Continued Fractions Unify Approximations, Identities, and Aperiodic Tilings — E8 Intelligence Research

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Continued fractions serve as a unifying structural backbone that connects rational approximations, algebraic identities, and aperiodic substitution tilings on cyclotomic fields. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.22225397

September 1, 2026Open Access

Continued Fractions Unify Approximations, Identities, and Aperiodic Tilings — E8 Intelligence Research

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Key Points

To demonstrate how continued fractions establish a unified mathematical framework connecting rational approximations, Ramanujan algebraic identities, and aperiodic substitution tilings.
Analyzed continued fraction expansions of fundamental constants, including pi and the golden ratio.
Evaluated algebraic conjugate pairs satisfying quartic polynomials derived from Ramanujan continued fraction identities.
Examined cyclotomic aperiodic substitution tilings using Perron–Frobenius eigenvalues and Pisot numbers in cyclotomic fields.
Continued fractions generate optimal rational approximations, including 22/7 for pi and the repeated unit expansion for the golden ratio.
Ramanujan continued fraction pairs satisfy identical degree-4 polynomial equations, establishing a direct connection between algebraic conjugates and fraction expansions.
Cyclotomic aperiodic substitution tilings exhibit inflation multipliers governed by Perron–Frobenius eigenvalues that align with continued fraction properties.

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Implication

Theoretical analysis demonstrates links between continued fractions and aperiodic substitution tilings, indicating a shared mathematical framework across number theory and geometry.

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Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a9935a0636c6408cfa7e0f7 https://doi.org/10.5281/zenodo.22225397

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Continued Fractions Unify Approximations, Identities, and Aperiodic Tilings — E8 Intelligence Research

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Continued Fractions Unify Approximations, Identities, and Aperiodic Tilings — E8 Intelligence Research

Key Points

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Implication

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