The mysteries of Venn diagrams

venn-7 copy.jpg Above, a Venn diagram I made visually depicting every combination of Dwarf from the fairy tale. Not included: Snow White.

venn-5.jpgThe other day I was tying to make some multi-set Venn diagrams with polar symmetry, which it turns out is harder than you'd think and has ties with prime number theory. It's an obscure but important area of combinatorics. Everybody knows a 3-set one (n=3), and I made a 5-set one with ellipses (pictured to the right). After that, I was stumped. That led me to a great web section maintained by Professors Frank Ruskey and Mark Weston. The Dwarf diagram at the top is based on that work, with Sleepy a little more opaque so you can see the shape. Ruskey and Weston display lots of lovely diagrams, including the elusive n=7 minimum vertex Venn diagram and the remarkable n=11 Venn diagram.

A Survey of Venn Diagrams


  1. The biostatistican A.W.F Edwards wrote a fascinating (yes, really) book on multi-set Venn diagrams called “Cogwheels of the Mind”

  2. I’m not sure I get it. Isn’t the Venn diagram used to illustrate overlapping sets? As the Seven Dwarfs don’t have common parts why would they be diagrammed in this way?

    Many moons ago I did a set of diagrams for an annual report that I was told turned out to be Venn diagrams (I had never heard of it before). They were illustrating the overlap of various engine product markets, ie: a “TYPE 2” engine served the Marine and Truck Market, but not the Generator Market, while a “Type 3” served the Generator and Truck Market, etc. As I recall most were two sets creating a third (the overlap). I guess I need to learn more, ’cause they do look cool. Thanks for the article Boing Boing – I’m gonna go dig up my Tufte books and get to the bottom of this…

    1. You can have a set of 1. This Venn diagram illustrates all possible combinations of 7 sets, each of which happens to contain 1 item (a Dwarf) in this case. Let’s say this is a chart of the Dwarfs whom Snow White likes. If Snow White likes all the Dwarfs except Grumpy, she could show where that combination exists on this diagram. If she likes Sleepy, Doc, and Bashful but no others, she can point to where that combination exists on this diagram. Same for every other possible combination of Dwarfs. Or it could be a chart of who has the day off. Every combination of Dwarfs available to work is represented on the diagram.

  3. This Venn diagram describes who has slept with whom, including any combination thereof. Snow White’s name has been removed from the center, in another Disney lawsuit.

  4. the n=11 diagram is a beautiful piece of work, truly. one of those things where, at first glance, you know it’s not random, but it takes a lot of careful study to figure out exactly what it means. and from such a simple question!

  5. …but when you mix all colors on the dwarven wheel, you are supposed to end up with (Snow) White…

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