Wikipedia:Featured picture candidates/3-adic integers with dual colorings

3-adic integers with dual colorings

Voting period is over. Please don't add any new votes. Voting period ends on 21 Mar 2012 at 02:42:03 (UTC)

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

Reason

I think the real value of this diagram is aesthetic; it "illustrates the subject in a compelling way, making the viewer want to know more". Don't get me wrong: there's a lot of relevant information packed into the diagram, but it's hard to describe what it all means, and that compromises its EV. It wouldn't be an ideal image to illustrate an idea in the body of an article, but it serves well as a lead image for p-adic number.

(A similar diagram, 2-adic integers with dual colorings.svg, is used for Pontryagin duality, but it's not as pretty.)

Articles in which this image appears

p-adic number

FP category for this image

Sciences/Mathematics

Creator

Melchoir

• Support as nominator --Melchoir (talk) 02:42, 12 March 2012 (UTC)
• Oppose Firstly, the image must be in the article for a week. At any rate, the image is pretty, and it does make me want to know more - but I don't think the EV is there at the moment. It isn't immediately clear what one is looking at. After a very quick skim over that paper I'm guessing that we are looking at an embedding of the (compact) group of 3-adic integers into $\mathbb {R} ^{2}$ under the addition operation of the ring described in the wiki article. The details of this embedding are not immediately clear to me. I can't see how the group operation works, and I don't know if this embedding has any interesting properties, like if there is some sort of isometry between the p-adic absolute value and the euclidean metric, for example. I can't see anywhere describing how to interpret the colourful circle shapes presumably representing corresponding characters on the discrete Prüfer 3-group. I also don't know what the light grey circles represent - they don't appear in the cited paper. JJ Harrison (talk) 06:05, 12 March 2012 (UTC)
  • Thanks for the feedback; I've expanded the caption to answer two of your questions concerning the meaning of the visual elements. For the other questions, I'm not sure how the diagram could be improved to better convey that information. There are theoretical challenges:
    1. There's no representation of the 3-adic integers that would make their group structure obvious by its inherent symmetry. It's possible for the Dihedral group article, for example. But to do it for the p-adic integers, we'd pretty much need a counterexample to the Hilbert–Smith conjecture. Part of the group operation could be acted out in an animation, but that would have to be a separate diagram.
    2. It would also be impossible to have an isometry between the p-adic metric and the Euclidean metric. (The chosen embedding is, however, Hölder continuous and measure-preserving, so there's that.) The metric structure is actually present in the diagram: the light grey circles are precisely the open balls in the 3-adic metric.
  • The image has been in the article for three weeks, so that shouldn't be a problem. Melchoir (talk) 10:18, 12 March 2012 (UTC)
    • Got my months confused :). I'll respond to the above when I'm at home and have a bit more time. JJ Harrison (talk) 01:21, 13 March 2012 (UTC)

Not Promoted --Makeemlighter (talk) 02:42, 21 March 2012 (UTC)