The wiki article on p-adic numbers has this wonderfully charming and pretty graphic:

This is supposed to represent "the 3-adic integers, with selected corresponding characters on their Pontryagin dual group."

And that's the only explanation provided. I've looked, but have been unable to find anything more detailed. So, I'm turning to MathStack: can someone break down this beautiful kaleidoscope?

Edit: The article on Pontryagin duality has a similar image for the 2-adic integers.

(also unsure of the best way to tag this...)

A character is a continuous group homomorphism $(G,+)\to\Bbb T$ (with $\Bbb T\subset\Bbb C^\times$ the circle group).

The dual group $\widehat{G}$ is the space of all characters on $G$ equipped with pointwise multiplication.

Let $\psi$ be a character of $\Bbb Z_p$. Since $\Bbb Z\subset\Bbb Z_p$ is dense, the values $\psi(\Bbb Z)$ determine $\psi$, and as $\Bbb Z=\langle 1\rangle$ this in turn means that $\psi$ is determined by $\psi(1)$. Since $p^r\to0$ in $\Bbb Z_p$, the values $\psi(p^r)=\psi(1)^{p^r}$ must converge to $\psi(0)=1$. If $\psi(1)$'s complex phase were not of $2\pi\Bbb Q$, then $\psi(1)^{p^r}$'s phase would never settle down - furthermore it must be $p$-torsion mod $2\pi$ to settle down, so $\psi(1)$ is some $p$-power root of unity. Thus $\widehat{\Bbb Z_p}\cong\Bbb Z(p^\infty)$ via $\psi\leftrightarrow\psi(1)$ (with $\Bbb Z(p^\infty)$ the Prüfer $p$-group).

(KCd has a nice related blurb on the character group of $\Bbb Q$, which motivates the adeles.)

Note this is analogous to $\Bbb Z$ and $\Bbb R/\Bbb Z$ being a dual pair, in view of the fact $\,\Bbb Z(p^\infty)\cong\Bbb Q_p/\Bbb Z_p$.

The group $\Bbb Z(p^\infty)$ can be thought of as $\Bbb Z[p^{-1}]/\Bbb Z$ under addition, so every element of the Prüfer group may be represented by the rationals expressible finitely as $0.\square\square\square\cdots\square$ in base $p$.

The topology of $\Bbb Z_p$ is that of a (countably infinite depth $p$-ary rooted) "tree": draw a point, then draw $p$ child nodes from that point, then $p$ child nodes from that point, and so on. The $p$-adic integers will be all "leaves" (infinite paths through the tree from the root). The metric balls are obtained by picking a node on the tree and collecting all leaves that run through that node.

(More on this in the note pictures of ultrametric spaces.)

An equivalent way of representing the topology of the $p$-adics is used here. Draw one big ball, then draw $p$ balls inside, then draw $p$ balls inside each of those, and so on indefinitely. To select an integer from $\Bbb Z_p$, make an infinite sequence of selections of these balls, one choice representing each digit of $\Bbb Z_p\ni x$'s $p$-adic expansion. In your picture, the gray nest of balls represents $\Bbb Z_3$.

A number of three-adic integers are chosen from the gray urn, and correspond to colored circles wreathed around the outside. Each one of these outer colored circles represents the Prüfer $3$-group, in particular each "leaf" is an element. The biggest leaves are $0/3,1/3,2/3$ and the second biggest leaves are $1/9,2/9,4/9,5/9,7/9,8/9$ (so basically the rationals $k/9$ with $0\le k<9$ not counting the ones already listed, $0/3,1/3,2/3$). The $r$th biggest leaves correspond the the rationals expressible as $0.\square\cdots\square$ with $r$ digits in base $p$, not already listed (i.e. with last square nonzero).

The idea of "duality" is not only do elements of $\Bbb Z(p^\infty)$ act as characters on $\Bbb Z_p$, but conversely elements of $\Bbb Z_p$ act as characters on $\Bbb Z(p^\infty)$. To each $p$-adic integer $x\in\Bbb Z_p$, the colors of the leaves $a\in\Bbb Z(p^\infty)$ (on the associated circle outside) correspond to the value of $x$ applied to $a$ as a character, which will always end up being a $p$-power root of unity (as seen above).

The image description page has much more detail:

The compact group of 3-adic integers (black points), with selected elements labeled by the corresponding character on the Pontryagin dual group (the discrete Prüfer 3-group) (colored discs).

Counter-clockwise from the right, the labeled elements are 0, 9, 3, 1, −1/8, −1/2, 1/4, −1/4, 1/2, 1/8, −1, −3, and −9. Open the SVG directly in your browser to get tooltips on group elements.

Each colored disc is tied to a 3-adic integer, $$x\in\mathbb{Z}_3$$, and it represents the corresponding character on the Prüfer 3-group, $$\chi_x : \mathbb{Z}(3^\infty) \to \mathbb{R}/\mathbb{Z}$$, defined by $$\chi_x(q) = x q$$. The circle group $$\mathbb{R}/\mathbb{Z}$$ used is a color wheel where $$0 =$$ red, $$1/3 =$$ green, and $$2/3 =$$ blue.

For details on the embedding of the 3-adic integers, see Chistyakov, D. V. (1996), "Fractal geometry for images of continuous embeddings of $$p$$-adic numbers and solenoids into Euclidean spaces", Theoretical and Mathematical Physics 109 (3): 1495–1507. . The particular mapping used is $$\Upsilon_s^{(\infty)}$$, defined in Definition 3 and depicted in Figure 1.12.

Does that help? I can address any further questions if that description doesn't cover them.