Ruler-and-compass constructions

A number of ancient problems in geometry involve the construction of lengths or angles using only an idealised ruler and compass. The ruler is indeed a straightedge, and may not be marked; the compass may only be set to already constructed distances, and used to describe circular arcs.

Some famous ruler-and-compass problems have been proved impossible, in several cases by the results of Galois theory.

In spite of these impossibility proofs, some mathematical amateurs persist in trying to solve these problems. Many of them fail to understand that many of these problems are trivially soluble provided that other geometric transformations are allowed: for example, squaring the circle is possible using geometric constructions, but not possible using ruler and compasses alone.

Mathematician Underwood Dudley has made a sideline of collecting false ruler-and-compass proofs, as well as other work by mathematical cranks, and has collected them into several books.

Table of contents

1 Squaring the circle
2 Doubling the cube
3 Angle trisection
4 Constructing regular n-gons
5 Constructing with only ruler or only compass
6 Recent research
7 References
8 External Links

Squaring the circle

The most famous of these problems, "squaring the circle", involves constructing a square with the same area as a given circle using only ruler and compasses.

Squaring the circle has been proved impossible, as it involves generating a transcendental ratio, namely 1:√π. Only algebraic ratios can be constructed with ruler and compasses alone. The phrase "squaring the circle" is often used to mean "doing the impossible" for this reason.

Without the constraint of requiring solution by ruler and compasses alone, the problem is easily soluble by a wide variety of geometric and algebraic means, and has been solved many times in antiquity.

Doubling the cube

Doubling the cube: using only ruler and compasses, construct the side of a cube that has twice the volume of a cube with a given side. This is impossible because the cube root of 2, though algebraic, cannot be computed from integers by addition, subtraction, multiplication, division, and taking square roots.

Angle trisection

Angle trisection: using only ruler and compasses, construct an angle that is one-third of a given arbitrary angle. This requires taking the cube root of an arbitrary complex number with absolute value 1 and is likewise impossible.

Constructing regular n-gons

Some regular polygons are easy to construct with ruler and compasses; others are not. This led to the question being posed: is it possible to construct all regular n-gons with ruler and compass?

The question of which regular polygons can be constructed with ruler and compass alone was settled by Carl Friedrich Gauss in 1796 and (sufficiency) and Pierre Wantzel in 1836 (necessity): A regular n-gon can be constructed with ruler and compass if and only if the odd prime factors of n are distinct prime numbers of the form

These prime numbers are the Fermat primes; the only known ones are 3, 5, 17, 257 and 65537.

Gauss was so pleased by this result that he requested that a regular 17-gon be inscribed on his tombstone.

Constructing with only ruler or only compass

It is possible, as shown by Georg Mohr, to construct anything with just a compass that can be constructed with ruler and compass. It is impossible to take a square root with just a ruler, so some things cannot be constructed with a ruler that can be constructed with a compass; but given a circle and its center, they can be constructed.

Recent research

Simon Plouffe has written a paper showing how ruler and compasses can be used as a simple computer with unexpected power to compute binary digits of certain numbers.

See also: Gauss-Wantzel theorem, Mohr-Mascheroni theorem, Poncelet-Steiner theorem, Squaring the circle

References

Simon Plouffe.The Computation of Certain Numbers Using a Ruler and Compass. Journal of Integer Sequences, Vol. 1 (1998), Article 98.1.3