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This research examines how quaternion orbits of the Coxeter-Weyl group generate H4 polytopes with distinct scaling properties.

July 27, 2026Open Access

Quaternion Orbits of W(H4) Generate H4 Polytopes with Golden Ratio Scaling — E8 Intelligence Research

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Randomized trial reveals H4 polytope transformations in quaternion orbits, implying geometric harmony.

This research examines how quaternion orbits of the Coxeter-Weyl group generate H4 polytopes with distinct scaling properties.
Generated H4 polytope vertices using quaternion forms and analyzed their transformation under the W(H4) Coxeter group with an order of 14400.
Investigated mathematical relationships between vertex coordinates, specifically utilizing the golden ratio and its properties.
Applied 3D projections to maintain icosahedral symmetry with dual polytope scaling factors.
Vertices of H4 polytopes were generated as quaternions with coordinates involving the golden ratio, achieving harmonious proportions.
Found that the dual relationship scales by the square of the golden ratio, connecting geometric harmony to a factor of approximately 2.618.
Demonstrated that quaternion representations and group action maintain integrity in 3D space while preserving symmetry.

Andrew Stewart Caldin (2026) studied this question.

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