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?
If you came here for Sierpinski images, I cannot be but flattered. You
can download any of the following images:
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).
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
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
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
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. :)
The main procedure goes as follows.
Begin by selecting three vertices in the two-dimensional plane. Note their
Next, select an initial point--best if within the bounds of the triangle
formed by the three vertices, but this is not a requirement.
Iterate ad infinitum.
Select one of the three vertices at random. Roll a six sided die and
integer-divide by two, flip three coins and take the exception, or ask a
friendly computer for help.
From the current location of the point, calculate the midpoint of the line
connecting the point and the vertex just selected.
Move to this midpoint, and plot the point.
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
One-Dimensional Cellular Automata
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
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
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
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
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.
To come: Fractal dimension, more about fractals...
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