{"id":5222,"date":"2018-12-29T16:22:22","date_gmt":"2018-12-29T23:22:22","guid":{"rendered":"http:\/\/theoryofeverything.org\/theToE\/?p=5222"},"modified":"2018-12-31T11:36:10","modified_gmt":"2018-12-31T18:36:10","slug":"integer-compositions-gray-code-and-the-fibonacci-sequence","status":"publish","type":"post","link":"https:\/\/theoryofeverything.org\/theToE\/2018\/12\/29\/integer-compositions-gray-code-and-the-fibonacci-sequence\/","title":{"rendered":"Integer Compositions, Gray Code , and the Fibonacci Sequence"},"content":{"rendered":"<p>I was prompted by Chris Birke to take a look at <a href=\"https:\/\/digitalcommons.georgiasouthern.edu\/cgi\/viewcontent.cgi?article=1012&amp;context=etd\">this very interesting paper<\/a> by Linus Lindroos. Chris is working with my E8 to H4 Folding Matrix for optimizing computation using network theory and hyper-dimensional geometric structures.<\/p>\n<p>I was able to validate the paper with some quick Mathematica code.<\/p>\n<p>Below is the code that sets up the generation of the tables, including various binary, numeric, and symbolic (graph) Compositions.<br \/>\n<a href=\"http:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2018\/12\/Fibbinary-Gray-Compositions.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-5224\" src=\"http:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2018\/12\/Fibbinary-Gray-Compositions.png\" alt=\"\" width=\"829\" height=\"2102\" srcset=\"https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2018\/12\/Fibbinary-Gray-Compositions.png 829w, https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2018\/12\/Fibbinary-Gray-Compositions-197x500.png 197w, https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2018\/12\/Fibbinary-Gray-Compositions-404x1024.png 404w\" sizes=\"auto, (max-width: 829px) 100vw, 829px\" \/><\/a><\/p>\n<p>Notice the last statement (above) lists the odd FibBinaries (up to N=200) as related to the Floor of the n*Phi^2-1, with n=1 to 77.<\/p>\n<p>Below is a listing of the first 100 N integers represented by their odd and even FibBinaries, along with the binary and Gray code Compositions.<br \/>\n<a href=\"http:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2018\/12\/Odd-Even-Fibinnaries-1.png\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2018\/12\/Odd-Even-Fibinnaries-1.png\" alt=\"\" width=\"796\" height=\"2192\" class=\"aligncenter size-full wp-image-5228\" srcset=\"https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2018\/12\/Odd-Even-Fibinnaries-1.png 796w, https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2018\/12\/Odd-Even-Fibinnaries-1-182x500.png 182w, https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2018\/12\/Odd-Even-Fibinnaries-1-372x1024.png 372w\" sizes=\"auto, (max-width: 796px) 100vw, 796px\" \/><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>I was prompted by Chris Birke to take a look at this very interesting paper by Linus Lindroos. Chris is working with my E8 to H4 Folding Matrix for optimizing computation using network theory and hyper-dimensional geometric structures. I was able to validate the paper with some quick Mathematica code. Below is the code that &hellip; <a href=\"https:\/\/theoryofeverything.org\/theToE\/2018\/12\/29\/integer-compositions-gray-code-and-the-fibonacci-sequence\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Integer Compositions, Gray Code , and the Fibonacci Sequence<\/span> <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":[3],"tags":[14],"class_list":["post-5222","post","type-post","status-publish","format-standard","hentry","category-physics","tag-math"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/5222","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/comments?post=5222"}],"version-history":[{"count":5,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/5222\/revisions"}],"predecessor-version":[{"id":5234,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/5222\/revisions\/5234"}],"wp:attachment":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/media?parent=5222"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/categories?post=5222"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/tags?post=5222"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}