Penrose Tilings as Projections of 5D Lattice via A4 Root System — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Penrose tilings arise as 2D projections of the 5D hypercubic lattice, with vertex configurations encoded by the A4 root system; the search results confirm the tiling's aperiodic fivefold symmetry but the lattice-projection link is implicit in the references, not explicitly derived in the snippets. MATH: Penrose tiling uses two rhombi (angles 36°/144° and 72°/108°), with edge lengths in ratio 1:1 (both rhombi share equal edge length). The inflation factor is the golden ratio φ = (1+√5)/2 ≈ 1.618, and the area ratio of thick to thin rhombus is φ. The vertex configurations in the 2D projection correspond to the 5D hypercubic lattice Z⁵ projected onto a 2D plane perpendicular to the (1,1,1,1,1) direction; the relevant root system is A4 (rank 4, 20 roots, Weyl group order 120). The projection yields a set of 5 basis vectors in 2D at angles 0°, 72°, 144°, 216°, 288° — the pentagonal star. The golden ratio appears as φ = 2cos(36°) = (1+√5)/2, and its inverse φ⁻¹ = φ−1 ≈ 0.618. The
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
- MERLIN SCIENCE — A_n Lattice Projections Unify Rhombus and Triangle Tilings via Voronoi — E8 Intelligence Research
- A_n Lattice Projections Unify Rhombus and Triangle Tilings via Voronoi-Delone Cells — E8 Intelligence Research
- MERLIN SCIENCE — Golden-Ratio Slices: Penrose Tiling from 5D Lattice Projections — E8 Intelligence Research
- A5 Symmetry Unites Viral Capsids and Penrose Tilings — E8 Intelligence Research
- Fivefold Symmetry, Penrose Tilings, and the Golden Ratio Link — E8 Intelligence Research
BibTeX
@misc{caldin2026penrose,
title = {{Penrose Tilings as Projections of 5D Lattice via A4 Root System — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23430986},
url = {https://zenodo.org/records/23430986},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Penrose Tilings as Projections of 5D Lattice via A4 Root System — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/17
PB - Zenodo
DO - 10.5281/zenodo.23430986
UR - https://zenodo.org/records/23430986
AB - FINDING: Penrose tilings arise as 2D projections of the 5D hypercubic lattice, with vertex configurations encoded by the A4 root system; the search results confirm the tiling's aperiodic fivefold symmetry but the lattice-projection link is implicit in the references, not explicitly derived in the snippets. MATH: Penrose tiling uses two rhombi (angles 36°/144° and 72°/108°), with edge lengths in ratio 1:1 (both rhombi share equal edge length). The inflation factor is the golden ratio φ = (1+√5)/2 ≈ 1.618, and the area ratio of thick to thin rhombus is φ. The vertex configurations in the 2D projection correspond to the 5D hypercubic lattice Z⁵ projected onto a 2D plane perpendicular to the (1,1,1,1,1) direction; the relevant root system is A4 (rank 4, 20 roots, Weyl group order 120). The projection yields a set of 5 basis vectors in 2D at angles 0°, 72°, 144°, 216°, 288° — the pentagonal star. The golden ratio appears as φ = 2cos(36°) = (1+√5)/2, and its inverse φ⁻¹ = φ−1 ≈ 0.618. The Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -