E8 Geometry and Elliptic Torsion: A Comparative Study — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The E8 root system (Gosset 4_21 polytope) is a central object in 8D geometry, with 240 vertices and 6720 edges, and its Fourier transform and rotations are visualized; separately, elliptic curve torsion (specifically 3-torsion) is studied via quadratic twists, but no direct link between E8 and elliptic curve torsion is established in these videos. | MATH: E8 root system: 240 roots, 6720 edges (4_21 polytope); rank 8, Coxeter number 30, Weyl group order 696,729,600. Fourier transform of E8 yields a dual lattice structure (self-dual, unimodular). Elliptic curve: quadratic twists with 3-torsion — the torsion subgroup E(Q)[3] ≅ Z/3Z or (Z/3Z)², related to the j-invariant and modular forms of weight 2. | CONNECTION: E8 is the exceptional root system linked to the golden ratio via its Cartan matrix eigenvalues (2cos(π/5) = φ ≈ 1.618, 2cos(2π/5) = 1/φ ≈ 0.618) — the E8 Dynkin diagram contains A4 subdiagrams whose eigenvalues involve φ. The 240 roots project to icosidodecahedral and i
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026geometry,
title = {{E8 Geometry and Elliptic Torsion: A Comparative Study — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635268},
url = {https://zenodo.org/records/23635268},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - E8 Geometry and Elliptic Torsion: A Comparative Study — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635268
UR - https://zenodo.org/records/23635268
AB - FINDING: The E8 root system (Gosset 4_21 polytope) is a central object in 8D geometry, with 240 vertices and 6720 edges, and its Fourier transform and rotations are visualized; separately, elliptic curve torsion (specifically 3-torsion) is studied via quadratic twists, but no direct link between E8 and elliptic curve torsion is established in these videos. | MATH: E8 root system: 240 roots, 6720 edges (4_21 polytope); rank 8, Coxeter number 30, Weyl group order 696,729,600. Fourier transform of E8 yields a dual lattice structure (self-dual, unimodular). Elliptic curve: quadratic twists with 3-torsion — the torsion subgroup E(Q)[3] ≅ Z/3Z or (Z/3Z)², related to the j-invariant and modular forms of weight 2. | CONNECTION: E8 is the exceptional root system linked to the golden ratio via its Cartan matrix eigenvalues (2cos(π/5) = φ ≈ 1.618, 2cos(2π/5) = 1/φ ≈ 0.618) — the E8 Dynkin diagram contains A4 subdiagrams whose eigenvalues involve φ. The 240 roots project to icosidodecahedral and i Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -