KNOTT-1.2 — Golden Rectangles, √5 Rectangles, & Whirling-Square Gnomonic Spirals

Flanking a unit square ABCD on both sides via its base-midpoint circle produces a 1 × √5 rectangle WEFN containing right golden rectangle AEFD; recursive whirling-square cuts in gnomon BEFC share perpendicular main and secondary diagonals [AF] ⊥ [EC] (spread s = 1) meeting at pole P = (Φ²/√5, Φ/√5).

Curated Source & Catalog Provenance

Catalog ID	KNOTT-1.2
Curated Source	Dr. Ron Knott, Phi in 2D Geometry and Trigonometry (#root5)
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_rectangles_and_spiral() (0 sp.Float atoms)
GEOMETOR Concordance	geometor/euclid, geometor/divine, geometor/pacioli

Constructive Synthesis & Exact Proof

Extending the midpoint-circle construction (KnottPhi2D) symmetrically to both sides of unit square ABCD centered on base midpoint M, circle (M, C) of radius \(\sqrt{5}/2\) meets base line [AB] at both E (\(x = \Phi\)) and W (\(x = -\phi\)).

• Root-5 & Golden Rectangles: Because \(\Phi + \phi = \sqrt{5}\), segment WE spans exact width \(\sqrt{5}\), framing Luca Pacioli’s and Jay Hambidge’s \(1 \times \sqrt{5}\) dynamic-symmetry rectangle WEFN (Hambidge1920) partitioned into central unit square ABCD and two flanking \(\phi \times 1\) golden rectangles, including right gnomon BEFC.

• Orthogonal Whirling-Square Diagonals: Subdividing BEFC recursively via golden cuts G2G1, H1H2, and J1J2 aligns every successive square corner along the two diagonals AF (slope \(1/\Phi = \phi\)) and EC (slope \(-1/\phi = -\Phi\)). Since \(\phi \cdot (-\Phi) = -1\), lines [AF] and [EC] meet at exact spread \(s = 1\) (perpendicularity) at the self-similar spiral eye P \(= (\Phi^2/\sqrt{5},\; \Phi/\sqrt{5})\).

Machine-Verified Exact Symbolic Invariants

Invariant	Exact Symbolic Value
golden rect width	\(\frac{1}{2} + \frac{\sqrt{5}}{2}\)
sqrt5 rect width	\(\sqrt{5}\)
diagonal spread	\(1\)
pole x	\(\frac{1}{2} + \frac{3 \sqrt{5}}{10}\)
pole y	\(\frac{\sqrt{5}}{10} + \frac{1}{2}\)
gnomon area	\(- \frac{1}{2} + \frac{\sqrt{5}}{2}\)