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.

Curated Source & Catalog Provenance

Catalog ID	KNOTT-1.3
Curated Source	Dr. Ron Knott, Phi in 2D Geometry and Trigonometry (#isoc)
Source Attribution	© 1996–2025 Dr. Ron Knott, University of Surrey — Explore Dr. Knott’s original exposition
Exact SymPy Validator	geometor.knott.flat_geometry.construct_golden_triangles_and_gnomons() (0 sp.Float atoms)
GEOMETOR Concordance	geometor/euclid, geometor/elements, geometor/wildberger, geometor/divine

Constructive Synthesis & Exact Proof

In Elements IV.10 (EuclidElements), Euclid constructs an isosceles triangle having each base angle double the apex angle; Dr. Ron Knott (KnottPhi2D) develops its paired acute/obtuse gnomon recursion as the backbone of pentagonal geometry.

• Acute Golden Triangle & Bisector Deflation: Over unit base AB, sweeping radius \(\Phi\) from A and B fixes apex C of acute golden triangle ABC, whose legs AC and BC are equal with quadrance \(Q = \Phi^2 = \Phi + 1\). Intersecting leg BC with unit circle (A, B) at D makes bisector AD and base AB equal (\(AD = AB = CD = 1\)), leaving \(BD = \Phi - 1 = \phi\).

• Self-Similar Cascade & Exact Spreads: Triangle ABC splits into obtuse golden gnomon \(\triangle CAD\) (\(36^\circ\text{–}36^\circ\text{–}108^\circ\)) and daughter acute golden triangle ABD of area ratio \(\Phi : 1\). Sweeping circle (D, B) onto AC at E (DE \(= \phi\)) repeats the deflation ad infinitum, with Norman J. Wildberger’s exact pentagonal spreads \(s(36^\circ) = (5 - \sqrt{5})/8\) and \(s(72^\circ) = (5 + \sqrt{5})/8\) (Wildberger2005).

Machine-Verified Exact Symbolic Invariants

Invariant	Exact Symbolic Value
Q AC	\(\frac{\sqrt{5}}{2} + \frac{3}{2}\)
Q AD	\(1\)
Q BD	\(\frac{3}{2} - \frac{\sqrt{5}}{2}\)
Q BE	\(\frac{7}{2} - \frac{3 \sqrt{5}}{2}\)
spread 36 deg	\(\frac{\sqrt{5}}{3 \sqrt{5} + 7} + \frac{5}{6 \sqrt{5} + 14}\)
spread 72 deg	\(\frac{\sqrt{5}}{\sqrt{5} + 3} + \frac{5}{2 \sqrt{5} + 6}\)
area ratio ABC to DAB	\(\frac{2 \sqrt{2 \sqrt{5} + 5}}{- \sqrt{5} \sqrt{2 \sqrt{5} + 5} + 3 \sqrt{2 \sqrt{5} + 5}}\)