File:3-adic integers with dual colorings.svg

English: 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, ${\displaystyle x\in \mathbb {Z} _{3}}$, and it represents the corresponding character on the Prüfer 3-group, ${\displaystyle \chi _{x}:\mathbb {Z} (3^{\infty })\to \mathbb {R} /\mathbb {Z} }$, defined by ${\displaystyle \chi _{x}(q)=xq}$. The circle group ${\displaystyle \mathbb {R} /\mathbb {Z} }$ 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 p-adic 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 1.12.