About fifteen people met for the Math Club this Monday - we had three generations this time! Justin and Ryan's grandmother joined the fun. I am now plotting and scheming to involve our men more when we resume the club after the summer break. We will skip the next few meetings because many people are traveling and such during the summer. I will let people know when we resume the Club.

The main topic of the May 19th meeting was COMPLEXITY. This is an umbrella notion, connecting many diverse topics and areas of mathematics and life in general. Humanity, as a whole, tends to crave complexity and to build increasingly more complex cultures in time. Individual humans, as we all know too well, also strive for simplicity, or else risk being overwhelmed and overloaded by the collective complexity created by their civilizations. When we are too tired, we just shun complexity and refuse it, but when we are OK, we deal with increasing complexity by creating well-organized systems.

From closet shelving to theories of quantum mechanics, systems help us a lot. Systems combine the complexity of all the parts with the simplicity of the organization of the whole. We group things or ideas together, sort them, label them, connect them by relationships, build taxonomies and, in general, systematize and order them. It is currently believed that humans are born with quite a few system-making cognitive tools (such as sorting and category-creating abilities for "same and different", and quickly acquire more as they develop. After all, by the age of two or three healthy kids gain amazing understanding of their native languages, and languages are quite complex.

Mathematics is the part of our culture where systems for dealing with complexity are especially formal, well-structured and often very elegant. Mathematicians also tend to express their systems in very specialized and involved jargons. One of my goals in life is to make these beautiful and useful mathematical systems available to the general public without the need to study math jargons for years upon years. To achieve that, I try to express mathematical systems using metaphors already widespread. Ironically, this approach also helps people to learn the formal math jargon.

For the name activity, now traditional for the start of the club, we counted how many different letters are in each person's name. That's not very complex, or interesting, so we counted how many two-letter combinations we can make out of the letters. Club members came up with two major ways of making sure they listed all the combinations - in other words, with two systems for organizing the complexity of many combinations. For example, if we use the name "Martha", the two ways will look thus:

m, a, r, t, h - 5 different letters

ma, mr, mt, mh
ar, at, ah
rt, rh
th
Making for 4+3+2+1=10 combinations

and

ma, mr, mt, mh
am, ar, at, ah
rm, ra, rt, rh
tm, ta, tr, th
hm, ha, hr, ht
Since we decided that order does not matter (ma=am), and every combination repeats twice, this makes for (5*4)/2=10 combinations, too - just organized differently.

I introduced two more ways of systematizing combinations: a combinatorial table, and a polygon. Probably the best-known example of the combinatorial table is the Pythagoras times table. For a funky example of using polygons for counting combinations, here is a polygon for combining seven deadly sins (e.g. gluttony+sloth=Saturday, and greed+pride=status symbols). Name polygons Club members made were beautiful. Of course, the next question was - what does it have to do with pies?! Club members decided that the systems we developed for letter combinations will also work quite well for pie fillings. If you have five fillings and you are making different pies with two of the fillings, you can make ten different pies, and you can organize them using lists, tables, or polygons. I am glad the Club arrived at these systems for counting combinations, because we stopped at a sort of a cliffhanger with pie fillings last time, not figuring much out. I'd like to thank Kalli, Martha's mom, for bringing the "millions of pie combinations" idea to our attention. I'd like to use the opportunity to thank all club members for their creativity and ideas!

We then went outside and worked on handshakes, sometimes way too vigorously! When two people meet, there is one handshake between them. When three people meet, there are three handshakes (A, B and C meeting - AB, AC and BC). Again, to solve the number of handshakes problem for any group of people, we can use the same systems for organizing combinations, as the Club members saw. Of course, I placed all these activities back to back, so the question of what makes them similar was easy and almost rhetorical. As people get more experienced in math, they are able to re-use the systems they develop without such scaffolding by mentors. So, for example, if a fashion expert claims you can make a lot of different outfits with five skirts and four blazers, you can see for yourself if the twenty possible combinations will be enough for your complexity needs. We also played Broken Telephone outside.

After elevenses, which we did not cook this time around, we re-assembled around the projector screen, because an argument broke out if an animal cracker was a llama or a camel. We looked up animal crackers in Wikipedia. Using shadows to project our cookies next to their pictures on the wall, we determined that the good people at Stauffer's consider that animal a camel, after all. The argument reminded of a classic meme "Llama llama duck", which we proceeded to watch, listen and attempt to sing.

Meme, a word designed after "gene" and pronounced similarly, is a little piece of information, like a joke or a video, that gets distributed in the culture all by itself, without any efforts of "official" information distribution networks such as newspapers or television. Here is an obligatory Wikipedia article. Some people also compare memes with viruses, for example, using "viral marketing" techniques to advertise their products. I am working on a project that will help to create and collect some math-rich memes. Meanwhile, we listened to another (non-math) meme song, a Korean advert for eggs. I will try to line up some math video memes by the next time we meet, because it is fun to sing together to a video. Maybe we should do some math karaoke?!

For the rest of the math club, we created fractals. They are self-similar pictures: if you zoom in, you will see the same structure, and you can zoom in forever. We made fractals by substitutions (a more complex fragment for a simpler fragment) and by branching. Here is an example of a slightly spooky branching fractal:

As for substitution fractals, we played with several online programs doing substitutions step-by-step, and Club members were simply mesmerized. Here is the Koch curve fractal we first made by hand, and then with the software. And here is another famous substitution fractal, Serpienski's Gasket. You need Java to run these applets. Fractals were known for a while, but the science studying them really took off with computer development, because making something like that by hand, as we saw during the Club, is very time-consuming and thus difficult or impossible for humans. Still, we made a few fractals (or, strictly speaking, sketches of a few levels of fractals, since a real fractal is always infinite) by hand - using geometric figures, ducks, eggs, fancy daggers from Runescape, or our own hands. Fractal, to me, is an excellent symbol of complexity and simplicity. The incredible beauty of the structure with infinitely many levels is created and organized by a simple substitution procedure, or a short, elegant equation describing it. Here is one of the largest Internet gallery of fractal art - enjoy! People have also been experimenting with using the idea of a fractal for music composition. Here is a library of some fractal music sites.

At the end, we did a demonstration with soda and Mentos. Mentos have huge surface areas for being that small, because their surfaces are kind of like fractals. That's why the surface touches many molecules at once when we drop a Mentos tablet into the bottle of soda, and produces an explosive reaction: a geyser. I was expecting larger geysers, but it was fun enough. The minus was that we did not have enough bottles and tablets for everybody to participate, and I don't like the whole "sage on the stage" model for activities. Anyway, it was fun enough, and here is a video of how soda-mentos geysers can look

Have a great summer - we will meet again soon! If you find good math for the Club, keep it and bring it!