Theon's Recurrence for √2: A Fibonacci-Like Sequence Converging to the Silver Ratio — E8 Intelligence Research

Abstract

FINDING: Theon of Smyrna's recurrence for √2 is a Fibonacci-like sequence generating successive approximations to √2, with a distinct recurrence relation that converges to the silver ratio (1+√2). MATH: Theon's recurrence: Let (p_n, q_n) be pairs such that p_1 = 1, q_1 = 1, and p_{n+1} = p_n + 2q_n q_{n+1} = p_n + q_n Then p_n / q_n → √2 as n → ∞. This is a linear recurrence with characteristic equation λ² = 2λ + 1, giving eigenvalues 1+√2 (silver ratio) and 1-√2. The ratio of successive terms (p_n / p_{n-1}) converges to 1+√2 ≈ 2.414, not the golden ratio 1.618. The recurrence is a special case of the Pell equation: p_n² - 2q_n² = ±1. CONNECTION: The silver ratio (1+√2 ≈ 2.414) is geometrically linked to octagonal symmetry (regular octagon side-to-diagonal ratio) and to the crystallographic root system B₂ (square lattice with diagonal). The recurrence mirrors the Fibonacci recurrence but with a factor 2, reflecting the difference between golden (pentagonal) and s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com