{"id":1766,"date":"2015-11-01T01:00:14","date_gmt":"2015-11-01T01:00:14","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=1766"},"modified":"2016-01-31T22:26:03","modified_gmt":"2016-01-31T22:26:03","slug":"balaban-11-cage","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2015\/11\/01\/balaban-11-cage\/","title":{"rendered":"Balaban 11-Cage"},"content":{"rendered":"
\"Balaban<\/a>

Balaban 11-Cage – F\u00e9lix de la Fuente<\/p><\/div>\n<\/div>\n

Félix de la Fuente<\/a> is an architect and dedicated amateur mathematician in love with discrete geometry, polytopes and combinatorics. This picture created by him shows a graph called the Balaban 11-cage<\/a>. <\/p>\n

A (3,11)-graph<\/b> is a simple graph<\/a> where every vertex has 3 neighbors and the shortest cycle has length 11. A (3,11)-cage<\/b> is a (3,11)-graph with the least possible number of vertices. The Balaban 11-cage<\/b> is the unique (3,11)-cage. <\/p>\n

The Balaban 11-cage has 112 vertices and 168 edges. It was discovered by Balaban in 1973, and its uniqueness was proved by McKay and Myrvold in 2003. <\/p>\n

\n
\"Fano<\/a>

Fano Plane<\/p><\/div>\n<\/div>\n

Puzzle:<\/b> The Fano plane<\/a> has 168 symmetries, and it contains 28 triangles and also 28 ways of selecting a point and a line that are not incident to each other. 112 is 4 times 28. Is there a way to build the Balaban 11-cage starting from the Fano plane? One obstacle is that the Balaban 11-cage has a symmetry group of order 64.<\/p>\n

The layout for this picture comes from here:<\/p>\n

• P. Eades, J. Marks, P. Mutzel and S. North, Graph-drawing contest report<\/a>, TR98-16, Mitsubishi Electric Research Laboratories, December 1998.<\/p>\n

A German Wikicommons user named Gunther<\/a> drew the picture of the Fano plane, put it on Wikicommons<\/a>, and released it into the public domain.<\/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 shows part of a graph called the Balaban 11-cage<\/a>. A (3,11)-graph<\/b> is a simple graph where every vertex has 3 neighbors and the shortest cycle has length 11. A (3,11)-cage<\/b> is a (3,11)-graph with the least possible number of vertices. The Balaban 11-cage is the unique (3,11)-cage. <\/p>\n

<\/div>\n","protected":false},"author":66,"featured_media":2164,"comment_status":"open","ping_status":"closed","sticky":true,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[21,22,2],"tags":[],"class_list":["post-1766","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-combinatorics","category-graphs","category-images-library"],"jetpack_featured_media_url":"https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/11\/balaban_11-cage3.png","jetpack_shortlink":"https:\/\/wp.me\/p42Vmc-su","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/posts\/1766","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=1766"}],"version-history":[{"count":19,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/posts\/1766\/revisions"}],"predecessor-version":[{"id":2287,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/posts\/1766\/revisions\/2287"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/media\/2164"}],"wp:attachment":[{"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/media?parent=1766"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/categories?post=1766"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/tags?post=1766"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}