File:2-adic integers with dual colorings.svg

English: The compact group of 2-adic integers (black points), with selected elements labeled by the corresponding character on the Pontryagin dual group (the discrete Prüfer 2-group) (colored discs).

Counter-clockwise from the right, the labeled elements are 0, 4, 2, −3, 1, −1/7, −1/3, 1/3, 1/7, −1, 3, −2, and −4. Open the SVG directly in your browser to get tooltips on group elements.

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

For details on the embedding of the 2-adic integers, see Chistyakov, D. V. (1996), “Fractal geometry for images of continuous embeddings of p-adic numbers and solenoids into Euclidean spaces”, in Theoretical and Mathematical Physics, volume 109, issue 3, DOI:10.1007/BF02073866, pages 1495–1507 The particular mapping used is ${\displaystyle \Upsilon _{s}^{(\infty )}}$, defined in Definition 3 and depicted in Figure 1.