Sierpinski Triangle

[Sierpinski Triangle]

Without a doubt, Sierpinski's Triangle is at the same time one of the most interesting and one of the simplest fractal shapes in existence. Because one of the neatest things about Sierpinski's triangle is how many different and easy ways there are to generate it, I'll talk first about how to make it, and later about what is special about it. Construction of the Sierpinski Triangle is easy, and there are many methods for its generation. We go right to the construction; if you have read this already, or if you are a boring stiff uninterested in creating your own Sierpinski triangle, you can skip to the What's a Fractal? section.

If you came here for Sierpinski images, I cannot be but flattered. You can download any of the following images:
Geometric Construction
The most conceptually simple way of generating the Sierpinski Triangle is to begin with a (usually, but not necessarily, equilateral) triangle (first figure below). Connect the midpoints of each side to form four separate triangles, and cut out the triangle in the center (second figure). For each of the three remaining triangles, perform this same act (third figure). Iterate infinitely (final figure).

[Sierpinski Construction]

The result, as you can see, is the Sierpinski triangle. The geometric construction of the Sierpinski triangle is the most intuitive way to generate this fascinating fractal; however, it is only the tip of the Sierpinski iceberg.

As a total aside, I have found that methodically drawing the Sierpinski triangle during boring lectures greatly relieves stress. If more boring lectures are anticipated, draw a huge one, like one that spans an entire sheet of regular paper drawn to painstaking detail. After a few lectures your boredom will be greatly relieved, your stress will go down, chicks (or hunks, as the case may be) will dig you, and you'll end up with a really, really impressively detailed (and large) Sierpinski Triangle which people will be really impressed with. They will say things like "Man, that's cool!" and "Whoa, how'dja do that!" and "Man, you must have been really bored."

Iterated Function System
The concept of the Iterated Function System was invented by creative fractal genius Michael Barnsley. An IFS is composed of a few simple elements: transformations and probabilities. The transformations are usually expressed as matrices--one translational matrix and one transformational matrix. This sounds a bit dry, but actually the concept is a bit more interesting than all that. For the Sierpinski Triangle, the transformations are simple, and are better expressed as rules in a recursive procedure:

The main procedure goes as follows.
Iterate ad infinitum.

That's it! Sierpinski's Triangle. To get any sort of worthwhile picture, of course, hundreds to thousands of points need to be plotted. In fact, usually the first twenty or so points are off track and need not be plotted. However, the algorithm outlined above is amazingly simple to do, and incredibly easy to implement on any machine. I have written Sierpinski programs for every graphing calculator under the sun, as well as for most desktop computers as well. Try it yourself in BASIC or on a graphing calculator. You'll surprise yourself at how easy it is to impress people with a graphing calculator. :)

Pascal's Triangle
Recall Pascal's Triangle, in which the outer edges are filled with ones and each inner element is the sum of its upper adjecent elements. This is a fascinating creature; Pascal's Triangle has always seemed to pop up in the strangest places: number theory, probability theory, even polynomial expansion. Well, it turns up in chaos theory as well. Look at the twelve-row expansion of Sierpinski's Triangle shown below:


Note that the triangle is divided into rectangular grids, with each grid occupied by one integer. Looks pretty innoculous by itself, nothing new, but any familiar to Pascal's Triangle--any math team member, any math puzzle lover--will know that nothing about Pascal's Triangle is innoculous. We are looking in this case, at the patterns of even and odd numbers in Pascal's Triangle. We know that they must be regular; so let us take a look at just how regular. Take each grid square, and color it black if it contains an odd number, and leave it uncolored if it contains an even number. See what appears:


The beginnings of Sierpinski's Triangle, clear as day.

And not only that, but we know that the pattern must continue further down. The fact that two odd numbers or two even numbers sum to an even number while an even and an odd sum to an odd number guarantee that this pattern must continue indefinitely.

And all this from something you never thought you'd see outside of basic Algebra.

One-Dimensional Cellular Automata
Under construction.

Boolean Algebra
Under construction.
For now, just plot y = x AND t or y = x OR t, as t varies.

What's a Fractal?

The Sierpinski Triangle raises all sorts of little questions that relate to topics in chaos theory not covered in the last few pages. For example, the Sierpinski Triangle is a canonical example of a shape known as a fractal. But soft, you ask, pray tell, what is a fractal?

Most simply, a fractal is a geometric construction that is self-similar at different scales. This is rather dry. More clearly, a fractal shape will look almost, or even exactly, the same no matter what size it is viewed at.

This is a pretty unintuitive concept. But let us look at the Sierpinski Triangle. The first step in the geometric construction of the Sierpinski Triangle involved splitting a triangle up into three other triangles. When we look at the finished Sierpinski Triangle, we can zoom in on any of these three sub-triangles, and it will look exactly like the entire Sierpinski Triangle itself. In fact, we can zoom in to any depth we would like, and always find an exactl replica of the Sierpinski Triangle.

This is deep. This is very deep.

What is the significance of fractals? They have numerous implications in mathematics and topography, as well as in computer graphics applications.

How Long is the Coast?

A long, long time ago, fractal god Benoit Mandelbrot posed a whimsical question: How long is the coastline of Britain? His mathematical colleagues were miffed, to say the least, at such annoying waste of their amazing computation powers on this insignificant minutae. They told him to look it up.

Of course, Mandelbrot had a reason for his peculiar question. Quite an interesting reason. Look up the coastline of Britain yourself, in some encyclopedia. Whatever figure you get, it is wrong. Quite simply, the coastline of Britain is infinite.

You protest that this is impossible. Well, consider this. Consider looking at Britain on a very large-scale map. Draw the simplest two-dimensional shape possible, a triangle, which circumscribes Britain as closely as possible. The perimeter of this shape approximates the perimeter of Britain.

However, this area is of course highly inaccurate. Increasing the amount of vertices of the shape going around the coastline, and the area will become closer. The more vertices there are, the closer the circumscribing line will be able to conform to the dips and the protrusions of Britain's rugged coast.

There is one problem, however. Each time the number of vertices increases, the perimeter increases. It must increase, because of the triangle inequality. Moreover, the number of vertices never reaches a maximum. There is no point at which one can say that a shape defines the coastline of Britain. After all, exactly circumscribing the coast of Britain would entail encircling every rock, every tide pool, every pebble which happens to lie on the edge of Britain.

Thus, the coastline of Britain is infinite.

Under construction.
To come: Fractal dimension, more about fractals...

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