Sunday, 29 August 2010

John Conway's "Game of Life"

Recently I became very interested in cellular automaton. Have you, the reader ever come across John Conway's "Game of Life"?

It is a very simple game. The game is played on a two dimensional grid. Each cell in the grid can have two states: alive or dead. Every cell has eight neighbouring cells and the states of these cells affect the state of the central cell. Each step the fate of a cell is determined by the following rules.

  • Any live cell with fewer than two live neighbours dies, as if caused by under-population.
  • Any live cell with more than three live neighbours dies, as if by overcrowding.
  • Any live cell with two or three live neighbours lives on to the next generation.
  • Any dead cell with exactly three live neighbours becomes a live cell, as if by reproduction


To better visualize this situation look at the following pictures. The "game" is played on a grid that looks like the one below. At the beginning of the game the player chooses which cells to populate. this is called the initial state of the grid. At each step (sometimes referred to as a generation) the cells surrounding these populated cells and the cells themselves are changed according to the rules given above.






The above rules may appear to be strikingly simple. But the beauty of this game lies in the fact that extremely complicated patterns can be created using these simple rules. I'll start simple. Say you populate three horizontal rows and then allow the game to start. Here is the result.












If we start out with a shape that is a bit more complex.








These are however unimpressive when compared to the highly complex patterns that have been discovered. For example:


















The game of life is not just a game. The complexity produced by these simple rules are a prime example of how complicated structures can evolve from simple physical systems. In particular the arguments put forward by creationists and supporters of intelligent design can be refuted. Since the human body contains billions of atoms it is possible for this group of atoms to develop complex behaviour such as locomotion, and thought and an intelligence.

The patterns that I have shown in this post are the simplest of all discovered. High power computers have been used to generate highly complex structures that exhibit extremely complex behaviour.

Sunday, 15 August 2010

A Thought Experiment

I have been pondering over this thought experiment for a few weeks.
Here is the experiment:

Imagine a perfectly spherical ball and imagine that the inside of the ball was a mirror. Imagine that you are standing inside this mirror ball with a portable light source such as a flashlight or a fluorescent light bulb (I'm trying to be green here!). You turn on the light source to illuminate the inside of the sphere. What would you see?

This thought experiment seems to have a deceptively simple solution as mirrors are something we are very familiar with. I started off with thinking about what would happen if we were in a cube in which the inner walls were were mirrors. Mirrors placed parallel to each other produce an infinite number of images. So I would see infinite copies of myself each successively smaller. But what of the lighting conditions? At first I thought that the room would be very bright. But after a bit of thought I realized that my surroundings would seem very dark indeed. I realized that I would be very brightly illuminated but the parallel universes on the other side of the mirror would be very dark except for my reflection.

Now how do I complete the analogy and carry the argument on to a perfectly spherical mirror? It is very difficult to develop an intuition for curved mirrors. In this case my best guess is that I would see infinite reflections of myself but the reflections would be distorted.

Now what if I turn off the light? This part of the solution is fun to figure out and alltogether less mysterious. The light that is inside the mirror would keep bouncing around until all the radiation is absorbed by my skin and clothes. So when I turn off the light I guess I would still see myself but as the radiation is absorbed I would slowly get dimmer and dimmer until all the radiation is absorbed.

But what if I just keep the light shining? In a ruby laser light bounces back and forth between two mirrors until it gets so intense that it escapes from one of the mirrors which is only partially silvered. So I think that would mean that if I kept the light shining I would slowly get brighter and brighter until the intensity of the radiation is so high that I will have to close my eyes to protect them from permanent damage.

All this is assuming that the mirrors are perfectly reflective. In reality no mirror is perfectly reflective and always absorbs some of the radiation.

Now this situation is very hypothetical and my reasoning may not be correct. So this post is open for comment and discussion. I would like to hear the opinions of others and have a discussion in the comments page. Maybe that will clear up some of the grey areas.

Friday, 13 August 2010

Two Mathematical Jokes

An infinite number of mathematicians walk into a bar. The first goes up to the bartender and says, "I'll have a pint of lager, please." Each next one says, "and I'll have half of what he's having." The bartender says, "You're all idiots," and pulls two pints.


Person 1: What's the integral of 1/cabin?
Person 2: A log cabin.
Person 1: No, a houseboat – you forgot to add the c!

Hilarious.

Note to laymen: You need to be quite familiar with advanced math to understand these jokes. So do not panic if you find yourself scratching your head over this one.