{"id":8421,"date":"2026-09-28T08:59:10","date_gmt":"2026-09-28T15:59:10","guid":{"rendered":"https:\/\/theoryofeverything.org\/theToE\/?p=8421"},"modified":"2026-09-30T12:48:25","modified_gmt":"2026-09-30T19:48:25","slug":"the-miracle-octad-generator-mog-the-leech-lattice-and-the-monster","status":"publish","type":"post","link":"https:\/\/theoryofeverything.org\/theToE\/2026\/09\/28\/the-miracle-octad-generator-mog-the-leech-lattice-and-the-monster\/","title":{"rendered":"The Miracle Octad Generator (MOG), the Leech Lattice, and the Monster"},"content":{"rendered":"\n

This section of the post visualizes the Miracle Octad Generator (MOG) used in the generation of the Steiner octad system related to the binary (ternary) Golay codes. For more information, please see Wikipedia (WP) here<\/a> and here<\/a> (a 2010 post on Internet Archive for Steven H. Cullinane’s site).<\/p>\n\n\n\n

For the companion Mathematica Notebook (NB) file, see here<\/a>. Click the PNG image below for SVG.<\/p>\n\n\n\n

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MOG Visualizations and code<\/figcaption><\/figure>\n\n\n\n

This section is an analysis and visualization of the Leech lattice’s related binary (ternary) Golay code. Please see here<\/a> (WP) for more information.<\/p>\n\n\n\n

For the companion Mathematica Notebook (NB) file, see here<\/a>. Click the PNG image below for SVG.<\/p>\n\n\n\n

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Leech lattice binary and ternary Golay code<\/figcaption><\/figure>\n\n\n\n

This section visualizes the 16D Barnes-Wall and 24D Leech Lattices. Much better visualizations from David Madore<\/a> (Gro-Tsen<\/a>) are here<\/a>. He uses GAP and SAGE to tease out the group theoretic symmetries.<\/p>\n\n\n\n

Octonion based Barnes-Wall and Leech Lattice visualizations based on my implementation of work from Geoffrey Dixon and Robert Wilson:<\/p>\n\n\n

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Octonion defined Barnes-Wall to 16D B82<\/sup> projection<\/figcaption><\/figure>\n<\/div>\n\n
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Octonion defined 24D Leech lattice to 8D E8 Petrie projection with all 196560 vertices in 2D and 3D, noting the similarity to the 6720 vertex rectified E8 in the same projection (above).<\/figcaption><\/figure>\n<\/div>\n\n
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Octonion defined Leech lattice to 24D E83<\/sup> Petrie projection with 20 increments of 1000 vertices for every 10000 out of 196560<\/figcaption><\/figure>\n<\/div>\n\n
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Same as above with 50% of the inner edges filtered out<\/figcaption><\/figure>\n<\/div>\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

This section of the post visualizes the Miracle Octad Generator (MOG) used in the generation of the Steiner octad system related to the binary (ternary) Golay codes. For more information, please see Wikipedia (WP) here and here (a 2010 post on Internet Archive for Steven H. Cullinane’s site). For the companion Mathematica Notebook (NB) file, … Continue reading The Miracle Octad Generator (MOG), the Leech Lattice, and the Monster<\/span> →<\/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":[4,59,14],"class_list":["post-8421","post","type-post","status-publish","format-standard","hentry","category-physics","tag-e8","tag-leech","tag-math"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t