GC-07.06 · Exact φ-Limit Resolution of the Hooper–Curry Fibonacci Dissection

Discoverer(s): William Hooper, Paul Curry, Alexander Bogomolny | Date: 1794 | Field: \(\mathbb{Q}(\phi)\) | Loci: 1 (Lemoine \(S = 3\))

Replace integer Fibonacci lengths \((3, 5, 8, 13)\) in the \(8 \times 8 = 64\) vs. \(5 \times 13 = 65\) dissection fallacy by exact powers \((1, \phi, \phi^2, \phi^3)\). In rectangle ABCD of width \(\phi^3\) and height \(\phi\), internal vertices Q \(= (\phi, 1/\phi)\) and P \(= (\phi^2, 1)\) lie identically on diagonal AC, forming the 4-point golden chain \(|AQ|/|QP| = |AP|/|AQ| = \phi\).
GC-07.06 — Exact φ-Limit Resolution of the Hooper–Curry Fibonacci Dissection AC: x*(-sqrt(5)/2 - 1/2) + y*(2 + sqrt(5)) = 0 Golden major AQ = sqrt(3) Golden minor QP = (-sqrt(3) + sqrt(15))/2 C = (2 + sqrt(5), 1/2 + sqrt(5)/2) D = (0, 1/2 + sqrt(5)/2) E = (sqrt(5)/2 + 3/2, 0) F = (1/2 + sqrt(5)/2, 1/2 + sqrt(5)/2) B = (2 + sqrt(5), 0) A = (0, 0) P = (sqrt(5)/2 + 3/2, 1) Q = (1/2 + sqrt(5)/2, -1/2 + sqrt(5)/2) C D E F B A P Q

Given & Constructive Steps

Given: Rectangle ABCD with A \(= (0,0)\), B \(= (\phi^3, 0)\), C \(= (\phi^3, \phi)\), and D \(= (0, \phi)\).

step Mark golden cuts E \(= (\phi^2, 0)\) on AB and F \(= (\phi, \phi)\) on DC.
step Erect vertical cuts EP of height \(1\) to P \(= (\phi^2, 1)\) and FQ of length \(1\) to Q \(= (\phi, 1/\phi)\).
step Draw diagonal AC, which passes with zero area gap through both Q and P.

Exact Symbolic Proof & Certificate

For Fibonacci numbers, Cassini’s identity \(F_{n-1}F_{n+1} - F_n^2 = (-1)^n\) produces a thin parallelogram of area \(\pm 1\) because slopes \(F_{n-2}/F_n\) and \(F_{n-1}/F_{n+1}\) differ slightly. Replacing \(F_k\) by \(\phi^k\) gives \(\phi \cdot \phi^3 - (\phi^2)^2 = 0\) and identical slopes \(\text{slope}(A,Q) = (1/\phi)/\phi = 1/\phi^2 = \text{slope}(A,P) = \text{slope}(A,C)\). Along AC, abscissas \((0, \phi, \phi^2, \phi^3)\) yield \(|AQ|/|QP| = \phi / 1 = \phi\).

Machine-Verified Symbolic Certificate

Invariant

Exact Symbolic Value

Primary Golden Ratio

AQ \(/\) QP \(= \frac{1}{2} + \frac{\sqrt{5}}{2}\) on collinear triple (A, Q, P)

Exact Segment Lengths

\(|AQ| = \sqrt{3}\), \(|QP| = \frac{- \sqrt{3} + \sqrt{15}}{2}\)

Rational Trigonometry Quadrances

\(Q(A, Q) = 3\), \(Q(Q, P) = \frac{9}{2} - \frac{3 \sqrt{5}}{2}\), \(Q_{\text{long}}/Q_{\text{short}} = \frac{\sqrt{5}}{2} + \frac{3}{2} = \phi^2\)

Algebraic Number Field

\(\mathbb{Q}(\phi)\)

Géométrographie Loci

2 given points · 1 lines · 0 circles · 3 polygons (elementary loci \(= 1\), Lemoine \(S = 3\))

Automated Divine Scan

geometor.divine.golden detected 8 exact collinear golden section(s) and 1 connected golden chain(s) in this model:

  • Section (A, Q, P): long \(|AQ|^2 = 3\) vs. short \(|QP|^2 = \frac{9}{2} - \frac{3 \sqrt{5}}{2}\)

  • Section (A, Q, C): long \(|QC|^2 = \frac{3 \sqrt{5}}{2} + \frac{9}{2}\) vs. short \(|AQ|^2 = 3\)

  • Section (A, P, C): long \(|AP|^2 = \frac{3 \sqrt{5}}{2} + \frac{9}{2}\) vs. short \(|PC|^2 = 3\)

  • Section (Q, P, C): long \(|PC|^2 = 3\) vs. short \(|QP|^2 = \frac{9}{2} - \frac{3 \sqrt{5}}{2}\)

  • Section (A, E, B): long \(|AE|^2 = \frac{3 \sqrt{5}}{2} + \frac{7}{2}\) vs. short \(|EB|^2 = \frac{\sqrt{5}}{2} + \frac{3}{2}\)

  • Section (F, P, B): long \(|PB|^2 = \frac{\sqrt{5}}{2} + \frac{5}{2}\) vs. short \(|FP|^2 = \frac{5}{2} - \frac{\sqrt{5}}{2}\)

  • Section (D, F, C): long \(|FC|^2 = \frac{3 \sqrt{5}}{2} + \frac{7}{2}\) vs. short \(|DF|^2 = \frac{\sqrt{5}}{2} + \frac{3}{2}\)

  • Section (D, Q, E): long \(|DQ|^2 = \frac{\sqrt{5}}{2} + \frac{5}{2}\) vs. short \(|QE|^2 = \frac{5}{2} - \frac{\sqrt{5}}{2}\)

Connected Golden Chains:

  • Chain (A — Q — P — C) spanning 2 overlapping golden sections (progression > <)

Historical Commentary & Attribution

First published by William Hooper in Rational Recreations (1794) and generalized by magician Paul Curry (1953), the missing-square paradox vanishes solely at the irrational fixed point \(\phi\).