{"id":1055,"date":"2012-04-06T14:05:52","date_gmt":"2012-04-06T19:05:52","guid":{"rendered":"http:\/\/sites.williams.edu\/Morgan\/?p=1055"},"modified":"2012-04-06T14:31:19","modified_gmt":"2012-04-06T19:31:19","slug":"efficient-containers-of-balls","status":"publish","type":"post","link":"https:\/\/sites.williams.edu\/Morgan\/2012\/04\/06\/efficient-containers-of-balls\/","title":{"rendered":"Efficient Containers of Balls"},"content":{"rendered":"<p><a href=\"http:\/\/www.susqu.edu\/brakke\/evolver\/examples\/williams\/williams-2-balls.htm\"><img loading=\"lazy\" decoding=\"async\" class=\"alignright size-medium wp-image-1057\" src=\"https:\/\/sites.williams.edu\/Morgan\/files\/2012\/04\/Screen-shot-2012-04-06-at-2.59.28-PM-300x296.png\" alt=\"\" width=\"101\" height=\"101\" srcset=\"https:\/\/sites.williams.edu\/Morgan\/files\/2012\/04\/Screen-shot-2012-04-06-at-2.59.28-PM-300x296.png 300w, https:\/\/sites.williams.edu\/Morgan\/files\/2012\/04\/Screen-shot-2012-04-06-at-2.59.28-PM-150x150.png 150w, https:\/\/sites.williams.edu\/Morgan\/files\/2012\/04\/Screen-shot-2012-04-06-at-2.59.28-PM.png 511w\" sizes=\"auto, (max-width: 101px) 100vw, 101px\" \/><\/a><a href=\"http:\/\/arxiv.org\/abs\/1111.3092v1\">Karoly Bezdek<\/a> has proved that a partition of space into convex polyhedra of bounded diameters containing unit balls has average surface area at least 24\/\u221a3 ~ 13.86 and conjectures that the minimum is 12\u221a2 ~ 16.97, given by rhombic dodecahedra. <a href=\"http:\/\/www.susqu.edu\/brakke\/evolver\/examples\/williams\/williams-2-balls.htm\">Ken Brakke<\/a> has computed that if one allows arbitrary cells containing unit balls, the rhombic dodecahedra still beat the Kelvin (~17.83) and <a href=\"http:\/\/en.wikipedia.org\/wiki\/Weaire\u2013Phelan_structure\">Weaire-Phelan<\/a> (~21.15) foams, but they can morph into a &#8220;draped Williams cell&#8221; (thumbnail at right from nice picture at <a href=\"http:\/\/www.susqu.edu\/brakke\/evolver\/examples\/williams\/williams-2-balls.htm\">Brakke<\/a>) with average area about 16.957, conjectured to be optimal.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Karoly Bezdek has proved that a partition of space into convex polyhedra of bounded diameters containing unit balls has average surface area at least 24\/\u221a3 ~ 13.86 and conjectures that the minimum is 12\u221a2 ~ 16.97, given by rhombic dodecahedra. Ken Brakke has computed that if one allows arbitrary cells containing unit balls, the rhombic [&hellip;]<\/p>\n","protected":false},"author":269,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[14043],"tags":[],"class_list":["post-1055","post","type-post","status-publish","format-standard","hentry","category-math"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/1055","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/users\/269"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/comments?post=1055"}],"version-history":[{"count":7,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/1055\/revisions"}],"predecessor-version":[{"id":1063,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/1055\/revisions\/1063"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/media?parent=1055"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/categories?post=1055"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/tags?post=1055"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}