Why you can’t make a Venn Diagram with 4 Circles.

It’s 11 AM. I just walked into my eighth combinatorics class of the semester, which happened to fall on Halloween. Having woken up a whole six hours early to catch the earliest bus to campus, I came to the realization that there was no need for a Halloween costume. The exhaustion etched on my face seemed to fulfill that role perfectly, instilling great fear in my peers. I took my first sip of coffee for the day. Ahh, finally.

The professor then posed the question, “Can you draw a Venn diagram that represents four sets?” Presented with such a spooky suggestion (ok, I’ll stop), the class pondered for a bit and then came up with these three variations.

This:

This:

And even this:

Having only seen Venn diagrams made with circles in our lifetime, these were some valiant attempts! However, the professor then posed another question: “How many pieces does a Venn diagram with n sets have?"

Doing some induction and set theory presented the following results:

n = 1

n = 2

n = 3

Conjecture: A Venn diagram of n sets has 2ⁿ — 1 pieces.

Therefore for n = 4 sets, 2⁴ — 1 = 15 pieces.

We can already phase out the second and third example, since they are incompatible with our set theory based on intuition:

Instead, let’s take a deep dive into the first example, as it’s the first result when you search for “Venn diagram of 4 sets” on Google.

google image results for “venn diagram of 4 sets”

If we simply count the pieces, we can see that it doesn’t meet the requirement of 15 pieces and therefore isn’t representative of a set of 4. But… that’s an anticlimactic ending. So, I propose another way to approach this problem that is more interesting.

Proof: Using VC Dimension

Wait, what is VC dimension? Is it a part of the MCU or something I’m not aware of? Well… I’d like to direct you to a great article that explains it well, recommended by my professor:

Measuring the Power of a Classifier With VC Dimension

But the TL;DR of VC dimension is that it is

“used in data science to measure how expressive a collection is for the purpose of being a data classifier. in our case, a circle is expressive enough to classify data of 2 or 3 sets, but not expressive enough for 4 sets” — my professor

I took it a step further and provided a geometrical demonstration that showcases why you can’t use a circle to “shatter” (meaning that a model can classify a set of points perfectly) a set of 4 points (or 4 sets of data in this case).

Let’s first start with 3 points and show that a circle is expressive enough for the data. Let a triangle represent the 3 sets of data, with each vertex being 1 set of data.

Geometric demonstration of VC dimension using a circle (classifier) and triangle (set of 3 points)

See how the circle is able to isolate each vertex of the triangle perfectly, then can isolate 2 adjacent vertices from the final vertex, and finally include all of the vertices. No case is left untouched. This showcases that a circle is expressive enough to shatter 3 sets of data.

Now, let a square represent 4 sets of data, and the circle fares well at the start until the case of the circle shattering opposing vertices without including all vertices in the process.

Geometric demonstration showcasing incompatibility of classifier (circle) to fully classify a set of 4 points (square)

This shows that it is not expressive enough to shatter sets of 4 or more.

But what can shatter 4 sets of data then? Well, I’m glad you asked! If you scrolled down further after searching “Venn diagram of 4 sets,” you would find the correct one that uses an ellipse.

Correct representation of a 4-set Venn diagram via Google images

In this case, it is able to fully shatter all of the vertices in the square!

Geometric demonstration showcasing compatibility of classifier (ellipse) to fully classify a set of 4 points (square)

In fact, you can use this method to verify if you can make a Venn diagram using any shape, of any set amount. Just identify how many sets you have, and use the corresponding shape with the same number of vertices. Then, pick your Venn diagram shape of choice and figure out if you can fully shatter the first shape. Done! Here are some more Venn diagrams of higher set amounts that are pretty neat:

n=5 Venn diagram made using ellipses

n=6 Venn diagram made using triangles

Symmetric n=7 Venn diagram. I don’t know the shape they used though…

Bonus!: Using Planar Graphs and Theory to Create New Venn Diagrams

If all you came here for was to know why you can’t create Venn diagrams with circles for 4 sets or more, you can stop here.

But… if you are interested in graph theory and even more Venn diagrams, you can stick around and see something even cooler.

Now, to understand why planar graphs and Venn diagrams are related, I’d like to direct you to a great article by combinatorics.org:

A Survey of Venn Diagrams: Graphs Associated with Venn Diagrams

Again, the TL;DR of this is that every Venn diagram can be constructed as a Venn dual, which is a function of the Venn diagram that is a maximal bipartite planar graph. Such a Venn dual can be constructed from a plain old Venn diagram by following these steps:

1. Create a vertex at each face in the Venn diagram.
2. Create a vertex to represent the universal set, as it’s necessary to account for the entire space outside of the circles in the original Venn diagram. This face is often referred to as the “external” or “infinite” face.
3. Connect each vertex with an edge if they share an adjacent face.

from combinatorics.org, a visual demonstration of venn dual

Using the example above, we can see that the Venn dual for a 3-circle Venn diagram is correct. But notice that they added 3 vertices outside of the Venn diagram and connected them to their corresponding face. For what I’m about to show you, let’s just have 1 vertex outside of the diagram, with each vertex that shares a common boundary connecting to that single one. Here’s a visual:

my venn dual creation for n=3 venn diagram

Now, if you don’t know some graph theory, I’ll give a brief layman’s rundown:

• Planar graph: Dots and lines connected together, but no edges overlap.
• Eulerian trail & path: Imagine you’re connecting dots on a page and you can’t lift up your pencil. An Eulerian trail is you connecting all the dots, but you can draw over what you’ve already written and over old dots. An Eulerian path is when you can’t draw over what you’ve already written (and hence over old dots also).

Now, here’s the surprise I’ve been keeping from you:

Now you’ve created a new Venn diagram using an existing one using the definition of a Venn dual and some graph theory! Pretty cool, huh?