of the sets must be visually displayed…
(While we cannot verify whether or not there exist cats that are also Chalkdust readers and/or tea drinkers, the Venn diagram insists we show all possible intersections.)
It is actually mathematically impossible to draw a Venn diagram exclusively with circles for more than three sets. If we add a fourth set below, no matter how you move the four circles around, you can never find a region that isolates only the intersection of diagonally opposite sets—cats and tea drinkers (or Chalkdust readers and accordion players). Formal mathematical proof is LeFt aS An eXeRcIsE fOr ThE rEaDeR.
As you can see then on the right, for high numbers of sets (‘high numbers’ $=$ greater than three), unfortunately some loss of symmetry in the diagrams is unavoidable. John Venn experimented with ellipses for the four-set case in an attempt to cling onto some diagrammatic elegance and symmetry. He also devised Venn’s constructions which gave a construction for Venn diagrams for any number of sets. These constructions started with the three-set circular Venn diagram and added arc-like shapes which weaved between each of the previous sets to create every possible logical intersection. These constructions quickly become quite dizzying (see the loopiness of Venn’s construction for six sets):
So what can we do if we don’t want to weave back and forth so dizzyingly for higher numbers of sets? Enter: Edwards–Venn diagrams. Anthony William Fairbank Edwards (1935–) constructed a series of Venn diagrams for higher numbers of sets. He did this by segmenting the surface of a sphere.
For example—as you can see on the right—three sets can be easily represented by taking three hemispheres of the sphere at right angles ($x = 0$, $y = 0$ and $z = 0$). He then added a fourth set to the representation, by taking a curve similar to the seam on a tennis ball, which winds up and down around the equator, and so on.
The resulting sets can then be projected back to a plane, to give cogwheel diagrams, with increasing numbers of teeth for more and more sets represented:
So finally, to conclude this article, I present to you my new and improved introductory (actually-a-Venn) diagram:
While I appreciate the poetic implication that the realms of my respective research groups (fluid dynamics and behavioural genomics) weave in and out of my PhD project, which is nested neatly in the centre, in reality the fact is that many many of these intersections are quite empty.
Please consider this article in support of my upcoming petition to make it mandatory for academics to introduce themselves at the start of every talk with a Venn (or Euler) diagram.
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- Michael Wendl dissects some variants of the magic separation, a self-working card trick.