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2002 05 24

I'm going to go on break for the next two weeks. The reason is that I just bought this new book called A New Kind of Science, by Stephen Wolfram. It's an utterly brilliant book which I feel will likely launch an important branch of science for the coming centuries. It's a massive 1200-page tome and I want these next two weeks to read it.

In preparation for reading it, last week I programmed a number of experiments with cellular automata and have been toying with them. The above image has been my favorite result so far. It was created with a series of simple rules applied to the single white pixel at the very top of the image. To color a pixel in the next row, you take the pixel above and the two pixels diagonally above it and give each a number. White gets 2, grey gets 1, and black gets 0. The sum of these three numbers determines the color of the new pixel, where 0, 1, 4, 5, makes black, 2 makes grey, 3 and 6 make white. This is the result of iterating this rule over 900 rows.

It really seems amazing that such a complex and interesting picture is somehow in the nine rules and the single pixel at the top. In addition, there are 3^9 possible sets of rules on the analogy of these. Trying them randomly produces extremely varried results, and many are just as complicated as the one here (although this is my favorite so far).

2002 06 03

I'm still on hiatus this week. Don't let this one comic fool you.

I like the nice reading break I'm taking. I finished several books on this break, including volume 1 of Euclid's Elements, which I greatly enjoyed, The Golden Compass, by Phillip Pullman, a truely GREAT fantasy book, Gone with the Wind, Journy to the Center of the Earth, and I'm about 3/4 done with A New Kind of Science.

The book is fascinating so far, but it doesn't quite live up to its hype. It is an advocation of a new paradigm of science for looking at the world, which tries to think of the world in terms of simple programs rather than mathematics. The first third of the book explains results of many experiments on very simple programs, and the rest of it explains the implications of these results for science. Some of the discoveries about simple programs are: that very simple rules can generate very complex behavior and even pure randomness; that beyond a certian threshhold of the complexity of the rules, the behavior that they generate is not necessarilly more or less complex than simplier or more complicated rules; and that randomness can be generated intrinsically in simple systems, without random initial conditions.

It is clear from Wolfram's arguments that these principles will likely have an effect on any scientist's outlook. However, they would be much easier to digest if his book were not written in one of the most annoying styles I've ever read. It's terribly obnoxious, patronizing, and repetitive, full of jeers at previous failures, and with many allusions to how great the author is. I knew something was up when I first heard the title, but I'm still amazed at how annoying it can be. The entire first chapter is nothing but self-glorification. However, this gets better as the book goes on and the author is able to give better and better concrete examples of how well his thinking works on real things.

2002 06 07

I don't have a comic today but this is just so cool I want to show everyone! This is the result of hours and hours of obsession, part of the reason I haven't had any comics in awhile. Last weekend, I saw an example in A New Kind of Science of a cellular automaton that would generate the square of any natural number entered into it. A cellular automaton is basically a grid of however many demensions you want, where each square at each step is updated by whatever rules you choose, based on the colors of the squares surrounding it. Oh yeah, I showed you one before, so you know what it is.

Anyway, this cellular automaton could compute the squares of integers. Each square in it used only the 3 squares directly above it to determine its color, using 8 colors. I thought that was really cool, so I decided to try making one myself.

My first attempt used 9 colors, and for some reason I didn't save the rules to that one, but I soon found how to reduce it to 8 colors. Here's a picture of how it calculates the squares of the numbers from 1 to 5. This one requires an input of n black squares followed by an orange square and returns a row of n^2 black squares. It follows basically the same idea as Wolfram's version, so far as I can tell by looking at the picture. The idea is, for a given n, to add n together n times. I stayed up all Sunday night making this. I couldn't stop myself.

These are my next two attempts. In the left one, I was able to alter the design slightly to make it work for only 7 colors, and in the left one I compacted it some more for only 6. These two take an input of n black squares and return a row of n^2 red squares. You can see how each time I was able to improve the design to allow for fewer colors. Many hours of frustration later, however, I was still unable to make this basic design work for 5 colors. I worked and worked, and almost had it, but each time something would screw up right at the very end, and any attempt to fix it would invariably cause it to go completely haywire. However, Wednesday afternoon, I suddenly thought of a different design that I thought could work much more easily.

THIS was the result. It's pretty obvious from looking at this that it is rather different from the previous 3. It's based on the fact that the nth square is always the sum of the first n odd numbers. So for example 1^2=1, 2^2=1+3, 3^2=1+3+5, 4^2=1+3+5+7, etc. It's easy to prove if you draw a picture. Anyway, the expanding blue region effectively counts off odd numbers, while the region on the left grows larger until it gets too close to the blue region. Each number is entered as a row of black, followed by a blue and a yellow. Isn't it cool!?

Incedentally, in the course of these experiments, I discovered another use for my synesthesia. All the colors in the pictures are the colors I have for the numbers that I used to designate them when I was programming. It was very helpfull because I could look at the picture and immediately see the rule I had to change or add because the colors so strongly suggest the numbers that represented them.

Don't worry, I PROMISE there'll be a regular week of comics next week! I'm ALMOST done with A New Kind of Science. It's really an amazing book, and it gets more amazing with each chapter. Every chapter has some amazing new revelation about something. The only problem is that it is at times very obnoxious and repetitive. I'd have made more comics if it wasn't for Mathematica, the program I used to make these. It's the easiest way I know of to program, and it's more addictive than a nintendo.

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