Compass and straightedge

Compass and straightedge constructions played an important role in the history of mathematics. Some constructions accomplished by the ancients led to revolutionary developments in abstract mathematics. Other construction problems posed in antiquity remain unsolved even today.

Description

A compass-and-straightedge construction is a diagram drawn freehand with only the aid of a compass and a ruler from which the markings have been erased, also known as a "straight edge". Part of the finished diagram should display the solution to a particular given problem. For example, if the problem is "Trisect a line," the solution might consist of the given line with three Xs constructed over it at even intervals.

The art of compass-and-straightedge construction was invented by the Ancient Greeks in the centuries before Christ. The Greeks were not only excellent mathematicians, but also accomplished navigators — so it perhaps seemed natural to them that their mathematical drawings should involve the use of compasses. The straightedge was a later embellishment; straight edges were plentiful in the ancient world (for example: sword edges, oars, and the foundations of buildings), and the concept of a straight edge was pleasing and aesthetic to the ancient Greek mind. Many popular constructions were collected in the Elements of Euclid.

In modern times, compass-and-straightedge constructions were rediscovered by Euler, who resurrected them from the pages of his namesake's Elements and found that their simple geometric truths were a pleasant diversion from the calculus wars then raging between the Newtonian British and the Leibnizian German mathematical communities. In the twentieth century, compass-and-straightedge methods have become a popular form of recreational mathematics, and the technique is taught as a one-week unit in many middle school math classes.

Classical unsolved problems

The Ancident Greeks solved all their problems with compass and straightedge. There were, however, three problems which they could not solve: doubling the cube, squaring the circle, and trisecting the angle. Pythagoras himself was said to have remarked that trisecting the angle was the hardest thing he had ever attempted. Archimedes attempted a "rationalistic" approach to trisecting the angle, in which he repeatedly bisected the angle until the number of divisions became a multiple of 3; however, this does not work for most angles, and in any event it is only physically possible to bisect most angles seven or eight times, so Archimedes' approach does not scale.

These problems can't be solved with compass and straightedge; the analysis can be done with the help of Abstract Algebra. In short, starting with a fixed length L, any two points that can be drawn by intersecting cicles and lines based on already existing points must lie at a distance that can be expressed as x L, where x is a number whose expression uses only basic arithmetic operations and square roots - reciprocally, for every number x that can be thus expressed, it's possible to construct a line segment of length x L.

Squaring the circle is equivalent to drawing a line segment of length <math>\pi L\,</math>, which can't be done, because <math>\pi\,</math> is not an algebraic number.

Doubling the cube and trisecting an angle are equivalent to solving irreducible cubic equations, whose solutions can't be expressed with only square roots - cubic roots (of the real number 2, in the doubling of the cube, or of complex numbers, in the case of the trisection) are required.