Golden Ratio Coordinates Unify Icosahedron and Dodecahedron Geometry — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The icosahedron and dodecahedron are constructed from golden-ratio coordinates (±1, ±φ, 0) and permutations, with the golden ratio φ = (1+√5)/2 as the fundamental irrational length; the pentagon's diagonal-to-side ratio is φ, proving φ irrationality via geometric construction. | MATH: φ = (1+√5)/2 ≈ 1.6180339887; φ⁻¹ = φ−1 ≈ 0.6180339887; φ² = φ+1 ≈ 2.6180339887; icosahedron vertices: (0, ±1, ±φ), (±1, ±φ, 0), (±φ, 0, ±1) — 12 vertices, 20 triangular faces; dodecahedron dual: 20 vertices, 12 pentagonal faces; pentagon diagonal/side = φ; Bilinski dodecahedron: 12 golden rhombi with diagonal ratio φ; symmetry group A5 (order 60) — rotational symmetry of both solids; truncated icosidodecahedron has 62 faces (30 squares, 20 hexagons, 12 decagons) with vertex figure 4.6.10. | CONNECTION: φ and φ⁻¹ appear as edge ratios in the golden rhombi (diagonal ratio = φ); the A5 symmetry group is isomorphic to the icosahedral group, which is a subgroup of the crystallographic point group Ih (
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 Coordinates Unify Icosahedron and Dodecahedron Geometry — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23490827},
url = {https://zenodo.org/records/23490827},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Golden Ratio Coordinates Unify Icosahedron and Dodecahedron Geometry — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/20
PB - Zenodo
DO - 10.5281/zenodo.23490827
UR - https://zenodo.org/records/23490827
AB - FINDING: The icosahedron and dodecahedron are constructed from golden-ratio coordinates (±1, ±φ, 0) and permutations, with the golden ratio φ = (1+√5)/2 as the fundamental irrational length; the pentagon's diagonal-to-side ratio is φ, proving φ irrationality via geometric construction. | MATH: φ = (1+√5)/2 ≈ 1.6180339887; φ⁻¹ = φ−1 ≈ 0.6180339887; φ² = φ+1 ≈ 2.6180339887; icosahedron vertices: (0, ±1, ±φ), (±1, ±φ, 0), (±φ, 0, ±1) — 12 vertices, 20 triangular faces; dodecahedron dual: 20 vertices, 12 pentagonal faces; pentagon diagonal/side = φ; Bilinski dodecahedron: 12 golden rhombi with diagonal ratio φ; symmetry group A5 (order 60) — rotational symmetry of both solids; truncated icosidodecahedron has 62 faces (30 squares, 20 hexagons, 12 decagons) with vertex figure 4.6.10. | CONNECTION: φ and φ⁻¹ appear as edge ratios in the golden rhombi (diagonal ratio = φ); the A5 symmetry group is isomorphic to the icosahedral group, which is a subgroup of the crystallographic point group Ih ( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -