Sierpinski Triangle
Posted on May 06, 2019 in Maths
Sierpinski triangle is a fractal with a unit shape of a (equilateral) triangle. It's named after a polish mathematician Waclaw Sierpinski.
There are copule straightforward algorithms to generate this structure. One is to start with a solid triangle and repeatedly remove a similar triangle from the center (i.e. one with the line joining mid-points as the sides).
But, there is another, a bit more interesting algorithm where it really isn't all that trivial to see how and why it generates Sierpinski triangle.
It is also pretty simple, straightforward and goes like this:
- Take three points(vertices) in a plane to form a triangle.
- Randomly select any point(tracer) inside the triangle and consider that your current position.
- Randomly select any one of the three vertex points.
- Move half the distance from your current position to the selected vertex.
- Plot the current position.
- Repeat from step 3.
This is what I am trying to demonstrate here. I also want to get a sense of the number of iterations it takes for a satisfactory Sierpinski Triangle looking shape to emerge.
Steps 4 - 6 is done automatically, just hit the play button below.
Iterations: 0
Arrowhead curve
Okay, here's a bonus. This is another way to create Sierpinski Triangle. It's called arrowhead curve. It is pretty interesting if this is the first time you're seeing this.
Once again, we start with something that looks nothing like Sierpinski Triangle. But slowly the shape begins to appear. This essentially creates Sierpinski Triangle out of line.
Of course, Sierpinski Triangle is the limit of this process as you take infinite order of this recursion. That makes this an infinite line which, isn't all that amazing to say, I guess. Every figure can be made with line of infinite length, or with infinite points.
But still, it's amazing to see the shape emerge.
Try to look for pattern and see what the steps are. It's more subtle than you might think at first.
The End.