Iwasawa Continued Fractions in Hyperbolic 3-Space via Diamond Lattice — E8 Intelligence Research

FINDING: Iwasawa continued fractions generalize to hyperbolic 3-space via ideal tetrahedra, with the diamond lattice emerging as a natural discrete structure for encoding geodesic approximations. | MATH: Iwasawa decomposition \(G=NAK\) for \(SL(2,C)\); continued fraction maps on \(H^3\) via Farey tessellation by ideal tetrahedra; diamond lattice = two interpenetrating fcc lattices, coordination number 4, bond angle \(109.47^∘\) (\(cos⁻¹(-1/3)\)); tetrahedral dihedral angle \((1/3)≈ 70.53^∘\) (complementary to \(109.47^∘\)); the regular ideal tetrahedron in \(H^3\) has all dihedral angles \(60^∘\), volume \(≈ 1.01494\) (Lobachevsky function \(Λ(π/3)\)). | CONNECTION: The diamond lattice is the 3D analogue of the honeycomb (hexagonal) lattice — both are the unique "deepest" sphere packings in their dimensions. The tetrahedral angle \(109.47^∘\) relates to the golden ratio via \(cos(109.47^∘) = -1/3\), and Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com