Consider the pattern of the parity of the binomial numbers. Color the even numbers white and odd numbers black. You get a Sierpinski Triangle.
We will prove this using spacetime-geometric reasoning. It would help a lot if you have done some geometry of special relativity.
Define the parity function $f(t, n) = C(t, n) \mod 2$, defined on $\mathbb{N}\times\mathbb{Z}$.
We can consider $f(t, n)$ as encoding a 1-d celluar automata, with $t \ge 0, n \in \mathbb{Z}$. Then we have the evolution rule:
$$ f(t+1, n) = f(t, n) + f(t, n-1) \mod 2$$
This is just the Rule 60 celluar automata.
Then it's easy to prove the following:
