Unifying Stern-Brocot, Farey Sequences, and Tessellation via PSL(2,Z) — E8 Intelligence Research

FINDING: The Stern-Brocot tree, Farey sequences, and the Farey tessellation form a single combinatorial-geometric object whose symmetry group is PSL(2,Z), connecting rational approximation to hyperbolic tiling. | MATH: Stern-Brocot tree = binary tree of mediants (a/b ⊕ c/d = (a+c)/(b+d)); Farey sequence F_n = sorted reduced fractions in [0,1] with denominator ≤ n; Farey tessellation = ideal triangles in upper half-plane with vertices at rationals ∪ {∞}; PSL(2,Z) acts transitively on oriented edges of this tessellation; the tree is the Cayley graph of PSL(2,Z) modulo its congruence subgroup Γ(2); key invariant: the modular group's fundamental domain has area π/3; continued fraction map T(x) = {1/x} is the shift on the Stern-Brocot tree's boundary (the real line). | CONNECTION: The Farey tessellation's edge lengths in hyperbolic metric correspond to the golden ratio φ = (1+√5)/2 when considering the orbit of the ideal triangle (0,1,∞) under the generator S: z→-1/z; the cusp width 1 and t Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com