KNOTT-4 — Algebraic Powers of Φ, Continued Fractions & Base-Φ Phigits¶
Linearization of all integer powers of Φ into Fibonacci multiples, the unit-volume Phibonacci brick, extremal continued fraction convergents F_{n+1}/F_n and Hurwitz’s √5 irrationality barrier, and George Bergman’s non-adjacent base-Φ (‘phigit’) numeral system.
Curated from Dr. Ron Knott’s University of Surrey archive (© 1996–2025 Dr. Ron Knott, University of Surrey).
KNOTT-4.1 — Fibonacci Power Linearization (Φⁿ = FₙΦ + Fₙ₋₁) & The Phibonacci Brick
Every integer power n ∈ Z of the Golden Ratio reduces to the linear form Φⁿ = Fₙ Φ + Fₙ₋₁ and (−φ)ⁿ = Fₙ₊₁ − Fₙ Φ = Fₙ₋₁ − Fₙ φ, while a rectangular Phibonacci brick of edges (1, φ, Φ) has unit volume 1, surface area 4Φ, and integer space diagonal 2.
KNOTT-4.2 — Extremal Continued Fraction [1; 1, 1, …], Convergents Fₙ₊₁/Fₙ, & Nested Radicals
The all-ones simple continued fraction Φ = [1; 1, 1, …] has rational convergents cₖ = Fₖ₊₁/Fₖ with exact alternating error cₖ − Φ = (−φ)ᵏ/Fₖ, saturating Hurwitz’s irrationality constant √5 via \(F_k^2 |c_k - \Phi| \to 1/\sqrt{5}\).
KNOTT-4.3 — Bergman’s Base-Φ (‘Phigit’) Numeral System
Under the carry law Φⁿ = Φⁿ⁻¹ + Φⁿ⁻² (100_Φ = 011_Φ), every positive integer N ≥ 1 admits a unique finite minimal non-adjacent {0, 1} expansion in powers of Φ; excluding infinite 011-tails yields a unique 00-free maximal form and exactly Fₖ valid finite phigit forms for every Fibonacci number Fₖ.
Collection Concordance¶
Catalog ID |
GEOMETOR Catalog Entry |
Curated Source Anchor |
|---|---|---|
Fibonacci Power Linearization (Φⁿ = FₙΦ + Fₙ₋₁) & The Phibonacci Brick |
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Extremal Continued Fraction [1; 1, 1, …], Convergents Fₙ₊₁/Fₙ, & Nested Radicals |
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Bergman’s Base-Φ (‘Phigit’) Numeral System |