What does this research mean for the field?

The Coxeter-Weyl group W(H₄) acting on quaternionic vertices decomposes into W(A₄) and W(H₃) orbits, with the golden ratio and its square serving as fundamental ratios in the H₄ root systems and geometric harmony. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To analyze the quaternionic construction of H₄ polytopes and characterize the decomposition of W(H₄) Coxeter-Weyl group orbits into W(A₄) and W(H₃) subgroups.

September 8, 2026Open Access

Quaternionic H₄ Polytopes: Branching Under W(A₄) and W(H₃) Subgroups — E8 Intelligence Research

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Authors

ACAndrew Stewart Caldin

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Implication

Theoretical analysis demonstrates branching decompositions and golden ratio relationships in quaternionic H₄ root systems, highlighting geometric harmonies across Coxeter-Weyl reflection subgroups.

Key Points

To analyze the quaternionic construction of H₄ polytopes and characterize the decomposition of W(H₄) Coxeter-Weyl group orbits into W(A₄) and W(H₃) subgroups.
Constructed H₄ polytope vertices in quaternion coordinates using root combinations defined by {±1, ±φ, ±φ²}.
Derived branching rules decomposing W(H₄) group orbits into W(A₄) and W(H₃) subgroups and evaluated Coxeter numbers, spectral gaps, and Cartan matrix properties.
Branching rules decomposed W(H₄) orbits into subgroups corresponding to Coxeter numbers h(H₄) = 30, h(H₃) = 10, and h(A₄) = 5.
The golden ratio (φ ≈ 1.618) governs H₄ root geometries, with the squared ratio φ² ≈ 2.618 appearing in Cartan matrix entries and dominant root system norms.
Identified primitive eigenvalues of e^{2πi/30} for h = 30 within the Coxeter element spectral gap, yielding a real part of cos(12°) = φ√(3−φ)/2 ≈ 0.951.

Cite This Study

Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a9fd77658e84d0ff5b462cb https://doi.org/10.5281/zenodo.22472673

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Implication

Theoretical analysis demonstrates branching decompositions and golden ratio relationships in quaternionic H₄ root systems, highlighting geometric harmonies across Coxeter-Weyl reflection subgroups.

Quaternionic H₄ Polytopes: Branching Under W(A₄) and W(H₃) Subgroups — E8 Intelligence Research

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Key Points

Cite This Study

Quaternionic H₄ Polytopes: Branching Under W(A₄) and W(H₃) Subgroups — E8 Intelligence Research

Discussion

Member takes

Implication

Key Points

Cite This Study

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