The period between about 475 B.C. and 325 B.C. saw enormous growth
in theoretical
mathematics led by many great mathematicians in the Greek sphere of
influence.
These advances were primarily in five subject areas:
1. The theory of numbers.
2. Metric geometry focusing primarily on the development of formulas for the
computation of areas and volumes of a wide variety of geometric figures.
3. The development of non-metric geometry focusing primarily on
geometric constructions
with compass and straightedge.
4. The theory of music (an investigation led by the Pythagoreans).
5. The theory of reasoning, mathematical proof and axiomatics. The
publication
of Euclid’s Elements certainly served as a culmination of these
investigations.
In their investigations of geometric constructions, these mathematicians were able to achieve many remarkable feats but were frustrated in their attempts to solve the following geometric construction problems:
Although the Greeks believed that these constructions were impossible, they were not able to prove their impossibiility. These problems drew the attention of many famous mathematicians over the centuries until they were finally proved to be impossible in the nineteenth century.
Proving that a certain geometric construction is impossible seems an especially daunting task. Of course, it is not sufficicient to simply fail to find such a construction. Even if many famous mathematicians fail to find a certain geometric costruction that does not imply that no such construction exists. The key to these impossibility proofs lies in a correspondence that exists betwwen geometric constructions and sets of real numbers. We will develop this correspondence later in this part of Unit 1 and in Unit 2..
In our discussion of the construction of a regular decagon in Part 3, we first described a geometric construction that claimed to construct, for a given line segment AB, an isosceles triangle ABP with a vertex angle at A of 36 degrees, and with equal sides of length |AB|. Once this triangle is constructed, it is an easy matter to proceed to construct a regular decagon and a regular pentagon of radius AB.
When you carried out the construction of the isosceles triangle ABP in Geometer’s Sketchpad and measured the angle at A in 1.3.2 Just Do It!, you found that the angle at A did indeed measure 36 degrees. As we pointed out in the Answer to that problem, such a measurement may not prove that the given construction is valid. We did provide a rigorous algebraic-geometric proof of the validity of the construction. We did this by introducing a rectangular coordinate system with the given line segment AB as the segment joining (0, 0) and (1, 0), and then following the construction algorithm algebraically by identifying the coordiates of each successive point constructed by the algorithm. The end result of this analysis was that the cosine of the vertex angle A of the triangle PBA is given by
We then showed, again by an algebraic-geometric argument that the value of the cosine of 36 degrees has the same exact value, and so the measure of angle A is exactly 36 degrees as required.
This algebraic-geometric approach to proving that the proposed construction of an angle of 36 degrees is valid is also the key to proving that certain geometric constructions are impossible. Here is the general idea behind the algebraic-geometric approach to proving that a given geometric construcion is either poosible or impossible:
In Steps 1 and 2, , we might identify given points by their coordinates, given line segments by their length or by the coordinates of their endpoints, and given circles by the coordinates of their center and their radius. For example, if the objective is to construct a regular n-gon with a given line segment AB as a radius and the point A as its center, then we might describe the given segment by the real numbers that are the coordinates of its two endpoints. The resulting polygon might then be described by the real numbers that are the coordinates of the endpoint C of an adjacent radii AC, or by the length of the side BC , in terms of the real number coordinates of the given objects. If there is a geometric construction of the required sort, then it is possible to obtain the real numbers that determine the final figure by successively “updating” these real numbers through the steps of the contruction to arrive at the real numbers that determine the final figure. (That is precisely what we did in our proof that the proposed construction of the regular decagon was correct.) On the other hand, if we can show that no sequence of updates of the real numbers determining the given figure can result in the real numbers that determine the final figure of the construction, then we can conclude that there is no such geometric construction.
For example, in the Squaring The Circle Problem, the relative positions of the given circle C and the desired square S are unimportant, only the equality of their areas is significant. Consequently, the given circle can be determined by a single real number r, its radius, and the desired square S by another real number s, its side length.
The circle C and the square S have the same area if and only if
Thus, if we assume that the circle has radius 1, the Squaring the
Circle Problem
reduces to the following problem:
In the following problem, you will find a similar reformulation of the problem of duplicating a cube.
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Let’s begin with three important observations about the angle trisection problem:
This can be verified by considering the following diagram:
Observation 1) has the following important consequece:
Explanation of 2): Certainly, if a line segment of length cos(m) can be constructed given a line segment of length 1, then an angle of measure m can be constructed by 1) and that angle would trisect the angle of measure 3m. Conversely, if the angle of measure 3m can be trisected given a line segment of length 1, then an angle of measure m can be constructed (by composing the trisection construction and the construction of the angle of measure 3m) and so by 1) a line segment of length cos(m) can be constructed given a line segment of length 1.
Angles of measure 180 degree and 90 degrees can be trisected because angles of mesure 60 degrees and 30 degrees are constucted by constructing an equilateral triangle and subdividing into two congruent right triangles with acute angles of mesure 30 degrees and 60 degrees. However, we will show that:
According to 2), we can prove 3) by showing that given a line segment of length 1, it is not possible to construct a line segment of length cosine of 20 degrees. To prove that, we proceed as follows:
a) Find a polynomial equation with rational number coefficients that has the cosine of 20 degree as one of its roots.
b) Prove that if x = r is a root of a polynomial equation of the sort decribed in a), then a segment of length r cannot be constructed given a segment of length 1.
You are asked to do Part a) in Problem 1 of Assignment 1.4. Part b) is a consequence of a general result about constructible roots of cubic polynomials that will be discussed in Part 3 of Unit 2.
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