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%.

Curated Source & Catalog Provenance

Catalog ID	KNOTT-1.5
Curated Source	Dr. Ron Knott, Phi in 2D Geometry and Trigonometry (#PhiRightAng)
Source Attribution	© 1996–2025 Dr. Ron Knott, University of Surrey — Explore Dr. Knott’s original exposition
Exact SymPy Validator	geometor.knott.flat_geometry.construct_kepler_triangle() (0 sp.Float atoms)
GEOMETOR Concordance	geometor/wildberger, geometor/euclid, geometor/divine

Constructive Synthesis & Exact Proof

In an October 1597 letter to Michael Mästlin, Johannes Kepler combined the Pythagorean theorem with the Golden Ratio to isolate the unique right triangle whose sides form a geometric progression (KnottPhi2D).

• Euclidean Geometric-Mean Construction: Along diameter segment from D \((-1, 0)\) through B \((0, 0)\) to E \((\Phi, 0)\), Thales semicircle (O, E) cuts the vertical axis at C \((0, \sqrt{\Phi})\) (Elements VI.13). In Kepler triangle DBC, short leg BD, long leg BC, and hypotenuse DC (which is equal to golden segment BE, joined by chord CE) have exact quadrances \((1, \Phi, \Phi^2)\) satisfying both \(Q_{BC}^2 = Q_{BD} Q_{DC}\) and \(1 + \Phi = \Phi^2\).

• Coincidence of Cosine & Tangent and Near-Quadrature: At acute apex C, \(\cos\theta_C = \tan\theta_C = \sin\theta_D = 1/\sqrt{\Phi}\) (\(\cos^2\theta_C = \tan^2\theta_C = \phi\)). Erecting square BGFC (via F and G) of perimeter \(4\sqrt{\Phi} \approx 5.08808\) alongside circle (M, C) centered at hypotenuse midpoint M of circumference \(\pi\Phi \approx 5.08320\) exhibits the celebrated circle–square perimeter near-quadrature highlighted by Dr. Knott (\(4 / (\pi\sqrt{\Phi}) \approx 1.000959\), within \(0.096\%\)).

Machine-Verified Exact Symbolic Invariants

Invariant	Exact Symbolic Value
Q leg short	\(1\)
Q leg long	\(\frac{1}{2} + \frac{\sqrt{5}}{2}\)
Q hypotenuse	\(\frac{\sqrt{5}}{2} + \frac{3}{2}\)
spread acute D	\(\frac{1}{\sqrt{5} + 3} + \frac{\sqrt{5}}{\sqrt{5} + 3}\)
spread apex C	\(\frac{3 - \sqrt{5}}{2}\)
cos sq equals tan sq C	\(- \frac{1}{2} + \frac{\sqrt{5}}{2}\)
square perimeter	\(4 \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}\)
circle circumference	\(\pi \left(\frac{1}{2} + \frac{\sqrt{5}}{2}\right)\)
quadrature ratio	\(\frac{4}{\pi \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}\)