Weyl Groups as Discrete Symmetry Engines Linking Icosahedral H₃ to E₈ and Langlands — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Weyl groups of root systems (including icosahedral H₃/H₄) are the discrete symmetry engines of Lie theory, with super-Weyl generalizations extending to Lie superalgebras; the icosahedral group A₅ ≅ H₃ Weyl group is the exceptional finite reflection group linked to E₈ via folding and to geometric Langlands through root data. | MATH: Weyl group W = ⟨s_α : α ∈ Δ⟩, s_α(β) = β − 2⟨β,α⟩/⟨α,α⟩·α; Cartan matrix A_ij = 2⟨α_i,α_j⟩/⟨α_j,α_j⟩; H₃ has Coxeter matrix m_ij = 5 (icosahedral), order |W(H₃)| = 120; H₄ order = 14400; E₈ root system has 240 roots, Weyl group order = 696,729,600; super Weyl groups (arXiv:2401.11068) are quotients of classical Weyl groups by parity/odd root actions. | CONNECTION: Icosahedral symmetry is the rank-3 Weyl group H₃ — its golden ratio φ = (1+√5)/2 = 1.618 appears in the Cartan integers: for adjacent simple roots, 4cos²(π/5) = φ+2 ≈ 3.618, and the squared length ratio of long/short roots in non-crystallographic H₃ is φ² = 2.618. The H₃ root system coordi
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026weyl,
title = {{Weyl Groups as Discrete Symmetry Engines Linking Icosahedral H₃ to E₈ and Langlands — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635567},
url = {https://zenodo.org/records/23635567},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Weyl Groups as Discrete Symmetry Engines Linking Icosahedral H₃ to E₈ and Langlands — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635567
UR - https://zenodo.org/records/23635567
AB - FINDING: Weyl groups of root systems (including icosahedral H₃/H₄) are the discrete symmetry engines of Lie theory, with super-Weyl generalizations extending to Lie superalgebras; the icosahedral group A₅ ≅ H₃ Weyl group is the exceptional finite reflection group linked to E₈ via folding and to geometric Langlands through root data. | MATH: Weyl group W = ⟨s_α : α ∈ Δ⟩, s_α(β) = β − 2⟨β,α⟩/⟨α,α⟩·α; Cartan matrix A_ij = 2⟨α_i,α_j⟩/⟨α_j,α_j⟩; H₃ has Coxeter matrix m_ij = 5 (icosahedral), order |W(H₃)| = 120; H₄ order = 14400; E₈ root system has 240 roots, Weyl group order = 696,729,600; super Weyl groups (arXiv:2401.11068) are quotients of classical Weyl groups by parity/odd root actions. | CONNECTION: Icosahedral symmetry is the rank-3 Weyl group H₃ — its golden ratio φ = (1+√5)/2 = 1.618 appears in the Cartan integers: for adjacent simple roots, 4cos²(π/5) = φ+2 ≈ 3.618, and the squared length ratio of long/short roots in non-crystallographic H₃ is φ² = 2.618. The H₃ root system coordi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -