Angle Trisection - Proof of Impossibility

Proof of Impossibility

The problem of constructing an angle of a given measure is equivalent to construct two segments such the ratio of their length is because one may pass from one solution to the other by a compass and straightedge construction. It follows that, given a segment that is sought as having a unit length, the problem of angle trisection is equivalent to construct a segment whose length is the root of a cubic polynomial — since by the triple-angle formula, This allows to reduce the original geometric problem to a purely algebraic problem.

One can show that every rational number is constructible and that every irrational number which is constructible in one step from some given numbers is a solution of a polynomial of degree 2 with coefficients in the field generated by these numbers. Therefore any number which is constructible by a series of steps is a root of a minimal polynomial whose degree is a power of 2. Note also that radians (60 degrees, written 60°) is constructible. We now show that it is impossible to construct a 20° angle; this implies that a 60° angle cannot be trisected, and thus that an arbitrary angle cannot be trisected.

Denote the set of rational numbers by Q. If 60° could be trisected, the degree of a minimal polynomial of cos(20°) over Q would be a power of two. Now let y = cos(20°).

Note that cos(60°). Then by the triple-angle formula, and so . Thus, or equivalently . Now substitute, so that . Let .

The minimal polynomial for x (hence cos(20°)) is a factor of . Because is degree 3, if it is reducible over by Q then it has a rational root. By the rational root theorem, this root must be 1 or −1, but both are clearly not roots. Therefore is irreducible over by Q, and the minimal polynomial for cos(20°) is of degree 3.

So an angle of 60° = (1/3)π radians cannot be trisected.

Many people (who presumably are unaware of the above result, misunderstand it, or incorrectly reject it) have proposed methods of trisecting the general angle. Some of these methods provide reasonable approximations; others (some of which are mentioned below) involve tools not permitted in the classical problem. The mathematician Underwood Dudley has detailed some of these failed attempts in his book The Trisectors.