A couple of days ago, in a post about Venn Diagrams, i happened to state, in an off-the-cuff manner, that you can't have a four-circle Venn diagram. Thankfully, that much is true. Had I said four-category Venn diagram, there might've been a rumble. Watch out when mathematicians rumble. They know the odds, and while they know that they may not always be ever in their favor, they know when they are in their favor, so back up, pup.
Anyway, Venn diagrams are possible with more than three categories. The trick is being careful in drawing them so that they every possibility, ever intersection, is represented. With four circles, there won't be enough overlap. If A, B, C, and D are four overlapping circles, then odds are you'll be missing a region of only AC or BD or both. Now, it is possible to create one with ovals (aka "ellipses"), but again, placement is key. You need to be exact or something will be missing, or, more likely if I'm free-drawing, there will be two non-contiguous spaces representing the same intersection. That's like someone picking up Staten Island, moving it somewhere in the middle of Pennsylvania, but still saying it's part of New York. Silly.
My other problem with it is readability. I guess you can get used to it, but it doesn't have the same "flow" as a two- or three-circle diagram has. You need to identify which ovals a given region is contained within and then what those ovals represent. Maybe it's just me.
Another method, much cooler, actually, is to go concave. (I wanted to make a "Con-Caveman" joke here, but it didn't read well. Pity.) This has the benefit of starting with the tried-and-true three-circle Venn diagram. Then a sort of rainbow is added to snake its way around the various intersections without completely covering any of them, like this:
Other tricks can be used to make larger ones, and some numbers seem to be easier than others (although I don't know if it's easier to make them or just easier to read and understand them after someone else created it.)
I will identify my biggest problem with these oversized Venns: their undersized parts. The sections are just too small to be useful. If you're doing some kind of programming and you just need a bucket, well, okay. However, if you are trying to categorize things in a math class, it's just a waste of time. The exercise in creativity is making one. Using one would be an exercise in futility.
Note: Comments on twitter, particularly by William Ricker, prompted this further investigation. Comments are welcome here, there, Facebook, Google+ or anywhere else, but not everyone follows all those things. (Many days, I don't!) So comments, with no registration required, are appreciated. Just remember, if you comment as "Anonymous", you won't get any responses in email, and we'll probably not take you as seriously as someone with a face (or an avatar) who isn't linking to some spammy website.
EDIT: Corrected William's name. Not my fault! There should've been a red squiggly under it. It's not like I wear my glasses when I type on the computer any more! My fault for not just typing "Bill". Or would it be "Blil"?