{"id":989,"date":"2014-08-14T01:00:45","date_gmt":"2014-08-14T01:00:45","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=989"},"modified":"2015-07-29T00:50:35","modified_gmt":"2015-07-29T00:50:35","slug":"733-honeycomb-meets-plane-at-infinity","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2014\/08\/14\/733-honeycomb-meets-plane-at-infinity\/","title":{"rendered":"{7,3,3} Honeycomb Meets Plane at Infinity"},"content":{"rendered":"
\n
\"{7,3,3}<\/a>

{7,3,3} Honeycomb Meets Plane at Infinity – Roice Nelson<\/p><\/div>\n<\/div>\n

 <\/p>\n

This picture by Roice Nelson<\/a> shows the boundary of the {7,3,3} honeycomb, the shape featured in our last article:<\/p>\n

\u2022 {7,3,3} honeycomb<\/a>.<\/p>\n

The black circles are holes, not contained in the boundary of the {7,3,3} honeycomb. There are infinitely many holes, and the actual boundary, shown in white, is a fractal with area zero.<\/p>\n

The {7,3,3} honeycomb lives in 3-dimensional hyperbolic space, a space that can be fit inside a ball using the Poincar\u00e9 ball model<\/a>. By the ‘boundary’ of the {7,3,3} honeycomb, we mean the set of points on the surface of the Poincar\u00e9 ball that are limits of points in the {7,3,3} honeycomb.<\/p>\n

Roice Nelson used stereographic projection to draw part of the surface of the Poincar\u00e9 ball as a plane: the plane at infinity<\/b>. So, the black region on the outside of the picture is also a hole in the boundary of the {7,3,3} honeycomb.<\/p>\n

The boundary of the {7,3,3} honeycomb is topologically interesting, because it is homeomorphic to the Sierpinski carpet. See also:<\/p>\n

\u2022 Sierpinski carpet<\/a>.<\/p>\n

We can see this using a result proved by Gordon Whyburn in 1958. A continuum<\/a><\/b> is a nonempty connected compact metric space. Suppose $X$ is a continuum embedded in the plane. Suppose the complement $\\mathbb{R}^2 – X$ has countably many connected components $C_1, C_2, C_3, \\dots$ and suppose:<\/p>\n

\u2022 the diameter of $C_i$ goes to zero as $i \\to \\infty$;<\/p>\n

\u2022 the boundary of $C_i$ and the boundary of $C_j$ are disjoint if $i \\ne j$;<\/p>\n

\u2022 the boundary of $C_i$ is a simple closed curve for each $i$;<\/p>\n

\u2022 the union of the boundaries of $C_i$ is dense in $X$.<\/p>\n

Then $X$ is homeomorphic to the Sierpinski carpet!<\/p>\n

To apply this result, note that the all the black circles in this picture, and also the black region on the outside, are the connected components $C_i$.<\/p>\n

In general, for any $n \\ge 7$, the boundary of the {$n$,3,3} honeycomb is homeomorphic to the Sierpinski carpet. For more details, see Section 7 of this paper:<\/p>\n

\u2022 Danny Calegari and Henry Wilton, 3-manifolds everywhere<\/a>.<\/p>\n

They prove that spaces of this sort occur very often inside the Gromov boundaries of hyperbolic groups.<\/p>\n

Roice Nelson, the creator of this image, has a blog with lots of articles about geometry, and he makes plastic models of interesting geometrical objects using a 3d printer:<\/p>\n

\u2022 Roice<\/a>.<\/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 picture by Roice Nelson shows the boundary of the {7,3,3} honeycomb<\/a>. The black circles are holes, not contained in the boundary of the {7,3,3} honeycomb. There are infinitely many holes, and the actual boundary, shown in white, is a fractal with area zero.<\/p>\n

<\/div>\n","protected":false},"author":66,"featured_media":1035,"comment_status":"open","ping_status":"closed","sticky":true,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[14,9,2],"tags":[],"class_list":["post-989","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-fractals","category-honeycombs","category-images-library"],"jetpack_featured_media_url":"https:\/\/blogs.ams.org\/visualinsight\/files\/2014\/08\/733_plane_at_infinity.png","jetpack_shortlink":"https:\/\/wp.me\/p42Vmc-fX","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/posts\/989","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=989"}],"version-history":[{"count":21,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/posts\/989\/revisions"}],"predecessor-version":[{"id":1042,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/posts\/989\/revisions\/1042"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/media\/1035"}],"wp:attachment":[{"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/media?parent=989"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/categories?post=989"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/tags?post=989"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}