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.

Curated Source & Catalog Provenance

Catalog ID	KNOTT-1.6
Curated Source	Dr. Ron Knott, Phi in 2D Geometry and Trigonometry (#penrose)
Source Attribution	© 1996–2025 Dr. Ron Knott, University of Surrey — Explore Dr. Knott’s original exposition
Exact SymPy Validator	geometor.knott.flat_geometry.construct_penrose_tiles() (0 sp.Float atoms)
GEOMETOR Concordance	geometor/divine, geometor/wildberger

Constructive Synthesis & Exact Proof

Roger Penrose (Penrose1974, Gardner1977) reduced aperiodic five-fold planar tiling to two golden prototile families cataloged by Dr. Ron Knott (KnottPhi2D): P2 (John Conway’s kite and dart) and P3 (thick and thin golden rhombs).

• P2 Kite & Dart Dissection: Inside thick \(72^\circ\text{–}108^\circ\) rhomb \(ABCD\) whose four sides AB, BC, CD, and DA are equal of length \(\Phi\), long diagonal AC has length \(\Phi^2 = \Phi + 1\). Marking K along AC so AK, BK, and DK are equal to \(1\) (leaving kite spine KC \(= \Phi\)) partitions the rhomb into convex kite KBCD (two acute golden triangles) and non-convex dart ABKD (two obtuse golden gnomons) in exact area ratio \(\Phi : 1\).

• P3 Thick & Thin Golden Rhombs: Adjoining thin \(36^\circ\text{–}144^\circ\) rhomb AFEB along shared edge AB (with edges AF, FE, and EB of length \(\Phi\)) yields interior spreads \(s(72^\circ) = (5+\sqrt{5})/8\) vs. \(s(36^\circ) = (5-\sqrt{5})/8\) and the identical area ratio \(\sin 72^\circ / \sin 36^\circ = 2\cos 36^\circ = \Phi\).

Machine-Verified Exact Symbolic Invariants

Invariant	Exact Symbolic Value
Q rhomb side	\(\frac{\sqrt{5}}{2} + \frac{3}{2}\)
Q dart short edge	\(1\)
Q kite spine KC	\(\frac{\sqrt{5}}{2} + \frac{3}{2}\)
kite to dart area ratio	\(\frac{1}{2} + \frac{\sqrt{5}}{2}\)
thick to thin rhomb area ratio	\(\frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{\sqrt{10} \sqrt{5 - \sqrt{5}} + 3 \sqrt{2} \sqrt{5 - \sqrt{5}}} + \frac{3 \sqrt{2} \sqrt{\sqrt{5} + 5}}{\sqrt{10} \sqrt{5 - \sqrt{5}} + 3 \sqrt{2} \sqrt{5 - \sqrt{5}}}\)
spread thick 72	\(\frac{\sqrt{5}}{\sqrt{5} + 3} + \frac{5}{2 \sqrt{5} + 6}\)
spread thin 36	\(\frac{\sqrt{5}}{4 \sqrt{5} + 12} + \frac{5}{4 \sqrt{5} + 12}\)