{"id":105,"date":"2012-07-17T07:18:01","date_gmt":"2012-07-17T12:18:01","guid":{"rendered":"http:\/\/daylateanddollarshort.com\/bloog\/?p=105"},"modified":"2012-07-17T07:24:16","modified_gmt":"2012-07-17T12:24:16","slug":"extending-a-theorem-of-barlotti","status":"publish","type":"post","link":"http:\/\/daylateanddollarshort.com\/bloog\/extending-a-theorem-of-barlotti\/","title":{"rendered":"Extending a Theorem of Barlotti"},"content":{"rendered":"<p>Barlotti&#8217;s Theorem states that an &#8220;affinely regular&#8221; polygon is the vertex sum of two regular polygons of the same type. This result can be interpreted as a straightforward decomposition of 2&#215;2 matrices, which in turn can be extended to a decomposition of <em>d<\/em>x<em>d<\/em> matrices, which immediately gives rise to the (Extended) Barlotti Theorem for Multiple Dimensions.<\/p>\n<p>My paper &#8212;aptly entitled <a title=\"&quot;An Extension of a Theorem of Barlotti to Multiple Dimensions&quot; (PDF)\" href=\"http:\/\/daylateanddollarshort.com\/mathdocs\/An-Extension-of-a-Theorem-of-Barlotti.pdf\">&#8220;An Extension of a Theorem of Barlotti to Multiple Dimensions&#8221;<\/a>&#8212; discusses generalizing the notion of vertex sum to &#8220;point sum&#8221;, so that the Extended Barlotti provides such statements as\u00a0<em>any ellipsoid is the point sum of three spheres<\/em>.<\/p>\n<p>The Extended Barlotti Theorem implies that spectral realizations of a graph (see <a title=\"&quot;Spectral Realizations of Graphs&quot; at the Bloog\" href=\"http:\/\/daylateanddollarshort.com\/bloog\/spectral-realizations-of-graphs\/\">this post<\/a>) form an additive basis of\u00a0<em>all<\/em>\u00a0realizations of that graph; that is, any realization can be expressed as the vertex sum of (harmonious) spectral realizations.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Barlotti&#8217;s Theorem states that an &#8220;affinely regular&#8221; polygon is the vertex sum of two regular polygons of the same type. This result can be interpreted as a straightforward decomposition of 2&#215;2 matrices, which in turn can be extended to a decomposition of dxd matrices, which immediately gives rise to the (Extended) Barlotti Theorem for Multiple [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9],"tags":[],"class_list":["post-105","post","type-post","status-publish","format-standard","hentry","category-harmonious-figures"],"_links":{"self":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/105","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/comments?post=105"}],"version-history":[{"count":3,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/105\/revisions"}],"predecessor-version":[{"id":114,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/105\/revisions\/114"}],"wp:attachment":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/media?parent=105"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/categories?post=105"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/tags?post=105"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}