Plimpton 322 Decoded: A Sexagesimal Table of Regular Reciprocal Pairs Generating Pythagorean Triples — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Plimpton 322 is a sexagesimal table of regular reciprocal pairs (p, q) generating Pythagorean triples, with a reconstruction algorithm that predicts missing rows and reveals a systematic sequence of ratios. | MATH: Pythagorean triples (a, b, d) with a² + b² = d²; sexagesimal (base-60) regular numbers p, q (i.e., p, q with prime factors only 2, 3, 5); generating formula: a = p² − q², b = 2pq, d = p² + q² (or variants with scaling); Bruins' scheme uses p/q from reciprocal pairs; the "factor 12" algorithm proposes b = M·Q_M, with M, Q_M integers, yielding a data-driven family of algorithms. The tablet lists 15 rows, with ratios like (d/a)² or (b/d)² forming a decreasing sequence of squared secants/tangents. | CONNECTION: The ratios in Plimpton 322 are not the golden ratio directly, but the underlying structure is base-60 — the same sexagesimal system that makes 0.618, 0.786, 1.618 expressible as exact finite sexagesimal fractions (e.g., 0.618 = 37;04,48 in base-60; 0.786 = 47;09,
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
- Babylonian Tablet Plimpton 322: Exact Sexagesimal Pythagorean Triples from 45° to 30° — E8 Intelligence Research
- Plimpton 322 Refined: Reconstructing Missing Pairs in Babylonian Sexagesimal Triples — E8 Intelligence Research
- Plimpton 322: A Hidden Sexagesimal Trigonometric Table — E8 Intelligence Research
- Plimpton 322 Decoded: A Sexagesimal-Optimized Table of Squared Secants — E8 Intelligence Research
- Babylonian Tablet Plimpton 322: Ancient Sexagesimal Trigonometry Predates Greeks — E8 Intelligence Research
BibTeX
@misc{caldin2026plimpton,
title = {{Plimpton 322 Decoded: A Sexagesimal Table of Regular Reciprocal Pairs Generating Pythagorean Triples — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23658205},
url = {https://zenodo.org/records/23658205},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Plimpton 322 Decoded: A Sexagesimal Table of Regular Reciprocal Pairs Generating Pythagorean Triples — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/26
PB - Zenodo
DO - 10.5281/zenodo.23658205
UR - https://zenodo.org/records/23658205
AB - FINDING: Plimpton 322 is a sexagesimal table of regular reciprocal pairs (p, q) generating Pythagorean triples, with a reconstruction algorithm that predicts missing rows and reveals a systematic sequence of ratios. | MATH: Pythagorean triples (a, b, d) with a² + b² = d²; sexagesimal (base-60) regular numbers p, q (i.e., p, q with prime factors only 2, 3, 5); generating formula: a = p² − q², b = 2pq, d = p² + q² (or variants with scaling); Bruins' scheme uses p/q from reciprocal pairs; the "factor 12" algorithm proposes b = M·Q_M, with M, Q_M integers, yielding a data-driven family of algorithms. The tablet lists 15 rows, with ratios like (d/a)² or (b/d)² forming a decreasing sequence of squared secants/tangents. | CONNECTION: The ratios in Plimpton 322 are not the golden ratio directly, but the underlying structure is base-60 — the same sexagesimal system that makes 0.618, 0.786, 1.618 expressible as exact finite sexagesimal fractions (e.g., 0.618 = 37;04,48 in base-60; 0.786 = 47;09, Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -