Archive for the ‘Games’ Category

Reflections on Trisecting the Angle

Thursday, March 12th, 2020

I’m not a mathematician by training, but the language and (for want of a better term) the sport of geometry has always had a special appeal for me. I wasn’t a whiz at algebra in high school, but I aced geometry. As a homeschooling parent, I had a wonderful time teaching geometry to our three kids. I still find geometry intriguing.

When I was in high school, I spent hours trying to figure out how to trisect an angle with compass and straightedge. I knew that nobody had found a way to do it. As it turns out, in 1837 (before even my school days) French mathematician Pierre Wantzel proved that it was impossible for the general case (trisecting certain special angles is trivial). I’m glad I didn’t know that, though, since it gave me a certain license to hack at it anyway. Perhaps I was motivated by a sense that it would be glorious to be the first to crack this particular nut, but mostly I just wondered, “Can it be done, and if not, why not?”

Trisecting the angle is cited in Wikipedia as an example of “pseudomathematics”, and while I will happily concede that any claim to be able to do so would doubtless rely on bogus premises or operations, I nevertheless argue that wrestling with the problem honestly, within the rules of the game, is a mathematical activity as valid as any other, at least as an exercise. I tried different strategies, mostly trying to find a useful correspondence between the (simple) trisection of a straight line and the trisection of an arc. My efforts, of course, failed (that’s what “impossible” means, after all). Had they not, my own name would be celebrated in different Wikipedia articles describing how the puzzle had finally been solved. It’s not. In my defense, I hasten to point out that I never was under the impression that I had succeeded. I just wanted to try and to know either how to do it or to know the reason why.

My failed effort might, by many measures, be accounted a waste of time. But was it? I don’t think it was. Its value for me was not in the achievement but in the striving. Pushing on El Capitan isn’t going to move the mountain, either, but doing it regularly will provide a measure of isometric exercise. Similarly confronting an impossible mental challenge can have certain benefits.

And so along the way I gained a visceral appreciation of some truths I might not have grasped as fully otherwise.

In the narrowest terms, I came to understand that the problem of trisecting the angle (either as an angle or as its corresponding arc) is fundamentally distinct from the problem of trisecting a line segment, because curvature — even in the simplest case, which is the circular — fundamentally changes the problem. One cannot treat the circumference of a circle as if it were linear, even though it is much like a line segment, having no thickness and a specific finite extension. (The fact that π is irrational seems at least obliquely connected to this, though it might not be: that’s just a surmise of my own.)

In the broadest terms, I came more fully to appreciate the fact that some things are intrinsically impossible, even if they are not obvious logical contradictions. You can bang away at them for as long as you like, but you’ll never solve them. This truth transcends mathematics by a long stretch, but it’s worth realizing that failing to accomplish something that you want to accomplish is not invariably a result of your personal moral, intellectual, or imaginative deficiencies. As disappointing as it may be for those who want to believe that every failure is a moral, intellectual, or imaginative one, it’s very liberating for the rest of us.

Between those obvious extremes are some more nuanced realizations. 

I came to appreciate iterative refinement as a tool. After all, even if you can’t trisect the general angle with perfect geometrical rigor, you actually can come up with an imperfect but eminently practical approximation — to whatever degree of precision you require. By iterative refinement (interpolating between the too-large and the too-small solutions), you can zero in on a value that’s demonstrably better than the last one every time. Eventually, the inaccuracy won’t matter to you any more for any practical application. I’m perfectly aware that this no longer pure math — but it is the very essence of engineering, which has a fairly prominent and distinguished place in the world. Thinking about this also altered my appreciation of precision as a pragmatic real-world concept. 

A more general expression of this notion is that, while some problems never have perfect solutions, they sometimes can be practically solved in a way that’s good enough for a given purpose. That’s a liberating realization. Failure to achieve the perfect solution needn’t stop you in your tracks. It doesn’t mean you can’t get a very good one. It’s worth internalizing this basic truth. And only by wrestling with the impossible do we typically discover the limits of the possible. That in turn lets us develop strategies for practical work-arounds.

Conceptually, too, iterative refinement ultimately loops around on itself and becomes a model for thinking about such things as calculus, and the strange and wonderful fact that, with limit theory, we can (at least sometimes) achieve exact (if occasionally bizarre) values for things that we can’t measure directly. Calculus gives us the ability (figuratively speaking) to bounce a very orderly sequence of successive refinements off an infinitely remote backstop and somehow get back an answer that is not only usable but sometimes actually is perfect. This is important enough that we now define the value of pi as the limit of the perimeter of a polygon with infinitely many sides.

It shows also that this is not just a problem of something being somehow too difficult to do: difficulty has little or nothing to do with intrinsic impossibility (pace the Army Corps of Engineers: they are, after all, engineers, not pure mathematicians). In fact we live in a world full of unachievable things. Irrational numbers are all around us, from pi to phi to the square root of two, and even though no amount of effort will produce a perfect rational expression of any of those values, they are not on that account any less real. You cannot solve pi to its last decimal digit because there is no such digit, and no other rational expression can capture it either. But the proportion of circumference to diameter is always exactly pi, and the circumference of the circle is an exact distance. It’s magnificently reliable and absolutely perfect, but its perfection can never be entirely expressed in the same terms as the diameter. (We could arbitrarily designate the circumference as 1 or any other rational number; but then the diameter would be inexpressible in the same terms.)

I’m inclined to draw some theological application from that, but I’m not sure I’m competent to do so. It bears thinking on. Certainly it has at least some broad philosophical applications. The prevailing culture tends to suggest that whatever is not quantifiable and tangible is not real. There are a lot of reasons we can’t quantify such things as love or justice or truth; it’s also in the nature of number that we can’t nail down many concrete things. None of them is the less real merely because we can’t express them perfectly.

Approximation by iterative refinement is basic in dealing with the world in both its rational and its irrational dimensions. While your inability to express pi rationally is not a failure of your moral or rational fiber, you may still legitimately be required — and you will be able — to get an arbitrarily precise approximation of it. In my day, we were taught the Greek value 22/7 as a practical rational value for pi, though Archimedes (288-212 BC) knew it was a bit too high (3.1428…). The Chinese mathematician Zhu Chongzhi (AD 429-500) came up with 355/113, which is not precisely pi either, but it’s more than a thousand times closer to the mark (3.1415929…). The whole domain of rational approximation is fun to explore, and has analogical implications in things not bound up with numbers at all.

So I personally don’t consider my attempts to trisect the general angle with compass and straightedge to be time wasted. It’s that way in most intellectual endeavors, really: education represents not a catalogue of facts, but a process and an exercise, in which the collateral benefits can far outweigh any immediate success or failure. Pitting yourself against reality, win or lose, you become stronger, and, one hopes, wiser. 

Freedom to fail

Thursday, March 31st, 2011

The previous entry on this blog was about failure not being an option — and I subscribe to that. Failure in an ultimate sense is something we should never choose for ourselves: the universe or some other person may well cause us to fail but we should not elect to fail in a final sense. Nevertheless, failure, and the freedom to fail in the short run without disastrous long-term consequences, is essential to learning. I have taught students with a whole range of abilities and inclinations over the years; there have been some who have been afraid to venture on anything, lest they fail to complete it to some arbitrary standard of perfection. Others tear into the subject with giddy abandon, making mistakes freely and without compunction. Of the two groups, it is invariably the latter that gets the job done. The students in the former group are frozen by fear or reverence for some external standard of excellence or perfection, and they really cannot or will not transcend that fear.

It may seem odd that, while I consider education to be one of the more important activities one can engage in throughout life, it’s actually the model of the game that speaks most directly to what’s going on here. The Dutch historian Johan Huizinga, in a marvelous little book called Homo Ludens, explores the notion of game and gaming in historical cultures. He identifies a number of salient features — but chief among them are two facts: first, that the universe of the game is somehow set apart, a kind of sacred precinct, and, second, that what goes on there does not effectively leave that arena. I think the same can be said of education — and, interestingly, the idea of education as a game is of long standing: the Roman word that most commonly was applied to the school was ludus, which is also the most common word for game or play.

Who doesn’t know at least one student who loves to play games, and who may be remarkably expert in them, but still has difficulty engaging the subjects he or she is nominally studying seriously? In my experience, it’s more the norm than the exception. I’ve heard people decry that fact as a sign of the sorry state into which the world has fallen — but I don’t think that’s all, or even most, of the picture. One of the things that sets games apart from other learning activities is that in a game, one is encouraged, or even required, to try things, in the relative certainty that, at first at least, one is going to make an awful mess of most of them. That’s okay. You get to do it again, and again, and again, if need be.

Within the bounds of the game, one is free to fail. Even there, one should not choose to fail: doing that subverts the game as nothing else ever could. But even if one is trying to win, failure comes easily and frequently, but without serious penalty. The consequence, though, is that students learn quickly enough how not to fail. The idea that one must get everything right the first time is nonsense. The creeping fear that one needs to score 100 on every quiz is nonsense. Even the belief that the highest grade signifies the best education is nonsense. Sure, I have had some students who got extraordinarily high grades and were very engaged with the material; I have had some students who were completely disengaged and got miserable scores. But those are the easy cases, and they are relatively few. The mixed cases are interesting and hard. I’ve had a few who operated the system in order to get good scores, but never really closed with the material. They walked away with a grade — though usually not the best grade — and little else. I wish it were possible to prevent tweaking the system this way, but it often is not. In the end, though, like the student at UCLA Christe recounted in the previous post, they achieved a real failure because they chose it: they sacrificed the substance of their education in order to win a favorable report on the education. It’s a bad trade — yet another instance of the means becoming autonomous.

I have also had other students — probably more of them than in any of the other groups — who thrashed about, and had real difficulty with the material, but kept bashing at it, and wound up making real strides, and in a meaningful sense winning the battle. Christe talked about how a baby learning to walk is taught by the unforgiving nature of gravity. That’s true enough. Gravity is exacting: its rules never waver, and so it may be unforgiving in that regard. It’s also very forgiving in another sense, however. Falling once or even a thousand times doesn’t keep you down or make you more likely to fall the next time. Every time you fall, assuming you haven’t injured yourself critically, you are free to get up again and keep on trying. And perhaps you have learned something this time. If not, give it another go.

Children learning to speak succeed with such amazing speed not in spite of but because of their abundant mistakes. They are forming concepts about the language, and testing and refining them by playing with it so recklessly. A child who learns that “I walked” is a way of putting “I walk” into the past will quite reasonably assume that “I runned” is a way of putting “I run” into the past. This may be local and small-scale setback when it comes to identifying the right verb form for the task: it most definitely is not failure in a larger sense. It’s a triumph. Sure, it’s incorrect English. It is, nevertheless, the vindication of that child’s language-forming capacity, and the ability to abstract general principles from specific instances. He or she will eventually learn about strong verbs. But such engagement with what one wants to say, and such fearlessness in expressing it, is rocket fuel for the mind. The child learns to speak the way a devoted gamer learns a game — through immersion and unquestioning involvement, untainted by the slightest fear of the failure that invariably, repeatedly attends the enterprise.

When I first started teaching Greek I and II online about fifteen years ago, I came up with what seemed to me a rather innovative plan for the final for the course. Over the years since I haven’t altered it much, because of all the things I’ve ever done as a teacher, it seems to have been one of the most successful. Though in recent years Sarah Miller Esposito has taken Greek I and II over from me, I believe that she’s still doing roughly the same thing, too. I set the final up as a huge, exhaustive survey of virtually everthing covered in the course — especially the mechanical things. All the declensions, all the conjugations, all the pronoun forms, and so on, became part of that final exam. It took many hours to complete. I eventually even gave up having other exams throughout the year. Everything (in terms of grade) could hang from the final.

Everything for a year depending on a final? For a high school student? This sounds like a nightmare. I’ve had parents balk and complain — but seldom students: not when they’ve been through it and seen the results. Here’s the trick: the student was allowed to take that exam throughout the summer, as many times as he or she wanted. It could be taken with the book in the lap, with an answer sheet propped up next to the computer; students could discuss the contents with one another, or ask me for answers (though they seldom needed to: I put the number of the relevant section in the book next to each question). The results of each pass could be reviewed, and each section could be retaken as many times as desired. The only requirement was this — the last time any given section of the exam (I think there are eighteen sections, some of them worth several hundred points each) was taken, it had to be taken under exam conditions: closed book, with no outside sources. The final version had to come in by Sept. 1. Students were free to complete it at any point prior: most of them didn’t. Why should they? They were playing the game, and improving their scores. They actually rather liked it. Especially after I was able to get these exam segments running under the Moodle, so that scoring was instantaneous and painless (frankly there’s little that’s as excruciating for a teacher to grade by hand as accented polytonic Greek), they did it a lot. They’d take each segment four, five, perhaps even ten times.

The results of this were, from a statistical point of view, probably ridiculous. It tended to produce a spread of scores ranging from a low of about 98.3 to a high of about 99.9. Nobody left without an A. “What kind of grade inflation is this?” one might ask. But the simple (and exhilarating) fact was that they all came back to class in the fall ready to perform like A students. They had the material down cold — and they hadn’t forgotten it all over the summer either. This is not just my own assessment: they went on to win national competitions, and to gain admission to some of the most prestigious universities in the country — where at least some of them tested into upper division classics courses right away. If that’s grade inflation, so be it. I like to think rather that it’s education inflation. We could use a little more of that. I don’t really take credit for it myself — it’s not that I was such a brilliant teacher. I’m not even primarily a Hellenist — I’m a Latinist. But I credit the fact that they became engaged with it as if with a game.

We live in a society with a remarkably strong gaming culture; but most historical societies have had the same thing. We have surviving games from Egypt and Greece and Rome; chess comes from ancient India and Persia, and go (probably the only game to match chess for complexity from simplicity) from ancient China and Japan. We have ancient African games, and ancient Native American games. Today the videogame industry is a multibillion dollar affair. Board games, card games, sporting equipment, and every other form of game equipment is marketed and consumed with a rare zeal. These products find buyers even in a downturn economy, because they appeal to something very fundamental about who we are. Even while the educational establishment seems to be ever more involved in protecting the fragile ego and self-image of the learner, our games don’t tell us pretty lies. They don’t tell us that we’ll win every time. They tell us we’ll fail and have to keep trying if we want to win. I really think that people savor that honesty, and that the lesson to be learned from it is enormously significant.

I know that there are a lot of things that people have had to say against games, and certainly an undue or inappropriate preoccupation with them may not be a good thing. Nevertheless, they are genuine part of our God-given nature, and they form, I would argue, one of our most robust models for learning. In games we are free to fail: and that freedom fosters the ability to learn, which is ultimately the legitimate freedom to win. If we can extract any lesson from our games, and perhaps apply it more broadly to the sphere of learning, I think we all will benefit.