{"id":3505,"date":"2015-08-27T16:06:02","date_gmt":"2015-08-27T23:06:02","guid":{"rendered":"http:\/\/theoryofeverything.org\/MyToE\/?p=3505"},"modified":"2015-12-27T18:35:10","modified_gmt":"2015-12-27T18:35:10","slug":"quasicrystals-and-geometry","status":"publish","type":"post","link":"http:\/\/theoryofeverything.org\/theToE\/2015\/08\/27\/quasicrystals-and-geometry\/","title":{"rendered":"QuasiCrystals and Geometry"},"content":{"rendered":"<p>I found a 1995 book <a href=\"http:\/\/english.360elib.com\/datu\/Q\/EM093592.pdf\">(PDF online) &#8220;QuasiCrystals and Geometry&#8221; by Marjorie Senechal<\/a>. There were some very nice diffraction patterns that match rectified 4_21 E8 Polytope 12,18, and 30-gon projections. See my overlays below:<\/p>\n<p>12-gon Rectified E8<br \/>\n<a href=\"http:\/\/theoryofeverything.org\/MyToE\/?attachment_id=3507\" rel=\"attachment wp-att-3507\"><img decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/MyToE\/wp-content\/uploads\/2015\/08\/quasi-Rect421-12gon0b.png\" alt=\"quasi-Rect421-12gon0b\" width=\"500\" class=\"alignnone size-full wp-image-3507\" \/><\/a><br \/>\nDiffraction<br \/>\n<a href=\"http:\/\/theoryofeverything.org\/MyToE\/?attachment_id=3506\" rel=\"attachment wp-att-3506\"><img decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/MyToE\/wp-content\/uploads\/2015\/08\/quasi-Rect421-12gon1b.png\" alt=\"quasi-Rect421-12gon1b\" width=\"500\" class=\"alignnone size-full wp-image-3506\" \/><\/a><br \/>\nOverlay<br \/>\n<a href=\"http:\/\/theoryofeverything.org\/MyToE\/?attachment_id=3508\" rel=\"attachment wp-att-3508\"><img decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/MyToE\/wp-content\/uploads\/2015\/08\/quasi-Rect421-12gon2b.png\" alt=\"quasi-Rect421-12gon2b\" width=\"500\" class=\"alignnone size-full wp-image-3508\" \/><\/a><\/p>\n<p>18-gon Rectified E8<br \/>\n<a href=\"http:\/\/theoryofeverything.org\/MyToE\/?attachment_id=3509\" rel=\"attachment wp-att-3509\"><img decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/MyToE\/wp-content\/uploads\/2015\/08\/quasi-Rect421-18gon0b.png\" alt=\"quasi-Rect421-18gon0b\" width=\"500\" class=\"alignnone size-full wp-image-3509\" \/><\/a><br \/>\nDiffraction<br \/>\n<a href=\"http:\/\/theoryofeverything.org\/MyToE\/?attachment_id=3527\" rel=\"attachment wp-att-3527\"><img decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/MyToE\/wp-content\/uploads\/2015\/08\/quasi-Rect421-18gon1b.png\" alt=\"quasi-Rect421-18gon1b\" width=\"500\" class=\"alignnone size-full wp-image-3527\" \/><\/a><br \/>\nOverlay<br \/>\n<a href=\"http:\/\/theoryofeverything.org\/MyToE\/?attachment_id=3514\" rel=\"attachment wp-att-3514\"><img decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/MyToE\/wp-content\/uploads\/2015\/08\/quasi-Rect421-18gon2b.png\" alt=\"quasi-Rect421-18gon2b\" width=\"500\" class=\"alignnone size-full wp-image-3514\" \/><\/a><\/p>\n<p>30-gon Rectified E8<br \/>\n<a href=\"http:\/\/theoryofeverything.org\/MyToE\/?attachment_id=3511\" rel=\"attachment wp-att-3511\"><img decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/MyToE\/wp-content\/uploads\/2015\/08\/quasi-Rect421-30gon0b.png\" alt=\"quasi-Rect421-30gon0b\" width=\"500\" class=\"alignnone size-full wp-image-3511\" \/><\/a><br \/>\nDiffraction<br \/>\n<a href=\"http:\/\/theoryofeverything.org\/MyToE\/?attachment_id=3510\" rel=\"attachment wp-att-3510\"><img decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/MyToE\/wp-content\/uploads\/2015\/08\/quasi-Rect421-30gon1b.png\" alt=\"quasi-Rect421-30gon1b\" width=\"500\" class=\"alignnone size-full wp-image-3510\" \/><\/a><br \/>\nOverlay<br \/>\n<a href=\"http:\/\/theoryofeverything.org\/MyToE\/?attachment_id=3512\" rel=\"attachment wp-att-3512\"><img decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/MyToE\/wp-content\/uploads\/2015\/08\/quasi-Rect421-30gon2b.png\" alt=\"quasi-Rect421-30gon2b\" width=\"500\" class=\"alignnone size-full wp-image-3512\" \/><\/a><\/p>\n<p>and another 12-gon rectified E8 pattern overlay from<a href=\"http:\/\/arxiv.org\/abs\/1306.3316\"> M. &#038; N. Koca&#8217;s in &#8220;12-fold Symmetric Quasicrystallography from affine E6, B6, and F4&#8221;<\/a>:<br \/>\n<a href=\"http:\/\/theoryofeverything.org\/MyToE\/?attachment_id=3531\" rel=\"attachment wp-att-3531\"><img decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/MyToE\/wp-content\/uploads\/2015\/08\/koca0b.png\" alt=\"koca0b\" width=\"500\" class=\"alignnone size-full wp-image-3531\" \/><\/a><\/p>\n<p><a href=\"http:\/\/theoryofeverything.org\/MyToE\/?attachment_id=3532\" rel=\"attachment wp-att-3532\"><img decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/MyToE\/wp-content\/uploads\/2015\/08\/koca1b.png\" alt=\"koca1b\" width=\"500\" class=\"alignnone size-full wp-image-3532\" \/><\/a><\/p>\n<p><a href=\"http:\/\/theoryofeverything.org\/MyToE\/?attachment_id=3533\" rel=\"attachment wp-att-3533\"><img decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/MyToE\/wp-content\/uploads\/2015\/08\/koca2b.png\" alt=\"koca2b\" width=\"500\" class=\"alignnone size-full wp-image-3533\" \/><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>I found a 1995 book (PDF online) &#8220;QuasiCrystals and Geometry&#8221; by Marjorie Senechal. There were some very nice diffraction patterns that match rectified 4_21 E8 Polytope 12,18, and 30-gon projections. See my overlays below: 12-gon Rectified E8 Diffraction Overlay 18-gon Rectified E8 Diffraction Overlay 30-gon Rectified E8 Diffraction Overlay and another 12-gon rectified E8 pattern &hellip; <a href=\"http:\/\/theoryofeverything.org\/theToE\/2015\/08\/27\/quasicrystals-and-geometry\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">QuasiCrystals and Geometry<\/span> <span class=\"meta-nav\">&rarr;<\/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":[3],"tags":[11,4,19,34,14,35],"class_list":["post-3505","post","type-post","status-publish","format-standard","hentry","category-physics","tag-2d","tag-e8","tag-fano","tag-h4","tag-math","tag-quasicrystals"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 4.9.9 - aioseo.com -->\n\t<meta name=\"description\" content=\"I found a 1995 book (PDF online) &quot;QuasiCrystals and Geometry&quot; by Marjorie Senechal. There were some very nice diffraction patterns that match rectified 4_21 E8 Polytope 12,18, and 30-gon projections. See my overlays below: 12-gon Rectified E8 Diffraction Overlay 18-gon Rectified E8 Diffraction Overlay 30-gon Rectified E8 Diffraction Overlay and another 12-gon rectified E8 pattern\" \/>\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<meta name=\"author\" content=\"jgmoxness\"\/>\n\t<link rel=\"canonical\" href=\"http:\/\/theoryofeverything.org\/theToE\/2015\/08\/27\/quasicrystals-and-geometry\/\" \/>\n\t<meta name=\"generator\" content=\"All in One SEO (AIOSEO) 4.9.9\" \/>\n\t\t<meta property=\"og:locale\" content=\"en_US\" \/>\n\t\t<meta property=\"og:site_name\" content=\"Visualizing a Theory of Everything! - Dedicated to the pursuit of beauty and Truth in Nature!\" \/>\n\t\t<meta property=\"og:type\" content=\"article\" \/>\n\t\t<meta property=\"og:title\" content=\"QuasiCrystals and Geometry - Visualizing a Theory of Everything!\" \/>\n\t\t<meta property=\"og:description\" content=\"I found a 1995 book (PDF online) &quot;QuasiCrystals and Geometry&quot; by Marjorie Senechal. There were some very nice diffraction patterns that match rectified 4_21 E8 Polytope 12,18, and 30-gon projections. See my overlays below: 12-gon Rectified E8 Diffraction Overlay 18-gon Rectified E8 Diffraction Overlay 30-gon Rectified E8 Diffraction Overlay and another 12-gon rectified E8 pattern\" \/>\n\t\t<meta property=\"og:url\" content=\"http:\/\/theoryofeverything.org\/theToE\/2015\/08\/27\/quasicrystals-and-geometry\/\" \/>\n\t\t<meta property=\"article:published_time\" content=\"2015-08-27T23:06:02+00:00\" \/>\n\t\t<meta property=\"article:modified_time\" content=\"2015-12-27T18:35:10+00:00\" \/>\n\t\t<meta name=\"twitter:card\" content=\"summary\" \/>\n\t\t<meta name=\"twitter:title\" content=\"QuasiCrystals and Geometry - Visualizing a Theory of Everything!\" \/>\n\t\t<meta name=\"twitter:description\" content=\"I found a 1995 book (PDF online) &quot;QuasiCrystals and Geometry&quot; by Marjorie Senechal. There were some very nice diffraction patterns that match rectified 4_21 E8 Polytope 12,18, and 30-gon projections. See my overlays below: 12-gon Rectified E8 Diffraction Overlay 18-gon Rectified E8 Diffraction Overlay 30-gon Rectified E8 Diffraction Overlay and another 12-gon rectified E8 pattern\" \/>\n\t\t<script type=\"application\/ld+json\" class=\"aioseo-schema\">\n\t\t\t{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"BlogPosting\",\"@id\":\"http:\\\/\\\/theoryofeverything.org\\\/theToE\\\/2015\\\/08\\\/27\\\/quasicrystals-and-geometry\\\/#blogposting\",\"name\":\"QuasiCrystals and Geometry - Visualizing a Theory of Everything!\",\"headline\":\"QuasiCrystals and Geometry\",\"author\":{\"@id\":\"http:\\\/\\\/theoryofeverything.org\\\/theToE\\\/author\\\/jgmoxness\\\/#author\"},\"publisher\":{\"@id\":\"http:\\\/\\\/theoryofeverything.org\\\/theToE\\\/#person\"},\"image\":{\"@type\":\"ImageObject\",\"url\":\"http:\\\/\\\/theoryofeverything.org\\\/MyToE\\\/wp-content\\\/uploads\\\/2015\\\/08\\\/quasi-Rect421-12gon0b.png\",\"@id\":\"http:\\\/\\\/theoryofeverything.org\\\/theToE\\\/2015\\\/08\\\/27\\\/quasicrystals-and-geometry\\\/#articleImage\"},\"datePublished\":\"2015-08-27T16:06:02-07:00\",\"dateModified\":\"2015-12-27T18:35:10-07:00\",\"inLanguage\":\"en-US\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/theoryofeverything.org\\\/theToE\\\/2015\\\/08\\\/27\\\/quasicrystals-and-geometry\\\/#webpage\"},\"isPartOf\":{\"@id\":\"http:\\\/\\\/theoryofeverything.org\\\/theToE\\\/2015\\\/08\\\/27\\\/quasicrystals-and-geometry\\\/#webpage\"},\"articleSection\":\"Physics, 2D, E8, Fano, H4, Math, QuasiCrystals\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"http:\\\/\\\/theoryofeverything.org\\\/theToE\\\/2015\\\/08\\\/27\\\/quasicrystals-and-geometry\\\/#breadcrumblist\",\"itemListElement\":[{\"@type\":\"ListItem\",\"@id\":\"http:\\\/\\\/theoryofeverything.org\\\/theToE#listItem\",\"position\":1,\"name\":\"Home\",\"item\":\"http:\\\/\\\/theoryofeverything.org\\\/theToE\",\"nextItem\":{\"@type\":\"ListItem\",\"@id\":\"http:\\\/\\\/theoryofeverything.org\\\/theToE\\\/topics\\\/physics\\\/#listItem\",\"name\":\"Physics\"}},{\"@type\":\"ListItem\",\"@id\":\"http:\\\/\\\/theoryofeverything.org\\\/theToE\\\/topics\\\/physics\\\/#listItem\",\"position\":2,\"name\":\"Physics\",\"item\":\"http:\\\/\\\/theoryofeverything.org\\\/theToE\\\/topics\\\/physics\\\/\",\"nextItem\":{\"@type\":\"ListItem\",\"@id\":\"http:\\\/\\\/theoryofeverything.org\\\/theToE\\\/2015\\\/08\\\/27\\\/quasicrystals-and-geometry\\\/#listItem\",\"name\":\"QuasiCrystals and Geometry\"},\"previousItem\":{\"@type\":\"ListItem\",\"@id\":\"http:\\\/\\\/theoryofeverything.org\\\/theToE#listItem\",\"name\":\"Home\"}},{\"@type\":\"ListItem\",\"@id\":\"http:\\\/\\\/theoryofeverything.org\\\/theToE\\\/2015\\\/08\\\/27\\\/quasicrystals-and-geometry\\\/#listItem\",\"position\":3,\"name\":\"QuasiCrystals and Geometry\",\"previousItem\":{\"@type\":\"ListItem\",\"@id\":\"http:\\\/\\\/theoryofeverything.org\\\/theToE\\\/topics\\\/physics\\\/#listItem\",\"name\":\"Physics\"}}]},{\"@type\":\"Person\",\"@id\":\"http:\\\/\\\/theoryofeverything.org\\\/theToE\\\/#person\",\"name\":\"jgmoxness\"},{\"@type\":\"Person\",\"@id\":\"http:\\\/\\\/theoryofeverything.org\\\/theToE\\\/author\\\/jgmoxness\\\/#author\",\"url\":\"http:\\\/\\\/theoryofeverything.org\\\/theToE\\\/author\\\/jgmoxness\\\/\",\"name\":\"jgmoxness\"},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/theoryofeverything.org\\\/theToE\\\/2015\\\/08\\\/27\\\/quasicrystals-and-geometry\\\/#webpage\",\"url\":\"http:\\\/\\\/theoryofeverything.org\\\/theToE\\\/2015\\\/08\\\/27\\\/quasicrystals-and-geometry\\\/\",\"name\":\"QuasiCrystals and Geometry - Visualizing a Theory of Everything!\",\"description\":\"I found a 1995 book (PDF online) \\\"QuasiCrystals and Geometry\\\" by Marjorie Senechal. 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See my overlays below: 12-gon Rectified E8 Diffraction Overlay 18-gon Rectified E8 Diffraction Overlay 30-gon Rectified E8 Diffraction Overlay and another 12-gon rectified E8 pattern","inLanguage":"en-US","isPartOf":{"@id":"http:\/\/theoryofeverything.org\/theToE\/#website"},"breadcrumb":{"@id":"http:\/\/theoryofeverything.org\/theToE\/2015\/08\/27\/quasicrystals-and-geometry\/#breadcrumblist"},"author":{"@id":"http:\/\/theoryofeverything.org\/theToE\/author\/jgmoxness\/#author"},"creator":{"@id":"http:\/\/theoryofeverything.org\/theToE\/author\/jgmoxness\/#author"},"datePublished":"2015-08-27T16:06:02-07:00","dateModified":"2015-12-27T18:35:10-07:00"},{"@type":"WebSite","@id":"http:\/\/theoryofeverything.org\/theToE\/#website","url":"http:\/\/theoryofeverything.org\/theToE\/","name":"Visualizing a Theory of Everything!","description":"Dedicated to the pursuit of beauty and Truth in Nature!","inLanguage":"en-US","publisher":{"@id":"http:\/\/theoryofeverything.org\/theToE\/#person"}}]},"og:locale":"en_US","og:site_name":"Visualizing a Theory of Everything! - Dedicated to the pursuit of beauty and Truth in Nature!","og:type":"article","og:title":"QuasiCrystals and Geometry - Visualizing a Theory of Everything!","og:description":"I found a 1995 book (PDF online) &quot;QuasiCrystals and Geometry&quot; by Marjorie Senechal. There were some very nice diffraction patterns that match rectified 4_21 E8 Polytope 12,18, and 30-gon projections. See my overlays below: 12-gon Rectified E8 Diffraction Overlay 18-gon Rectified E8 Diffraction Overlay 30-gon Rectified E8 Diffraction Overlay and another 12-gon rectified E8 pattern","og:url":"http:\/\/theoryofeverything.org\/theToE\/2015\/08\/27\/quasicrystals-and-geometry\/","article:published_time":"2015-08-27T23:06:02+00:00","article:modified_time":"2015-12-27T18:35:10+00:00","twitter:card":"summary","twitter:title":"QuasiCrystals and Geometry - Visualizing a Theory of Everything!","twitter:description":"I found a 1995 book (PDF online) &quot;QuasiCrystals and Geometry&quot; by Marjorie Senechal. There were some very nice diffraction patterns that match rectified 4_21 E8 Polytope 12,18, and 30-gon projections. See my overlays below: 12-gon Rectified E8 Diffraction Overlay 18-gon Rectified E8 Diffraction Overlay 30-gon Rectified E8 Diffraction Overlay and another 12-gon rectified E8 pattern"},"aioseo_meta_data":{"post_id":"3505","title":null,"description":null,"keywords":null,"keyphrases":null,"primary_term":null,"canonical_url":null,"og_title":null,"og_description":null,"og_object_type":"default","og_image_type":"default","og_image_url":null,"og_image_width":null,"og_image_height":null,"og_image_custom_url":null,"og_image_custom_fields":null,"og_video":null,"og_custom_url":null,"og_article_section":null,"og_article_tags":null,"twitter_use_og":false,"twitter_card":"default","twitter_image_type":"default","twitter_image_url":null,"twitter_image_custom_url":null,"twitter_image_custom_fields":null,"twitter_title":null,"twitter_description":null,"schema":{"blockGraphs":[],"customGraphs":[],"default":{"data":{"Article":[],"Course":[],"Dataset":[],"FAQPage":[],"Movie":[],"Person":[],"Product":[],"ProductReview":[],"Car":[],"Recipe":[],"Service":[],"SoftwareApplication":[],"WebPage":[]},"graphName":"","isEnabled":true},"graphs":[]},"schema_type":"default","schema_type_options":null,"pillar_content":false,"robots_default":true,"robots_noindex":false,"robots_noarchive":false,"robots_nosnippet":false,"robots_nofollow":false,"robots_noimageindex":false,"robots_noodp":false,"robots_notranslate":false,"robots_max_snippet":null,"robots_max_videopreview":null,"robots_max_imagepreview":"large","priority":null,"frequency":null,"local_seo":null,"limit_modified_date":false,"created":"2022-04-06 04:16:37","updated":"2025-06-10 15:44:59"},"aioseo_breadcrumb":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"http:\/\/theoryofeverything.org\/theToE\" title=\"Home\">Home<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">\u00bb<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"http:\/\/theoryofeverything.org\/theToE\/topics\/physics\/\" title=\"Physics\">Physics<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">\u00bb<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\tQuasiCrystals and Geometry\n\t\t<\/span><\/div>","aioseo_breadcrumb_json":[{"label":"Home","link":"http:\/\/theoryofeverything.org\/theToE"},{"label":"Physics","link":"http:\/\/theoryofeverything.org\/theToE\/topics\/physics\/"},{"label":"QuasiCrystals and Geometry","link":"http:\/\/theoryofeverything.org\/theToE\/2015\/08\/27\/quasicrystals-and-geometry\/"}],"_links":{"self":[{"href":"http:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/3505","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/comments?post=3505"}],"version-history":[{"count":1,"href":"http:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/3505\/revisions"}],"predecessor-version":[{"id":3887,"href":"http:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/3505\/revisions\/3887"}],"wp:attachment":[{"href":"http:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/media?parent=3505"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/categories?post=3505"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/tags?post=3505"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}