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.
Curated from Dr. Ron Knott’s University of Surrey archive (© 1996–2025 Dr. Ron Knott, University of Surrey).
KNOTT-2.1 — Exact Radical Sines, Cosines, Tangents & Spreads of 18°, 36°, 54°, 72°
On the unit circle, cos 36° = sin 54° = Φ/2 and sin 18° = cos 72° = φ/2 differ by exactly 1/2 and multiply to 1/4, while the pentagonal spreads s(36°) = (5 − √5)/8 and s(72°) = (5 + √5)/8 form the quadratic roots of Wildberger’s five-fold spread factor 16 s² − 20 s + 5 = 0.
KNOTT-2.2 — The Complete 3° Constructible Angle Lattice (3°k for k = 1..29)
Superimposing pentagonal angles in Q(√5) with equilateral and half-right angles in Q(√2, √3) expresses sin(3°k), cos(3°k), tan(3°k), and spread s(3°k) in explicit nested square roots for all 29 acute multiples of 3°, while Gauss’s Fermat prime 17 = 2⁴ + 1 extends compass constructibility to cos(π/17).
Collection Concordance¶
Catalog ID |
GEOMETOR Catalog Entry |
Curated Source Anchor |
|---|---|---|
Exact Radical Sines, Cosines, Tangents & Spreads of 18°, 36°, 54°, 72° |
||
The Complete 3° Constructible Angle Lattice (3°k for k = 1..29) |