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).

Curated Source & Catalog Provenance

Catalog ID	KNOTT-2.2
Curated Source	Dr. Ron Knott, Exact Trig Values for Simple Angles (#otherexact)
Source Attribution	© 1996–2025 Dr. Ron Knott, University of Surrey — Explore Dr. Knott’s original exposition
Exact SymPy Validator	geometor.knott.trigonometry.construct_three_degree_lattice_model() (0 sp.Float atoms)
GEOMETOR Concordance	geometor/euclid, geometor/elements, geometor/wildberger

Constructive Synthesis & Exact Proof

Combining the pentagonal subfield \(\mathbb{Q}(\sqrt{5}, \sqrt{10 \pm 2\sqrt{5}})\) with the equilateral and half-right subfield \(\mathbb{Q}(\sqrt{2}, \sqrt{3})\) generates closed-form square-root radicals for all 29 acute multiples of \(3^\circ\) (KnottSimpleTrig).

• Euclid IV.16 Regular 15-Gon & 3° Generator: On unit circle (O, U), superimposing equilateral ray OP60 (\(60^\circ\)) and pentagonal ray OP72 (\(72^\circ\)) isolates the \(12^\circ\) regular pentadecagon arc, making chord P60P72 and origin-based chord UP12 equal. Equivalently, subtracting Ailles’ \(15^\circ\) ray OP15 from pentagonal ray OP18 (or bisecting OP12 through OP6 to OP3) yields the atomic constructible angle \(3^\circ = 18^\circ - 15^\circ\), whose chord P15P18 and unit-base chord UP3 are equal.

• Gauss’s Regular 17-Gon Bridge: While \(1^\circ\) cannot be constructed with compass and straightedge (requiring cube roots via angle trisection), Carl Friedrich Gauss (Gauss1801) proved that Fermat prime \(F_2 = 2^{2^2} + 1 = 17\) admits a four-step quadratic radical tower for \(\cos(\pi / 17)\), verified here symbolically via Chebyshev’s polynomial \(T_{17}(\cos(\pi / 17)) + 1 = 0\).

Complete 29-Angle 3° Constructible Lattice

Exact Radicals & Spreads for \(3^\circ k\) (\(k = 1\dots 29\))

Angle	\(\sin\theta\)	\(\cos\theta\)	Spread \(s(\theta) = \sin^2\theta\)
\(3^\circ\)	\(- \frac{\sqrt{6} \sqrt{2 \sqrt{5} + 10}}{16} - \frac{\sqrt{6}}{16} - \frac{\sqrt{2}}{16} + \frac{\sqrt{10}}{16} + \frac{\sqrt{2} \sqrt{2 \sqrt{5} + 10}}{16} + \frac{\sqrt{30}}{16}\)	\(- \frac{\sqrt{10}}{16} - \frac{\sqrt{6}}{16} + \frac{\sqrt{2}}{16} + \frac{\sqrt{2} \sqrt{2 \sqrt{5} + 10}}{16} + \frac{\sqrt{30}}{16} + \frac{\sqrt{6} \sqrt{2 \sqrt{5} + 10}}{16}\)	\(- \frac{\sqrt{5} \sqrt{2 \sqrt{5} + 10}}{32} - \frac{\sqrt{15}}{16} - \frac{\sqrt{3}}{16} + \frac{\sqrt{2 \sqrt{5} + 10}}{32} + \frac{1}{2}\)
\(6^\circ\)	\(- \frac{\sqrt{5}}{8} - \frac{1}{8} + \frac{\sqrt{3} \sqrt{10 - 2 \sqrt{5}}}{8}\)	\(\frac{\sqrt{3}}{8} + \frac{\sqrt{10 - 2 \sqrt{5}}}{8} + \frac{\sqrt{15}}{8}\)	\(- \frac{\sqrt{15} \sqrt{10 - 2 \sqrt{5}}}{32} - \frac{\sqrt{5}}{16} - \frac{\sqrt{3} \sqrt{10 - 2 \sqrt{5}}}{32} + \frac{9}{16}\)
\(9^\circ\)	\(- \frac{\sqrt{2} \sqrt{10 - 2 \sqrt{5}}}{8} + \frac{\sqrt{2}}{8} + \frac{\sqrt{10}}{8}\)	\(\frac{\sqrt{2}}{8} + \frac{\sqrt{10}}{8} + \frac{\sqrt{2} \sqrt{10 - 2 \sqrt{5}}}{8}\)	\(- \frac{\sqrt{5} \sqrt{10 - 2 \sqrt{5}}}{16} - \frac{\sqrt{10 - 2 \sqrt{5}}}{16} + \frac{1}{2}\)
\(12^\circ\)	\(- \frac{\sqrt{15}}{8} + \frac{\sqrt{3}}{8} + \frac{\sqrt{2 \sqrt{5} + 10}}{8}\)	\(- \frac{1}{8} + \frac{\sqrt{5}}{8} + \frac{\sqrt{3} \sqrt{2 \sqrt{5} + 10}}{8}\)	\(- \frac{\sqrt{15} \sqrt{2 \sqrt{5} + 10}}{32} - \frac{\sqrt{5}}{16} + \frac{\sqrt{3} \sqrt{2 \sqrt{5} + 10}}{32} + \frac{7}{16}\)
\(15^\circ\)	\(- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\)	\(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\)	\(\frac{1}{2} - \frac{\sqrt{3}}{4}\)
\(18^\circ\)	\(- \frac{1}{4} + \frac{\sqrt{5}}{4}\)	\(\frac{\sqrt{2 \sqrt{5} + 10}}{4}\)	\(\frac{3}{8} - \frac{\sqrt{5}}{8}\)
\(21^\circ\)	\(- \frac{\sqrt{30}}{16} - \frac{\sqrt{6}}{16} + \frac{\sqrt{2}}{16} + \frac{\sqrt{10}}{16} + \frac{\sqrt{2} \sqrt{10 - 2 \sqrt{5}}}{16} + \frac{\sqrt{6} \sqrt{10 - 2 \sqrt{5}}}{16}\)	\(- \frac{\sqrt{2} \sqrt{10 - 2 \sqrt{5}}}{16} + \frac{\sqrt{2}}{16} + \frac{\sqrt{6}}{16} + \frac{\sqrt{10}}{16} + \frac{\sqrt{30}}{16} + \frac{\sqrt{6} \sqrt{10 - 2 \sqrt{5}}}{16}\)	\(- \frac{\sqrt{15}}{16} - \frac{\sqrt{5} \sqrt{10 - 2 \sqrt{5}}}{32} - \frac{\sqrt{10 - 2 \sqrt{5}}}{32} + \frac{\sqrt{3}}{16} + \frac{1}{2}\)
\(24^\circ\)	\(- \frac{\sqrt{10 - 2 \sqrt{5}}}{8} + \frac{\sqrt{3}}{8} + \frac{\sqrt{15}}{8}\)	\(\frac{1}{8} + \frac{\sqrt{5}}{8} + \frac{\sqrt{3} \sqrt{10 - 2 \sqrt{5}}}{8}\)	\(- \frac{\sqrt{15} \sqrt{10 - 2 \sqrt{5}}}{32} - \frac{\sqrt{3} \sqrt{10 - 2 \sqrt{5}}}{32} + \frac{\sqrt{5}}{16} + \frac{7}{16}\)
\(27^\circ\)	\(- \frac{\sqrt{10}}{8} + \frac{\sqrt{2}}{8} + \frac{\sqrt{2} \sqrt{2 \sqrt{5} + 10}}{8}\)	\(- \frac{\sqrt{2}}{8} + \frac{\sqrt{10}}{8} + \frac{\sqrt{2} \sqrt{2 \sqrt{5} + 10}}{8}\)	\(- \frac{\sqrt{5} \sqrt{2 \sqrt{5} + 10}}{16} + \frac{\sqrt{2 \sqrt{5} + 10}}{16} + \frac{1}{2}\)
\(30^\circ\)	\(\frac{1}{2}\)	\(\frac{\sqrt{3}}{2}\)	\(\frac{1}{4}\)
\(33^\circ\)	\(- \frac{\sqrt{2} \sqrt{2 \sqrt{5} + 10}}{16} - \frac{\sqrt{6}}{16} - \frac{\sqrt{2}}{16} + \frac{\sqrt{10}}{16} + \frac{\sqrt{30}}{16} + \frac{\sqrt{6} \sqrt{2 \sqrt{5} + 10}}{16}\)	\(- \frac{\sqrt{30}}{16} - \frac{\sqrt{2}}{16} + \frac{\sqrt{6}}{16} + \frac{\sqrt{10}}{16} + \frac{\sqrt{2} \sqrt{2 \sqrt{5} + 10}}{16} + \frac{\sqrt{6} \sqrt{2 \sqrt{5} + 10}}{16}\)	\(- \frac{\sqrt{15}}{16} - \frac{\sqrt{2 \sqrt{5} + 10}}{32} - \frac{\sqrt{3}}{16} + \frac{\sqrt{5} \sqrt{2 \sqrt{5} + 10}}{32} + \frac{1}{2}\)
\(36^\circ\)	\(\frac{\sqrt{10 - 2 \sqrt{5}}}{4}\)	\(\frac{1}{4} + \frac{\sqrt{5}}{4}\)	\(\frac{5}{8} - \frac{\sqrt{5}}{8}\)
\(39^\circ\)	\(- \frac{\sqrt{6} \sqrt{10 - 2 \sqrt{5}}}{16} + \frac{\sqrt{2}}{16} + \frac{\sqrt{6}}{16} + \frac{\sqrt{10}}{16} + \frac{\sqrt{2} \sqrt{10 - 2 \sqrt{5}}}{16} + \frac{\sqrt{30}}{16}\)	\(- \frac{\sqrt{10}}{16} - \frac{\sqrt{2}}{16} + \frac{\sqrt{6}}{16} + \frac{\sqrt{2} \sqrt{10 - 2 \sqrt{5}}}{16} + \frac{\sqrt{30}}{16} + \frac{\sqrt{6} \sqrt{10 - 2 \sqrt{5}}}{16}\)	\(- \frac{\sqrt{5} \sqrt{10 - 2 \sqrt{5}}}{32} - \frac{\sqrt{3}}{16} - \frac{\sqrt{10 - 2 \sqrt{5}}}{32} + \frac{\sqrt{15}}{16} + \frac{1}{2}\)
\(42^\circ\)	\(- \frac{\sqrt{5}}{8} + \frac{1}{8} + \frac{\sqrt{3} \sqrt{2 \sqrt{5} + 10}}{8}\)	\(- \frac{\sqrt{3}}{8} + \frac{\sqrt{2 \sqrt{5} + 10}}{8} + \frac{\sqrt{15}}{8}\)	\(- \frac{\sqrt{15} \sqrt{2 \sqrt{5} + 10}}{32} + \frac{\sqrt{5}}{16} + \frac{\sqrt{3} \sqrt{2 \sqrt{5} + 10}}{32} + \frac{9}{16}\)
\(45^\circ\)	\(\frac{\sqrt{2}}{2}\)	\(\frac{\sqrt{2}}{2}\)	\(\frac{1}{2}\)
\(48^\circ\)	\(- \frac{\sqrt{3}}{8} + \frac{\sqrt{2 \sqrt{5} + 10}}{8} + \frac{\sqrt{15}}{8}\)	\(- \frac{\sqrt{5}}{8} + \frac{1}{8} + \frac{\sqrt{3} \sqrt{2 \sqrt{5} + 10}}{8}\)	\(- \frac{\sqrt{3} \sqrt{2 \sqrt{5} + 10}}{32} - \frac{\sqrt{5}}{16} + \frac{7}{16} + \frac{\sqrt{15} \sqrt{2 \sqrt{5} + 10}}{32}\)
\(51^\circ\)	\(- \frac{\sqrt{10}}{16} - \frac{\sqrt{2}}{16} + \frac{\sqrt{6}}{16} + \frac{\sqrt{2} \sqrt{10 - 2 \sqrt{5}}}{16} + \frac{\sqrt{30}}{16} + \frac{\sqrt{6} \sqrt{10 - 2 \sqrt{5}}}{16}\)	\(- \frac{\sqrt{6} \sqrt{10 - 2 \sqrt{5}}}{16} + \frac{\sqrt{2}}{16} + \frac{\sqrt{6}}{16} + \frac{\sqrt{10}}{16} + \frac{\sqrt{2} \sqrt{10 - 2 \sqrt{5}}}{16} + \frac{\sqrt{30}}{16}\)	\(- \frac{\sqrt{15}}{16} + \frac{\sqrt{10 - 2 \sqrt{5}}}{32} + \frac{\sqrt{3}}{16} + \frac{\sqrt{5} \sqrt{10 - 2 \sqrt{5}}}{32} + \frac{1}{2}\)
\(54^\circ\)	\(\frac{1}{4} + \frac{\sqrt{5}}{4}\)	\(\frac{\sqrt{10 - 2 \sqrt{5}}}{4}\)	\(\frac{\sqrt{5}}{8} + \frac{3}{8}\)
\(57^\circ\)	\(- \frac{\sqrt{30}}{16} - \frac{\sqrt{2}}{16} + \frac{\sqrt{6}}{16} + \frac{\sqrt{10}}{16} + \frac{\sqrt{2} \sqrt{2 \sqrt{5} + 10}}{16} + \frac{\sqrt{6} \sqrt{2 \sqrt{5} + 10}}{16}\)	\(- \frac{\sqrt{2} \sqrt{2 \sqrt{5} + 10}}{16} - \frac{\sqrt{6}}{16} - \frac{\sqrt{2}}{16} + \frac{\sqrt{10}}{16} + \frac{\sqrt{30}}{16} + \frac{\sqrt{6} \sqrt{2 \sqrt{5} + 10}}{16}\)	\(- \frac{\sqrt{5} \sqrt{2 \sqrt{5} + 10}}{32} + \frac{\sqrt{3}}{16} + \frac{\sqrt{2 \sqrt{5} + 10}}{32} + \frac{\sqrt{15}}{16} + \frac{1}{2}\)
\(60^\circ\)	\(\frac{\sqrt{3}}{2}\)	\(\frac{1}{2}\)	\(\frac{3}{4}\)
\(63^\circ\)	\(- \frac{\sqrt{2}}{8} + \frac{\sqrt{10}}{8} + \frac{\sqrt{2} \sqrt{2 \sqrt{5} + 10}}{8}\)	\(- \frac{\sqrt{10}}{8} + \frac{\sqrt{2}}{8} + \frac{\sqrt{2} \sqrt{2 \sqrt{5} + 10}}{8}\)	\(- \frac{\sqrt{2 \sqrt{5} + 10}}{16} + \frac{1}{2} + \frac{\sqrt{5} \sqrt{2 \sqrt{5} + 10}}{16}\)
\(66^\circ\)	\(\frac{1}{8} + \frac{\sqrt{5}}{8} + \frac{\sqrt{3} \sqrt{10 - 2 \sqrt{5}}}{8}\)	\(- \frac{\sqrt{10 - 2 \sqrt{5}}}{8} + \frac{\sqrt{3}}{8} + \frac{\sqrt{15}}{8}\)	\(- \frac{\sqrt{5}}{16} + \frac{\sqrt{3} \sqrt{10 - 2 \sqrt{5}}}{32} + \frac{\sqrt{15} \sqrt{10 - 2 \sqrt{5}}}{32} + \frac{9}{16}\)
\(69^\circ\)	\(- \frac{\sqrt{2} \sqrt{10 - 2 \sqrt{5}}}{16} + \frac{\sqrt{2}}{16} + \frac{\sqrt{6}}{16} + \frac{\sqrt{10}}{16} + \frac{\sqrt{30}}{16} + \frac{\sqrt{6} \sqrt{10 - 2 \sqrt{5}}}{16}\)	\(- \frac{\sqrt{30}}{16} - \frac{\sqrt{6}}{16} + \frac{\sqrt{2}}{16} + \frac{\sqrt{10}}{16} + \frac{\sqrt{2} \sqrt{10 - 2 \sqrt{5}}}{16} + \frac{\sqrt{6} \sqrt{10 - 2 \sqrt{5}}}{16}\)	\(- \frac{\sqrt{3}}{16} + \frac{\sqrt{10 - 2 \sqrt{5}}}{32} + \frac{\sqrt{5} \sqrt{10 - 2 \sqrt{5}}}{32} + \frac{\sqrt{15}}{16} + \frac{1}{2}\)
\(72^\circ\)	\(\frac{\sqrt{2 \sqrt{5} + 10}}{4}\)	\(- \frac{1}{4} + \frac{\sqrt{5}}{4}\)	\(\frac{\sqrt{5}}{8} + \frac{5}{8}\)
\(75^\circ\)	\(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\)	\(- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\)	\(\frac{\sqrt{3}}{4} + \frac{1}{2}\)
\(78^\circ\)	\(- \frac{1}{8} + \frac{\sqrt{5}}{8} + \frac{\sqrt{3} \sqrt{2 \sqrt{5} + 10}}{8}\)	\(- \frac{\sqrt{15}}{8} + \frac{\sqrt{3}}{8} + \frac{\sqrt{2 \sqrt{5} + 10}}{8}\)	\(- \frac{\sqrt{3} \sqrt{2 \sqrt{5} + 10}}{32} + \frac{\sqrt{5}}{16} + \frac{\sqrt{15} \sqrt{2 \sqrt{5} + 10}}{32} + \frac{9}{16}\)
\(81^\circ\)	\(\frac{\sqrt{2}}{8} + \frac{\sqrt{10}}{8} + \frac{\sqrt{2} \sqrt{10 - 2 \sqrt{5}}}{8}\)	\(- \frac{\sqrt{2} \sqrt{10 - 2 \sqrt{5}}}{8} + \frac{\sqrt{2}}{8} + \frac{\sqrt{10}}{8}\)	\(\frac{\sqrt{10 - 2 \sqrt{5}}}{16} + \frac{\sqrt{5} \sqrt{10 - 2 \sqrt{5}}}{16} + \frac{1}{2}\)
\(84^\circ\)	\(\frac{\sqrt{3}}{8} + \frac{\sqrt{10 - 2 \sqrt{5}}}{8} + \frac{\sqrt{15}}{8}\)	\(- \frac{\sqrt{5}}{8} - \frac{1}{8} + \frac{\sqrt{3} \sqrt{10 - 2 \sqrt{5}}}{8}\)	\(\frac{\sqrt{3} \sqrt{10 - 2 \sqrt{5}}}{32} + \frac{\sqrt{5}}{16} + \frac{\sqrt{15} \sqrt{10 - 2 \sqrt{5}}}{32} + \frac{7}{16}\)
\(87^\circ\)	\(- \frac{\sqrt{10}}{16} - \frac{\sqrt{6}}{16} + \frac{\sqrt{2}}{16} + \frac{\sqrt{2} \sqrt{2 \sqrt{5} + 10}}{16} + \frac{\sqrt{30}}{16} + \frac{\sqrt{6} \sqrt{2 \sqrt{5} + 10}}{16}\)	\(- \frac{\sqrt{6} \sqrt{2 \sqrt{5} + 10}}{16} - \frac{\sqrt{6}}{16} - \frac{\sqrt{2}}{16} + \frac{\sqrt{10}}{16} + \frac{\sqrt{2} \sqrt{2 \sqrt{5} + 10}}{16} + \frac{\sqrt{30}}{16}\)	\(- \frac{\sqrt{2 \sqrt{5} + 10}}{32} + \frac{\sqrt{3}}{16} + \frac{\sqrt{15}}{16} + \frac{\sqrt{5} \sqrt{2 \sqrt{5} + 10}}{32} + \frac{1}{2}\)

Machine-Verified Exact Symbolic Invariants

Invariant	Exact Symbolic Value
lattice count	\(29\)
cos 3 deg	\(- \frac{\sqrt{10}}{16} - \frac{\sqrt{6}}{16} + \frac{\sqrt{2}}{16} + \frac{\sqrt{2} \sqrt{2 \sqrt{5} + 10}}{16} + \frac{\sqrt{30}}{16} + \frac{\sqrt{6} \sqrt{2 \sqrt{5} + 10}}{16}\)
sin 3 deg	\(- \frac{\sqrt{6} \sqrt{2 \sqrt{5} + 10}}{16} - \frac{\sqrt{6}}{16} - \frac{\sqrt{2}}{16} + \frac{\sqrt{10}}{16} + \frac{\sqrt{2} \sqrt{2 \sqrt{5} + 10}}{16} + \frac{\sqrt{30}}{16}\)
spread 3 deg	\(- \frac{\sqrt{5} \sqrt{2 \sqrt{5} + 10}}{32} - \frac{\sqrt{15}}{16} - \frac{\sqrt{3}}{16} + \frac{\sqrt{2 \sqrt{5} + 10}}{32} + \frac{1}{2}\)
spread 12 deg	\(- \frac{\sqrt{15} \sqrt{2 \sqrt{5} + 10}}{32} - \frac{\sqrt{5}}{16} + \frac{\sqrt{3} \sqrt{2 \sqrt{5} + 10}}{32} + \frac{7}{16}\)
spread 15 deg	\(\frac{1}{2} - \frac{\sqrt{3}}{4}\)
fifteen gon chord quadrance	\(- \frac{\sqrt{3} \sqrt{2 \sqrt{5} + 10}}{4} - \frac{\sqrt{5}}{4} + \frac{9}{4}\)
cos pi over 17	\(\sqrt{\frac{\sqrt{17}}{32} + \frac{\sqrt{34 - 2 \sqrt{17}}}{32} + \frac{\sqrt{- 2 \sqrt{2 \sqrt{17} + 34} - \sqrt{34 - 2 \sqrt{17}} + 3 \sqrt{17} + 17}}{16} + \frac{15}{32}}\)