Plimpton 322: Ancient Babylonian Unit Circle Parametrization via Regular Reciprocals — E8 Intelligence Research

FINDING: Plimpton 322 encodes a complete sexagesimal parametrization of the unit circle via regular number reciprocals, yielding exact Pythagorean triples and lattice points. | MATH: The tablet lists 15 rows of (sexagesimal) values: \( (b, l, d) \) with \( d^2 = l^2 + b^2 \). The underlying parametrization is \( x = \frac{2t}{1+t^2}, \; y = \frac{1-t^2}{1+t^2} \) for \( t = \frac{p}{q} \) (regular numbers in base 60). Equivalently, \( (l, b, d) = (2pq, p^2 - q^2, p^2 + q^2) \). The sexagesimal reciprocals \( p/q \) and \( q/p \) appear as the two diagonals of the tablet's columns. The "missing pairs" in Bruins' reconstruction are explained by a systematic generation using \( p, q \) both regular (i.e., \( p,q \in \{2^a 3^b 5^c\} \)). | CONNECTION: The regular numbers form a multiplicative lattice in \( \mathbb{Q}^+ \) — a rank-3 free abelian group with generators 2, 3, 5. The unit circle parametrization maps this lattice to rational points on \( S^1 \), which are dense in the circle — Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com