Catalog of Golden Ratio Constructions

A machine-verified symbolic catalog of 45 geometric constructions and theorems across 7 geometric families, certifying 129 exact golden sections and 33 connected golden chains in 100% exact SymPy algebra.

Cataloging Ethos & Scholarly Provenance

Mathematical theorems, geometric incidences, and algebraic proportions are universal facts of nature and logic. Over two decades on Cut-the-Knot (Golden Ratio in Geometry, © 1996–2018 Alexander Bogomolny), Bogomolny curated an extraordinary constellation of classical and modern Golden Ratio constructions contributed by geometers around the world—spanning Euclid, Claudius Ptolemy, Johannes Kepler, Émile Lemoine, H. E. Huntley, George Odom, Kurt Hofstetter, Michel Bataille, Tran Quang Hung, Bùi Quang Tuân, Jerzy Kocik, and many others.

In the GEOMETOR Digital Scriptorium tradition—alongside the companion Ron Knott Golden Ratio Catalog (geometor/knott, cataloging Ron Knott’s Fibonacci Numbers and the Golden Section)—this codex catalogs each universal geometric fact in GEOMETOR’s canonical symbolic format:

1. 100% Exact Symbolic Modeling: Every construction is built from scratch in geometor.model with zero floating-point approximations over \(\mathbb{Q}(\phi)\), \(\mathbb{Q}(\sqrt{2}, \phi)\), \(\mathbb{Q}(\sqrt{3}, \phi)\), \(\mathbb{Q}(\sqrt{\phi})\), and cyclotomic pentagonal extensions.

2. Rational Trigonometry Certification: Primary ratios are certified via exact quadrance identities \(Q_{\text{long}} / Q_{\text{short}} = \phi + 1 = \phi^2\) and Lemoine Géométrographie locus metrics.

3. Automated Divine Discovery: Every model is scanned by geometor.divine.golden to uncover secondary golden sections and connected golden chains.

4. Original Prose & Interactive Vector Plates: All expositions, proofs, and theme-adaptive .euclid-graphic SVGs are newly authored for GEOMETOR with bidirectional :g: hover highlighting and full bibliographic attribution.

Seven Geometric Families

GC-01 · Classical & Minimal-Step Compass Constructions

Euclid II.11, Lemoine–Hofstetter 5-step records, Mascheroni 5-circle compass-only, and rusty compass

GC-02 · Equilateral Triangles & Regular Hexagons

George Odom’s midline chord, reflected equilaterals, Bataille’s hybrid, and butterfly chords

GC-03 · Squares, Lattice Grids & Golden Crosses

Inscribed squares, 2×2 and 3×3 lattice grids, Suppa’s incircles, the Greek Cross, and rotated squares

GC-04 · Right Triangles, Egyptian 3–4–5 & Kepler Proportions

Huntley’s 3–4–5 bisector circle, Kepler’s geometric-progression right triangle, and Niemeyer’s linkage

GC-05 · Pentagons, Pentagrams, Affine Invariance & Stars

Ptolemy’s Almagest pentagon, Prasolov’s irregular equi-area pentagon, six-pentagon star, and Wu Xing

GC-06 · Circles, Apollonian Windows, Yin–Yang & Sangaku

Infinite rational secants, tangent equal circles, Kocik’s Golden Window, Yin–Yang, and Sangaku packings

GC-07 · Rhombi, Trapezoids, Mixtilinear Circles & Dissections

2:1 and golden rhombi, 1:3 area-bisecting trapezoids, mixtilinear circles, and Hooper–Curry resolution

Comparative Synthesis & Scholarly Apparatus

• Comparative Table of All 45 Constructions — Complete metrics across all 45 constructions (number fields, elementary loci, Lemoine simplicity, and detected golden chains).

• Bibliography & Archival Sources — Primary treatises, journal articles, and Cut-the-Knot archival citations.

• Index of Geometers & Contributors — Biographical index of classical and contemporary geometers.

• Glossary of Geometric & Algebraic Terms — Technical glossary of constructive and Rational Trigonometry invariants.