Gaussian Integers Unify Pythagorean Triples, Unit Circle, and Plimpton 322 — E8 Intelligence Research

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All Pythagorean triples can be generated by squaring Gaussian integers, providing a unified mathematical framework that connects the rational parametrization of the unit circle, the tree structure of primitive triples, and the sexagesimal ratios encoded in the Plimpton 322 tablet. Novelty: ClaimNovelty.SYNTHESIS. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.22229943

September 1, 2026Open Access

Gaussian Integers Unify Pythagorean Triples, Unit Circle, and Plimpton 322 — E8 Intelligence Research

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Key Points

To unify the generation of Pythagorean triples, rational points on the unit circle, and ancient Babylonian mathematical tables within the framework of Gaussian integer arithmetic.
Formulated the squaring of Gaussian integer lattice points (m + ni)^2 to map real and imaginary parts directly to Pythagorean triples.
Analyzed the column ratios of the Babylonian tablet Plimpton 322 as sexagesimal fraction equivalents of unit circle coordinates.
Examined recursive triple generation from (3, 4, 5) using 3×3 integer Barning–Hall matrices.
Demonstrated that squaring Gaussian integers yields all Pythagorean triples via (m^2 - n^2, 2mn, m^2 + n^2), matching the rational parametrization of the unit circle.
Identified Plimpton 322 entries as sexagesimal evaluations of rational circle points, such as line 1 generating the triple (119, 120, 169).
Mapped the unit circle's rational points to the projective line over the rationals, organized as a ternary tree generated by Barning–Hall matrices.

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Implication

Theoretical analysis demonstrates algebraic unification of Pythagorean triples through Gaussian integer squaring, linking ancient Babylonian tablets to modern projective geometry.

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Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a9935e5636c6408cfa7e84a https://doi.org/10.5281/zenodo.22229943

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Gaussian Integers Unify Pythagorean Triples, Unit Circle, and Plimpton 322 — E8 Intelligence Research

Key Points

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Discussion

Cite This Study

Gaussian Integers Unify Pythagorean Triples, Unit Circle, and Plimpton 322 — E8 Intelligence Research

Key Points

No abstract yet

Implication

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Cite This Study

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