Papers › The Isomorphism of H₄ and E₈
The Isomorphism of H₄ and E₈
J. G. Moxness
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This paper gives an explicit isomorphic mapping from the 240 real ℝ⁸ roots of the E₈ Gosset 4₂₁ 8-polytope to two golden ratio scaled copies of the 120 root H₄ 600-cell quaternion 4-polytope using a traceless 8×8 rotation matrix 𝕌 with palindromic characteristic polynomial coefficients and a unitary form e^(i𝕌). It also shows the inverse map from a single H₄ 600-cell to E₈ using a 4D↪8D chiral left↔right mapping function, φ scaling, and 𝕌⁻¹. This approach shows that there are actually four copies of each 600-cell living within E₈ in the form of chiral H_(4L)⊕φH_(4L)⊕H_(4R)⊕φH_(4R) roots. In addition, it demonstrates a quaternion Weyl orbit construction of H₄-based 4-polytopes that provides an explicit mapping between E₈ and four copies of the tri-rectified Coxeter-Dynkin diagram of H₄, namely the 120-cell of order 600. Taking advantage of this property promises to open the door to as yet unexplored E₈-based Grand Unified Theories or GUTs.
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