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This research aims to illustrate how quaternion multiplication algebra generates the 600-cell and its dual, the 120-cell, using the Coxeter group H4.

July 27, 2026Open Access

Quaternion Algebra Generates 600-Cell and 120-Cell via Coxeter Group H4 — E8 Intelligence Research

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Finding demonstrates the algebraic generation of geometric structures in four-dimensional space, suggesting new insights into their properties.

This research aims to illustrate how quaternion multiplication algebra generates the 600-cell and its dual, the 120-cell, using the Coxeter group H4.
Utilized quaternion algebra to derive the properties of the 600-cell and 120-cell polyhedra.
Examined the relationships and symmetries of these structures through the Coxeter group H4.
Investigated the role of the golden ratio in the matrix entries of H4 and its implications on the root system.
The 600-cell consists of 600 tetrahedral cells and 120 vertices, while the 120-cell comprises 120 dodecahedral cells and 600 vertices.
Proved that the quaternion group Q8 is embedded within the H4 group, indicating a deeper symmetry relationship.
Illustrated that the golden ratio and its reciprocal are fundamental to the H4 root system and the construction of these polyhedra.

Andrew Stewart Caldin (2026) studied this question.

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Finding demonstrates the algebraic generation of geometric structures in four-dimensional space, suggesting new insights into their properties.