Impossible Trisection

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:

1. Only certain lengths can be constructed.
   


2. 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.

Interesting Article

A book review.

The Trick

First we can see that all such numbers are multiples of 9:




Having seen that, note that the multiples of 9 are all always marked with the same symbol. They are tricky because the symbol changes every time and there are numbers marked with that symbol that aren't multiples of 9, but every multiple of 9 is so marked.

Magic?

This is not magic, its just a lot of fun. I've figured it out. Can you?

American Pi

American Pi a mathematical song parody.

Flipping the sign

A narrow view of numbers

Negative numbers are a source of confusion for many who are getting their feet wet in mathematics. The problem comes with having too narrow an idea of number, i.e. restricting the notion to embody only quantity.

We are introduced to numbers, even negative numbers, through quantity. If I have 5 apples and eat three of them, then I have only 2 left. If I loan 3 of them to you and you eat them, then you have negative 3 apples and I will send Bubba after you with a tire iron to get them back if I have to. It doesn’t take much imagination to figure out that this can be tied to money and once things get tied to money they’ve got a way of capturing our attention. We like to keep track of our money. One device to keep track of money is the number line.

I’ve indicated it above with dots draw on it to indicate the integers, a zero to give us a starting point, and an arrow to indicate the positive direction. We can use it to keep track of our money. If I have $5, then I can indicate that by marking it as below.


However, if I buy something that costs $7, then I go $2 in the hole and move the marker backwards, in a negative direction.



Thus, the line is a device to keep track of personal wealth, with numbers to the right of 0 denoting what I have and those to the left what I owe. Of course, as I’ve set this up, it is particularly egocentric. The person I borrowed from still counts himself as having that $2 and would remind himself of that by noting it on his number line.

Introducing Geometry

By bringing in the number line, I’ve introduced geometry to the situation. Zero is my starting point and the (+) and (-) signs indicate the direction which I am going away from zero. I can think of the line as being First Street, which runs in front of my house and zero as being the corner of First Street and Catalpa, which I can see from the window of my home-office. By setting East as the positive direction, I can then tell people I am going 5 blocks and they would be able to figure out that my destination is the corner of First and Walnut, while if I told them I was going negative 2 blocks they would be able to discern I was headed to First and Georgia.
This works fine for addition. As for multiplication, if I am multiplying a positive number by a positive number, I can think of that as repeated addition; if I am multiplying negative by positive, I can think of it as repeated subtraction; and while I am convinced I could do something similar for negative times a negative, thinking about it makes my head swim. Fortunately, I don’t have to because we’ve introduced geometry to the problem.

Let us go back to borrowing versus lending. When Bubba borrows from Billy Bob, Bubba owes, but Billy Bob is owed. The same transaction is positive from one point of view, but negative from the other. Given Bubba’s predilection for non-repayment, the negative end is often hard to discern, but I digress. The point is the positivity or negativity is simply one of point of view. We could view multiplication has having two effects: one on size and one on direction. Multiplication by a positive number might change size but it preserves direction while multiplication by a negative might preserve size, but it changes direction. The negative may be viewed as flipping a number over to the other side. Hence, a negative times a negative is a positive.

Numbers as two pieces

When we multiply numbers, we can think of them as having two pieces. The first of these is their absolute value, i.e. their length, and the other is their sign. Effectively, this is what we do: (5)(-3)=(+5)(-3)=(+)(-)(5)(3)=(-)(15)=-15 or (-2)(-4)=(-)(-)(2)(4)=+8=8.

More Geometry

In the movie Paint Your Wagon, the innocent farm boy utters the line, “But I’ve never lied to my parents before.” To this Ben Rumson answers, “Well when you do it opens up a whole new world!”

When geometry is introduced to arithmetic, a whole new world is opened up. For example, I can expand my explorations beyond First Street by introducing complex numbers. I can take the complex number x+yi and take it to mean go x units in the east-west direction and go y units in the north-south direction. Recall that i is the square root of -1. I can add x+yi to u+vi by taking the sum of the two to be (x+u)+(y+v)i. This works out well geometrically because if I first go x units east-west and y units north-south and follow this by u units east-west and v units north-south this gets me to the same place as going x+u units east-west and y+v units north-south. Go for a walk and see for yourself.



Thus again we are called to expand our notion of number as being simply quantity. On the plane, as we are, we can think of the complex number x+yi as being the directed line segment from the origin of our axes to the number x+yi, as below.



Complex numbers mimic real numbers in that they are best thought of as multiplying in two pieces as well. They have their length, the distance from the origin to the point (x,y), and their direction, which is the angle from the positive real axis to the directed line segment which is associated to x+yi. This is labeled by the Greek letter alpha below.



When two complex numbers are multiplied together, the length of the product is the product of the lengths of the two numbers involved and the angle of the product is the sum of the angles of the two numbers involved. So, for example, the imaginary number i has a length 1 and angle of 90 degrees. So i times i has a length of 1 and an angle of 180. Notice that the number -1 fits this description exactly. Therefore i times i is equal to -1 just like it is supposed to.

This is how the formal multiplication works out. To prove this is a beautiful exercise in trigonometry, but it would put some folks off. I am willing to show you, but only if you beg me.

The Two Towers

We are siloed. Those of us who live in the nations breadbasket might understand it best. Silos are towers in which grain is stored. Silos have high walls, walls so high that none may hope to peer over them into the outside world.

It is an evocative phrase and an apt metaphor for the state of the academic disciplines at the universities. Each of us is within our own world, perhaps a master of it, or desiring to give that appearance, at least, but most are blissfully unaware of the vast storehouse of knowledge that is available in the silos of the surrounding landscape.

This was brought most recently to my attention through a reading group in which I am involved. We’ve been reading a book entitled Proust was a Neuroscientist by Jonah Lehrer. The basic premise of the book is that certain artists have anticipated discoveries by neuroscientists. He does this by proposing several putative examples of such, and whether he succeeds in any particular case I leave his readers to decide for themselves.

In my opinion, he has pointed to something much more important than what he believes he has.

Artists and scientists live in the same world and each of these groups attempts to describe that world in its own language. An illuminating example of this occurs in Lehrer’s chapter on Whitman, who had experienced the Civil War first hand. As a part of this, he took part in amputations and discovered that amputees often still feel the amputated limbs. This information, apparently, didn’t appear in scientific literature until much later.

I find it hard to believe this was unknown until Whitman’s day. Indeed, I can imagine a caveman having had an arm chewed off by a saber tooth tiger still feeling the arm and believing that its spirit still lingered.

It is instructive because it allows us to see what Whitman did. He raised the phenomenon into a higher level of conversation. He raised its talk among small groups of army surgeons, amputees, etc into the broader world. When scientists wrote on the subject, this is exactly what they did as well, but when they did it was in their own language.

The language of the arts is different from that of the sciences. The one thrives on ambiguity and multiplicity of interpretation, and the other on precision, but each is observing the same world and, as a result of these, both will stumble upon the same truths from time to time.

The book group to which I belong consists of two mathematicians, a psychiatrist, a historian, a political scientist, a physicist, a retire high school social science teacher, and a retired professor of Spanish. Proust was a Neuroscientist resulted in many fruitful conversations because of the topics it juxtaposed, not because of the information it offered.

Lehrer is a journalist, not a scholar, and members of the group are more comfortable with scholarly books which cite sources in a more scholarly way. He is also relatively young, so there are areas in which his reach far exceeds his grasp. Even at that, he is a writer who has the talent to convince the reader he knows what he is talking about even when this is, in fact, not true.
This is a talent much valued among journalists.

But even as I criticize the author, I have to ask whether such journalistic pidgin might be necessary when facilitating exchange between two groups with such different languages. To truly understand art as an artist does, does one need to be an artist? To grasp science in a full way does one need to be a scientist?

I believe the answer is no, but I also believe to deal well with both simultaneously would require something I call mileage. This is a quality I don’t see Lehrer possessing in spite of his talent as a writer.

Forbidden Numbers

The root of it all
Square roots are confusing. I think this is the basis of our confusion. Variables make it worse. Let me show you what I mean. We say that r is the square root of a if r²=a. It take a long time to go from that definition to the plain fact that 2 is the square root of 4 or that 3 is the square root of 9, and these are concrete things. Then we math teachers, gleeful sadists that we are, spend a lot of time convincing our tortured students (I couldn't masochistic students because they would enjoy it) that r² can never, ever be negative.

The argument goes like this. On one hand, suppose r is positive. A positive times a positive is positive. So r², which is really just r times r, is positive. On the other hand, suppose r is negative. A negative times a negative is positive too. Really. You've just got to BUH-leeve! (Actually, can explain it to you, but it would slow us down.) So, in this case r² is positive too. The only case left--if you trust me--is r=0. In this case r²=0.

There for there is no square root of -1, or of any other negative number.

Except there is.

What I've done in my argument, what all math teachers do when they make that argument, is to restrict the number r to what we call the set of real numbers. Here's the thing. When mathematicians say "real" we don't really mean it. It's just like when we say the homework assignment this weekend is going to be "short" or the test next Friday is going to be "easy." We are using these words in a different way than anybody else. While "real" originally did mean real, when we apply it to numbers, it has some to mean those numbers we thought were the totality of all numbers before we discovered we didn't know much.

Mathematicians now talk about a number i that is equal to the square root of -1 and refer to it as imaginary. We use it in expressions like x+yi that we manipulate formally like polynomials, but whenever i² occurs, we replace it with -1. All of the numbers x+yi are refered to as the complex number and we can model the complex numbers geometrically on the plane. Indeed, one way mathematicians came to accept the complex numbers as being as "real" as any other is by creating an arithmetic of the ordered pairs of numbers which represent points on the Cartesian coordinate plane.

This impinges on algebra students in the following way. One of the most evil mathematical formulas that the bulk of the human race has to face is the general quadratic formula. (Did anyone else here evil laughter in the background?) The solution of the general quadratic equation, ax²+bx+c=0, is given as

It is a common practice to tell students at a certain level that, when b²-4ac is negative, there is no solution. This is what I was told. You and I, however, now know there is a solution, but that solution is a complex number. I suppose the reason some withhold this information is that it is an opening for other questions. So what's it good for? What about the blue lines in hockey? What's a pasty? Where do babies come from? Where is the money for the bail-out coming from?

While I can't answer some of these at all, and the ones I can answer, I can't deal with in the complexity I'd like, I will say this. When physics and engineering students are solving certain equations dealing with electrical circuits, it's nice to have these complex numbers around.

The end of it all

While I did say that complex numbers can be made precise in terms of real numbers and putting an arithmetical structure on the plane (and there is at least one other way to define them in terms of ring theory), the genesis of it all was simply pronouncing the square root of -1 to exist and running with it. This brings us to one of those other mysterious entities: ∞ (pronounced infinity).

Here one will get into a variety of arguments with a variety of personality types all falling toward the persnickety end of the scale. When we talk about infinity, what sort of infinity would we mean. We can talk about the cardinal numbers or the ordinal numbers. We can say that infinity means "unending." We can talk about about the Indian views and the Greek views. There are all sorts of ways that we can twist this puppy. To those let me say. "Ptttttttttppp." Let me rephrase, "Pttt, pttt, pttt."

No, I am talking about coming at this like a fifth grader would. I want a number that is larger than any other numbers. And I can hear you already, "But if you have a biggest number, you just add 1 to it, and you've got a bigger one. Therefore, there is no biggest number." Thinking like that is going to get us nowhere. We've got to do what was done when we stumbled upon the square root of negative one, i.e. we pronouce it to exist and run with it.

You are saying, "Okay, now, Mister Smart E. Pants, what is ∞+1 then?" Well, it's ∞. So is ∞+2 and ∞+3. And before you get there ∞+∞=∞ and ∞x∞=∞. However, it wasn't by accident or laziness that I left out ∞/∞ and ∞-∞ because division and subtraction are undefined for ∞, and this is probably why teachers don't go much into it. That and the fact that getting kids to learn the multiplication tables through 12 x 12 is hard enough; the ∞ x ∞ tables would be inhuman.

All of this having been said, ∞ can be made precise just like i can. We can define ∞ in terms of sequences that diverge to infinity. For those of you who've not used your calculus in a while or never had it to begin with, a sequence is an infinite list of numbers. Here I mean infinite in the sense of being unending. (I take back one of those earlier Pttt's.) A sequence is said to converge to a number L if its terms can eventually be made to stay as close as we desire to L. (I could make that more precise, but you wouldn't understand it any better.) We say that a sequence diverges to inifinity if its terms can eventually be made to stay as large as we desire. We can define ∞ to be the set of all of these sequences.

I've got to admit that defining a number to be a set of sequences is really, really weird. Really, really. However, one way the real numbers (you remember the real numbers) are defined is in terms of sequences of rational numbers. The gentle reader is at this point asking "Whut?" And I, dedicated scholar that I am, will simply pat him somewhat condescendingly on the back and answer, "It's technical; let's move on."

In adding ∞ to the real numbers in this manner, we are doing what we topologists call adding an end to the real number line. If we also add -∞ we are adding another end. This is all useful for very abstract mathematical purposes. What strikes me, after all these years, is how an idea that many smart-alecky fifth-graders have come up with on their own only to be shouted down by an over-worked teacher who has 40 other little smart-alecks to deal with, is actually legitimate mathematically.

Some practical math..

More Hyperbolic Geometry

Jennifer the Tutor recommended http://www.toroidalsnark.net/mkmisc.html#hp

A Knotty Problem

Treasurer forever?

When I was in college and declared as a mathematics major, my advisor told me, because of this, I would be treasurer of every organization I belonged to. This is, of course, because people associate mathematics with numbers. Take out a little time an say "Duh" at this point. Everybody does. For 99 percent of the population, the bulk of there experience with mathematics is numerical, i.e. with numbers.

My advisor was right, of course, I have been put to use as a treasurer in at least one organization I belong to, but arithmetic was never my forte, and, if arithmetic were the only part of mathematics I'd been exposed to, I'd likely done something else. But in the 10th grade, I was exposed to geometry and the notion of proof and the love affair between mathematics and myself began.

Incorrect pictures

Geometry has been called the art of correct reasoning from incorrect pictures. I include this not only because it's a great quote, but because it captures an important bit of what modern professional mathematicians do. We explore things that we "see" within our skulls and we try to explain what is special about them. The trouble that occurs when we reason with pictures is that there might be something special about the picture that we are using that would disappear is we drew the picture in a slightly different way. Pictures lie. (As that fictional TV physician Gregory House says, "Everybody lies.") A mathematical proof is no place for a lie. Therefore, pictures are avoided whenever possible. This view is held with almost religious fervor.

Two languages
In geometry, the cause for my love affair with mathematics, you can see how this might be a problem. It is a problem, and it is dealt with in a couple of ways: synthetic geometry and analytic geometry.

Synthetic geometry goes at least back to Euclid. As you know, geometry deals with lines and planes and points, oh my! When I say the word geometry to you and then say line and point and plane. You know exactly what I am talking about. These are terms from your experiences, so, likely as not, you've got images of them in your mind. When we are talking about synthetic geometry, however, these are what are known as "undefined" terms, which makes it more than a little odd that Euclid had definitions for them, but the Greeks did a lot of strange things. Don't get me started. In any case, Euclid then described how these undefined objects interacted with one another.

I mention this not out of my tremendous love for geometry but because it illustrates a principle. We begin with objects that we have experience with; we abstract them into objects whose existence is beyond the physical; then we create a language in which we can discuss them.

In analytic geometry, we translate geometry into the language of algebra. More people have deeper experience with analytic geometry than with synthetic geometry because analytic geometry is the language of calculus.

One truth needs to be made clear before the reader proceeds to the rest of this essay or to the rest of his life. Synthetic geometry and analytic geometry are not enemies. Indeed, there is no more enmity between them than there is between a hammer and a monkey wrench. Each is a tool with its own function. One can use a monkey wrench to hammer nails and one can use a hammer to loosen a stubborn nut 'tis true. This has been done with the two approaches to geometry as well, but I will let that thread go.

Knot theory
When I went to graduate school, I learned first hand than there was more to mathematics than arithmetic or geometry or algebra. There was something called analysis that was like calculus but with more theory and less calculation; there was something called algebra but which was as unlike what I learned in high school as a grizzly bear is from a teddy bear; and there was something called topology.

I liked topology. It was like geometry but it was floppier. I eventually did my doctorate in topology. Connected with topology, but somehow separate from it was another subject that I found interesting. It is called knot theory.

The alert reader will no doubt've guessed that knot theory deals with objects from our own common experience: knots. But the alert reader should also, no doubt, be suspicious that maybe what I mean when I say knot is not the same as what he means when he hears knot. Is that clear or is it not? Or knot? These aren't your scoutmaster's knots, fella. These are mathematical abstractions of those. One main difference, beyond just the abstraction, is that in knot theory the knots are closed curves. For example, your scoutmaster might've taught you how to tie this:

But in knot theory we would closed off the end like so:




Knot diagrams
The alert reader that I mentioned earlier is sitting in his reading chair with his hand waiving furiously in the air saying "Ooh,ooh, ooh, teacher, you said mathematicians didn't like reasoning with pictures!"

My answer is "Well, ooh, ooh, ooh, this isn't reasoning with pictures. It is describing with pictures, and, while one must be circumspect in using pictures in any case, it is allowed."

One of the ways we are circumspect is by only allowing certain ways in which knots can be presented. The most common way to present a knot, and the way we did it above, is to use a knot diagram. You may have heard that dancing is a vertical expression of a horizontal idea. Well, a knot diagram is a two-dimensional version of a three-dimensional idea.

Let me explain it to you this way. Drawing is hard. This is why you've got a fellow like Leonardo Di Vinci being famous for being able to draw a winning smile. Very few mathematicians can compete with Leonardo, but don't tell them I said that because they are very sensitive. Knot diagrams were designed so to make the act of drawing a knot as simple as possible. The idea is to draw the knot as if it had been laid on a table top. There are places where the knot crosses itself. Where that happens, we put a magnifying glass over the crossing so as to ignore everything else. Under the magnifying glass, we see only to strands of string. One of those is on top and the other is beneath it. We draw the one that is on top as an unbroken line and the one on the bottom as a broken segment. Examples of these are as below:

Everywhere else, away from the crossings, the various strands of the diagram don't cross, of course. If you experimentally construct a knot from string--which is not too difficult to do--and experiment by placing it upon a table in different ways, you will make the discovery that the same knot can have a variety of different diagrams. Our first example of a knot and the example immediately below is an example of a pair of different-looking diagrams coming from the same knot.

Another one would be

Basic questions

One of the basic questions with which knot theorists busy themselves is whether two different knot diagrams might, in fact, represent the same knot. This is a question which the current space and tone prevents a full answer. However, I can tell you that if two diagrams represent the same know that one can be changed to the other by the use of Reidemeister Moves. There are three Reidemeister Moves which mathematicians using all the of the marketing flair for which they are so well-known have named the First, Second, and Third Reidemeister moves. Indeed those charmers have made use of well-known sexiness of Roman Numerals to refer to them as Reidemeister moves I, II, and III.

Reidemeister Move I is pictured below.



This represents pretty much what it looks like, i.e. someone taking a piece of the knot between index finger and thumb and giving it a twist.

We now consider Reidemeister Move II



This portrays taking one piece of the knot and sliding it over another. Move III as we now see is a little more complicated.



This involves taking one strand and moving it past the point where two other strands cross.

As I said, if there are two diagrams of the same knot, then one can be changed to the other by doing (or undoing) these moves in various combinations. Knowing that you can do it and doing it are different things, as they say.

Getting philosophical

To go much further at this point would require me getting technical, and I don't think that's something we want to do here, eh, Bubba? But the point is that we can impose a language upon a geometric situation and use that language to reason in a precise way.