Unifying Stern-Brocot, Farey Sequences, and Tessellation via PSL(2,Z) — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
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
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BibTeX
@misc{caldin2026unifying,
title = {{Unifying Stern-Brocot, Farey Sequences, and Tessellation via PSL(2,Z) — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23464952},
url = {https://zenodo.org/records/23464952},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Unifying Stern-Brocot, Farey Sequences, and Tessellation via PSL(2,Z) — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/19
PB - Zenodo
DO - 10.5281/zenodo.23464952
UR - https://zenodo.org/records/23464952
AB - 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
ER -