{"id":1857,"date":"2006-05-07T11:10:33","date_gmt":"2006-05-07T19:10:33","guid":{"rendered":"http:\/\/www.ogre.nu\/wp\/?p=1857"},"modified":"2006-05-07T14:02:17","modified_gmt":"2006-05-07T22:02:17","slug":"when-grown-ups-play-with-blocks","status":"publish","type":"post","link":"https:\/\/bendwavy.org\/wp\/?p=1857","title":{"rendered":"when grown-ups play with blocks"},"content":{"rendered":"<p>I&#8217;ve redone the Wikipedia page on <a href=\"http:\/\/en.wikipedia.org\/wiki\/Andreini_tessellation\">convex uniform tilings of Euclidean 3-space<\/a>.<\/p>\n<p>It occurs to me that one could enumerate the convex uniform tilings of flat, spherical and hyperbolic 3-spaces by an approach similar to what I&#8217;ve used to find fullerenes.  First make a list of the vertex figures of convex uniform polyhedra: these are polygons which share the property that their corners lie on a circle.  Then use a spiral search to build irregular polyhedra from these polygons.  Whenever such a polyhedron&#8217;s vertices all lie on a sphere, you have the vertex figure of a candidate solution (some of which will fail for other reasons).  The size of the sphere tells you whether and which way the relevant space is curved.<\/p>\n<p>Has this been done?<\/p>\n","protected":false},"excerpt":{"rendered":"<p>I&#8217;ve redone the Wikipedia page on convex uniform tilings of Euclidean 3-space. It occurs to me that one could enumerate the convex uniform tilings of flat, spherical and hyperbolic 3-spaces by an approach similar to what I&#8217;ve used to find &hellip; <a href=\"https:\/\/bendwavy.org\/wp\/?p=1857\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[16,12],"tags":[],"class_list":["post-1857","post","type-post","status-publish","format-standard","hentry","category-eye-candy","category-mathematics"],"_links":{"self":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/1857","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1857"}],"version-history":[{"count":0,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/1857\/revisions"}],"wp:attachment":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1857"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1857"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1857"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}