The Isomorphism of $H_4$ and $E_8$

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.