Tetrahedron's Circumradius Ratio: √6/4, Not Golden Ratio — E8 Intelligence Research

FINDING: The regular tetrahedron's circumradius-to-edge ratio is √6/4 ≈ 0.612372, a value that sits remarkably close to the golden-ratio-derived 0.618 (φ−1) but is not exactly it — a distinction that matters for any claimed "harmonic" link. The Wolfram proof and the inscribed/circumscribed sphere derivations confirm this via standard solid geometry. MATH: - For a regular tetrahedron with edge length *a*: - Circumradius R = a·√6/4 - Inradius r = a·√6/12 - R/r = 3 (exact integer ratio) - R/a = √6/4 ≈ 0.6123724357 - The extended law of sines for any triangle: a/sin(A) = 2R — generalizes to tetrahedra via the circumsphere. - The quantum walk paper (arXiv:1411.3958) uses graph-theoretic transport on regular lattices — its mathematical core involves edge-state localization and dispersion relations, but no explicit tetrahedral circumradius appears; the connection to the geometric proof is tangential (both involve regular structures, but no shared constant). CONNECTIO Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com