{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-isomorphism-of-h-4-and-e-8","title":"The Isomorphism of $H_4$ and $E_8$","arxiv_id":"2311.01486","date":"2023-11-02","proceeding":null,"authors":["J. G. Moxness"],"abstract":"This paper gives an explicit isomorphic mapping from the 240 real $\\mathbb{R}^{8}$ roots of the $E_8$ Gosset $4_{21}$ 8-polytope to two golden ratio scaled copies of the 120 root $H_4$ 600-cell quaternion 4-polytope using a traceless 8$\\times$8 rotation matrix $\\mathbb{U}$ with palindromic characteristic polynomial coefficients and a unitary form $e^{\\text {i$\\mathbb{U}$}}$. It also shows the inverse map from a single $H_4$ 600-cell to $E_8$ using a 4D$\\hookrightarrow$8D chiral left$\\leftrightarrow$right mapping function, $ \\varphi$ scaling, and $\\mathbb{U}^{-1}$. This approach shows that there are actually four copies of each 600-cell living within $E_8$ in the form of chiral $H_{4L}$$\\oplus$$\\varphi H_{4L}$$\\oplus$$H_{4R}$$\\oplus$$\\varphi H_{4R}$ roots. In addition, it demonstrates a quaternion Weyl orbit construction of $H_4$-based 4-polytopes that provides an explicit mapping between $E_8$ and four copies of the tri-rectified Coxeter-Dynkin diagram of $H_4$, namely the 120-cell of order 600. Taking advantage of this property promises to open the door to as yet unexplored $E_8$-based Grand Unified Theories or GUTs.","url_abs":"https://arxiv.org/abs/2311.01486v2","url_pdf":"https://arxiv.org/pdf/2311.01486v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-isomorphism-of-h-4-and-e-8","repo_url":"https://github.com/jgmoxness/The_Isomorphism_of_H4_and_E8","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}