So things got wacky in seventh grade today—the good kind of wacky. Lots went on, but during a chunk of class we rocked a discussion about Euler diagrams and Venn diagrams. Here’s the handout I gave them. We talked about the ways in which Euler diagrams and Venn diagrams are the same and different. I shared some about Euler and about Venn and about Branko Grünbaum and about Khalegh Mamakani and Frank Ruskey.

We also made a diagram of Euler diagrams and Venn diagrams, which is of course an Euler diagram itself. So meta! (See below.) Some kids were so charmed by this idea.

The other Euler diagram below brought up the issue of why one might prefer to use an Euler diagram rather than a Venn diagram in a particular instance. For example, it’s nice to be able to show that animals and minerals are disjoint categories, and that four-legged animals are a subset of all animals. If you instead try to apply a Venn structure to these three categories, some funny things happen. Not wrong things—right things, in fact!—but for some purposes they are certainly distracting things. We had fun coming up with examples. A particularly fun moment was when the lightbulbs went on in several kids’ heads about what has four legs but isn’t an animal (or mineral). We tweaked “chair” to “wooden chair” once we added in “marble chair” for the intersection of “four-legged” and “mineral.”

And of course, who could fail to delight in the four-legged mineral-animal that is the Horse Rock Pokémon?!

This is a sensible scheme. Ours? Less sensible, but more laughable. |

An Euler diagram of Euler diagrams! It’s Venn-ception! I hope Bertrand Russell doesn’t see this… |

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I totally see how Euler Diagrams are a subset of Venn Diagrams, but if a Venn diagram has empty regions, they can indicate subsets equally well, no?

Couldn’t the relation be summarized by a Venn diagram where one circle is Venn, the other is Euler, and the Venn-not-Euler is empty? I’m playing now.

You can totally do that. The information in any Euler diagram can be expressed as a Venn diagram where some of the regions are empty. But this will often be clunky—the fact that Euler diagrams are more streamlined and flexible is, I’d say, a big part of their appeal. If we’re talking about making simple, satisfying (with an eye to necessity and sufficiency) presentations of information, Euler’s often the way to go.

Though of course Venn diagrams have their own many charms.

I had the thought that an Euler diagram can be thought of as a reduction of a Venn diagram, and that this reduction process is kind of like a homotopy. The empty regions just get sucked up.

Thanks, as always, for reading and commenting!