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The Sierpiński Triangle

A pattern that shows up everywhere in chaos and fractal theory.

Brief Explanation

This page shows what happens when you draw points at “random”. That is, there is a random element, but the overall result is very patterned. The basic idea is that you click the buttons, below the box, in order from top to bottom.

The button draws a single dot at random somewhere in the box. The dot is larger than all the later ones, so you can see it.
The buttons draw a new dot halfway from the first dot to point A, B, or C. It will be harder to see because it is the smallest possible dot on your computer screen. Try each of them, squint your eyes, and find the new point.
The button does the same as the blue ones, but picks A, B, or C at random. Click this button many times. Do you see any pattern? Probably not yet. Why should you? It’s random, right?
But of course you get impatient, and your clicking finger gets tired! Try the buttons. Surprise! A more detailed explanation follows below the box and buttons.

Try It Yourself

Pick a starting point at random, and draw it large. (So you can more easily see it.)
Draw another point, halfway between the previous point and A.
Draw another point, halfway between the previous point and B.
Draw another point, halfway between the previous point and C.
Draw another point, halfway toward A, B, or C, chosen randomly.
Has the effect of clicking the red button 100 times.
Has the effect of clicking the red button 1000 times.
Has the effect of clicking the red button 10,000 times.

Remarks

What you are doing each time you click a button is iterating a function. What that means is just repeating it, each time plugging in the last output into the new input. This is a kind of recursion. The sequence of points is called an orbit. Depending on the rule you use for the function, the orbit can go in all sorts of ways. Michael Barnsley first explored this, and called it the chaos game.

The usual way to get a picture of the Sierpiński triangle is to start with a large equilateral triangle, and remove a smaller triangle from the middle. Then do the same thing with the smaller remaining triangles. And then again with the remaining ones, etc.

The set of points you get when you have done this an infinite number of times (yes, that’s a tricky issue, that infinity) is called the Sierpiński triangle, after the Polish mathematician Wacław Sierpiński who investigated it in 1915.

This set is an attractor for the function we are iterating on this page: the “Go halfway to A, B, or C” function. That means that no matter where we start, the orbit of points always gets closer and closer to the set. So close that, as far as the computer image can render, we are actually on it. Now, why that should be, or more generally how to find the attractors for arbitrary functions, is a big and deep topic!

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