Mathematical Catalog

Six thematic collections cataloging 22 Golden Ratio, trigonometric, polyhedral, phigit, Sturmian, and Fibonacci–Lucas structures from Dr. Ron Knott’s Fibonacci Numbers, the Golden Section and the Golden String (© 1996–2025 Dr. Ron Knott, University of Surrey), each proven with 100% exact SymPy algebra and rendered as an interactive .euclid-graphic SVG.

KNOTT-1 — 2D Golden Geometry & Constructions

Planar compass-and-straightedge divisions in extreme and mean ratio, golden and root-5 rectangles with orthogonal gnomonic spiral poles, self-deflating golden triangles and gnomons, pentagonal star progressions and overhand paper-strip knots, Kepler’s geometric-progression right triangle, and Penrose aperiodic P2/P3 prototiles.

KNOTT-2 — Exact Pentagonal & Constructible Trigonometry

Closed-form quadratic radicals and Rational Trigonometry spreads for 18°, 36°, 54°, and 72°, extended by Euclid IV.16 pentagon-equilateral superposition across all 29 constructible multiples of 3° in the first quadrant and contrasted with Gauss’s Fermat-prime regular 17-gon cosine.

KNOTT-3 — 3D Golden Geometry of Solids

Three-dimensional golden coordinates spanning the regular icosahedron and dodecahedron from three mutually orthogonal golden rectangles, Euclid XIII.17’s inscribed cube and five-cube compound, and the acute and obtuse golden rhombohedra of icosahedral quasicrystals.

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.

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.

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.

Master 22-Entry Concordance

ID	Title	Knott Anchor	Symbolic Validator
KNOTT-1.1	Internal & External Golden Section Constructions on a Segment	#cons	construct_golden_section_segment()
KNOTT-1.2	Golden Rectangles, √5 Rectangles, & Whirling-Square Gnomonic Spirals	#root5	construct_golden_rectangles_and_spiral()
KNOTT-1.3	Golden Triangles (72°–72°–36°) & Golden Gnomons (36°–36°–108°)	#isoc	construct_golden_triangles_and_gnomons()
KNOTT-1.4	Regular Pentagon, Pentagram, Decagon & Paper-Knot Pentagon	#pent	construct_pentagon_and_paper_knot()
KNOTT-1.5	Kepler’s Golden Right Triangle (1, √Φ, Φ) & Circle-Square Near-Quadrature	#PhiRightAng	construct_kepler_triangle()
KNOTT-1.6	Penrose Kites & Darts (P2) and Thick & Thin Golden Rhombs (P3)	#penrose	construct_penrose_tiles()
KNOTT-2.1	Exact Radical Sines, Cosines, Tangents & Spreads of 18°, 36°, 54°, 72°	#3654	construct_pentagonal_trig_model()
KNOTT-2.2	The Complete 3° Constructible Angle Lattice (3°k for k = 1..29)	#otherexact	construct_three_degree_lattice_model()
KNOTT-3.1	Three Orthogonal Golden Rectangles → Regular Icosahedron & Dodecahedron	#coords	construct_orthogonal_golden_rectangles_solids()
KNOTT-3.2	Inscribed Cube & Compound of Five Cubes in a Dodecahedron	#cubeinDod	construct_cube_in_dodecahedron_model()
KNOTT-3.3	Acute & Obtuse Golden Rhombohedra (3D Quasicrystal / Ammann Tiles)	#quasicrystalsPhi	construct_golden_rhombohedra_model()
KNOTT-4.1	Fibonacci Power Linearization (Φⁿ = FₙΦ + Fₙ₋₁) & The Phibonacci Brick	#powers	construct_phi_powers_model()
KNOTT-4.2	Extremal Continued Fraction [1; 1, 1, …], Convergents Fₙ₊₁/Fₙ, & Nested Radicals	#contfract	construct_convergents_model()
KNOTT-4.3	Bergman’s Base-Φ (‘Phigit’) Numeral System	#canonical	construct_phigit_scale_model()
KNOTT-5.1	The Infinite Golden String (101101011011…), Rabbit Tree & Cutting-Line Slope φ	#PhiPlot	construct_golden_string_cutting_model()
KNOTT-5.2	Zeckendorf’s Theorem (Canonical Fibonacci Base Representation)	#zeck	construct_zeckendorf_staircase_model()
KNOTT-5.3	Wythoff’s Array, Beatty’s Complementary Sequences (⌊nΦ⌋, ⌊nΦ²⌋), & Stolarsky’s Array	#wythoff	construct_wythoff_beatty_model()
KNOTT-6.1	Fibonacci Generator of Primitive Pythagorean Triples	#pythfib	construct_fibonacci_pythagorean_model()
KNOTT-6.2	Cassini’s Identity (Fₙ₋₁Fₙ₊₁ − Fₙ² = (−1)ⁿ) & The 64 ↔ 65 / 63 Jigsaw Dissections	#jigsaw1	construct_fibonacci_jigsaw_model()
KNOTT-6.3	Binet, Lucas, Catalan, d’Ocagne, Honsberger & Fibonomial Power Identities	#defs	construct_lucas_binet_model()
KNOTT-6.4	Fibonacci Arctangent Telescoping Formulae for π	#piandfib	construct_hayashi_arctan_grid_model()
KNOTT-6.5	Golden Angle (360° / Φ² ≈ 137.50776°) & Parastichy Bridges	#seeds	construct_parastichy_lattice_model()