Euler's 1770s Extremal Ellipse Conditions: Minimal Area and Perimeter Through Fixed Points — E8 Intelligence Research

Abstract

FINDING: Project Euler is a repository of computational number theory challenges; the only mathematically substantive item is Euler's 1770s work on extremal ellipses through fixed point sets. | MATH: Minimal-area/perimeter ellipse through n points — relates to Löwner-John ellipsoids; Euler's papers E563, E691, E692 derive variational conditions for ellipse parameters (semi-axes a,b, rotation θ) satisfying ∂(area)/∂(parameter)=0 and ∂(perimeter)/∂(parameter)=0, with perimeter given by complete elliptic integral E(k) where k²=1−(b/a)². | CONNECTION: Ellipse eccentricity e=√(1−(b/a)²) — for minimal perimeter ellipses through symmetric point sets, extremal ratios b/a often approach √(1−e²) values; the golden ratio φ=1.618 appears in degenerate cases where points lie on a circle (a=b, e=0) or collinear (e→1). No direct 0.382/0.618/0.786/2.618 or base-60 link found in the cited material. | DEPTH: 3 — The Euler ellipse papers are historically interesting but standard convex geometry; Project Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com