{"id":7830,"date":"2022-08-10T06:34:17","date_gmt":"2022-08-10T13:34:17","guid":{"rendered":"https:\/\/theoryofeverything.org\/theToE\/?p=7830"},"modified":"2026-02-23T08:46:59","modified_gmt":"2026-02-23T15:46:59","slug":"visualizing-quaternion-generated-dual-to-the-snub-24-cell-2","status":"publish","type":"post","link":"https:\/\/theoryofeverything.org\/theToE\/2022\/08\/10\/visualizing-quaternion-generated-dual-to-the-snub-24-cell-2\/","title":{"rendered":"Visualizing Quaternion Generated Dual to the Snub 24-cell"},"content":{"rendered":"\n

I did a Mathematica (MTM) analysis of several important papers here<\/a> and here<\/a> from Mehmet Koca, et. al. The resulting MTM output in PDF format is here<\/a> and the .NB notebook is here<\/a>.<\/p>\n\n\n

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\"\"<\/a>
3D Visualization of the outer hull of the 144 vertex Dual Snub 24 Cell, with vertices colored by overlap count:
* The (42) yellow have no overlaps.
* The (51) orange have 2 overlaps.
* The (18) tetrahedral hull surfaces are uniquely colored.<\/figcaption><\/figure>\n<\/div>\n\n
\n
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The Dual Snub 24-Cell with less opacity<\/figcaption><\/figure>\n<\/div>\n\n\n

What is really interesting about this is the method to generate these 3D and 4D structures is based on Quaternions (and Octonions with judicious selection of the first triad={123}). This includes both the 600 Cell and the 120 Cell and its group theoretic orbits. The 144 vertex Dual Snub 24 Cell is a combination of those 120 Cell orbits, namely T'(24) & S’ (96), along with the D4 24 Cell T(24).<\/p>\n\n\n

\n
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3D Visualization of the outer hull of the alternate 96 vertex Snub 24 Cell (S’)<\/figcaption><\/figure>\n<\/div>\n\n
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Visualization of the concentric hulls of the Alternate Snub 24 Cell<\/figcaption><\/figure>\n<\/div>\n\n
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Various 2D Coxeter Plane Projections with vertex overlap color coding.<\/figcaption><\/figure>\n<\/div>\n\n
\n
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3D Visualization of the outer hull of M(192) as one of the W(D4) C3 orbits of the 120-Cell (600)<\/figcaption><\/figure>\n<\/div>\n\n
\n
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3D Visualization of the outer hull of N(288) that are the 120-Cell (600) Complement of
the W(D4) C3 orbits T'(24)+S'(96)+M (192)<\/figcaption><\/figure>\n<\/div>\n\n
\n
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3D Visualization of the outer hull of the 120-Cell (600) generated using T’<\/figcaption><\/figure>\n<\/div>\n\n
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3D Visualization of the outer hull of the 120-Cell (600) generated using T<\/figcaption><\/figure>\n<\/div>\n\n
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3D Animation of the 5 quaternion generated 24-cell outer hulls consecutively adding to make the 600-Cell.<\/figcaption><\/figure>\n<\/div>\n\n
\n
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3D Animation of the 5 quaternion generated 600-cell outer hulls consecutively adding to make the 120-Cell.<\/figcaption><\/figure>\n<\/div>","protected":false},"excerpt":{"rendered":"

I did a Mathematica (MTM) analysis of several important papers here and here from Mehmet Koca, et. al. The resulting MTM output in PDF format is here and the .NB notebook is here. What is really interesting about this is the method to generate these 3D and 4D structures is based on Quaternions (and Octonions … Continue reading Visualizing Quaternion Generated Dual to the Snub 24-cell<\/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":[8,13,4,34,14,16],"class_list":["post-7830","post","type-post","status-publish","format-standard","hentry","category-physics","tag-3d","tag-4d","tag-e8","tag-h4","tag-math","tag-physics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/7830","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=7830"}],"version-history":[{"count":2,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/7830\/revisions"}],"predecessor-version":[{"id":7838,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/7830\/revisions\/7838"}],"wp:attachment":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/media?parent=7830"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/categories?post=7830"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/tags?post=7830"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}