Iwasawa Continued Fractions: Encoding Diamond Lattice Symmetry via Bianchi Groups — E8 Intelligence Research

FINDING: Iwasawa continued fractions generalize Euclidean algorithms to hyperbolic 3-space, with ideal tetrahedra as fundamental domains — directly encoding the diamond lattice's symmetry via the Bianchi group PSL(2, O₃). | MATH: Iwasawa decomposition: SL(2,C) = NAK, with N = {[[1,z],[0,1]]}, A = {[[e^t,0],[0,e^-t]]}, K = SU(2). Continued fraction map on upper half-space H³: z ↦ (z - ⌊z⌋)⁻¹, where ⌊z⌋ is the nearest Gaussian integer. Ideal tetrahedra in H³ have dihedral angles (π/3, π/3, π/3) for the regular case — volume = 3·Λ(π/3) ≈ 1.01494 (Lobachevsky function Λ(θ) = -∫₀^θ ln|2sin t| dt). Diamond lattice = face-centered cubic with basis (0,0,0), (1/4,1/4,1/4) — its symmetry group Fd3̄m (space group #227) contains the tetrahedral point group Td (order 24). The Bianchi group PSL(2, O₃) (O₃ = Eisenstein integers Z[ω], ω = e^{2πi/3}) acts on H³ with ideal tetrahedral fundamental domains whose vertices lie at cusps — these tetrahedra tile H³ and project to the diamond lattice's tetrahed Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com