KNOTT-1.1 — Internal & External Golden Section Constructions on a Segment

Given a finite segment AB, midpoint-circle compass-and-straightedge constructions divide AB internally at G so AB : AG = AG : GB = Φ and extend AB externally to E so AE : AB = AB : BE = Φ, erecting Golden Rectangle AEFD.

Curated Source & Catalog Provenance

Catalog ID	KNOTT-1.1
Curated Source	Dr. Ron Knott, Phi in 2D Geometry and Trigonometry (#cons)
Source Attribution	© 1996–2025 Dr. Ron Knott, University of Surrey — Explore Dr. Knott’s original exposition
Exact SymPy Validator	geometor.knott.flat_geometry.construct_golden_section_segment() (0 sp.Float atoms)
GEOMETOR Concordance	geometor/euclid, geometor/elements, geometor/divine, geometor/fundamentals

Constructive Synthesis & Exact Proof

Dr. Ron Knott opens his 2D Golden Geometry survey (KnottPhi2D) by pairing the two complementary compass-and-straightedge problems of Euclid (EuclidElements, Books II.11 and VI.30): extending a given straight segment externally in extreme and mean ratio, and dividing the same segment internally.

1. External Golden Extension: Starting from given segment AB of length \(2\) and erecting square ABCD, bisect the base at M so segment MC has exact quadrance \(Q(M, C) = 1^2 + 2^2 = 5\). Sweeping guide circle (M, C) onto line [AB] beyond B determines E with AE \(= 1 + \sqrt{5} = 2\Phi\) and BE \(= \sqrt{5} - 1 = 2\phi\), completing Golden Rectangle AEFD above F.

2. Internal Golden Division: Joining A to midpoint H of \(BC\) produces AH of length \(\sqrt{5}\). Unit circle (H, B) cuts AH at K (\(AK = \sqrt{5} - 1 = 2\phi\)), and circle (A, K) transfers that length onto the base at internal golden cut G. Thus major internal segment AG and external overhang BE are equal (\(2\phi\)), leaving minor segment GB \(= 2\phi^2\) and fixing the projective cross-ratio \((A, E; G, B) = \phi^2 = 2 - \Phi\).

Machine-Verified Exact Symbolic Invariants

Invariant	Exact Symbolic Value
AB	\(2\)
AG	\(\sqrt{6 - 2 \sqrt{5}}\)
GB	\(\sqrt{14 - 6 \sqrt{5}}\)
AE	\(\sqrt{2 \sqrt{5} + 6}\)
internal ratio AB over AG	\(\frac{1 + \sqrt{5}}{2}\)
external ratio AE over AB	\(\frac{\sqrt{2 \sqrt{5} + 6}}{2}\)
cross ratio A E G B	\(\frac{3}{2} - \frac{\sqrt{5}}{2}\)