{"id":8167,"date":"2026-08-04T11:14:00","date_gmt":"2026-08-04T18:14:00","guid":{"rendered":"https:\/\/theoryofeverything.org\/theToE\/?p=8167"},"modified":"2026-08-04T12:59:36","modified_gmt":"2026-08-04T19:59:36","slug":"e8-and-h4-gram-matrix-visualizations","status":"publish","type":"post","link":"http:\/\/theoryofeverything.org\/theToE\/2026\/08\/04\/e8-and-h4-gram-matrix-visualizations\/","title":{"rendered":"E8 and H4 Gram Matrix Visualizations"},"content":{"rendered":"\n

Beautiful MatrixPlot’s of E8 and H4. The 240 E8 vertices (+120\/-120 roots) are sorted by the 9th row of the binary structure Pascal Triangle, excluding the 2nd & 2nd to last entries of +\/- 8 (i.e. {1,8<\/mark><\/s>,28,56,70=(35\/-35),-56,-28,-8<\/mark><\/s>,-1<\/strong>}, where the 112 integer D8 are 56\/-56<\/strong> and the 128 half-integer BC8 are {1,38,35,-35,-28,-1<\/strong>}). <\/p>\n\n\n\n

Below is the E8 Gram matrix of ALL roots (not the typical representation as the Cartan matrix using only the simple roots):<\/p>\n\n\n

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E8 Gram matrix<\/figcaption><\/figure>\n<\/div>\n\n\n

H4+H4\u03c6 8D Gram matrix created using the dot product of E8 against the 8×8 E8 to H4 folding matrix U<\/strong>:<\/p>\n\n\n

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H4+H4\u03c6 8D Gram matrix<\/figcaption><\/figure>\n<\/div>\n\n\n

H4\u03c6 4D Gram matrix created as 120 H4 4D vertices:<\/p>\n\n\n

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H4\u03c6 4D Gram matrix<\/figcaption><\/figure>\n<\/div>\n\n\n

Compare this to the Cartan matrix produced as the Gram matrix of only the 8 E8 simple roots:<\/p>\n\n\n

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E8 Gram Weights (in Pascal Triangle order)<\/figcaption><\/figure>\n<\/div>\n\n\n

Then compare to the Cartan matrix produced as the Gram matrix of only the 8 E8 simple roots:<\/p>\n\n\n

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E8 Cartan matrix<\/figcaption><\/figure>\n<\/div>","protected":false},"excerpt":{"rendered":"

Beautiful MatrixPlot’s of E8 and H4. The 240 E8 vertices (+120\/-120 roots) are sorted by the 9th row of the binary structure Pascal Triangle, excluding the 2nd & 2nd to last entries of +\/- 8 (i.e. {1,8,28,56,70=(35\/-35),-56,-28,-8,-1}, where the 112 integer D8 are 56\/-56 and the 128 half-integer BC8 are {1,38,35,-35,-28,-1}). Below is the E8 … Continue reading E8 and H4 Gram Matrix Visualizations<\/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":[],"class_list":["post-8167","post","type-post","status-publish","format-standard","hentry","category-physics"],"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