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The Sierpinski Gasket

© Copyright 2002, Jim Loy

A Sierpinski gasket is also called a Sierpinski sieve. On the left and right, we see a series of Sierpinski gaskets (drawn using Fractint and Paint Shop Pro), discovered by Waclaw Sierpinski. The first order would just be a straight line segment. Here, I show orders 2 through 7. You should probably see how each new order is built from the previous one. The true Sierpinski gasket is the limit of infinitely many of these steps. Instead of lines, we can also build it with dark triangles (or any other object).

The Sierpinski gasket is related to The Yanghui Triangle (usually called Pascal's triangle), below. I have drawn hexagons around the odd numbers. That pattern is identical to that of the Sierpinski gasket, forever.

There is an interesting experiment called "the chaos game," in which random (presumably chaotic) chance produces great order. On the left, we see a picture. We draw three (or more) points (the vertices of a triangle, which doesn't have to be equilateral or isosceles), labeled 1, 2, and 3. Then we choose a starting point S, at random (the one I chose is not within the triangle). Then we begin the game. We proceed to choose random numbers, 1, 2, or 3 (with dice or whatever). Each random number defines a new point halfway between the latest point and the point toward which our random number directs us. For example, my first random number was a 1; so I drew a point halfway between S and 1. Then I got another random 1, then 3, 2, 1, and 3. After drawing 6 points, I perceive no obvious pattern. With a computer, it is easier to continue to choose many more points.

Below, we have three pictures of the chaos game. The first has 50 dots. The second has 160 dots, and starts to show a pattern. And the third has 10,000 dots, and the pattern is clear, the Sierpinski gasket. Surprised?

Why would this "game" produce the Sierpinski gasket? Well, if our first point within the triangle is on the gasket, then a little thought shows that all subsequent points will also be on the gasket. But, if the first point is not on the gasket, then it is easy to show that none of the subsequent points will be on the gasket! None! And most first points will not be on the gasket, as the area of the gasket is zero. I didn't show it here, but if we start in one of the white areas of the gasket, the picture after a few hundred points looks just like the Sierpinski gasket. Why?

Pretend that the first point is within the largest white area. Then the second point will be within one of the second largest white areas. And the third will be within one of the third largest white areas, and so on. There will never be more than one point drawn within any of the infinitely many white areas. That is very sparse. So the white areas will stay white. After a few points, all of the remaining points will be so close to the gasket that we cannot tell whether they are on the gasket or not. We know they are not on the gasket. But they will look like they are right on the gasket. And our result will look just like the Sierpinski gasket. How about that?

Addendum #1:

By adding rounded corners to the defining curve, we get a nonintersecting curve that traverses the gasket from one corner to another. Mandelbrot called this a Sierpinski arrowhead.

Addendum #2:

Here is a cellular automaton (see Life). We start with one black square at the very top of the graph. And then we apply the rules shown on the upper left. Each three adjacent squares horizontally determines the color of the one square directly below them. For example, three white squares produce a white square below them, and two white squares followed by one black square produce a black square. In this simple version, three of the eight rules are never used. So, using five simple rules, our one black square grows into the delightfully complicated Sierpinski gasket. And it continues to grow forever, if we keep supplying graph paper.

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