Rotationally symmetric Venn diagrams

No doubt you will have seen a Venn diagram. They are a wonderful way of presenting logical information. For example, they allow us to illustrate the fact that centaurs lie in the union of objects with male torsos and horse legs (Figure 1).

Figure 1. Not all male torsos are connected to horse legs and vice versa. However, we see that centaurs do lie in the intersection.

Recently there has been an upsurge in using Venn diagrams as way of illustrating jokes, or song titles. My personal favourite explains where the platypus fits in the animal kingdom (Figure 2).

Figure 2. Although not scientifically sound it does show that the keyboard guitar and platypus can be defined as the intersection of two other sets.

Of course you are not restricted to two sets of objects. A Venn diagram can be made of any number of sets. For example Figure 3 illustrates the some of the lyrics from the song “The Joker” by the Steve Miller Band.

Figure 3. A seven set intersection diagram illustrating the characteristics of certain famous people.

Technically, Figure 3 illustrates an Euler diagram and not a Venn diagram. A Venn diagram contains every single possible intersection between all combinations of the sets, whereas an Euler diagram only shows the intersections you are interested in. For example, in Figure 3 there is no section where only grinners and jokers intersect (this could possible contain Heath Ledger).

When dealing with two or three sets the obvious Euler diagram is also a Venn diagram (Figure 4). Interestingly, they both also have rotational symmetry. This will be considered in more detail in the next article.

Figure 4. Two and three set Venn diagrams.

However, when we get to four circles, things are not so easy anymore and the basic Euler diagram (Figure 5, left) is no longer a Venn diagram. However, by removing the restriction that the groups have to be circles we can once again produce a four set Venn diagram (Figure 5, right). Sadly though, we have lost the pleasing rotational symmetry.

Figure 5. If only circular shapes are used we cannot create a Venn diagram, only an Euler diagram (left). However, by generalizing the set's shape, we can produce a Venn diagram once more (right).

Next time I will present another part of the discussion with Barry Cipra and we will see under what conditions Venn diagrams can have rotational symmetry.