My Dearly Beloved, the professional philosopher, mentioned after reading the last comics review that one thing to protest in the Too Much Coffee Man strip — showing Venn diagram cartoons and Things That Are Funny as disjoint sets — was that the Venn diagram was drawn wrong. In philosophy, you see, they’re taught to draw a Venn diagram for two sets as two slightly overlapping circles, and then to black out any parts of the diagram which haven’t got any elements. If there are three sets, you draw three overlapping circles of equal size and again black out the parts that are empty.
I granted that this certainly better form, and indispensable if you don’t know anything about what sets, intersections, and unions have any elements in them, but that it was pretty much the default in mathematics to draw the loops that represent sets as not touching if you know the intersection of the sets is empty. That did get me to wondering what the proper way of doing things was, though, and I looked it up. And, indeed, according to MathWorld, I have been doing it wrong for a very long time. Per MathWorld (which is as good a general reference for this sort of thing as I can figure), to draw a Venn diagram reflecting data for N sets, the rules are:
- Draw N simple, closed curves on the plane, so that the curves partition the plane into 2N connected regions.
- Have each subset of the N different sets correspond to one and only one region formed by the intersection of the curves.
Partitioning the plane is pretty much exactly what you might imagine from the ordinary English meaning of the world: you divide the plane into parts that are in this group or that group or some other group, with every point in the plane in exactly one of these partitions (or on the border between them). And drawing circles which never touch mean that I (and Shannon Wheeler, and many people who draw Venn diagram cartoons) are not doing that first thing right: two circles that have no overlap the way the cartoon shows partition the plane into three pieces, not four.
I can make excuses for my sloppiness. For one, I learned about Venn diagrams in the far distant past and never went back to check I was using them right. For another, the thing I most often do with Venn diagrams is work out probability problems. One approach for figuring out the probability of something happen is to identify the set of all possible outcomes of an experiment — for a much-used example, all the possible numbers that can come up if you throw three fair dice simultaneously — and identify how many of those outcomes are in the set of whatever you’re interested in — say, rolling a nine total, or rolling a prime number, or for something complicated, “rolling a prime number or a nine”. When you’ve done this, if every possible outcome is equally likely, the probability of the outcome you’re interested in is the number of outcomes that satisfy what you’re looking for divided by the number of outcomes possible.
If you get to working that way, then, you might end up writing a list of all the possible outcomes and drawing a big bubble around the outcomes that give you nine, and around the outcomes that give you a prime number, and those aren’t going to touch for the reasons you’d expect. I’m not sure that this approach is properly considered a Venn diagram anymore, though, although I’d introduced it in statistics classes as such and seen it called that in the textbook. There might not be a better name for it, but it is doing violence to the Venn diagram concept and I’ll try to be more careful in future.
The Mathworld page, by the way, provides a couple examples of Venn diagrams for more than three propositions, down towards the bottom of the page. The last one that I can imagine being of any actual use is the starfish shape used to work out five propositions at once. That shows off 32 possible combinations of sets and I can barely imagine finding that useful as a way to visualize the relations between things. There are also representations based on seven sets, which have 128 different combinations, and for 11 propositions, a mind-boggling 2,048 possible combinations. By that point the diagram is no use for visualizing relationships of sets and is simply mathematics as artwork.
Something else I had no idea bout is that if you draw the three-circle Venn diagram, and set it so that the intersection of any two circles is at the center of the third, then the innermost intersection is a Reuleaux triangle, one of those oddball shapes that rolls as smoothly as a circle without actually being a circle. (MathWorld has an animated gif showing it rolling so.) This figure, it turns out, is also the base for something called the Henry Watt square drill bit. It can be used as a spinning drill bit to produce a (nearly) square hole, which is again pretty amazing as I make these things out, and which my father will be delighted to know I finally understand or have heard of.
In any case, the philosophy department did better teaching Venn diagrams properly than whatever math teacher I picked them up from did, or at least, my spouse retained the knowledge better than I did.