More About Constructible Numbers and Figures

This series of posts deals with determining which geometric figures are "constructible", that is, can be formed using only a compass and straightedge.

The set of constructible real numbers, or those numbers whose absolute value is equal to the length of a line segment constructible with compass and straightedge, has already been shown to be of the type called a field. In addition, this field contains all of the rational numbers. But what others, if any, does it contain?

It may be noted that the circle has not featured in any of the constructions thus far. In fact, by use of a circle, one can construct a segment whose length is the (positive) square root of that of a given segment. The construction is illustrated below.



A segment of length a^1/2 is constructed by first drawing segments of length 1 and a on end (1). Then, a circle is drawn with the combined segment (of length 1 + a) as a diameter (2). Finally, a perpendicular is erected from the diameter at the point of intersection of the two segments to the circle, and two other segments are drawn connecting the point of intersection of the perpendicular and the circle to the endpoints of the diameter (3). The resulting figure has three triangles, the largest partitioned into the two smaller by the perpendicular. One need only observe that the large triangle is a right triangle, as one of its angles subtends a semicircular arc, and since this large triangle shares a side and an angle with each of the two smaller right triangles, it is similar to each. The two smaller right triangles are then also perpendicular to each other, and so the ratio of 1 to the length of the perpendicular must be equivalent to the ratio of the same length to a. The length of the perpendicular is thus the square root of a.

Thus the field of constructible numbers includes any number that can be derived from a finite sequence of additions, subtractions, multiplications, divisions, and square roots from the unit length 1. In fact, these are all of the constructible numbers. To see this, note first that the construction of segments in the plane involves only the intersections of lines and/or circles. The general equation for a line is ax + by = c, a linear equation, and the general equation for a circle is (x - h)² + (y - k)² = r². It is clear that in solving for the intersection of these two types of functions, the highest degree one could encounter for the intersection points to satisfy is 2, i.e., a quadratic. Finally, by the quadratic formula and the distance formula, which each involve only square roots, the most general type of number one can construct can be seen to be one involving nested square roots, a conclusion in agreement with the previous result.

To illustrate the power of this new concept, we know turn to some applications. First, we relate our result to the geometric construction problems posed at the beginning of this series of posts:

Example 1:
The problem of "squaring the circle" was shown to require the constructibility of the length π^1/2. This condition is equivalent to the constructibility of π and is therefore impossible, as π is what is called a transcendental irrational number; there is no polynomial with integer (or rational, by extension) coefficients with π as a root.
Example 2:
"Doubling the cube" was shown to require the constructibility of a segment of length 2^1/3. This problem is impossible as well, because though there is a polynomial with this number as a root, namely x³ - 2 = 0, this polynomial is of degree 3, not 2, and cannot be factored in any way to reduce its degree. Please note that even though a number such as 2^1/4 is the root of the degree 4 polynomial x⁴ - 2 = 0, the substitution y = x² reduces it to two polynomials of degree 2, and this number is thus constructible.
Example 3: To illustrate the applicability of this concept to figures that actually are constructible, consider the equilateral triangle. Since all three sides are of the same (arbitrary) length, the ability to draw an equilateral triangle depends on its angles, all of which are 60°. To equate the construction of an angle to the construction of a segment, we use trigonometry:



The above figure illustrates that the constructbility of the angle 60° is equivalent to that of the segments of lengths cos(60°) = 1/2 and sin(60°) = 3^1/2/2. If they are given, a right triangle can be drawn with legs of these lengths, thereby giving the angle. Since 3^1/2/2 involves only a square root, it is constructible, and 1/2 obviously is. Thus the equilateral triangle can be drawn with compass and straightedge as well.

The problem of constructibility played a greater role in ancient times than it does today. The standards that constitute "existence" for a mathematical object, though still debated, are much looser than in the time of the Ancient Greek mathematicians. For example, we now accept cubic curves, for example, as perfectly reasonable mathematical objects, even though they cannot be constructed with compass and straightedge (in fact, an arbitary point on one of these curves may not be constructible). The problem now mainly serves as a mathematical curiosity, and as an example of how one can calculate the power, in this case the constructing power, of certain systems in mathematics.

Sources: A First Course in Abstract Algebra by John B. Fraleigh, Constructible Number on Wikipedia