Doubling the cube

Solved, 1837GeometryAlgebra

Statement

Given the edge of a cube of side length $a$ , construct in finitely many steps using only an idealized compass and an unmarked straightedge the edge length $x = \sqrt[3]{2}\,a$ of a second cube whose volume $x^3 = 2a^3$ is twice that of the given cube.

Proved impossible by Pierre Laurent Wantzel in 1837. Setting $a = 1$ , doubling the cube requires constructing the real number $x = \sqrt[3]{2}$ , which is a root of the polynomial $x^3 - 2 = 0$ . By Eisenstein's criterion for the prime $p = 2$ (or rational root inspection), $x^3 - 2$ is irreducible over $\mathbb{Q}$ , so the field extension $[\mathbb{Q}(\sqrt[3]{2}):\mathbb{Q}]$ has degree $3$ . Wantzel proved that any length constructible with an unmarked straightedge and compass lies in an iterated quadratic extension of $\mathbb{Q}$ of degree $2^k$ ; since $3$ does not divide $2^k$ , $\sqrt[3]{2}$ is not constructible.

While impossible with an unmarked straightedge and compass alone, $\sqrt[3]{2}$ can be constructed exactly when the allowed operations are extended to solve cubic equations. Classical Greek geometers gave solutions using three-dimensional intersecting cylinders, cones, and tori (Archytas of Tarentum, c. 400 BCE), conic sections (Menaechmus), the cissoid of Diocles, the conchoid of Nicomedes, or a marked straightedge (neusis). In modern origami mathematics, Peter Messer (1986) showed that folding a square sheet of paper divided into three equal strips constructs $\sqrt[3]{2}$ directly as a ratio of segments along the sheet's edge.

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
Thomas Little Heath (1921). A History of Greek Mathematics, Volume 1: From Thales to Euclid
David S. Richeson (2019). Tales of Impossibility: The 2000-Year Quest to Solve the Mathematical Problems of Antiquity

History of the problem

430 BCEAccording to Greek tradition recounted by Eratosthenes, citizens of Delos consult the oracle at Delphi during a plague and are told to double the size of Apollo's cubical altar; Hippocrates of Chios reduces the problem to finding two mean proportionals $x, y$ satisfying $a : x = x : y = y : 2a$ .
350 BCEMenaechmus discovers conic sections while solving Hippocrates's proportion $a/x = x/y = y/(2a)$ , constructing $\sqrt[3]{2}\,a$ as the intersection of the parabolas $x^2 = ay$ and $y^2 = 2ax$ (or a parabola and the rectangular hyperbola $xy = 2a^2$ ).
1637René Descartes publishes La Géométrie, translating geometric constructions into algebraic equations and classifying doubling the cube and angle trisection as "solid" problems of degree $3$ solvable by intersecting a circle with a fixed parabola.René Descartes
1837Pierre Laurent Wantzel proves that every length constructible with compass and straightedge has algebraic degree $2^k$ over $\mathbb{Q}$ , rigorously establishing that $\sqrt[3]{2}$ cannot be constructed.