Part I, Chapter II — Critical & Symbolic Notes¶
Klein’s Monodromy Method vs. Wantzel’s Real Cubic $4u^3 - 3u = \cos\varphi$
Function-Theoretic Monodromy vs. Real Trigonometric Cubics¶
Characteristically for the author of Lectures on the Icosahedron (1884) and the Erlangen Program, Klein proves the impossibility of angle trisection in § 3 via Riemann-surface monodromy: as $z = e^{i\varphi}$ winds once around the unit circle, the algebraic function $z^{1/3}$ undergoes a 3-cycle $(x_1\; x_2\; x_3)$, immediately proving irreducibility over $\mathbb{C}(\cos\varphi, \sin\varphi)$ without computing a single trigonometric identity!
To complement Klein’s generic function-field proof with Wantzel's (1837) concrete numerical counterexample over $\mathbb{Q}$, apply the triple-angle cosine identity $\cos\varphi = 4\cos^3(\varphi/3) - 3\cos(\varphi/3)$ at $\varphi = \frac{\pi}{3} = 60^\circ$ ($\cos 60^\circ = \frac{1}{2}$). Setting $u = 2\cos 20^\circ$ yields
By the Rational Root Theorem, the only candidate rational roots of $u^3 - 3u - 1 = 0$ are $u = \pm 1$, neither of which vanishes ($(\pm 1)^3 - 3(\pm 1) - 1 \in \{-3, +1\}$). Hence $u^3 - 3u - 1$ is irreducible over $\mathbb{Q}$ of degree $3 \neq 2^h$, proving that even the specific constructible angle $60^\circ$ cannot be trisected with straightedge and compasses. Both x**3 - 2 and u**3 - 3*u - 1 are verified symbolically in geometor.klein.constructibility.analyze_constructibility().
Cross-Corpus Links in GEOMETOR¶
Pappus’s Plane, Solid, and Linear Classification (`geometor/pappus`): In Synagoge Books III & IV, Pappus reproved the failure of planar geometers who tried to solve solid problems (cube duplication and angle trisection) by plane loci (lines and circles) instead of conic sections. Klein’s Chapter II supplies the exact algebraic proof that Pappus’s ancient intuition lacked (and Chapter V exhibits the exact conic and conchoid/cissoid constructions).
Euclid’s Angle Bisection vs. Trisection (`geometor/euclid`): Compare quadratic angle bisection (Elements I.9, solving $2\cos^2(\varphi/2) - 1 = \cos\varphi$ of degree $2^1$) with cubic trisection of degree $3$.