From Graph Cliques to Minimal Ellipses: Euler's Dual Mathematical Legacy — E8 Intelligence Research

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Project Euler's hardest problems involving cyclic figurate sets are graph-theoretic NP-complete searches, and Euler's historical extremal ellipse papers solve minimal-area and perimeter ellipses through fixed points, linking these concepts to crystallographic symmetries. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.22633360

September 7, 2026Open Access

From Graph Cliques to Minimal Ellipses: Euler's Dual Mathematical Legacy — E8 Intelligence Research

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Key Points

To examine theoretical connections between NP-complete combinatorial challenges in Project Euler and Leonhard Euler's historical papers on extremal ellipses.
Analyzed the computational complexity of Project Euler problems, specifically modeling prime pairing and cyclic figurate sets as graph-theoretic clique decision problems.
Evaluated Euler's variational formulations (papers E563, E691, and E692) for determining minimal-area and minimal-perimeter ellipses passing through fixed points using Euler–Lagrange equations.
Demonstrated that Project Euler clique problems represent NP-complete computational searches, with figurate number formulations mapping directly to root-system polytope counts.
Identified that extremal ellipses passing through more than five fixed points lack closed-form solutions and introduce elliptic integrals.
Showed that resolving extremal ellipses across symmetric point sets recovers dihedral crystallographic point-group symmetries.

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Implication

Mathematical analysis reveals structural parallels between graph clique searches and extremal ellipses, suggesting deep connections across geometry, number theory, and symmetry.

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Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a9fd7c658e84d0ff5b46b65 https://doi.org/10.5281/zenodo.22633360

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From Graph Cliques to Minimal Ellipses: Euler's Dual Mathematical Legacy — E8 Intelligence Research

Key Points

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Implication

Discussion

Cite This Study

From Graph Cliques to Minimal Ellipses: Euler's Dual Mathematical Legacy — E8 Intelligence Research

Key Points

No abstract yet

Implication

Discussion

Cite This Study

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