Wednesday, December 20, 2006

A Method of Trisecting Angles


Folks, as promised, here is the method for dividing any angle into three equal parts. That's how it looks to me anyway, and until someone demonstrates that the sections are not equal (rather than simply pointing to the damn proof and saying, "Look, it's impossible!") I will trust my eyes on this one. Try it yourself and see...

Trisection of angle using only compass and straightedge.

1-Draw arbitrary angle A-O-A' (no greater than 180°. If greater than 180°, bisect, work as follows, then rejoin.)

2-Draw bisector, B, of angle through O.

3-Draw arbitrary-sized circle P; on bisector B, with center P, and tangent to inside of angle A-O-A':

4-Draw parallels, C and D, to bisector B, and tangent to circle P'

5-Draw parallels, E and E'; through point P, to angle A-O-A'

6-Draw parallels, F and F'; to angle A-O-A'; tangent to circle P'

7-Draw line, from intersection of lines F and C at G, connecting with intersection of lines D and F' at H, crossing B at J.

8-Draw circle O' with center at O, through J and new points K, L, M, N.

9-Draw circles J, L, and M' around points J, L, and M, same size as circle P:

10-Draw trisector X-O, from O, tangent to both circles J'and L'.

11-Draw trisector Y-O, from O, tangent to both circles J'and M'.

12-One may also draw hexsectors Z and Z' through points L and M from O. Line D-O is also a hexsector.

I have found it possible to extend this method, enabling one to 5-sect an angle (pentasect, for word techies), 7-sect, or any odd-numbered division of an angle.

It seems evident to me that by measuring with compass, the angles are indeed equally 1/3 the original angle. Still, a proper mathematical proof is required. As mentioned earlier, this would entail an investigation into Wantzel's proof to determine where he went wrong. (Readers can download that paper here.)

Alfred King Aldin. "A Pythagoreanism." Philosophic Research and Analysis, Fall 1971, pp. 6-7.
(This is an obscure journal coming out of Boulder, Colorado, and is has been defunct for some time now. He also published a response to readers in the issue of Late Winter 1973.)

M.L. Wantzel (1837). "Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas". Journal de Mathématiques Pures et Appliquées 1 (2): 366–372.
(Yes, dear research students, I copied that link right off Wikipedia! And I enjoyed it!)

(Image scanned and posted on the Internet with evil delight from Mr. Aldin's original article.)

Tuesday, December 12, 2006

On "Difficulty"


At last I'm on vacation! Time to try and get some work done - but first some long-postponed blogging...

One thing that always annoyed me is to be told that something is difficult. "Mathematics is very hard," some people say. Even teachers fall into this trap.

Why is something hard? Who says it's hard? I don't find it easy, but that doesn't mean it's hard for everyone. Some people even find math easy; I wish I were one of those folks. The

How many times have you heard a story about somebody who was interested in something, went ahead and figured it out, and then heard after the fact that he wasn't supposed to be able to do it? This is worth considering.

What makes something difficult? Not the material itself. It's entirely relative to the individual, or culture, or species, depending on what the nature of the challenge is. And then it's largely a matter of practice.

For human beings, it's very hard to walk on two legs - at first. Once they get it, though, they're off and running before you know it. For other critters, they're off and running once they hit the ground.

Answering fundamental questions about the world - now that's pretty hard, always has been. Just when you think you've got it, something or somebody comes along and messes up your tidy little theory. And it's back to square one.

For some cultures, it's hard to imagine an impersonal God. I'm not talking about believing in one, just the idea of one. And the reason for that is they're used to conceiving of a personal God with a face (sometimes literally).

For some individuals, math is hard. Maybe it'll always be that way, but maybe they'll get it and even be pretty good at it. It all depends on the person and the teachers they get - blockheads or angels.

Some things are supposed to be impossible for anybody. There's even proofs to show you that the difficulty isn't just in your head. Take trisection, for example. The task is simple: for any angle, divide it into three equal angles. Easy, right?

Wrong. People have beat their heads against the wall for centuries in an effort to figure out a way to do this. But in 1836 a proof was published by a mathematician named Wantzel that this task is not only difficult but impossible. Mathematicians have been grinning smugly ever since. (You can read all about it here and here. Archimedes cheated - see here.)

The only trouble is, there seems to be a method that works. I found one in an obscure journal years ago, and it obeys the rules of the game. Now I'm not a mathematician, but I know what I see, and this method looks very convincing to me. I've seen many other attempts, but they are either obviously wrong or inapplicable beyond a certain angle. But this method apparently lacks both these shortcomings.

(Yes, I do have a copy of it. Just not at the time of this writing. I will try and post it later.)

What have I learned from this?

1. I need to learn math.
2. I need to learn math to do, among other things,
a. understand that damn proof,
b. see what's wrong with it,
c. prove that this method works.

Is there any practical benefit to this exercise? No, not if you're really asking, "Can it fix your car?" or "Will it make you money?" But those aren't the only benefits around. Consider these:

1. A hidden assumption will be exposed.
2. We will have yet another example showing that we can do more than we think.
3. How we think greatly influences our abilities.

One thing about inquiry is that, to a large degree, it concerns finding limitations. That's what world records are for. Kids do it all the time: How far can I jump? How many pieces of chewing gum can I stuff into my mouth? How much can I get away with before Teacher smacks me? Grown-ups too: What happens if I mix these chemicals together? How long can I ride a unicycle? If I leave this number out, can I pay less on my income tax? The basic question underlying all these is, How far can you go?

If you don't try, you'll never find out. Even when the proofs say it can't be done, it's worth asking, Why not?

(Image joyfully pirated from http://mathworld.wolfram.com/AngleTrisection.html)

Originality, creativity, and philosophy


Everybody knows Steve Martin from the movies; some still remember that he did stand-up comedy. Not so many know that he studied philosophy. He said that philosophy was one of the best things for thinking creatively. Was he right? At first this might sound odd: after all, aren’t philosophers just those dead white males who analyze concepts? How can that help you be creative? I want to show that Steve Martin indeed was right, maybe more so than he knew at the time he said it.

I should preface this by saying that although I practice philosophy, and am writing this, I’m not doing so in an effort to justify my existence. I don’t need to do that. What I’m doing is showing what the arrows point to, and further how weird it is.

I taught a course in doing philosophical research this semester. It’s a first-year course for BA students, and it gave me a lot of food for thought for the process of doing research – as well as for how to teach it. I’m not sure if my approach was a success; we’ll see with the final papers and feedback forms.

As stimulating as it was for me, I wasn’t able to get all my thoughts out. This is just an attempt to take down some of those so that they will be of further use, either to my students or myself.

The Place of Originality and Creativity in Research. One day I harped on the word “originality” as the most overworked in our working vocabulary, and I still stand by that. The unfortunate thing is that I may have been perceived as being against originality – or its allied word, creativity.

I’m not against either entity, really; what I can’t stand is the overuse of the words. There’s a big difference, and it’s a shame we have no substitute. Which, paradoxically, is not so bad. It forces us to come to terms with what those words really mean. Hopefully then we’ll be more judicious about using them in the future.

Let’s look at them one by one. Each of these is often used to mean that some product or person is unique and positive; we usually save them for solutions to problems and artistic fields. “An original solution to the problem of universals,” says the blurb on a book cover. “The author has written an ingenious book that is sure to stimulate the reader.” What does this tell us? – Buy the book! It’s a gem of an answer shining out among tired, old worn-out ones.

Originality. Such a tired word (yawn). Go back to the etymology, and find what it said: origō, point or place of beginning. As far as the activity is concerned, ask this: who is the source of the questions being asked? Who is in the driver’s seat of the investigation? Nobody can die for you, and nobody can think for you. In fact, nobody can do anything for you. If you are compelled to do something “– or else,” you are still the one choosing to act.

What does it mean when we say a writer has done “original research?” Most of us think of research as mousing around in the library, poring over piles of books. That’s supposed to be unoriginal. But that’s untrue. All those books and note-taking are driven by the questions you ask. Every corporation has an R&D department; are they reading dusty tomes? No, they’re doing experiments. They’re testing whatever it is they’re interested in, tweaking it here and there to see what happens; hopefully they can discover something and use it in some new machine or laundry detergent.

“Research,” according to my Oxford, means “a careful study of a subject, especially in order to discover new facts or information about it.” The idea is simple enough; knowing how that translates into action is the stuff of myth-busting.

In product development, research is for finding new things to sell. In philosophy, there’s something very similar going on. (Please don’t think I put them on the same level; work with me here.) Ideas are the stuff philosophers work with, which is why books are their medium. But don’t mistake the book for the thing itself – the idea is one thing, the words another. When you read up, you’re doing background research; that’s to see what has been done before. The reasons for this are various: to prevent reinventing the wheel, to prevent making the same mistakes. The aim is one – economy of effort. You don’t want to do what’s already been done, and you don’t goof the same way. Find another way to screw up.

Experts focus on what’s known, creators on the unknown. It only makes sense to know the difference so you don’t repeat things unnecessarily. Do your homework. You want to find out what is known, but also what is not yet known. This requires some thinking outside the box.

“Library research” is past-oriented; original research is future-oriented. They are not mutually exclusive.

In the course, what I focused on overtly was giving structure to the library efforts, the preliminary work. This is what is usually considered “research”, but I made it plain (perhaps too often, though) that this is only the half of it. When you study a concept you’re doing research. Analyzing that concept: you’re taking it apart to see how it works turning it over in your mind, asking lots of questions – this too is research.

Creativity. Another tired but starry-eyed word. From creāre, to bring forth, produce, to cause or grow. Thanks to Romanticism, this and “originality” are begging for a bullet in the head. (By the way, I don’t mean to slag the best of the best Romantics – Schiller, Herder, and the like. I mean the romantic notion in the most ordinary of ordinary senses. It’s an unfortunate fact: great minds do not move the world, mediocre interpretations of great minds move the world. The first pose bracing questions, the second proffers stultifying answers.)

As we understand it now, creative thinking has as its essence the questioning of basic assumptions – thinking outside the box.[1] A classic example is the nine-dot puzzle. If you’re not familiar with this, try it before reading on…

Did you get it? The key to solving this is to take it strictly on its terms. Four straight lines, connect the dots – that’s all. You don’t have to stick to horizontal or vertical lines, and the lines don’t have to end at a point. That’s all just assumption. Try it now and see how you fare…

Did you get it now? I hope so. You just saw through an assumption, and found a way out.

Children are often said to be naturally creative. There is a kind of reading into children’s activity here: if creativity has to do with questioning assumptions, children cannot be said to be creative. Why not? Because they simply haven’t formed assumptions yet. One thing I remember about growing up was that I wasn’t surprised by much. Why not? Since I was young, I hadn’t built up many expectations, and surprise always plays on expectations. Until I realized this, my memory of not being surprised had puzzled me. I’m not a very remarkable person, never was, and I wouldn’t find it unusual if others shared the same experience.

Among other things that the wrong kind of experience and education do, they teach kids what to accept without questioning. It needn’t be that way. One thing to do is simply observe what is right there before your eyes. (I said simple, not easy.)

If you ever stopped to think about a simple piece of technology, for example a car – I mean really stopped to think about what it took to get that thing in front of you – if you ever thought about it, you might find it rather amazing. First the manufacturing: the metal has to be mined and refined, then molded; the pieces have to be assembled just right. Then the marketing: somebody had to sell the damn thing, and somebody had to buy it. Now for the design: that car was designed by a person, or more likely, several people. Go back through the history: before that car was designed, there were others from the same company, other companies that were trying to muscle into the market; earlier cars weren’t as efficient, so they had to be improved on; before cars even existed we had horses and carriages, chariots. And to get the technology that made the car possible – well, you get the idea.

So you started with this old ’78 Peugeot which you wanted to junk, and ended up with (among other things) a headful of history. If you’ve got an ounce of sensitivity in your soul, you’ll appreciate what it took to get that rust-bucket on the road. Thinking outside the box.

Now look what at philosophy – what else is it but questioning the most basic assumptions of our world? There is a world, and we’re all in it. If we think about it at all, we just shrug our shoulders: yeah, so what? But wait – why should the world exist to begin with? I’m not demanding the extinction of the world, I’m calling for an explanation of it. We can comprehend this thing, and we naturally seek reasons for things; if we can comprehend the world, then of course we’ll want to know the reason for its being around at all.

Why is there a world? Is that a nonsensical question? Why? If you give me reasons for its being a nonsensical question (“Because it’s stupid” is not a reason), you have to know the limits of reasoning to back it all up. If you don’t, then your answer is not sufficient. That means that if you take the question seriously, you question the assumptions of that question – and those of your own answer. When was the last time you did that? Thinking outside the box.

What then is creative research? It’s explanation that calls the taken-for-granted into question, and seeks a more satisfying answer. Thinking outside the box.

* * * * *

So philosophers engage in creative thinking in the purest form. Does that mean we’re privileged human beings? Yes, with qualifications. (What’s a philosophical answer without qualifications?) Yes, we enjoy a privilege not everybody partakes in. Does that mean we’re some elite class? No, without qualification. I’d argue that every human being can do philosophy, and that everyone who asks a question such as “What’s it all about?” thinking philosophically. Sadly, many shake off the question like so much dust from their shoe. They find the question overwhelming, and hence decide they cannot answer it – or, worse, that it’s unanswerable. Maybe that’s so, but I’ve never seen a proof of that. And even if I did, I wouldn’t believe it. I’ve seen proofs undermined, proofs deemed irrefutable by hundreds of experts. (See the entry above.)

If we go back to the root of creativity, we find it has an organic meaning. This suggests a product is intimately connected to the source, to that which made it grow. That would explain why plagiarism strikes at the heart of a writer or artist. And it would suggest that questioning assumptions is itself a sort of generative act.

If we look at the aspect of producing, bringing into existence what wasn’t there before, we run up against the idea of naïveté. To question an assumption is to look at it without the rose-colored spectacles which keep us from seeing it in the first place – in other words, to see it for the first time. Like a child, who does not even know the assumption exists, except that we do know it. Uncovering an assumption can be an exhilarating experience; it is liberating to see it for what it is – and that we don’t have to accept it.

Immanuel Kant urged us, “Sapere aude![2] Dare to know! Buckminster Fuller urged us, “Dare to be naïve.”[3] Both are right. Both involve questioning. Knowing, questioning, being naïve: these are three points that yield a form. I put it this way:

Sapere aude! Quære aude! Nascī aude!



[1] Russell L. Ackoff & Sheldon Rovin. Beating the System: Using Creativity to Outsmart Bureaucracies. San Francisco: Berrett-Koehler, 2005, p. 25.

[2] Immanuel Kant. “What is Enlightenment?” Political Writings. Cambridge: Cambridge University Press, 1991, p. 54.

[3] Fuller, R. Buckmister. Synergetics. New York: Macmillan, 1982. Look here.

(Image daringly lifted from www.rhapsody.com/stevemartin)