KNOTT-5 — The Golden String, Zeckendorf Base, & Wythoff / Stolarsky Arrays¶
The infinite Sturmian Golden String (101101011011…) generated both by morphism substitution and by a lattice-cutting ray of slope φ, Zeckendorf’s canonical non-consecutive Fibonacci base, and the bijective Beatty, Wythoff, and Stolarsky interspersion arrays.
Curated from Dr. Ron Knott’s University of Surrey archive (© 1996–2025 Dr. Ron Knott, University of Surrey).
KNOTT-5.1 — The Infinite Golden String (101101011011…), Rabbit Tree & Cutting-Line Slope φ
Iterating the Rabbit substitution 1 ↦ 10, 0 ↦ 1 generates the infinite Sturmian Golden String dₙ = ⌊(n + 1)φ⌋ − ⌊nφ⌋, identically realized geometrically as the sequence of vertical (‘1’) and horizontal (‘0’) unit-lattice crossings of the ray y = φ x from the origin.
KNOTT-5.2 — Zeckendorf’s Theorem (Canonical Fibonacci Base Representation)
Every positive integer N ≥ 1 decomposes uniquely as a sum of non-consecutive Fibonacci numbers F_{k₁} + F_{k₂} + … + F_{kᵣ} (kᵢ ≥ kᵢ₊₁ + 2, kᵣ ≥ 2), producing a unique 11-free binary numeral in Fibonacci place values (…, 13, 8, 5, 3, 2, 1).
KNOTT-5.3 — Wythoff’s Array, Beatty’s Complementary Sequences (⌊nΦ⌋, ⌊nΦ²⌋), & Stolarsky’s Array
Because 1/Φ + 1/Φ² = 1, Beatty’s sequences Aₙ = ⌊nΦ⌋ and Bₙ = ⌊nΦ²⌋ = Aₙ + n partition Z_{>0} without overlap; seeding generalized Fibonacci rows via W(r, c) = ⌊(r + 1)Φ⌋ F_{c+2} + r F_{c+1} (Wythoff) or nearest-integer multiples [kΦ] (Stolarsky) places every positive integer in a unique (row, col) cell.
Collection Concordance¶
Catalog ID |
GEOMETOR Catalog Entry |
Curated Source Anchor |
|---|---|---|
The Infinite Golden String (101101011011…), Rabbit Tree & Cutting-Line Slope φ |
The Fibonacci Rabbit Sequence (The Golden String) (#PhiPlot) |
|
Zeckendorf’s Theorem (Canonical Fibonacci Base Representation) |
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Wythoff’s Array, Beatty’s Complementary Sequences (⌊nΦ⌋, ⌊nΦ²⌋), & Stolarsky’s Array |
General Fibonacci Series, Wythoff and Stolarsky Arrays (#wythoff) |