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Proof and Unsolved Problems
1.1. ANGLE TRISECTION AND BIRD MIGRATION
Our subject is the impossibility of several famous ruler and compass constructions—to construct an angle equal to one-third of a given angle, to construct a cube with twice the volume of a given cube, to subdivide a circle into any given number of equal parts. Interest in these problems remains high although they were all solved early in the nineteenth century. Unfortunately, this is not because the solutions are especially simple and beautiful. They are not simple, and their beauty is perhaps perceived only by mathematicians. Possibly interest in these problems remains high because the solutions contradict one's intuition and frustrate one's ego and because the solutions are much more difficult to explain than the problems are to state. Even so, that so many students and amateur mathematicians are captivated by these problems, especially the problem of angle trisection, is puzzling. Residents of the Great Lakes area exhibit much less interest in how migratory birds are able to return to their home territories, although at least as many Great Lakes area residents have observed migrating birds as have studied plane geometry. This is odd, since biologists do not yet have a satisfactory explanation of how many birds can find their way home to their birthplace across thousands of miles of land and ocean, whereas all mathematicians are agreed that the angle trisection problem is solved. Biologists do not receive dozens of letters each year with suggested explanations of bird migration, but mathematicians do receive dozens of letters each year containing, so it is claimed, constructions for trisecting any angle.
Junior high school and high school "angle trisectors" may be excused. It is less easy to excuse adult "angle trisectors." To be candid, almost none of the latter can be convinced he has not made a great discovery; or, if he can be convinced that the particular construction he offers fails, he cannot be persuaded to cease his search, that the search is fruitless. I use the pronoun "he" deliberately. There are, I have concluded, almost no female angle trisectors! In fact, I know of none. Some of the more persistent angle trisectors are, on the other hand, among the most respected males in our society. They are physicians engaged in general practice! Is this because a doctor is used to "being his own boss" and having everyone accept his opinions without question? Confirmed angle trisectors either do not understand the difference between an unsolved and a solved problem in mathematics, or they do not admit the validity of indirect proofs. In any case, no one, least of all a mathematician, can convince a confirmed angle trisector that it is impossible to trisect a 60º angle with unmarked straightedge and compasses alone and that a proof of this is a solution to the problem of angle trisection.
Our subject is mysterious to some people and challenging to most people. There is confusion in the minds of some people as to whether or not the classical construction problems are solved and what it means to assert that they have been solved. Therefore, it is now wise and proper to give a brief discussion of what a proof in mathematics is and what an unsolved problem is.
The standard by which practically all the world's mathematicians judge a proof is this: A proof is that which has convinced and now convinces the intelligent reader. Of course, one asks, who are the intelligent readers? The best answer I can give to that question is that within a given culture the intelligent readers of mathematical proofs are those people who are generally accepted to be mathematicians. Moreover, proof is relative: What is good mathematics in this culture in this age may not be considered good mathematics in this or another culture in a future age, just as today we consider much mathematics of past cultures and ages to be incomplete or incorrect. Next, attention should be focused on the point that a proof is an argument that has convinced and now concinces. The use of past and present tenses is deliberate. I maintain that an argument is not a proof until it has been articulated, heard or read, and, finally, found to be convincing, so convincing that there exist live men who are presently convinced of it. A mathematical proof is a temporal, communicable phenomenon in the minds of living men. Mathematical proofs are not arguments written on tablets of gold in Heaven (or on Earth); they are certain collections of thoughts that many people, intelligent readers, hold in common.
Saying this much is already to invite much philosophical dispute. To say more is to become more involved and technical than is proper here. What about the proofs in this little book? They are restatements of proofs accepted by a wide variety of mathematicians, those people who conjecture and prove theorems, and they have convinced several "intelligent readers." I hope they convince you too.
1.3. SOLVED AND UNSOLVED PROBLEMS
One must not confuse the impossibility of a geometric construction with an unsolved problem—or with the unsolvability of a problem! Consider the following example: to construct the longest straight line segment. The construction is impossible, and the problem is thereby solved. It is not an unsolved problem or an insoluble one. (For, suppose one were able to construct a line segment longer than any other. An axiom of plane geometry is that a straight line segment can be extended indefinitely beyond each of its end points. Thus the constructed "longest" segment would be extendable, and each of its extensions would be longer than itself so that it could not be the longest segment.) Another example is: to construct a square whose side length is a whole number of units and whose area is two square units. Clearly, there exists no such square, and a proof of this statement constitutes a solution to the problem. On the other hand, if we change our rules slightly and admit as candidates for solutions to our problem squares of any side length, then we can solve the problem affirmatively. (Given a straight line segment AB of unit length, we construct a second segment AC perpendicular to AB at A and also of unit length. Then BC is a side of a square of area 2. The length of BC is [square root of 2], which is not a whole number.) Analogously, if we change the classical rules and permit the use of another instrument in addition to an unmarked straightedge and compasses, then indeed we can trisect a 60º angle and any other angle as well. Such instruments are described in Chapter 6.
It is true, however, that there are unsolved mathematical problems. Here are some examples:
1. Conjecture: There exist infinitely many whole numbers n such that n and n + 2 are both prime. This conjecture is known as the twin prime problem. It is of unknown origin and old.
2. Consider the set S of all squares whose side lengths are integers. Is there a square in S with the remarkable property that it can be cut into square pieces, each piece having side length an integer and no two pieces being congruent? Yes. Such a remarkable square is illustrated in Fig. 1.1. Call the set of all such remarkable squares T.
Conjecture: Twenty-four is the least number of pieces of any square in T. Note that the square illustrated is composed of exactly 24 pieces.
3. Conjecture: Let S be the surface of a bounded, three-diniensional convex body. If passing through each point P of S there can be constructed two perpendicular planes, each of which cuts S in a circle, S is a sphere. This problem is due to H. T. Croft.
The prognosis for each of these conjectures is different. The third one is likely to be settled soon, but it appears to be difficult enough that this is uncertain. The second conjecture is also likely to be settled soon. However, one can say more: A procedure for solving the problem exists. This procedure involves examining an enormous number of special cases, so many that, at present, to examine all of them on a computer would take a long time, much more of expensive computer time than proving or disproving the conjecture is worth. When a fast enough machine with a large enough memory becomes available, and this should be soon, we shall be able economically to learn the truth of the conjecture or disprove it. The character of the first conjecture is vastly different. For perhaps two thousand years brilliant mathematicians have attempted to find a solution. Yet not even a partial solution has been found. Some number theorists do not believe that a solution can be reached with the mathematics known today. There even exist a few who suspect that the question, Are there infinitely many whole numbers n for which n and n + 2 are both prime?, is undecidable; that is, it may be impossible on the basis of the usual axioms governing whole numbers either to prove or disprove that there are infinitely many "twin primes"!
1.4. THINGS TO COME
We close this introduction with a sketch of the remainder of the book. We shall be concerned with the following problems:
1. Construction of an angle equal to one-third of a given angle.
2. Construction of a cube with twice the volume of a given cube.
3. Construction of a regular n-gon (n = 3, 4, 5, 6, 7, ...).
4. Construction of a square with the same area as a given circle.
In Chapter 5 we shall complete the proof that there exist cases in which the first two constructions are impossible to carry out with straightedge and compasses alone. In Chapter 11 we shall determine those natural numbers n for which it is impossible to construct a regular n-gon with straightedge and compasses alone; but we only discuss the difficulty of proving the impossibility of squaring a circle in this book. Alas, a proof of the impossibility of this fourth construction involves too intricate an analysis to present to anyone but an accomplished mathematician.
Our discussion of these construction problems is organized as follows:
i. The ground rules of geometric construction and their algebraic interpretation.
ii. Some of the history of these famous construction problems.
iii. Fields of real numbers: the rational field and its iterated quadratic extensions; constructible numbers.
iv. A study of cubic polynomials and its application to the problems of angle trisection and cube duplication.
v. Non ruler and compass trisections; approximate trisections.
vi. Factorization of polynomials; irreducible polynomials.
vii. An interlude: unique factorization.
viii. Algebraic fields, linear independence, basis, and degree.
ix. Application of the theory of algebraic fields and irreducibility to division of a circle into n equal parts—the impossible cases; why squaring a circle is difficult.CHAPTER 2
Ground Rules and their Algebraic Interpretation
2.1. CONSTRUCTED POINTS
What is meant by "geometric construction with straightedge and compasses alone" is something that we must clarify completely before we attempt to solve classical construction problems. This section is intended to provide such clarification.
In beginning each geometric construction we imagine that we are upon a plane of which two points O and A one unit apart are identified. We further imagine that we have ready for use an unmarked straightedge (which we shall call a ruler) with which straight line segments can be drawn and a pair of compasses (which we shall call a compass) with which circles can be drawn. In making a geometric construction with ruler and compass alone we are permitted only the operations described below, which are all made relative to the two points O and A given to us. Further, we are permitted to perform only a finite number of these operations in making any geometric construction with ruler and compass alone. In the specification of permissible construction operations below, O and A are defined to be constructed points.
THE PERMISSIBLE CONSTRUCTION OPERATIONS
1. With the ruler one may draw the line segment joining any two constructed points, and one may extend it as desired.
2. With the compass, with any constructed point as center, and with the distance between any two constructed points as radius, one may draw a circle.
2.1. DEFINITION. A line segment whose end points are constructed points is a constructed segment.
2.2. DEFINITION. Each circle that is drawn as described above is a constructed circle.
It is a consequence of the following definition that the class of constructed points may be enlarged indefinitely from the original constructed points O and A by means of permissible construction operations.
2.3. DEFINITION. A constructed point is
1. O or A, or
2. a point of intersection of two nonparallel constructed segments, or
3. a point of intersection of a constructed segment and a constructed circle, or
4. a point of intersection of two distinct constructed circles.
Several remarks are in order. We emphasize that a constructed point is a result of finite number of construction operations. An infinite number of construction operations are not allowed in a legal ruler and compass construction. Second, if two constructed segments are not parallel and do not intersect, one may extend them until they do intersect. Third, given a constructed segment and a constructed circle that do not intersect, one can extend the segment until it intersects the circle only if the radius of the circle is at least as big as the distance from the center of the circle to the line containing the segment. (There is one point of intersection possible if this distance is equal to the radius, two if it is smaller.) Fourth, we stress that exactly one pair of constructed points is given and that all the others are identified relative to the given pair. A point chosen at random is not identified relative to the given pair.
Why should we exclude random points and lines from our arguments concerning ruler and compass construction problems? The motivation is that if a random selection of a point or line enables one to complete a certain geometric construction, then so must a specific selection. For example, suppose it is required to construct the midpoint of OA. This construction may be performed as follows by choosing a point at random: Let Z be any point on OA which is closer to A than to O. With O and A as centers and radius [bar.OZ] construct two circles. The segment joining the points of intersection of these circles meets OA at its midpoint. But the random selection of Z in this construction is unnecessary. The specific choice of Z as A, which is a constructed point, will do.
We offer a second example. Let it be asked to divide OA into three equal parts. The familiar construction using a random line is: Draw a segment of any line l through O not containing A (see Fig. 2.1). Draw any circle with center O. Let X be the point where it intersects the segment. Draw a second circle with center X and radius [bar.XO]. Let the points where it meets l be Y and O. Draw a third circle with center Y and radius [bar.YX] = [bar.OX]. Let the points where it meets l be Z and X. Draw the segment ZA. Through X and Y construct segments parallel to ZA meeting CA at X' and Y', respectively. Then [bar.OX'] = [bar.X'Y'] = [bar.Y']. (The familiar construction of a line segment through a given point and parallel to a given segment is: Let the given segment be CD and the exterior point be P. Draw CP and extend it beyond P. We assume, of course, that P is not on CD extended. Draw the circle with center P and radius [bar.PC]. Call its point of intersection with CP extended P'. With P as center and [bra.CD] as radius, draw a circle. With P' as center and radius [bar.PD], draw a circle. Let D' be a point of intersection of these two circles. Then PD' is parallel to CD, since the triangles P'PD' and PCD are congruent, etc.; see Fig. 2.2.)
Excerpted from Ruler and the Round by Nicholas D. Kazarinoff. Copyright © 1970 Prindle, Weber & Schmidt, Incorporated. Excerpted by permission of Dover Publications, Inc..
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