Golden Ratio Absent in D3's Irreducible Representations — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The dihedral group D3 (symmetry group of an equilateral triangle) has irreducible representations whose character values and eigenvalue structures connect to the golden ratio via its order-3 rotational subgroup. | MATH: D3 has order 6, with irreps: trivial (1-dim), sign (1-dim), and standard 2-dim rep. For the 2-dim irrep, the rotation element r (order 3) has eigenvalues e^(±2πi/3) = -1/2 ± i√3/2. The character of r is -1. The golden ratio φ = (1+√5)/2 does NOT appear directly in D3's character table. However, the dihedral group D5 (order 10, pentagon symmetry) does: its 2-dim irreps have rotation eigenvalues e^(±2πi/5) and e^(±4πi/5), whose real parts are (√5-1)/4 ≈ 0.309 and -(√5+1)/4 ≈ -0.809 — related to φ via 2cos(2π/5) = (√5-1)/2 = 1/φ ≈ 0.618. | CONNECTION: The golden ratio emerges from D5, not D3. D3's geometry (equilateral triangle) has angles 60° — base-60 sexagesimal connection. The 2-dim irrep of D3 is the standard representation on R², whose rotation matrix has tr
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026golden,
title = {{Golden Ratio Absent in D3's Irreducible Representations — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23409784},
url = {https://zenodo.org/records/23409784},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Golden Ratio Absent in D3's Irreducible Representations — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/16
PB - Zenodo
DO - 10.5281/zenodo.23409784
UR - https://zenodo.org/records/23409784
AB - FINDING: The dihedral group D3 (symmetry group of an equilateral triangle) has irreducible representations whose character values and eigenvalue structures connect to the golden ratio via its order-3 rotational subgroup. | MATH: D3 has order 6, with irreps: trivial (1-dim), sign (1-dim), and standard 2-dim rep. For the 2-dim irrep, the rotation element r (order 3) has eigenvalues e^(±2πi/3) = -1/2 ± i√3/2. The character of r is -1. The golden ratio φ = (1+√5)/2 does NOT appear directly in D3's character table. However, the dihedral group D5 (order 10, pentagon symmetry) does: its 2-dim irreps have rotation eigenvalues e^(±2πi/5) and e^(±4πi/5), whose real parts are (√5-1)/4 ≈ 0.309 and -(√5+1)/4 ≈ -0.809 — related to φ via 2cos(2π/5) = (√5-1)/2 = 1/φ ≈ 0.618. | CONNECTION: The golden ratio emerges from D5, not D3. D3's geometry (equilateral triangle) has angles 60° — base-60 sexagesimal connection. The 2-dim irrep of D3 is the standard representation on R², whose rotation matrix has tr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -