The Sierpinski triangle

The Sierpinski triangle is one of the most basic types of geometric images known as fractals.

The Sierpinski triangle is a deterministic fractal. It has the basic properties that are common to all fractals like recurrence : whatever part of the triangle you take, if you magnify it, you will find exactly the same triangle in it again and again.
Here is a picture of the Sierpinski triangle : What is very interesting with the Sierpinski triangle is that you can generate it very easily in more than one way. In this report are described two different sorts of construction :

• The first one is a random algorithm called Chaos game where you iterate a point construction from a seed in any triangle. After an important number of iterations, the sequence of points will fill out the Sierpinski triangle. In fact the orbit of the seed is attracted to the Sierpinski triangle which is called a strange attractor.
• The second one uses a deterministic geometric construction algorithm. You take any triangle and use the midpoints of each vertices to create a new triangle that you remove from the first one. This leaves three triangles on which you iterate the process.

• The Sierpinski triangle is the rest of the points when you have iterated the removal to infiny.

There are many other ways to generate it and sometimes it happens in surprising ways like in the modulo 2 representation of the Pascal Triangle.
The same process can be also applicated to a square, it is called the Sierpinski carpet.