[Math Lair] The Three Construction Problems of Antiquity

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"One of the unnoticed good effects of television is that people now watch it instead of producing pamphlets squaring the circle"
—Underwood Dudley, Mathematical Cranks

There are three famous geometric construction problems originally proposed by the ancient Greeks. They consist of performing the following constructions using only a straightedge and compass:

Duplicating the Cube
Construct a cube having twice the volume of a given cube. This involves constructing the line the cube root of 2 times the length of a given line segment.
Trisecting an Angle
Devise a method by which any angle may be trisected.
Squaring the Circle
Construct a square equal in area to a given circle (or the reverse).

Note that the straightedge and compass may only be used as provided for in the first three of the five postulates in Euclid's Elements. In other words, the straightedge may only be used to connect two points or to extend a line; the compass may only be used to create a circle given a centre and a point on the circumference. The ancient Greeks were unable to solve these problems. Most likely, they realized that they were impossible, as they had resorted to finding approximations to these tasks as well as ways of performing these tasks using other equipment or by using the compass and straightedge in non-Euclidean ways.

The verdict on whether these three problems could be solved at all was not handed down until relatively recently. Not until the 19th century were these problems proven to be impossible, and it was by algebraic methods. It can be shown that the only lengths that can be constructed by means of compass and straightedge are those that are successions of square roots applied to rational numbers. These numbers correspond to algebraic equations of even degree. The cube root of 2, however, is the root of a third-degree equation, and cannot be the solution of an equation of even degree. Similarly, trisecting an angle involves finding a solution to cos 3θ = 4 cos3 θ − 3 cos θ which boils down to another third-degree equation.

It is harder to prove that the third problem cannot be solved. The solution involved showing that π is a transcendental number. Since a transcendental number cannot be the solution of any algebraic equation, this construction too is impossible.