Fractal Parametrization
Idea
Given a fractal, how do we
parametrize it?
Sierpinski n-gon
Taking advantage of
self-similarity, we present a simple parametrization, $S_n(t)$, of the
Sierpinski n-gon in
$\mathbb{C}$ based on our
digit function.
$$S_n(t)=\sum_{k\in\mathbb{Z}^-}R^k e^{i\frac{2 \pi D_n^k(t)}{n}},$$
for all $0\le t < 1$, where
$$R=2\left(1+\sum_{k=1}^{\left\lfloor\frac{n}{4}\right\rfloor}\cos\frac{2\pi k}{n}\right).$$
$$\text{Graph 1. Interactive graph of the $K^\text{th}$ partial sum of $S_n(t)$.}$$
Some interesting cases are when $n=3$: the
Sierpinski triangle, $n=4$: a
space-filling variation of the
Vicsek fractal, and $n\geq 5$: have boundaries that are
Koch curves.
Sierpinski carpet
With a simple modification to $S_8$, we may produce a parametrization, $S_C(t)$, of the
Sierpisnki carpet in
$\mathbb{C}$ based on our
digit function.
$$S_C(t)=\sum_{k\in\mathbb{Z}^-}3^k \left[e^{i\frac{2 \pi D_8^k(t)}{8}}\right],$$
for all $0\le t < 1$, where $[\cdot]$ is the
nearest integer function for
Gaussian integers.
$$\text{Graph 2. Interactive graph of the $K^\text{th}$ partial sum of $S_C(t)$.}$$
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