Simple City

Constructible Numbers

A sure route to mathematical fame is to resolve a problem that has stood open for centuries, defying the greatest minds of previous generations. In 1837, Pierre Wantzel’s seminal analysis of constructible numbers was enough to settle not just one, but an entire slew of the most famous problems in the subject, namely those relating to ruler-and-compass constructions.

As with so much in the history of mathematics, the topic had its origins in the empire of ancient Greece. The geometers of that period were interested not only in contemplating shapes in the abstract, but also in creating them physically. Initially, this was for artistic and architectural purposes, but later for the sheer challenge it posed. In time, mathematicians came to understand that the obstacles theyencountered in these ruler-and-compass constructions brought with them a great deal of mathematical insight. Nowhere was this more true than in the ancient enigma of squaring the circle, and what that revealed about the number (pi).

Classical problems

Greek geometers decided on a set of simple rules for building shapes, using only the simplest possible tools: a ruler and pair of compasses. The ruler is unmarked, so it can only be used for drawing straight lines, not for measuring length (therefore these are sometimes called straight-edge-and-compass constructions). The compass is used to draw circles, but it may only be set to a length that has already been constructed.

Today’s schoolchildren still learn how to use these devices to divide a segment of straight line into two equal halves and to bisect a given angle. These were two of the very first ruler-and-compass constructions. A more sophisticated technique allows a line to be trisected, that is, divided into three equal parts. What of trisecting an angle, though? Various approximate methods were discovered, which were accurate enough for most practical purposes, but no one could find a method which worked exactly. This proved a mystery, and gave the first hint that there was real depth beneath this question. But what does it mean if one task can be carried out by ruler and compass and another cannot?

The most famous of the ruler-and-compass problems, and indeed one of the most celebrated questions in mathematics, is that of squaring the circle. The question is this: given a circle, is it possible to create, by ruler and compass, a square which has exactly the same area? At the heart of this question lies the number (pi) (see page 54). The problem ultimately reduces to this: given a line 1 unit long, is it possible to construct by ruler and compass another line exactly (pi) units long?

Another classical problem was that of doubling the cube. This problem had its origins in a legend from around 430 BC. To overcome a terrible plague, the citizens of the island of Delos sought help from the Oracle of Apollo. They were instructed to build a new altar exactly twice the size as the original. At first they thought it should be easy: it could be done by doubling the length of each side. But that process leads to the volume of the altar increasing by a factor of 8 (since that is the number of smaller cubes that can fit inside the new one). To produce a cube whose volume is double that of the original, the sides need to be increased by a factor of ( sqrt[3]{2}) (that is the cube root of 2, just as 2 is itself the cube root of 8). The question of doubling the cube therefore reduces to this: given a line segment 1 unit long, is it possible to construct another exactly ( sqrt[3]{2}) units long?

Wantzel’s deconstruction

Working in the turbulent setting of France in the early 19th century, Pierre Wantzel turned these ancient questions over in his mind. He recognized that the form of many ruler-and-compass questions is the same. The key to them was this: given a line 1 unit long, which other lengths can be constructed? And which cannot? If a line of length (x) can be constructed, then Wantzel deemed (x) a constructible number. Setting aside the geometrical origins of these problems, he devoted himself to studying the algebra of constructible numbers. Some things were obvious: for example, if (a) and (b) are constructible, then so must be (a + b), (a – b), (a times b), and (a div b). But these operations do not exhaust the range of constructible numbers; Wantzel realized that it is also possible to construct square roots, such as (sqrt{a}).

His great triumph came in 1837, when he showed that everything constructible by ruler and compass must boil down to some combination of addition, subtraction, multiplication, division and square roots. Since (sqrt[3]{2}) is a cube root, and cannot be obtained via these algebraic operations, it followed immediately that the Delians’ ambition to double the cube was unattainable. A similar line of thought revealed the impossibility of trisecting an angle.

As for the greatest problem of all, squaring the circle, the final piece didn’t fall into place until 1882, when Ferdinand von Lindemann proved that (pi) is a transcendental number (see page 197). Then Wantzel’s work immediately implied the non-constructibility of (pi), and the impossibility of squaring the circle was finally established.