The Chaos Game

The goal of the Chaos Game is to improve students' geometric intuition and algorithmic thinking. Before playing The Chaos Game, you should understand the role of the chaos game in constructing fractals. See the book Fractals, especially Lessons 5-7, for more details.

How to play the Chaos Game. When you open the chaos game applet, you see a game board that consists of the Sierpinski triangle computed down to level 2, i.e., with nine smaller triangles. One of these triangles is colored green; this is the Target. You also see a point colored red at the lower right hand corner of the triangle. This is the Seed. Your goal is to move the Seed into the INTERIOR of the Target in as few moves as possible. Each time you make a "move" in the chaos game, this point will move to a new location in the game board. This becomes your current location.

The moves. There are three possible moves in this chaos game. If you click on the top (red) vertex, your current location moves half the distance to the topmost vertex. Simialrly, clicking on the lower left (blue) or lower right (green) vertex, moves the current location half the distance to that vertex. By a judicious choice of moves, you should be able to move your point into the INTERIOR of the Target in just 4 moves. This is indicated in the Best Score window. Your score (the number of moves you have made) is recorded in the window called Your Score.

In the upper left corner of the screen your successive moves are recorded as a series of red, green, and blue dots. When you succeed in moving your current location into the interior of the Target, you receive a message telling you so.

Resetting the Target. To replay the current game again (without changing Target, click on Try Again. To change to a new Target, click on Restart. The computer randomly selects a new Target.

The Algorithm. There is an algorithm for moving the starting point into the interior of the Target in exactly 4 moves, no matter what Target you start with. Your job is to discover this algorithm. You should be able to explain in advance how to move the Starting Point into the Target in a couple of sentences, no matter where the Target is located.

More difficult Targets.. If you click on the Medium, Hard, or Master buttons at the top of the game board, you find chaos games with a Target at deeper levels of the Sierpinski triangle. The Best Score panel is appropriately modified. The algorithm that you developed at the Novice level should help you maneuver through these more difficult games.

Other Chaos Games. The default chaos game is chaos game #1. If you click on 2 or 3 at the upper right corner, you can play other (more difficult) chaos games. The rules of games 2 and 3 involve rotations. In game 2, the rules are the same at the two lower vertices. That is, if you click on either blue or green, you move half the distance to the appropriate vertex. But if you click on red, you first move half the distance to the little red "x" and then rotate about the x by 180 degrees. That is, you flip around this vertex to the exact opposite side.

For game three, you move half the distance to the corresponding red x and then rotate by 180 degrees around it for each of the three different moves.

Bug Report: Apparently there is a bug in Game two. For a very few targets, the computer claims that you won when, in fact, you only hit the boundary of the target, not the interior. This will be fixed some time.... Thanks to Paul Strickland for pointing this out!

Go to The Chaos Game.

Created by Noah D. Goodman, Adrian Vajiac, and Robert L. Devaney, based on an idea of Kevin Lee.

For comments and suggestions write to Robert L. Devaney at