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 \(SL(2,\mathbb{C})\); continued fraction maps on \(\mathbb{H}^3\) via Farey tessellation by ideal tetrahedra; diamond lattice = two interpenetrating fcc lattices, coordination number 4, bond angle \(109.47^\circ\) (\(\cos^{-1}(-1/3)\)); tetrahedral dihedral angle \(\arccos(1/3)\approx 70.53^\circ\) (complementary to \(109.47^\circ\)); the regular ideal tetrahedron in \(\mathbb{H}^3\) has all dihedral angles \(60^\circ\), volume \(\approx 1.01494\) (Lobachevsky function \(\Lambda(\pi/3)\)). | 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 \(109.47^\circ\) relates to the golden ratio via \(\cos(109.47^\circ) = -1/3\), and Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com