{"id":608,"date":"2019-03-25T15:40:59","date_gmt":"2019-03-25T15:40:59","guid":{"rendered":"http:\/\/x-colony.com\/WorldPress\/?page_id=408"},"modified":"2019-04-16T19:48:47","modified_gmt":"2019-04-16T19:48:47","slug":"theorem-of-dual-tetrahedron","status":"publish","type":"page","link":"http:\/\/x-colony.com\/PoCo\/index.php\/activities\/theorems\/theorem-of-dual-tetrahedron\/","title":{"rendered":"Theorem of Dual Tetrahedron"},"content":{"rendered":"\n<p>The dual of a convex geometric solid is obtained by taking the center of gravity for all faces as vertices. The faces of the dual are associated with the original vertices: for each vertex consider the center of adjacent faces and use them to generate a face of the dual. Obviously, the convex hull of these vertices will be used (not the star polygons).<\/p>\n\n\n\n<p><strong>Theorem of Dual Tetrahedron<\/strong><\/p>\n\n\n\n<p>The dual of a regular tetrahedron has the side length equal to 1\/3 of the side length of the original tetrahedron.<\/p>\n\n\n\n<p><strong>Solution<\/strong><\/p>\n\n\n\n<p>Of course, we prefer a visual solution. <\/p>\n\n\n\n<p>First, we construct the dual (red) of a tetrahedron (green).<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"272\" height=\"185\" src=\"http:\/\/x-colony.com\/PoCo\/wp-content\/uploads\/2019\/03\/image-72-1.png\" alt=\"\" class=\"wp-image-415\"\/><figcaption>\t\t\t\t\t\t<\/figcaption><\/figure><\/div>\n\n\n\n<p>Secondly, we truncate the green tetrahedron (cutting edges at 1\/3 from each vertex). To convert this into a hull body, we have to generate its convex hull and to compact the faces of the resulting object.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"446\" height=\"147\" src=\"http:\/\/x-colony.com\/PoCo\/wp-content\/uploads\/2019\/03\/image-73-1.png\" alt=\"\" class=\"wp-image-418\" srcset=\"http:\/\/x-colony.com\/PoCo\/wp-content\/uploads\/2019\/03\/image-73-1.png 446w, http:\/\/x-colony.com\/PoCo\/wp-content\/uploads\/2019\/03\/image-73-1-300x99.png 300w\" sizes=\"(max-width: 446px) 85vw, 446px\" \/><figcaption>\n\t\t\t\t\t\t<\/figcaption><\/figure><\/div>\n\n\n\n<p>And last, we promote the dual to the default <strong>Build <\/strong>module and add copies of it to all missing corners of the previously constructed object, thus obtaining a shape similar to the original.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"192\" height=\"183\" src=\"http:\/\/x-colony.com\/PoCo\/wp-content\/uploads\/2019\/03\/image-74.png\" alt=\"\" class=\"wp-image-419\"\/><figcaption>\n\t\t\t\t\t\t<\/figcaption><\/figure><\/div>\n","protected":false},"excerpt":{"rendered":"<p>The dual of a convex geometric solid is obtained by taking the center of gravity for all faces as vertices. The faces of the dual are associated with the original vertices: for each vertex consider the center of adjacent faces and use them to generate a face of the dual. Obviously, the convex hull of &hellip; <a href=\"http:\/\/x-colony.com\/PoCo\/index.php\/activities\/theorems\/theorem-of-dual-tetrahedron\/\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> &#8220;Theorem of Dual Tetrahedron&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":80,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":[],"_links":{"self":[{"href":"http:\/\/x-colony.com\/PoCo\/index.php\/wp-json\/wp\/v2\/pages\/608"}],"collection":[{"href":"http:\/\/x-colony.com\/PoCo\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/x-colony.com\/PoCo\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"http:\/\/x-colony.com\/PoCo\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/x-colony.com\/PoCo\/index.php\/wp-json\/wp\/v2\/comments?post=608"}],"version-history":[{"count":3,"href":"http:\/\/x-colony.com\/PoCo\/index.php\/wp-json\/wp\/v2\/pages\/608\/revisions"}],"predecessor-version":[{"id":860,"href":"http:\/\/x-colony.com\/PoCo\/index.php\/wp-json\/wp\/v2\/pages\/608\/revisions\/860"}],"up":[{"embeddable":true,"href":"http:\/\/x-colony.com\/PoCo\/index.php\/wp-json\/wp\/v2\/pages\/80"}],"wp:attachment":[{"href":"http:\/\/x-colony.com\/PoCo\/index.php\/wp-json\/wp\/v2\/media?parent=608"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}