KNOTT-6 — Fibonacci-Lucas Identities, Pythagorean Triples, Jigsaw Paradoxes & π¶
Four-term Fibonacci generators of Pythagorean right triangles, Cassini’s alternating ±1 determinant realized as 64 ↔ 65 and 64 ↔ 63 four-piece dissections, Binet/Lucas/Catalan/d’Ocagne/Honsberger and Fibonomial power identities, Fibonacci arctangent telescoping series for π, and the Golden Angle parastichy lattice.
Curated from Dr. Ron Knott’s University of Surrey archive (© 1996–2025 Dr. Ron Knott, University of Surrey).
KNOTT-6.1 — Fibonacci Generator of Primitive Pythagorean Triples
Any four consecutive positive Fibonacci numbers (Fₙ, Fₙ₊₁, Fₙ₊₂, Fₙ₊₃) form the Pythagorean right triangle (Fₙ Fₙ₊₃, 2 Fₙ₊₁ Fₙ₊₂, F₂ₙ₊₃) of area Fₙ Fₙ₊₁ Fₙ₊₂ Fₙ₊₃ and inradius Fₙ Fₙ₊₁, primitive whenever 3 ∤ n and twice a primitive triple when 3 ∣ n.
KNOTT-6.2 — Cassini’s Identity (Fₙ₋₁Fₙ₊₁ − Fₙ² = (−1)ⁿ) & The 64 ↔ 65 / 63 Jigsaw Dissections
Cassini’s alternating identity Fₙ₋₁ Fₙ₊₁ − Fₙ² = (−1)ⁿ manifests geometrically when the four pieces of an 8 × 8 = 64 square (two 3 × 8 triangles and two 5 × (3, 5) trapezoids) rearrange into a 5 × 13 = 65 rectangle with a +1 parallelogram gap OR into a 63-unit notched block with a −1 parallelogram overlap.
KNOTT-6.3 — Binet, Lucas, Catalan, d’Ocagne, Honsberger & Fibonomial Power Identities
Binet’s closed forms Fₙ = (Φⁿ − (−φ)ⁿ)/√5 and Lₙ = Φⁿ + (−φ)ⁿ place (Lₙ, Fₙ) on the alternating Pell-Lucas hyperbolas Lₙ² − 5 Fₙ² = 4(−1)ⁿ with unit sector triangles, unify Catalan, d’Ocagne, and Honsberger identities, and annihilate k-th powers Fₙᵏ via signed Fibonomial coefficients.
KNOTT-6.4 — Fibonacci Arctangent Telescoping Formulae for π
By Cassini’s odd-index relation F₂ₙ₊₁² − 1 = F₂ₙ F₂ₙ₊₂, every even-indexed reciprocal Fibonacci angle splits as arctan(1/F₂ₙ) = arctan(1/F₂ₙ₊₁) + arctan(1/F₂ₙ₊₂), telescoping from arctan(1/F₂) = π/4 into Lehmer’s series π/4 = ∑_{k=1}^∞ arctan(1/F₂ₖ₊₁).
KNOTT-6.5 — Golden Angle (360° / Φ² ≈ 137.50776°) & Parastichy Bridges
Dividing a full turn in extreme and mean ratio yields the Golden Angle 2π / Φ² = 2π φ² ≈ 137.50776°, whose cylindrical primordia satisfy Fₖ φ² − Fₖ₋₂ = (−φ)ᵏ, pairing opposing left- and right-handed parastichy spirals in consecutive Fibonacci numbers (Fₖ, Fₖ₊₁).
Collection Concordance¶
Catalog ID |
GEOMETOR Catalog Entry |
Curated Source Anchor |
|---|---|---|
Fibonacci Generator of Primitive Pythagorean Triples |
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Cassini’s Identity (Fₙ₋₁Fₙ₊₁ − Fₙ² = (−1)ⁿ) & The 64 ↔ 65 / 63 Jigsaw Dissections |
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Binet, Lucas, Catalan, d’Ocagne, Honsberger & Fibonomial Power Identities |
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Fibonacci Arctangent Telescoping Formulae for π |
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Golden Angle (360° / Φ² ≈ 137.50776°) & Parastichy Bridges |