Exploring cellular automata 1

Conway’s game of life is maybe the most well known ruleset for cellular automata. Each cell in a grid has some possible set of states, and set of rules for changing state based on its own and it’s (usually) immediate neighbor cells. Time elapses with a number of discrete steps, sometimes called epochs. In Game of Life, a cell can either be “alive” or “dead” (on or off), and will live for the next time step only if it has one or two alive neighbors.

Some incredible patterns emerge, including spaceships, guns (creating gliders and spaceships), breeders (creating guns), other puffersrakes, long-lived methuselahs (this example creating some gliders).

Other rulesets lead to neat patterns. Here’s a fun one with many cell states. It lets you rate the results as it scans some starting parameter space.

So, we’re certainly not limited to one ruleset. Coming up with one is easy — finding if there are any neat patterns maybe not so much. It might be better to first explore simpler rules — ones in only 1 dimension. In this case, a linear space is segmented into cells, and each cell has only two neighbors. Given some configuration of three cells in a row, the middle cell’s state in the next epoch will be dependent on its own state as well as the two alongside. When considering how to change any cell, since there are only three cells to consider, and only two possible states for each, there are only $2^3 = 8$ arrangements to consider when changing any cell. Because of this small number, it’s no problem to display the rule of the automaton as a list of diagrams, each showing one of the eight starting arrangements and the resulting state of the middle cell in the next epoch.

With some initial state, for example:

The next line is produced by scanning over this line and matching the top row with the one from the rule icons.

Note that this wraps around, connecting the two edges. This process then continues indefinitely for successive lines. We’re free to pick any length of line and any arrangement of on and off cells as an initial condition, but once this is done, the rest is automated. This particular initial condition for this rule ends up looking like this:

This is kind of neat, but these processes don’t get to show off unless they’ve got some room to work with. Here’s the same rule applied to an initial state of a single black cell in a long row of white.

The repeating triangles are a pattern characteristic of some of the more interesting rules. Rules 30, 90, 110, and 184 are supposedly the most cited in scientific or mathematics articles.

Because there are only 8 arrangements of cells we need to specify to determine a 1-D ruleset, and each arrangement offers the choice of the middle cell turning on or off, there are $2^8 = 256$ possible rules of this type. You can see links to them all here. One could argue there are only 128 rules, 64, or even 32, since negating all the colors, taking a mirror image, or negating the mirror image doesn’t change the behavior much. This means each rule is one of a set of four that operate in the same way.

I’ve already linked to the Wolfram Atlas a few times. The images above were taken from there or made using Mathematica. Stephen Wolfram, of Mathematica fame, is also well-known for his work on cellular automata. He’s argued that these things may be more fundamental to nature than mathematics. It’s considered controversial. I haven’t read A New Kind of Science, a book where he lays this all out, but it’s free and online. Maybe worth checking out. It seems like some of the argument hinges on the fact that some of these automata are Turing complete — they can be set up to operate and produce results like any computer program.

I’d just like to note that my interest in cellular automata is based pretty much entirely on having fun. It’s 11 o’ clock on a Saturday night.

A step after checking out the 1-D rules is to explore the 2-D rules. Next I’ll show off a rule and try to do something interesting with it.

Pareto Principle

The fractal drawing period went well enough. I suggested to the students that they instruction sheets increased in difficulty (when in the same order as in the previous post). I expected most to go for the Sol LeWitt drawing. In fact, that was kind of a wash! I had really hoped to have a nice full sized Wall Drawing 797 poster made by the kids. Most went straight for the dragon fractal. I can’t really blame them, since it looks so cool, but my instructions were unclear enough to make it difficult for most. The biggest issue was that most students didn’t pick up that each “elbow line” needed to be drawn so that it connected opposite corners of the squares on a piece of graph paper. Their dragon drawings ended up looking flat and floppy, and after only a couple iterations there wasn’t a whole lot left to work with.

Most of the kids were not thrilled to be doing any mindfulness activity. Many asked if I could ask the administration to never have a mindfulness advisory day again. There were, however, a few who quietly and attentively worked on drawing the curves, and a couple did quietly exclaim that the dragon curve was pretty cool.

I’ll keep things in perspective. My goal wasn’t to make them experts on drawing these figures, but to at least expose them so that they can recognize the ideas if they happen to come across them later. I’d also hope that a seed of interest has been planted in at lease one student, so that they’d be motivated to read a bit more about these things on their own.

The ratio of interested kids to uninterested kids reminded me of something called the Pareto Principle. This rule of thumb asserts that, in a situation like this, I could expect 80% of the effort or participation to come from 20% of the students.

This is a really lousy model for the drawing activity, not only because effort isn’t a well-quantified value. Number of drawings, time spent with pencil to paper, some arbitrary standard for quality all could play into the measure of effort. In addition, the “4:1 for 1:4” Pareto rule is a goofy way of putting it; the statement that “20% do 80% of the work” can make one think that there are some students who are doing 4 or 5 times the work of another. In fact, it would mean that the hard-working students would be doing 16 times more than one from the larger population. This seems a bit high for a typical high school class.

The Pareto rule might work a bit better in the business example, “20% of your customers will give you 80% of the sales,” but any decent business would hopefully make predictions based on their particular situation and history. Perhaps most hover around this distribution — it doesn’t seem too wild of an idea that a few die hard regulars are the ones keeping any given bar afloat.

The real benefit of this rule of thumb is to give that sense of perspective. It would have been unrealistic for my goal to have the room be lit with energy, all of the kids scrambling around in excitement because they were given some drawing instructions.  To expect 1/5 of the students to be truly interested might sound pessimistic, but in retrospect it’s a good place to start when making expectations. It might also be a good place to start in a brand-new business.

The rule is an instance of a Pareto distribution, which is a generalization that would be able to tell you exactly how much each student is producing (rather than big groups), as well as describe groups of students who are producing work at different ratios than 4:1. The benefits of being able to describe your system (be it schoolchildren, sales, volunteer participation, whatever) with a Pareto distribution would not only lie in how accurate the shape is, but also in knowing exactly the parameters that fine-tune that shape.

The Pareto concept has other ways of showing itself. Zipf’s Law says that you order the words in the English language by how often they’re used, the Nth word in the list will be 1/N as popular as the 1st. So, you see a very small number of words showing up a very large percentage of the time in writing. Like the Pareto principle, this is a generalization, and different people, regions, and documents will have variations on word popularity. These variations allows for statistical stylometry, as well as making sure every book in the library isn’t identical.

The Internet %1 Rule (or 1-9-90 Rule) is one that suggests that only about 1% of users on a website actively create content for that site. This doesn’t mean 1% of people who read BBC.com are writing news articles, but it might mean that 1% of a news site’s readers are leaving comments. I’ve heard this referred to often in the realm of podcasts, where show hosts can expect about 1% of users to email in, or participate in a contest, etc.

Again, it’s key to note the error bars we’re willing to accept. If 2% rather than 1% of listeners responded to a call for podcast questions, the creators might not notice the difference. Using the Pareto principle as a very rough rule of thumb, as a suggestion for what to expect, is the way to make it work for you effectively.

Fractal Drawing for Mindfulness Activity

Tomorrow, our student advisory team is leading a school block dedicated to mindfulness. Each teacher leading an advisory was asked to propose an activity that would promote mindfulness.

I put together the following four sets of instructions for kids to sit and do focused drawing for the period.

Sol LeWitt: Wall Drawing 797

Hilbert Curve

Peano Curve

Dragon Curve

My experience with very short meditation sessions before and after kendo practices are the most direct form of regular mindfulness of which I’ve been a part, at least as a designated “activity.” I’m not prepared to lead 45 minutes of quiet meditation for high schoolers, though, and especially not if I have to keep them in pain sitting in seiza. I like to think my time hiking in the Adirondacks has been mindful, in that it involves focus, physical energy, awareness of one’s surroundings and situation, as well as the calming connection to nature, but I can’t take the kids out on a tough hike for a half hour in the middle of the school day. Exercise like running could be mindful, but it sure isn’t when I go, since I’m usually listening to a podcast or music while I jog. Really, the request to guide a session caught me off guard, since I don’t think I’m qualified to say or present much about mindfulness.

The other activities the teachers are leading include Drop Everything and Read, yoga, “games outside”, adult coloring books, and exercise. These all seem like they could be done well enough.

I figured that, since I’m preoccupied with math while at the school anyway, I might as well try to fit in some kind of math-related thing. The fractal curve drawing stuff is something I brought up while co-teaching a Math and Art elective at a previous school, and it had already been floating around in my head while thinking about leading a math team, or introducing some kind of alternative project for one of my current classes.

Sitting down and scratching away a pattern that follows a rule, and knowing it’s going to end up with a near image in the end sounds like a great relaxing time to me. I do have to remind myself that sometimes my brain is a runaway freight train, and people may not be so interested in thinking about Star Trek or Homestuck as much as I want to keep bringing them up. The idea that someone might not be as interested in drawing fractals for an hour as me is something I need to be aware of. I do think there’s a good chance this set of activities will activate the LEGO loving kid in some of them, though.

While sitting in a meeting, I realized the dragon fractal could be generated by recording a list of L and Rs, representing left and right 90-degree turns while drawing a line in a grid. Start with just an L. For each iteration, add an L to the end of the string, then write a mirror image of all of the previous letters: write them in reverse order, and replace L <—> R.

L

L L R

L L R L L R R

L L R L L R R L L L R R L R R

L L R L L R R L L L R R L R R L L L R L L R R R L L R R L R R

etc. These five strings correspond to the images at the top of the Dragon Curve drawing instructions. The mirror image behavior creates the self-similarity. It also made it more clear why the paper folding method creates the same figure.

Later I realized that this set of instructions is essentially the same as the Lindenmayer system of writing the dragon fractal. Let “F represent drawing a line forward, “+” a 90 degree counter clockwise turn, “-” a 90 degree clockwise turn, and “X” and “Y” as placeholders which add to the structure but don’t indicate a drawing instruction. Start with FX as your string of drawing instructions. For each iteration, replace

X –> X+YF+

and

Y –>−FX−Y

You can see the mirror image instruction here: the string to replace each “Y” can be recreated by taking the X replacement string, reversing the order, and swapping X <—> Y and “-” <—> “+”.

In the case that fractal drawing is too intimidating for anyone, I’ll include the Sol LeWitt page as well. I hope we can beast out a full posterboard with 797-like drawing tomorrow as a class taking turns. The MASS MoCA trip mentioned in the instructions was with a bunch of good friends in August, and they drove a conversation about the piece that day, and continued to think and play with it for some time afterwards (Phil and Ben’s blog posts, each with some neat insights and programming results, can be found in the links <— over there and <— over there).

We’ll see how tomorrow goes.