{"id":976,"date":"2014-10-01T01:00:39","date_gmt":"2014-10-01T01:00:39","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=976"},"modified":"2015-09-04T17:51:41","modified_gmt":"2015-09-04T17:51:41","slug":"2-adic-integers","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2014\/10\/01\/2-adic-integers\/","title":{"rendered":"2-adic Integers"},"content":{"rendered":"
\n
\"2-adic<\/a>

2-adic Integers – Christopher Culter<\/p><\/div>\n<\/div>\n

 <\/p>\n

This image created by Christopher Culter shows the compact abelian group of 2-adic integers<\/a> (black points), with selected elements labeled by the corresponding character on the Pontryagin dual group<\/a> (colored discs).<\/p>\n

Counterclockwise from the right, the labeled elements are <\/p>\n

$$ 0, 4, 2, \u22123, 1, \u2212\\frac{1}{7}, -\\frac{1}{3}, \\frac{1}{3}, \\frac{1}{7}, \u22121, 3, \u22122, -4 $$<\/p>\n

The Pontryagin dual of the group of 2-adic integers is the Pr\u00fcfer 2-group<\/a> $\\mathbb{Z}(2^\\infty)$. See our earlier article<\/p>\n

• Pr\u00fcfer 2-group<\/a> <\/p>\n

for an explanation of that. Each colored disc here is tied to a 2-adic integer, $x\\in\\mathbb{Z}_2$, and it represents a character <\/p>\n

$$ \\chi_x : \\mathbb{Z}(2^\\infty) \\to \\mathbb{R}\/\\mathbb{Z}$$<\/p>\n

defined by <\/p>\n

$$ \\chi_x(q) = x q.$$ <\/p>\n

Points in the circle $\\mathbb{R}\/\\mathbb{Z}$ are drawn using a color wheel where $0$ is red, $\\frac{1}{3}$ is green, and $\\frac{2}{3}$ is blue.<\/p>\n

For details on the embedding of the 2-adic integers in the plane, see:<\/p>\n

• D. V. Chistyakov, Fractal geometry for images of continuous embeddings of $p$-adic numbers and $p$-adic solenoids into Euclidean spaces<\/a>, Theoretical and Mathematical Physics<\/i> 109<\/b> (1996), 1495\u20131507 <\/p>\n

The particular mapping used is $\\Upsilon_s^{(\\infty)}$, defined in Definition 3 and depicted in Figure 1.12 of this paper.<\/p>\n

\n

Visual Insight<\/i> is a place to share striking images that help explain advanced topics in mathematics. I\u2019m always looking for truly beautiful images, so if you know about one, please drop a comment here<\/a> and let me know!<\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

This image created by Christopher Culter shows the compact abelian group of 2-adic integers<\/a> (black points), with selected elements labeled by the corresponding character on the Pontryagin dual group<\/a> (colored discs).<\/p>\n

<\/div>\n","protected":false},"author":66,"featured_media":1002,"comment_status":"open","ping_status":"closed","sticky":true,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[4,14,23,2],"tags":[],"class_list":["post-976","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebraic-number-theory","category-fractals","category-groups","category-images-library"],"jetpack_featured_media_url":"https:\/\/blogs.ams.org\/visualinsight\/files\/2014\/10\/2-adic_integers.png","jetpack_shortlink":"https:\/\/wp.me\/p42Vmc-fK","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/posts\/976","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/users\/66"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/comments?post=976"}],"version-history":[{"count":13,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/posts\/976\/revisions"}],"predecessor-version":[{"id":981,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/posts\/976\/revisions\/981"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/media\/1002"}],"wp:attachment":[{"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/media?parent=976"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/categories?post=976"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/tags?post=976"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}