Sexagesimal Mathematics: Proto-Trigonometry and Noncanonical Expansions in Babylonian Tablets — E8 Intelligence Research

FINDING: Sumerian/Babylonian sexagesimal (base-60) system encodes advanced computational methods, including proto-trigonometry on Plimpton 322, with noncanonical digit-set expansions extending integer representation theory. MATH: - Base-60 positional notation: \( ∑ₖ a_k 60^k \), \( a_k ∈ \{0,...,59\} \) - Plimpton 322: Pythagorean triples \( a^2 + b^2 = c^2 \) in sexagesimal, e.g., (119, 120, 169) → ratios \( b/a \) approximating secant/tangent values. - Noncanonical systems: For base \( b \), digit set \( D ⊂ Z \) with \( 0 ∉ D \), finite expansions \( ∑ᵢ₌₀^n d_i b^i \) — existence proven for certain \( D \) (arXiv:0804.2190). - Key constants: 60 (highly composite: \( 2^2 · 3 · 5 \)), 1/60 = 0.01666..., 1/3600 = 0.000277... CONNECTION: - Base-60 yields exact fractions for 1/2, 1/3, 1/4, 1/5, 1/6, 1/10, 1/12, 1/15, 1/20, 1/30 — all terminating. This mirrors crystallographic root systems (A₂, G₂) where 60° angles and hexagonal latt Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com