Content of type 'item'

KNOTT-1.1 — Internal & External Golden Section Constructions on a Segment

Given a finite segment AB, midpoint-circle compass-and-straightedge constructions divide AB internally at G so AB : AG = AG : GB = Φ and extend AB externally to E so AE : AB = AB : BE = Φ, erecting Golden Rectangle AEFD.

KNOTT-2.1 — Exact Radical Sines, Cosines, Tangents & Spreads of 18°, 36°, 54°, 72°

On the unit circle, cos 36° = sin 54° = Φ/2 and sin 18° = cos 72° = φ/2 differ by exactly 1/2 and multiply to 1/4, while the pentagonal spreads s(36°) = (5 − √5)/8 and s(72°) = (5 + √5)/8 form the quadratic roots of Wildberger’s five-fold spread factor 16 s² − 20 s + 5 = 0.

KNOTT-3.1 — Three Orthogonal Golden Rectangles → Regular Icosahedron & Dodecahedron

Three mutually perpendicular 2 × 2Φ golden rectangles have 12 corners (0, ±1, ±Φ) forming a regular icosahedron of edge 2, volume 20Φ²/3, and rational dihedral spread 4/9, while (±1, ±1, ±1) ∪ cyc(0, ±φ, ±Φ) forms a regular dodecahedron of edge 2φ, volume 8 + 4Φ, and dihedral spread 4/5.

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-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-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-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).

KNOTT-2.2 — The Complete 3° Constructible Angle Lattice (3°k for k = 1..29)

Superimposing pentagonal angles in Q(√5) with equilateral and half-right angles in Q(√2, √3) expresses sin(3°k), cos(3°k), tan(3°k), and spread s(3°k) in explicit nested square roots for all 29 acute multiples of 3°, while Gauss’s Fermat prime 17 = 2⁴ + 1 extends compass constructibility to cos(π/17).

KNOTT-3.2 — Inscribed Cube & Compound of Five Cubes in a Dodecahedron

Inside a regular dodecahedron of edge 2φ, the 60 face diagonals of length 2 = Φ · (2φ) partition into five congruent inscribed cubes of edge 2 and volume 8, with each of the 20 dodecahedral vertices belonging to exactly two cubes.

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-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-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-1.3 — Golden Triangles (72°–72°–36°) & Golden Gnomons (36°–36°–108°)

An isosceles acute golden triangle ABC with base AB = 1 and legs AC = BC = Φ is partitioned by base-angle bisector AD (with AD = CD = 1 and BD = φ) into an obtuse golden gnomon CAD and a self-similar acute golden triangle DAB with exact spreads s(36°) = (5 − √5)/8 and s(72°) = (5 + √5)/8.

KNOTT-3.3 — Acute & Obtuse Golden Rhombohedra (3D Quasicrystal / Ammann Tiles)

Triples of five-fold icosahedral vectors of quadrance Φ + 2 span acute (prolate) and obtuse (oblate) golden rhombohedra bounded by six congruent golden rhombi of diagonal ratio Φ : 1, with exact volumes 2Φ² and 2Φ in ratio Φ : 1 and 10 + 10 assembly into a rhombic triacontahedron of volume 20Φ³.

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.

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.

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-1.4 — Regular Pentagon, Pentagram, Decagon & Paper-Knot Pentagon

The five diagonals of a regular pentagon of side 1 enclose a similar inner pentagon of side φ² and cut each diagonal in the geometric progression φ² : φ : 1 : Φ; furthermore, folding a flat uniform strip of width w into a tight overhand knot locks parallel trapezoid edges into an exact regular pentagon of side w sec 18°.

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-1.5 — Kepler’s Golden Right Triangle (1, √Φ, Φ) & Circle-Square Near-Quadrature

A right triangle with side lengths (1, √Φ, Φ) has side quadrances (1, Φ, Φ²) simultaneously in geometric progression and Pythagorean sum 1 + Φ = Φ², satisfies cos θ = tan θ = 1/√Φ, and equates the perimeter 4√Φ of the square on leg √Φ with the circumference πΦ of the circle on diameter Φ to within 0.096%.

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).

KNOTT-1.6 — Penrose Kites & Darts (P2) and Thick & Thin Golden Rhombs (P3)

A thick 72°–108° golden rhomb of side Φ divides at unit distance K along its long diagonal of length Φ² into a convex Penrose kite KBCD and a non-convex Penrose dart ABKD with exact area ratio Φ : 1, matching the Φ : 1 area ratio between thick (72°) and thin (36°) Penrose golden rhombs.