文件: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 ${\displaystyle \chi_x(q) = x q}$. 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., Fractal geometry for images of continuous embeddings of p-adic numbers and p-adic solenoids into Euclidean spaces, Theoretical and Mathematical Physics, 1996, 109 (3): 1495–1507, arXiv:math/0202089 , doi:10.1007/BF02073866 . The particular mapping used is ${\displaystyle \Upsilon_s^{(\infty)}}$, defined in Definition 3 and depicted in Figure 1.12.