The Isomorphism of H_4 and E_8

This paper gives an explicit isomorphic mapping from the 240 real 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 8times8 rotation matrix U with palindromic characteristic polynomial coefficients and a unitary form e^{i\mathbb{U}}. It also shows the inverse map from a single H_4 600-cell to E_8 using a 4Dhookrightarrow8D chiral leftleftrightarrowright mapping function, φ scaling, and 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}oplusφH_{4L}oplusH_{4R}oplusφH_{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.