Constructible Numbers and Figures

The problem of constructible figures—determining which geometrical objects can be constructed using only the straightedge and compass—was a longstanding problem of mathematics, resolved in the early 19th century. It is closely related to the famous problems of squaring the circle (constructing a square of equal area to a given circle) doubling the cube (finding a cube with double the volume of a given cube), and trisecting the angle (constructing an angle with measure exactly a third of a given one), and actually encompasses these problems, as we shall see below.

The Ancient Greeks did not merely focus on the problem of determining which figures were constructible to categorize geometric objects—their standards of rigor were such that, if a curve or figure could not be constructed, they did not consider it to exist!

But, as we shall show, the problem of constructible figures can be reduced to determining what length line segments can be constructed. Thus the set of constructible figures is determined by a set of real numbers that corresponds to a set of line segments with these numbers as lengths. To illustrate this concept, we will reduce a few of the problems mentioned above to the problem of constructing a line segment of a given length.



The problem of squaring the circle can be reduced to finding a line segment that is one side of the square, i.e., constructing a line segment of length π^1/2. (More precisely, the construction requires the ability to construct a line segment whose length forms a ratio of π^1/2 to the radius of the circle. However, assuming the base segment to be of length 1, the problem reduces to the one above)



The problem of doubling the cube is technically in three-dimensions, but it depends on the ability, in plane geometry, to construct a line segment of length 2^1/3 (again, this is actually the ratio of the side of the larger cube to that of the smaller).

Many other construction problems can similarly be translated into the language of lengths of line segments or ratios of such. Now, the problem is to find what lengths can be constructed. First, we determine what sort of set the set of constructible lengths is. To do this, consider two lengths a and b that are given to be constructible. In other words, if one were forming geometric objects on a piece of paper, one would have, in addition to a compass and straightedge, objects of length a and b from which things can be measured. What other lengths can be obtained from these?



Clearly, given lengths a and b, one can place two lines of these respective lengths end to end, giving a line of length a + b.



Similarly, the length b - a can be constructed from lines of respective lengths a and b. Some care must be taken here, as this construction only yields a positive number for the length of b - a if the length b is greater than the length a. Alternatively, one could consider oriented line segments, or vectors, with initial and terminal points that can be "negative". Here we shall limit ourselves to regular line segments, but allow negative values to be members of the set of constructible lengths. Hence a real number x is a member of the set if its absolute value is the length of a constructible line segment.



Constructing a line segment whose length is the product of two given numbers is a little trickier. In the above figure, we assume a and b to be positive. First, mark a line segment of length a on a given ray, beginning at the endpoint of the ray. Then, draw any other ray out of the endpoint not coincident with the first ray (1). On this second ray, mark two segments beginning from the endpoint of lengths 1 and b. Draw a line segment, l, connecting the the other end of the line segment of length 1 to the end of the segment of length a (2). Finally, draw a line, k, through the end of the segment of length b parallel to l. The intersection of this line with the initial ray demarcates a line segment of length ab from the endpoint. This conclusion follows from the similar triangle law, as the ratio of 1 to a is the same as the ratio of b to ab.



A similar method, again using similar triangles and ratios involving their sides, brings about a segment of length a/b for positive a and b. Begin by marking a line segment of length a along a ray, and draw another ray sharing its endpoint with the first (1). On the second ray, draw a line segment of length b from the endpoint, and connect the opposite end of that segment with that of the segment of length a, forming l (2). Finally, draw the line segment k, beginning at the point on the second ray one unit away from the endpoint (3). The intersection of k with the first ray will define a segment of length a/b, as the ratio of b to a is the same as that of 1 to a/b.

Therefore, given lengths a and b, one can construct a + b, a - b, ab, and a/b. In other words, performing any of these operations on two members of the set of constructible lengths creates another constructible length. The set is said to be closed under these operations. In addition, it is interesting to note that the unit length 1 is needed to compute the product and quotient of lengths. Therefore, we assume all lengths to be in terms of the unit 1, which is also given. Finally, we assume the existence of the length 0, which is simply a point. (It is easy to confirm that the algebraic properties of 0 are satisfied by its geometric counterpart. If one takes b to be 0 in the diagram for division, the line l coincides with the first ray, and the line k, being parallel to l but through a different point, never intersects the first ray. This is consistent with division by zero being undefined.)

A set of this type, closed under addition, subtraction, multiplication, and division, and including 0 and 1, (though division by 0 is undefined) is called a field. Thus the set of constructible real numbers is a field. Any rational function of a and b is a member of this field, where a rational function of a and b is a quotient of polynomials f(a,b)/g(a,b) where both are of finite degree and g(a,b) does not equal 0. Thus the set of constructible real numbers contains all rational numbers, a rather intuitive conclusion. The question of what other numbers the set contains, and its consequences on the motivating problems discussed above, is addressed in the next post, coming April 15.

Sources: History of Mathematical Thought from Ancient to Modern Times, vol. 2, by Morris Kline, A First Course in Abstract Algebra by John B. Fraleigh