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.
Curated from Dr. Ron Knott’s University of Surrey archive (© 1996–2025 Dr. Ron Knott, University of Surrey).
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-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-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-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-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-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.
Collection Concordance¶
Catalog ID |
GEOMETOR Catalog Entry |
Curated Source Anchor |
|---|---|---|
Internal & External Golden Section Constructions on a Segment |
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Golden Rectangles, √5 Rectangles, & Whirling-Square Gnomonic Spirals |
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Golden Triangles (72°–72°–36°) & Golden Gnomons (36°–36°–108°) |
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Regular Pentagon, Pentagram, Decagon & Paper-Knot Pentagon |
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Kepler’s Golden Right Triangle (1, √Φ, Φ) & Circle-Square Near-Quadrature |
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Penrose Kites & Darts (P2) and Thick & Thin Golden Rhombs (P3) |