Aug 22, 2009 © Harry P. Schlanger
Using a simple algorithm and a die, this game is a mathematical experiment that creates a fractal pattern
of the well-known Sierpinski triangle.
The Sierpinski triangle
is a fractal
named after the Polish mathematician Waclaw Sierpinski (1882 - 1969) who described it in 1915. It is called
by different names such as Sierpinski gasket or Sierpinski sieve.
The Sierpinski pattern is mathematically generated and reproducible at any magnification or reduction;
such is the nature of a fractal.
Sierpinski sieve can be created by directly drawing a series of ever diminishing triangles, or
interestingly, by random mathematical experiment, as was originally devised by Michael Barnsley
(Fractals Everywhere, Elsevier, 1988). Barnsley informally coined his technique,
The Chaos Game.
Chaos Game Construction
To construct the chaos game, all one needs is pen, paper and a means to measure midpoints. Three points
are started that outline a triangle (Figure 1).
Fig 1. Chaos Game traced out by three points form the corners of a triangle.
Start the game by selecting a point at random.
These three points are labelled (1, 2), (3, 4), and (5, 6)
and this area becomes the "playing board" for the game.
To begin the game, a point is selected at random. This point can be anywhere within the triangle or
outside of it and is labelled P.
Throwing the Die
Now the player rolls a fair die, then proceeds halfway from point P to the point (or angle) labelled
with the rolled number, and plots a new point.
Assuming a 6 has been rolled, the player moves from point
P to the angle labelled (5, 6) and plots a new point (Figure 2), which become the new point P.
Repeating the above steps many times, a series of dots emerge, some of which never reach certain
regions within the large triangle's periphery.
Fig 2. Throwing a die and plotting a new point
To gain a sharper picture of the emerging pattern,
a "cloud" of these points are required. This is a reason why it would be preferable to generate
points on a computer.
The completed pattern formed is shown in the article's image above, called the Sierpinski triangle. It is an infinite
number of triangles continued within the larger triangle.
If one increases the resolution (i.e. making the playing board larger), one can see ever more triangles. This is feature is called self-similarity, in which the
same pattern is seen irrespective of scale. It is a well known and important characteristic of fractals.
Initial Conditions
Interestingly, the shape is not dependent on the initial point. No matter where one begins, the
Sierpinski triangle is created from the resultant pattern. This is despite the fact that two
“random” events are needed to play the game:
- Selection of the initial point
- Roll of the die
Thus at the local level, the points are always plotted in random order. The reason the Sierpinski
triangle always emerges is because the system reacts to the random events in a deterministic manner.
Local randomness and global determinism create the stable structure.
Computer Program
A
computer program has been written by the author to easily create a Sierpinski triangle. The program
steps are simple and can copied and pasted into an Access 2007 report module.
References:
- Fractal Market Analysis, Edgar E. Peters, Wiley & Sons NY, 1994
- Chaos, James Gleick, Penguin (Non-Classics), 2008 (for non-experts)
The copyright of the article The Chaos Game: How to Simply and Randomly Generate a Sierpinski Triangle is owned
by Harry P. Schlanger. Permission to republish in print or online must be granted by the author in writing.
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Aug 22, 2009 © Harry P. Schlanger
