Golden Ratio Squared Links Fibonacci Growth to Quasicrystal Tiling Self-Similarity — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Golden rhombus inflation/deflation scaling factor is φ², linking Fibonacci growth to quasicrystalline tiling self-similarity. | MATH: φ = (1+√5)/2 ≈ 1.618; φ² = φ+1 ≈ 2.618; inflation factor = φ²; deflation factor = 1/φ² ≈ 0.382. | CONNECTION: 0.382 and 2.618 are reciprocal harmonic ratios; φ² appears in Penrose tiling inflation rules; 0.382 = 1/φ² is the fundamental scaling step in quasicrystal diffraction patterns. | DEPTH: 8 FINDING: Golden simplices (elementary tetrahedral bricks) generate icosahedral quasicrystals via φ-based inflation rules. | MATH: Golden triangle angles: 36°, 72°, 72°; golden rhombus acute angle = arctan(2) ≈ 63.43°; obtuse angle = 180° - arctan(2) ≈ 116.57°; dihedral angles in golden tetrahedra involve φ. | CONNECTION: Icosahedral symmetry (order 60) is the largest finite crystallographic point group; base-60 appears in icosahedral rotation angles (72° = 360°/5); φ governs 5-fold symmetry in quasicrystals. | DEPTH: 9 FINDING: Limit of sin(φ^n π) as Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Materials Chemistry
Field: Materials Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7170202122
Cites:
Cited by:
Related to: