Penrose Tiling and Icosahedral Quasicrystal as Cut-and-Project Shadows — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Penrose tiling is the 2D cut-and-project shadow of a higher-dimensional lattice; its 3D analogue is the icosahedral quasicrystal, both generated by the same projection formalism. | MATH: Kite/dart inflation ratio φ = (1+√5)/2 = 1.6180339...; vertex configurations from de Bruijn pentagrid (5 families of parallel lines, indices k ∈ ℤ, sum of grid coordinates = 0 or 1); cut-and-project: E₈ (or ℤ⁵ ⊂ ℝ⁵) → ℝ² via irrational 2-plane with slope φ; acceptance window = projection of unit hypercube. | CONNECTION: φ = 1.618, φ² = 2.618, 1/φ = 0.618, 1/φ² = 0.382 — all present as tile edge ratios and inflation eigenvalues. Icosahedral symmetry = H₃/H₄ Coxeter group; H₄ is the symmetry of the 600-cell whose vertices are the 120 icosians (unit quaternions), the binary icosahedral group 2I ⊂ unit quaternions. | DEPTH: 9
FINDING: The icosian ring (Hurwitz quaternions) provides the exact algebraic coordinates for the 3D icosahedral quasicrystal — the same structure that underlies E₈ via the i
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
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BibTeX
@misc{caldin2026penrose,
title = {{Penrose Tiling and Icosahedral Quasicrystal as Cut-and-Project Shadows — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23658835},
url = {https://zenodo.org/records/23658835},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Penrose Tiling and Icosahedral Quasicrystal as Cut-and-Project Shadows — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/26
PB - Zenodo
DO - 10.5281/zenodo.23658835
UR - https://zenodo.org/records/23658835
AB - FINDING: Penrose tiling is the 2D cut-and-project shadow of a higher-dimensional lattice; its 3D analogue is the icosahedral quasicrystal, both generated by the same projection formalism. | MATH: Kite/dart inflation ratio φ = (1+√5)/2 = 1.6180339...; vertex configurations from de Bruijn pentagrid (5 families of parallel lines, indices k ∈ ℤ, sum of grid coordinates = 0 or 1); cut-and-project: E₈ (or ℤ⁵ ⊂ ℝ⁵) → ℝ² via irrational 2-plane with slope φ; acceptance window = projection of unit hypercube. | CONNECTION: φ = 1.618, φ² = 2.618, 1/φ = 0.618, 1/φ² = 0.382 — all present as tile edge ratios and inflation eigenvalues. Icosahedral symmetry = H₃/H₄ Coxeter group; H₄ is the symmetry of the 600-cell whose vertices are the 120 icosians (unit quaternions), the binary icosahedral group 2I ⊂ unit quaternions. | DEPTH: 9 FINDING: The icosian ring (Hurwitz quaternions) provides the exact algebraic coordinates for the 3D icosahedral quasicrystal — the same structure that underlies E₈ via the i Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -