The ancients practiced geometry with the tools they had on hand, the compass and the straight edge. While they were aware of more exotic curves like the parabola, the ellipse, and the hyperbola, they could draw circles and draw lines and that was it.
Geometry itself proceeds from the practical needs of construction. In order to build, one needs to lay out a plan which will, in turn, require certain geometric objects. One needs to be able to construct a line that is perpendicular to a given line or a line parallel to a given line. The question also arises of how one can start with an arbitrary angle and divide it into two angles equal in size to one another; this is called bisecting and angle.
At that point, it becomes a natural question whether one can divide an angle into three separate pieces using only the classical tools. This is called the problem of trisecting an angel, and it stood for two millennia before it was shown that it is impossible to solve.
I will give a description of its impossibility below, but let me say that many people find the statement that something is impossible to be a challenge. I think this is a positive testimony to the spirit of Man. Many times, when people say “impossible,” they mean “difficult.” That is not the case with the problem of trisection. Impossible means impossible.
The genius to the proof of impossibility is the connection of geometric constructions with numbers. The way this is done is connecting each line segment with its length. Our culture is so used to a system of lengths, weights, and measures that we take this part for granted, but in fact it is a key step. There has to be a way of measuring the line segment. The way this is done is that a particular line segment is designated as having a length of one unit, where that unit is whatever your favorite unit might be.
The association of geometry with numbers works so well because the arithmetic operations of addition, subtraction, multiplication, and division can be carried out though various geometric constructions. These four operations are known in the mathematics as the operations that can be performed in a field, that is to say an algebraic field is a set among whose elements these four operations can be carried out. A field is an abstract mathematical concept and there is a well-developed theory of fields. Let me hasten to add that this sort of field has no connection with the concept of vector fields that engineers and physicists are familiar with.
The proof of the impossibility of construction hinges on the following two facts:
- Only certain lengths can be constructed.
- The cosine of twenty degrees cannot be constructed.
Let me now comment on each of these.
As the four arithmetic operations can be carried out by geometric construction, the so-called constructible numbers do form a field. This field is generated by the number one, which is obtained from that segment we designate as having a length of one. It is not difficult to see that this field of constructible numbers must include the rational numbers. What might be more difficult to see is that the field of constructible numbers in its entirety may be built up by creating a series of fields each one obtained from the one below by adding square roots. The proof of this is technical, but let me say that it rests on the fact that the equation of a circle in analytic geometry is of degree two. Remember, our geometric constructions only allow a compass for making circles and a straight edge for drawing lines.
I make a big deal out of the fact that only allowing the addition of square roots and square roots of square roots and square roots of square roots of square roots and so on is possible because it is important to understanding why the cosine of twenty degrees cannot be constructed. The way this is said in the language of technical mathematics is that if F is a subfield of the constructible numbers, then F is an extension of the field of rational numbers of order two to the nth power.
In order to comment on the second fact above, I must introduce an equation from trigonometry. For any angle θ,
It is well known to every student of trigonometry that the cosine of sixty degrees is equal to one-half. Therefore, if we let x denote the cosine of twenty degrees and do a little algebra, the above yields
There are, no doubt, other equations that this particular value of x must satisfy, but it will, in particular, satisfy this one and that is important because this one has special properties.
A. It is irreducible over the rational numbers, i.e. it cannot be factored into rational factors.
B. It is of degree 3.
It is an exercise in high school algebra to show that A is true, and B is a simple observation. Their implication in the well developed theory of fields is that any field that contains x would have to be an order three extension of the rational numbers. Since three is not equal to two to the nth power for any integral value of n, it follows that the cosine of 20 degrees cannot be constructed.
This means a 20-degree angle cannot be constructed. This means a 60-degree angle cannot be trisected. This means there cannot be a general method for trisecting all angles because such an algorithm would enable one to trisect the 60-degree angle.
As you may have noticed, I’ve been forced to fuzz my exposition behind the clouds of technicality. If you would care to peer behind those clouds, I recommend Abstract Algebra: Theory and Applications by Judson. In an extraordinary stroke of luck it is available for free online at http://abstract.ups.edu/download.html.
