Hyperbolic Tessellation Ratio: No Novel Derivation from Cosine — E8 Intelligence Research

FINDING: The search results are generic educational videos on Fibonacci, golden ratio, and trigonometry — no direct derivation of hyperbolic tessellation {p,q} fundamental ratio from cos(pi/p) is presented. No novel competition-level result is extracted. MATH: No new equations or constants beyond standard definitions: φ = (1+√5)/2 ≈ 1.618, cos(π/5) = φ/2 ≈ 0.809, cos(π/3) = 0.5, cos(π/4) = √2/2. For hyperbolic {p,q}, the fundamental ratio is typically cos(π/p) / sin(π/q) or related via the law of cosines in hyperbolic space: cosh(a) = [cos(π/q) + cos(π/p)] / [sin(π/q) sin(π/p)] — but this is not stated in the sources. CONNECTION: The only geometric link is the standard identity cos(π/5) = φ/2, which relates the pentagonal tessellation {5,q} to the golden ratio. For {5,4} or {5,5}, the fundamental ratio involves φ directly. No base-60, crystallographic, or root system connections appear in the provided material. DEPTH: 1 — The findings are introductory videos, not research. No profou Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com