Glide Planes in Crystallography vs. Fermilab Lattice QCD Mass Ratios — E8 Intelligence Research

FINDING: Glide planes in crystallography combine reflection with half-lattice translation, generating non-symmorphic space-group symmetries; the Fermilab lattice QCD result is unrelated to the crystallographic search but yields heavy-quark mass ratios. MATH: Glide operation: \( (x,y,z) \rightarrow (\bar{x}, y+\tfrac{1}{2}, z+\tfrac{1}{2}) \) (e.g., \(n\)-glide) — translation vector \( \mathbf{t} = \tfrac{1}{2}(a+b) \) or \( \tfrac{1}{2}(a+c) \), etc. The square of a glide is a pure lattice translation: \( g^2 = T_{\mathbf{t}} \), with \( \mathbf{t} \) a half-lattice vector. Screw axes: rotation by \( 2\pi/n \) plus translation \( m/n \cdot \mathbf{c} \). Space groups: 230 total; 73 symmorphic, 157 non-symmorphic (contain glides/screws). CONNECTION: The half-translation \( \tfrac{1}{2} \) is the crystallographic analogue of the golden-ratio-derived 0.5 in Fibonacci phyllotaxis (0.618 = 1 − 0.382; 0.5 is the degenerate case of the golden angle \( 137.5^\circ = 2\pi(1-0.618) \)). Glid Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com