{"id":737,"date":"2018-02-26T05:45:46","date_gmt":"2018-02-26T13:45:46","guid":{"rendered":"http:\/\/puzzlezapper.com\/blog\/?p=737"},"modified":"2018-02-26T05:45:46","modified_gmt":"2018-02-26T13:45:46","slug":"flexible-pentominoes-on-rhombic-polyhedra","status":"publish","type":"post","link":"https:\/\/puzzlezapper.com\/blog\/2018\/02\/flexible-pentominoes-on-rhombic-polyhedra\/","title":{"rendered":"Flexible pentominoes on rhombic polyhedra"},"content":{"rendered":"<p>If you subdivide the faces of a <a href=\"http:\/\/mathworld.wolfram.com\/RhombicTriacontahedron.html\">rhombic triacontahedron<\/a> into 2\u00d72 grids, you can tile the polyhedron with two copies of each pentomino.<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/puzzlezapper.com\/blog\/wp-content\/uploads\/2018\/02\/5-omino-half-30-hedron.png\" alt=\"\" width=\"445\" height=\"425\" class=\"aligncenter size-full wp-image-739\" srcset=\"https:\/\/puzzlezapper.com\/blog\/wp-content\/uploads\/2018\/02\/5-omino-half-30-hedron.png 445w, https:\/\/puzzlezapper.com\/blog\/wp-content\/uploads\/2018\/02\/5-omino-half-30-hedron-300x287.png 300w\" sizes=\"auto, (max-width: 445px) 100vw, 445px\" \/><br \/>\nOne way of looking at this figure is as a tiling of the projective hemi-rhombic triacontahedron. The projective (also known as abstract) polyhedra can be formed by identifying the opposite faces of certain polyhedra with each other. So the projective hemi-cube has three square faces, and the projective hemi-rhombic triacontahedron has 15 rhombic faces. Stitching together the opposite sides of the unshaded area in the figure is a way to form this 15 face &#8220;polyhedron&#8221;.<\/p>\n<p>I came up with that one a couple of years ago, but I neglected to put up a blog post because I didn&#8217;t like the graphic enough. I suspect that it&#8217;d look really cool if the lines of the rhombic triacontahedron were properly projected onto a flat disk, but I don&#8217;t have the expertise to make that happen. I finally decided that it was worth sharing even if it doesn&#8217;t look as cool as it could.<\/p>\n<p>Below is another tiling of subdivided rhombi. The significance of this figure is that four copies could be used to cover a <a href=\"http:\/\/mathworld.wolfram.com\/RhombicHexecontahedron.html\">rhombic hexecontahedron<\/a>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/puzzlezapper.com\/blog\/wp-content\/uploads\/2018\/02\/5-omino-15-rhombi-2.png\" alt=\"\" width=\"341\" height=\"296\" class=\"aligncenter size-full wp-image-741\" srcset=\"https:\/\/puzzlezapper.com\/blog\/wp-content\/uploads\/2018\/02\/5-omino-15-rhombi-2.png 341w, https:\/\/puzzlezapper.com\/blog\/wp-content\/uploads\/2018\/02\/5-omino-15-rhombi-2-300x260.png 300w\" sizes=\"auto, (max-width: 341px) 100vw, 341px\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>If you subdivide the faces of a rhombic triacontahedron into 2\u00d72 grids, you can tile the polyhedron with two copies of each pentomino. One way of looking at this figure is as a tiling of the projective hemi-rhombic triacontahedron. The projective (also known as abstract) polyhedra can be formed by identifying the opposite faces of &hellip; <a href=\"https:\/\/puzzlezapper.com\/blog\/2018\/02\/flexible-pentominoes-on-rhombic-polyhedra\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Flexible pentominoes on rhombic polyhedra<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[66,127,128,203,10,200,202,201],"class_list":["post-737","post","type-post","status-publish","format-standard","hentry","category-recreational-mathematics","tag-flexible-polyforms","tag-flexible-polyrhombs","tag-flexominoes","tag-hemi-rhombic-triacontahedron","tag-pentominoes","tag-polyhedra","tag-rhombic-hexecontahedron","tag-rhombic-triacontahedron"],"_links":{"self":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/737","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/comments?post=737"}],"version-history":[{"count":6,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/737\/revisions"}],"predecessor-version":[{"id":745,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/737\/revisions\/745"}],"wp:attachment":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/media?parent=737"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/categories?post=737"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/tags?post=737"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}