Angle trisection with compass and straightedge

Solved, 1837GeometryAlgebra

Statement

Given an arbitrary angle $\theta$ , construct in finitely many steps using only an idealized compass and an unmarked straightedge two rays that divide $\theta$ into three equal angles of measure $\theta/3$ .

Proved impossible by Pierre Laurent Wantzel in 1837. Wantzel showed that a real number $\alpha$ is constructible by compass and straightedge from a unit segment if and only if $\alpha$ lies in a tower of quadratic field extensions of $\mathbb{Q}$ , which implies that the degree $[\mathbb{Q}(\alpha):\mathbb{Q}]$ of its minimal polynomial must be a power of $2$ . For $\theta = 60^\circ$ , the triple-angle identity $\cos(3\alpha) = 4\cos^3(\alpha) - 3\cos(\alpha)$ with $\cos(60^\circ) = 1/2$ shows that $x = 2\cos(20^\circ)$ satisfies $x^3 - 3x - 1 = 0$ . Since this cubic polynomial has no rational roots, it is irreducible over $\mathbb{Q}$ , so $[\mathbb{Q}(\cos 20^\circ):\mathbb{Q}] = 3 \neq 2^k$ , proving that $60^\circ$ cannot be trisected.

Although no general compass-and-straightedge trisection exists, special angles such as $90^\circ$ or $45^\circ$ can be trisected because $30^\circ$ and $15^\circ$ are constructible; more generally, a rational multiple of $\pi$ of the form $2\pi/n$ can be trisected if and only if $3$ does not divide $n$ or the regular $3n$ -gon is constructible by the Gauss–Wantzel theorem. Allowing richer tools restores general trisectability: Archimedes's marked-ruler neusis, intersections of conics (Pappus of Alexandria), mechanical linkages, or Huzita–Justin Axiom 6 in mathematical origami (which folds two points onto two lines simultaneously and solves arbitrary cubic equations) can trisect any given angle.

References

Pierre Laurent Wantzel (1837). Recherches sur les moyens de reconnaître si un problème de géométrie peut se résoudre avec la règle et le compas
Ian Stewart (2015). Galois Theory · DOI:10.1201/b18187
David S. Richeson (2019). Tales of Impossibility: The 2000-Year Quest to Solve the Mathematical Problems of Antiquity

History of the problem

430 BCEHippias of Elis invents a transcendental mechanical curve, the quadratrix (trisectrix), to trisect arbitrary angles after Greek geometers find that straightedge and compass alone do not yield a general construction.
250 BCEArchimedes shows that any angle can be trisected by a neusis construction using a straightedge with two marked points separated by the radius of a circle.Archimedes of Syracuse
1593François Viète connects angle trisection and neusis constructions to the trigonometric solution of irreducible cubic equations with three real roots (casus irreducibilis).
1837Pierre Laurent Wantzel publishes the first rigorous proof that an arbitrary angle cannot be trisected with an unmarked straightedge and compass, showing that $60^\circ$ leads to an irreducible cubic equation.