#### Sierpinski Triangle

"The Sierpinski Triangle is a fractal with the overall shape of an quilateral triangle, subdivided recursively into smaller equilateral triangles." - Read more

###### Formula
The Sierpinski Triangle can be calculated with the help of random numbers. The corresponding algorithm is also known as Chaos Game. The algorithm determines how to draw the triangle with the help of three two-dimensional points $$p_1,p_2,p_3$$, a random point $$v_1$$ and a random value $$r_n$$, where $$r_n$$ can be either $$1$$, $$2$$ or $$3$$.
The associated formula is $$v_{n+1}=\frac{1}{2}(v_n+p_{r_n})$$.
Each point $$v_{n+1}$$ is the exact midpoint of point $$v_n$$ and a randomly selected corner $$p_{r_n}$$ of the triangle.

To start the Chaos Game, select three corners of the Sierpinski Triangle by clicking at three locations in the black frame.

The Sierpinski Triangle can be calculated a lot faster, of course. A simple method recursively creates three new triangles for each existing triangle. The method then creates three new triangles for the just created ones. This could go on for ever, but, of course, it is limited to the size of your computer´s RAM.

The controls are the same as in the above script. Just select three corners.