In the book "A Course in $p$ Adic Analysis" by Alain M Robert, he describes how the $p$ adic integers can be expressed as fractals. For example, $\mathbb{Z}_3$ can be expressed as a Sierpińsky gasket shown below.
But the $p$ adic integers apparently can form more complex fractals. For example, here is an example of $\mathbb{Z}_5$:
How exactly does this work? The author does give explanations about the work behind the different models, but I am just confused how $\mathbb{Z}_p$ can be expressed in this way at an elementary level. What exactly is going on here intuitively?
Additionally, if someone could point me to resources that elaborate on this subject (ie $p$ adic integers and their relation to fractals) that would be amazing.
A $p$-adic integer can be written as $$x = \sum_{m=0}^\infty x_m p^m$$ where each $x_m \in \{0,1,\ldots,p-1\}$. If you want to map this into $\mathbb R^d$, take $p$ distinct vectors ${\bf v}_0, \ldots, {\bf v}_{p-1} \in \mathbb R^d$ and a constant $0 < c \le 1/p$, and map $$x \mapsto \sum_{m=0}^\infty c^m {\bf v}_{x_m}$$ The map will be continuous; at least if $c$ is small enough, it should be one-to-one (I'll leave the details to you).