{"id":7009,"date":"2025-03-28T08:41:57","date_gmt":"2025-03-28T15:41:57","guid":{"rendered":"https:\/\/theoryofeverything.org\/theToE\/?p=7009"},"modified":"2025-11-23T16:46:08","modified_gmt":"2025-11-23T23:46:08","slug":"e8-h4-and-the-mcgee-group-graph","status":"publish","type":"post","link":"https:\/\/theoryofeverything.org\/theToE\/2025\/03\/28\/e8-h4-and-the-mcgee-group-graph\/","title":{"rendered":"E8, H4 and the McGee Group Graph"},"content":{"rendered":"\n

In a recent Azimuth post<\/a>, John Baez discussed the McGee group and (3,7)-cage graph. A link to a conversation on MathOverflow<\/a> describes its connection to the Fano plane and the octonions via the Heawood graph (which he had discussed on his Visual Insight blog here<\/a>). <\/p>\n\n\n\n

So I thought it would be fun to show how the alternate graph of the McGee group shown on WP<\/a> is indeed the 24-cell with 64 of 96 edges removed being projected to the B4 Coxeter plane. So again, it is linked to E8=H42<\/sup> (as 4 4D Left (L)\/Right (R) golden ratio (\u03c6) scaled 600-cell=24-cell+snub 24-cell from the E8 to H4 rotation or folding matrix<\/a>). Another related post here<\/a> shows how the E8 Petrie projection decomposes into the two sets of 4 rings as H4 and H4\u03c6 600-cells. <\/p>\n\n\n\n

User:Leshabirukov’s Alternate McGee graph is here<\/a> and below with blue edges being the 4 dimensional axis:<\/p>\n\n\n

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Below are renderings from my hyperdimensional hypercomplex (quaternion\/octonion\/sedenion) VisibLie_E8 viewer:<\/p>\n\n\n

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McGee Group (3.,7)-cage graph as a 24-cell projected to the B4 Coxeter plane with 64 edges removed (and including 4 axis “edges” for a total of 24 vertices and 36 edges)<\/figcaption><\/figure>\n<\/div>\n\n\n

To get a view of how B4 projection is isomorphic to the Greg Egan animation, below is my animation overlaid on top of it:<\/p>\n\n\n

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The animation showing the isomorphism between Egan’s base image and the H4\u03c6 24-cell projected to the B4 Coxeter plane. <\/figcaption><\/figure>\n<\/div>\n\n\n

The first frame is Greg Egan’s base image in the Baez article. Each frame is 2 seconds. <\/p>\n\n\n\n

The second frame adds the XYZW axes annotations (4 pairs of red node links that are antipodal across the labeled axis letter (e.g. the nodes above\/below the black X line are connected by Y axis endpoints with p#’s 56 & 201, the nodes above\/below the black Y line are connected by X axis endpoints with p#’s 74 & 183). These red endpoints are all associated with the 16-cell (i.e. 4-orthoplex octahedron) within the 24-cell.<\/p>\n\n\n\n

The third frame differentiates the the inner(cyan) \/ outer (purple) octagon rings. These two rings are each 8 of 16 vertices of the 8-cell (i.e. 4-cube Tesseract) within the 24-cell. <\/p>\n\n\n\n

The 4th frame adds the vertex numbers (p and p*) split between #=1-128 and 257-# (as respectively palindromic 256-129) which are always antipodal in the X-Y axis of the B4 Coxeter plane projection (or any other Coxeter plane projection) as well as being antipodal within each of the 4 sets of 6 nodes in the Egan animation.<\/p>\n\n\n

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A static image for ease of analysis.<\/figcaption><\/figure>\n<\/div>\n\n
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24-cell with all of its 96 edges projected to the B4 Coxeter plane. Vertex numbers are from a canonical sort of E8 in Pascal triangle order (when including the 8 generator and 8 anti-generator vertices as 2-9 and 248-255 (respectively).<\/figcaption><\/figure>\n<\/div>\n\n\n

For anyone who studies deeply the E8 to H4 connections as I do, you know the L\/R H4 and H4\u03c6 24-cells have the same palindromic L\/R 4D elements in the E8 vertices as well. This means the E8 based McGee groups overlap identically, as shown here:<\/p>\n\n\n

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H4 and H4\u03c6 24-cells in B4 Coxeter plane projection<\/figcaption><\/figure>\n<\/div>\n\n\n

So, here is the full E8 in the same Coxeter B4 projection with vertex coloring based on the overlap counts. <\/p>\n\n\n

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\"\"\/<\/a><\/figure>\n<\/div>\n\n
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How the E8 H4 and H4\u03c6 snub 24-cells (as 4 \u03c0\/5 rotations of each of the \u03c6 scaled 24-cells) fill in this E8 projection with more McGee groups is more interesting and complex. The patterns for the removal of the specific edges are also interesting and inform the possible physics of E8 based unification theories.<\/p>\n\n\n\n

… and a McGee graph torus’ fascinating connection <\/a>to the world of music….<\/p>\n\n\n

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What the author calls the “Cholidean Harmony Structure”<\/figcaption><\/figure>\n<\/div>\n\n\n

Picture the same red WXYZ 4D axis node-pairs (not shown in this image) positioned at the centroid of the triads. It has 12 vertices and 24 edges. The full 24-cell McGee vertices has a 4D-hexadic structure with mirrored triadic pairs across edge-connected axis nodes.<\/p>\n\n\n\n

Analogously, if we view the McGee torus as 4 rings of 4-edges (plus 4 sets of 4 WXYZ quaternionic axis-connected edges), one could then view musical harmony as being represented with 3 (Complex or Imaginary) rings of 4 edges each forming the triads (and excluding the Real component and its axis connections).<\/p>\n\n\n\n

This difference might be understood by looking at the composition of the hyperdimensional hypercomplex functions (e.g. quaternion \/ octonion Exp \/ Cos \/ Sin \/ Octonionic Fourier Transform (OFT)<\/a> ) with respect to Im vs. Re differences.<\/p>\n\n\n\n

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

In a recent Azimuth post, John Baez discussed the McGee group and (3,7)-cage graph. A link to a conversation on MathOverflow describes its connection to the Fano plane and the octonions via the Heawood graph (which he had discussed on his Visual Insight blog here). So I thought it would be fun to show how … Continue reading E8, H4 and the McGee Group Graph<\/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":[4,34,14,16],"class_list":["post-7009","post","type-post","status-publish","format-standard","hentry","category-physics","tag-e8","tag-h4","tag-math","tag-physics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/7009","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=7009"}],"version-history":[{"count":58,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/7009\/revisions"}],"predecessor-version":[{"id":7525,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/7009\/revisions\/7525"}],"wp:attachment":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/media?parent=7009"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/categories?post=7009"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/tags?post=7009"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}