File:3-adic integers with dual colorings.svg: Difference between revisions

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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 [http://upload.wikimedia.org/wikipedia/commons/c/ce/3-adic_integers_with_dual_colorings.svg directly in your browser] to get tooltips on group elements.
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 [http://upload.wikimedia.org/wikipedia/commons/c/ce/3-adic_integers_with_dual_colorings.svg directly in your browser] to get tooltips on group elements.


Each colored disc is tied to a 3-adic integer, <math>x\in\mathbb{Z}_3</math>, and it represents the corresponding character on the Prüfer 3-group, <math>\chi_x : \mathbb{Z}(3^\infty) \to \mathbb{R}/\mathbb{Z}</math>, defined by <math>\chi_x(q) = x q</math>. The [[w:circle group|circle group]] <math>\mathbb{R}/\mathbb{Z}</math> used is a [[w:color wheel|color wheel]] where 0 = red, 1/3 = green, and 2/3 = blue.
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For details on the embedding of the 3-adic integers, see {{Citation |last=Chistyakov |first=D. V. |year=1996 |title=Fractal geometry for images of continuous embeddings of ''p''-adic numbers and solenoids into Euclidean spaces |journal=Theoretical and Mathematical Physics |volume=109 |issue=3 |pages=1495–1507 |doi=10.1007/BF02073866 |arxiv=math/0202089}}}}

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For details on the embedding of the 3-adic integers, see {{Citation |last=Chistyakov |first=D. V. |year=1996 |title=Fractal geometry for images of continuous embeddings of ''p''-adic numbers and solenoids into Euclidean spaces |journal=Theoretical and Mathematical Physics |volume=109 |issue=3 |pages=1495–1507 |doi=10.1007/BF02073866 |arxiv=math/0202089}}}}. The particular mapping used is <math>\Upsilon_s^{(\infty)}</math>, defined in Definition 3 and depicted in Figure 1.12.
|date=2012-02-17
|date=2012-02-17
|source={{own}}
|source={{own}}

Revision as of 10:00, 12 March 2012

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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, , and it represents the corresponding character on the Prüfer 3-group, , defined by . The circle group 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”, in Theoretical and Mathematical Physics, volume 109, issue 3, DOI:10.1007/BF02073866, pages 1495–1507
. The particular mapping used is , defined in Definition 3 and depicted in Figure 1.12.
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