Antikythera Mechanism's Gear Ratios Decode Lunar-Solar Cycles via Continued Fractions — E8 Intelligence Research

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The Antikythera Mechanism encodes lunar-solar cycles using gear ratios that function as convergents of continued fractions for astronomical periods, such as the 235/19 ratio for the Metonic cycle. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.22106041

August 26, 2026Open Access

Antikythera Mechanism's Gear Ratios Decode Lunar-Solar Cycles via Continued Fractions — E8 Intelligence Research

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Key Points

Investigate how the gear trains of the Antikythera Mechanism mathematically encode solar and lunar periodicities using continued fraction convergents.
Analyzed the astronomical parameters of the 235-synodic-month Metonic cycle, 76-year Callippic cycle, and 223-month Saros cycle.
Calculated gear ratios from mechanical fragments, including the 53-tooth gear and 64:38 pairing, and compared them against mathematical continued fraction expansions.
Identified the 235:19 gear ratio as a continued fraction convergent for the lunar month-to-year ratio (12.368266...), generating an error rate of only ~0.0003 days per year.
Determined that the 64:38 (32:19) gear ratio models the synodic-to-anomalistic month ratio of approximately 2.085, while the 53-tooth gear relates to the 235/19 lunation rate (53/19 ≈ 2.789).
Demonstrated that the 223-month Saros dial links 242 draconic and 239 anomalistic months over an 18-year, 11⅓-day span, reflecting a 223/235 (≈ 0.949) relation to the Metonic cycle.

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Implication

Mathematical analysis reveals Antikythera Mechanism gear ratios match continued fraction convergents of lunar-solar cycles, highlighting ancient computational precision.

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Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a91469ad15324a1df3aa6a5 https://doi.org/10.5281/zenodo.22106041

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Antikythera Mechanism's Gear Ratios Decode Lunar-Solar Cycles via Continued Fractions — E8 Intelligence Research

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Antikythera Mechanism's Gear Ratios Decode Lunar-Solar Cycles via Continued Fractions — E8 Intelligence Research

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