Strangely Attracted to Chaos


Iterators and the Chaos Game



To understand how fractals work, you need to learn a concept called iteration. Iteration is the process of repeating a procedure, called an iterator, over and over again to achieve a desired outcome.

You experience iteration when you play a game of checkers. On each turn, you examine the current state of the game, i.e. whose pieces are where. After some thought, you make a move you hope will bring you one step closer to your desired outcome, which is to capture all of your opponent's pieces and win the game. Unfortunately, your opponent also wants to win, and may confound your strategy with a clever move of his/her own.

Herein lies the fun of checkers: When your opponent moves, he/she changes the game state (which pieces are where), upon which you will base future moves. This is called feedback - your and your opponent's moves are "fed back" into the game state. This is why, although the rules of checkers are simple, a game of checkers can become extremely complicated after only a few moves.

The Chaos Game

   Let me introduce you to another game of iteration: You'll need a die, a piece of paper and a pencil. Draw three dots on the piece of paper as shown in the diagram to the left. Label each dot as shown. Place your pencil at the point labelled "1,2".
   The numbers assigned to each of the three dots represent the faces of a die (each point gets 2 faces of the die). Roll the die. Move your pencil halfway between its current position and the dot assigned to the number you rolled, and mark a new point.

Let's suppose you roll a 5. Move your pencil halfway between its starting position and dot "5,6".

Now your pencil is at a new position.
   Roll the die again and repeat (iterate) the process. Let's suppose you roll a 4. Move your pencil halfway beetween its current position and dot "3,4", and mark a new dot.
   Roll again, and move and mark. Suppose you roll another 4. Move halfway again. Now your pencil is closer to dot "3,4".
   Roll again, and move and mark.
   Let's suppose you continued to iterate this process - rolling, moving, marking. What would happen after 100 points? After 1000? After a million? Would you eventually fill the paper with a jumble of dots, or would something more interesting emerge?

Let's find out by having a computer plot thousands of dots for us. Click to the next page, Serpinski Solution, and watch closely.


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