The Amazing Sierpinski Triangle.
There are few branches of maths as awe-inspiring as fractals. Fractals are those never ending patterns where a small part is representative of the whole. The deeper you delve into a fractal the more you realise that you have only just begun and have not even scratched the surface. One such fractal is the Sierpinski Triangle.
Look at the animation above. No realise that every little triangle in the whole Sierpinski Triangle is as big and as detailed as the whole thing. You can zoom in to more and more detail forever. (The AutoCAD file for this be found here on GrabCAD – but it is not infinite)
But let’s start from the start.
What is the Sierpinski Triangle?
The Sierpinski Triangle begins with one solid triangle, as seen in Step 1 in this image.
In Step 2, we one triangle out of the middle of the first triangle, creating three triangles. Then in Step 3, we subtract one triangle out of each new triangle, creating 3 times as many triangles again. If we continue this forever, down to infinity, we will have a Sierpinski Triangle. Notice that each triangle we create in any of the steps is going to become a Sierpinski Triangle on its own. So each of the 243 tiny triangles in Step 6 will eventually become a complete Sierpinski Triangle on its own.
This is where it gets interesting because, with each step, the perimeter of the triangle multiplies by 1.5 and the area get 1/4 smaller. So essentially at the ∞th step, the perimeter is infinite and the area is zero.
How else can we construct the Sierpinski Triangle?
The next method of constructing the Sierpinski Triangle is by a recursive algorithm. In this algorithm, we take a straight line (as shown in Step 1 below) and replace it with three straight lines (as shown in Step 2). Then we repeat the process, once again, replacing al the straight lines with three smaller lines, alternating sides every time (as shown in Step 3).
Step 9 already begins to look like a Sierpinski Triangle. If we repeat these steps to infinity we will eventually get a perfect Sierpinski Triangle.
So with our first method, we created a Sierpinksi Triangle using ever decreasing area (2 dimensional) to create the triangle, and in our second method, we used ever increasing length (1 dimensional) to create it. Wierd. Not to mention that the Triangle’s area is zero and its length is infinite. So how many Dimensions does it have?
The Sierpinski triangle is 1.585 Dimensional
It turns out that the Sierpinski Triagle is neither 1 dimensional nor 2 dimensional but somewhere in-between. It happens to be 1.585 dimensional as it has Hausdorff dimension.
All in all the Sierpinski Triangle is a remarkable and mesmerizing geometric construction. Here is a series of videos explaining more about the Sierpinski Triangle.