The Math Less Traveled

I’ve finally gotten around to making a nice factorization diagram poster:

You can buy high-quality prints from Imagekind. (If you order soon you should have them before Christmas! =) I’m really quite happy with imagekind, the print quality is fantastic and the prices seem reasonable. You can choose among different sizes—I suggest 32"x20" ($26 on the default matte paper, + shipping), but the next smaller size (24"x15", $17) is probably OK too.

If you have ideas for variant posters you’d like to see, let me know (though I probably won’t be able to do anything until after the new year).

Note that Jeremy also has a variety of factorization diagram posters for sale.

For more information, including links to the source code, a high-resolution PNG, and other things, see this factorization diagrams page.

[This is part six in an ongoing series; previous posts can be found here: Differences of powers of consecutive integers, Differences of powers of consecutive integers, part II, Combinatorial proofs, Making our equation count, How to explain the principle of inclusion-exclusion?. However, this post is self-contained; no need to go back and read the previous ones just yet.]

“But”, I hear you protest, “Pi Day was ages ago!” Ah, but I didn’t say Pi Day, I said PIE Day. To clarify:

• Pi Day: a day on which to celebrate the not-so-fundamental circle constant, (March 14)
• Pie Day: a day on which to eat pie (every day)
• PIE Day: a day on which to explain the Principle of Inclusion-Exclusion (PIE). (That’s today!)

(Actually, I’m only going to begin explaining it today; it’s getting too long for a single blog post!) In any case, the overall goal is to finish up my long-languishing series on a combinatorial proof of a curious identity (though this post is self-contained so there’s no need to go back and reread that stuff yet). The biggest missing piece of the pie is… well, PIE! I’ve been having trouble figuring out a good way to explain it in sufficient generality—it’s one of those deceptively simple-seeming things which actually hides a lot of depth. Like a puddle which turns out to be a giant pothole. (Except more fun.)

In a previous post (now long ago) I asked for some advice and got a lot of great comments—if my explanation doesn’t make sense you can try reading some of those!

So, what’s the Principle of Inclusion-Exclusion all about? The basic purpose is to compute the total size of some overlapping sets.

To start out, here is a diagram representing some non-overlapping sets.

Each set is represented by a colored circle and labelled with the number of elements it contains. In this case, there are 25 people who like bobsledding (the red circle), six people who like doing laundry (the blue circle), and 99 people who like math (the green circle). The circles do not overlap at all, meaning that none of the people who like math also like laundry or bobsledding; none of the people who like doing laundry also like bobsledding or math; and so on. So, how many people are there in total? Well, that’s easy—just add the three numbers! In this case we get 130.

Now, consider this Venn diagram which shows three overlapping sets.

Again, I’ve labelled each region with the number of elements it contains. So there are two people who like bobsledding but not math or laundry; there are three people who like bobsledding and math but not laundry; there is one person who likes all three; and so on. It’s still easy to count the total number of elements: just add up all the numbers again (I get 14).

So what’s the catch?

The catch is that in many situations, we do not know the number of elements in each region! More typically, we know something like:

• The total number of elements in each set. Say, we might know that there are 7 people who like bobsledding in total, but have no idea how many of those 7 like math or laundry; and similarly for the other two sets.

• The total number of elements in each combination of sets. For example, we might know there are two people who like bobsledding and laundry—but we don’t know whether either of them likes math.

This is illustrated below for another instance of our ongoing example. The top row shows that there are sixteen people who like bobsledding in total, eleven who like laundry in total, and eighteen who like math—but again, these are total counts which tell us nothing about the overlap between the sets. (I’ve put each diagram in a box to emphasize that they are now independent—unlike in the first diagram in this post, having three separate circles does not imply that the circles are necessarily disjoint.) So is probably too many because we would be counting some people multiple times. The next row shows all the intersections of two sets: there are three people who like bobsledding and laundry (who may or may not like math), eight people who like bobsledding and math; and six people who like laundry and math. Finally, there is one person who likes all three.

The question is, how can we deduce the total number of people, starting from this information?

Well, give it a try yourself! Once you have figured that out, think about what would happen if we added a fourth category (say, people who like gelato), or a fifth, or… In a future post I will explain more about the general principle.

Do you enjoy writing beautiful code to produce beautiful artifacts? Have something cool to show off at the intersection of functional programming and visual art, music, sound, modeling, visualization, or design?

The deadline for submitting a paper has passed, but the Workshop on Functional Art, Music, Modeling and Design (FARM 2013) is currently seeking proposals for 10-20 minute demonstrations to be given during the workshop. For example, a demonstration could consist of a short tutorial, an exhibition of some work, or even a livecoding performance. Slots for demonstrations will be shorter than slots for accepted papers, and will not be published as part of the formal proceedings, but can be a great way to show off interesting work and get feedback from other workshop participants. A demonstration slot could be a particularly good way to get feedback on work-in-progress.

A demo proposal should consist of a 1 page abstract, in PDF format, explaining the proposed content of the demonstration and why it would be of interest to the attendees of FARM. Proposals will be judged on interest and relevance to the stated goals and themes of the workshop.

Submissions can be made via EasyChair.

I haven’t written here in quite a while—I’ve switched into “work on research for my dissertation really hard so that I can actually graduate” mode, and with a 21-month old in the mix that leaves very little time for blogging. Still, I do plan to continue writing some here, hopefully more as I get into some new rhythms of life. And I’ve got some exciting factorization diagrams-related projects up my sleeve, so you can look forward to that too.

For today though, just a link (I don’t remember where I first saw it):

This is a super-fun game with a very simple interface, where you are challenged to construct various things (polygons, circle packings, etc.) using only straightedge and compass, in the finest tradition of the ancient Greeks. It’s like eighth grade geometry class all over again, except fun. I’ve completed 26 of the 40 challenges so far — how many can you do?

Wearing my PhD-student hat, I’m helping organize a workshop, FARM, to be held in Boston this September. I thought it worth mentioning since some readers of this blog—especially those interested in the intersection of math, art, and programming—may find it interesting. The focus of the workshop is essentially using beautiful code to produce beautiful artifacts—whether art, music, or anything else. If that sounds interesting to you, you should consider submitting a paper, or planning to attend! See the website for more details.

Here’s something fun I was playing around with today. I generated 100 random points and connected some of them with lines. Can you figure out how I chose which lines to draw?

How about these? (Same points, different lines.)

Or these?

Here are some more. The first three groups in the grid below correspond to the pictures already shown above.