The Sierpiński Triangle

Understanding Fractals

In mathematics, Fractals are geometric objects which intuitively do not seem to have a well defined notion of dimension (which we expect to be a non-negative integer). Rigorously, after defining a more general notion of dimension such as Box counting dimension or Hausdorff Dimension, they are exactly the geometric objects with non-integer dimension.

The Sierpiński as an example of a fractal

The Sierpiński triangle is an example of a fractal, which can be constructed by the iterative removal of triangles as follows:

Using the Hausdorff notion of dimension, it can be proven that the Sierpiński triangle has dimension $\log(3)/\log(2) \approx 1.585,$ being nearly at the midpoint of being a one dimensional object (a curve), and a two dimensional object (a surface).

Abstract construction using Bernoulli shifts

A Bernoulli shift is a discrete-time stochastic process with i.i.d. random variables on a set of finite symbols.

For the case of constructing the Sierpiński triangle, we may think of our stochastic process as a sequence of three-sided coin flips, for some mysterious coin with three sides equally likely of landing faced up.

In this hypothetical scenario, we represent each side of the coin by $0,1,2$ and we represent our vertices as $v_0,v_1,v_2$. Given a large sample of these sequence of coinflips we now proceed as follows:

1. For each sequence of coinflips $X$, place a point $p_0(X)$ on $v_{x_0}$;
2. Define $p_{i+1}(X)$ to be the midpoint between $p_i(X)$ and $v_{x_{i+1}}$.

The probabilistic Sierpiński triangle of depth $N$ is defined to be the set of all points $p_N(X)$. The following is a visualization of this process using the python Library Manim.