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Friday, May 1, 2009

Imagine a Perfect Cube

This post grew from a comment I was going to write on my post Pi Whats, but as the comment grew and grew I felt it deserved its own post. It's about measurement and precision, and how whether you think they matter in mathematics depends on your perspective.

Do you think a string can be pi inches long?

You need to ask yourself this question:

Do you think a string can be 1 inch long? I mean EXACTLY 1 inch long?

If your answer is yes, a string can be exactly 1 inch long, then you automatically think it can be exactly pi inches long. Get a sphere whose diameter is exactly one inch, wrap the string around its circumference one time and cut it the exact length of the circumference, and the string is exactly pi inches long. If you are going to try to argue with me on this particular conclusion, then you are wrong. Pi is not some ill-defined Mysterioso of a number. It's very precisely defined. It's the circumference of any circle divided by its diameter. It's the circumference (in u units) of a circle whose diameter is 1 u.

If your answer is no, a string can't be exactly 1 inch long, then I'm okay with you saying it can't be exactly pi inches long, either.

The larger philosophical concept is that mathematics deals with ideal, perfect, abstract constructs that can really only exist in our minds. I can imagine a line segment of unit length. Cutting a physical piece of string with scissors to be exactly 1 inch long is much more treacherous terrain.

Scientists understand this, that's why they invented significant figures, and their calculations incorporate expected percent error. Whenever you measure something, you are necessarily rounding your measurement to the nearest level of precision on your measuring device. If you use a metric ruler with millimeter markings, and you measure something and say it's 2.6 centimeters, that means the thing you measured was closest to the marking at 2.6. It could be anywhere from 2.55 to 2.65 centimeters long.

Therefore, if I am thinking like a scientist or engineer (possibly using math but not thinking like a mathematician) dealing with only objects I can hold in my hand, I don't think you can point at a piece of string and say "that string is exactly one inch long".

But in math we don't often discuss these problems of measurement. Because in math we deal in abstract perfection that can really only exist in our minds. We don't stop every time we give a measurement in a math problem and talk about the imprecision inherent in measuring something. In the physical realm there is no perfect square. No perfect right angle. No perfect circle. No length that is the square root of 2. But in my mind, the string can be exactly one inch long because I say so. These things can exist in our imaginations. In my mind, I can picture a square with sides of unit length, and draw in a diagonal, and I have created a length that is exactly the square root of two. In my mind, I can reason that somewhere between a string that is clearly less than one inch long and a string that is clearly greater than one inch long, there is a place I could theoretically cut that would be exactly one inch long. So as a mathematician, I say the string can be one inch long.

I'm reminded of this litmus test from Out of the Labyrinth: Setting Mathematics Free:

Close your eyes and imagine a perfect wooden cube. Now saw this cube three times, along each of its three axes.

Question: did you see sawdust?

If so, you're no mathematician.


  1. It goes even further than that. It's precisely because mathematicians think in such abstractions (and are aware that they think in abstractions) that they can understand the nature of the abstractions better.

    To be a little more concrete, Euclid wrote the postulates in the Elements with the intuitive meaning most people have, that "these are things we can do". Given a center and a radius (one unit, for example) we can draw a circle on the board with that center and radius. But that's actually not the way we read the book anymore.

    Instead, we say that we're defining a formal system. We have things called "points", and things called "lengths", and things called "circles" (which have "centers" and "radii"). And given a point and a length, we can draw a circle with that center and radius. This is one of the basic pieces that go into the system, and any collection of definitions that satisfy this property (and the other four) will allow us to conclude everything else in book I.

    In particular, pairs of numbers constitute "points" in the plane. The distance function gives us a notion of "length", and a "circle" is defined as a locus of points (there's that word "locus" again). And yes, given a pair of numbers and a number there is a corresponding locus of points.

    The question you're addressing here is a non-mathematical question, that I find only someone thinking like a mathematician can even ask. That is, does the real, physical world satisfy the common notions and postulates of Euclid's book I? In particular, does the notion of length "exist" in the real world, and satisfy all the formal properties we speak about in mathematical terms?

    But this is a high school math class, which raises two points. First, this is a high school math class, and the subtleties involved in the question are far beyond the high-school question. Secondly, this is a high school math class, and the question is inherently extra-mathematical. So either way, it's outside of anything these students need to concern themselves with.


  2. Thank you for taking it further, unapologetic. You made my post way better. I feel lucky to count you as a reader and commenter.

    I don't necessarily agree that the basics of the philosophical dialog are beyond the abilities or interest of most high school students. Now, I do NOT spend more than a couple minutes, if any, on this when I am trying to introduce radians. But I do think there's a place for addressing it appropriately with high school students. No they probably don't need to know about it, yet, but it's interesting, and makes them consider "what is math, really", and makes them more educated humans than does being able to rattle off an exact value for cos(pi/6).

    But this post wasn't for them. It was for people who think a string can be 1 inch long, but not pi inches long. :-) The point was: it can be both, or it can be neither, but you have to pick...and I couldn't explain it in a couple paragraphs.

  3. I like this way of thinking, too. I point out something similar in the first week of my calculus class. In particular, I say, "When I present a problem, I am giving you infinite significant digits and I expect the same in your answer when possible." That's because they're so used to doing things in their calculators and giving the answers as decimals, sometimes rounded only to 2 places. I'd much rather have a fraction like 215pi/17 (as nasty as they seem to think that is).

    I also like to take time out once in a while to talk about the philosophy of what we're doing in this sense and unapologetic's sense, too. I've always found it interesting that we are taking a discrete world (I don't think we could have a 1inch string because of atoms/quarks quantized energy, etc.), model it with a continuous function, then discretize it again to get a handle on the function (riemann sums, for example), and then go back to continuous by taking infinite limits.

    On days like those, the kids know to just sit back and try to figure it out without exploding. In fact, they call them "blow your mind days." I don't think they completely understand what's going on, but I also can see that they love talking about the deeper meanings there. And that's where I disagree with unapologetic as well. At a time in their lives when they are most trying to figure themselves out (middle and high school), these kids who are good academically seem to enjoy a little philosophy talk and throwing another viewpoint into the mix of trying to find themselves in this world.

  4. Thanks for the post Kate. I like talking with the philosophy of math with my students too, as Dave talks about. The world is described by the rationals, but irrationals are a nice convenience to have to get the work done.

    I fully accept the "If a string is exactly one inch then..." argument, though I like to play with it. For me, pi is my straw person. The kids I deal with are comfortable with pi so I like to really point out how otherworldly and abstract it really is. How we never ever use pi to build anything, we just use rational approximations. That real numbers aren't all that real or at least that complex numbers are just as "real" as the "real" numbers.

    As for sawdust, I dual boot into Math and Physics :-)

  5. Either here or here, Martin Gardner had a take on a related story:

    An alien visitor from outter space wants to take back home a copy of the Encyclopaedia Brittanica, but it won't fit in his spaceship. SO he codes the entire text, by writing the ASCII code of every symbol in a very long string. He then adds "0." at the beginning, and has thus coded all the knowledge of mankind in a single rational number larger than 0 and smaller than 1. All he has to do, with his high precision extraterrestrial technology, is make a mark on a metal bar, that precisely divides the bar into two parts, whose ratio yields the desired rational number.

    I definitely see metal dust when I try to picture this...

  6. I get the point of your article (inch yes/pi no is intuitive to some but wrong). But there is a bit of automatic abstraction in your problem.

    First, what does it mean to be EXACTLY one inch long? Do we worry about a filiment of thread extending beyound the main body of the string? Do we measure to the last molecule? The orbit of the outermost electron? Being EXACTLY one inch long is already an abstraction.

    Second, just because you might be able to have a string that is one inch long does not mean you can have a sphere with a diameter of one inch - it is harder to construct a sphere (why not a circle?) than a piece of string.

    Third, and most importantly, the question, 'Can a string be pi inches long?" is a different question from "Can you know when you have a string that is pi inches long?" Take a piece of string .95 inches long. Stretch it to 1.05 inches. No doubt is was, at some moment, EXACTLY one inch long in any sense of the word.

    I once advised a fifth grade math club. We started each session by considering math problems. The object was to come up with an answer that was not the intended one but had an explanation that fully supported the unintended answer. The kids loved it, and were incredible in both their inventiveness and critical thinking. I would hate to see that wit and intellectual joy be subdued by high school math.

  7. Take a piece of string .95 inches long. Stretch it to 1.05 inches. No doubt is was, at some moment, EXACTLY one inch long in any sense of the word.I can doubt EXACTLY that assertion. You're essentially relying on the intermediate value theorem, which ultimately goes back to basic topological assumptions about the real line being a continuum. But there's no ironclad evidence of the existence of even countably infinite sets, let alone the uncountably infinite sets one would need for a continuum to exist in the real world. So it's conceivable that spacetime and everything in it is actually not continuous, and that the string manages to jump past being exactly 1cm long, just like squares of rational numbers jump past the number 2 without ever being exactly 2.

  8. Hi Anonymous - Those kinds of questions and objections are what thinking about this is supposed to provoke.

    It sounds like you think the string can not be 1" long, if you are considering filaments and molecules. :) Any measurement is really an approximation!

    I realize it was a leap to assume a 1" diameter sphere follows from the existence of a 1" string. Would it be better to consider a cylinder? I went with a 3D object because it's no mean feat to wrap a string around a flat planar circle, like a CD. But if you can have a 1" string, why not a 1" diameter? Why not be able to fabricate a 1" diameter sphere?

    As unapologetic has been demonstrating, I think these questions are really unanswerable to us mere mortals, and require an understanding of number theory and analysis from first principles. (Which I might have sort of had at one time, but no longer do. If anyone knows a math professor named Jean-Francois LaFont, don't tell him I said that.)

    There is an important place for intuition in math - the "real" math guys tend to poo-poo it - but really that's where it all came from. (In the beginning were the natural numbers, and then Grok tried to share 10 griddle cakes among himself and 2 of his clansmen, and all hell broke loose.) It's fun to consider the trouble you can get into if you rely on intuition alone.

    If you can have a 1" string and not know you do...can you really have a 1" string? If you don't know about it? :-)

    I would love to hear more about your fifth graders and that exercise where they came up with justifications for unexpected answers. What were the problems like? What did they come up with?

    Mr. Condescending, I can't tell if you are being condescending. :-)

  9. > The object was to come up with an answer that was not the intended one but had an explanation that fully supported the unintended answer. The kids loved it, and were incredible in both their inventiveness and critical thinking.

    Anonymous, this is great. I do hope you'll tell us more.

    Kate, I'm thinking your "There is an important place for intuition in math - the "real" math guys tend to poo-poo it" remark paints with too broad a brush. Unless I'm misunderstanding who you mean by 'real' math guys.

  10. There is an important place for intuition in math - the "real" math guys tend to poo-poo itTo the contrary. Where do you think we get started? Yes, there are people who think like you say, but they tend to be more the sort who use math in their jobs, but never got beyond differential equations into the creative end of the math pool.

  11. I think sometimes math does itself a disservice when it beats the public to death with rigor. Mind you I love rigor, I love the shiny solid beautiful towers of mathematical reasoning. Abstraction can be breathtakingly beautiful. But you have to know your audience.

    Physics is proof that you can relax the rigor and get results. They (physicists) have a strong sense of intuition about what they're modeling and while mathematicians may cringe at times, the physicists accomplish wonderful things. Of course, after play time we have to come in and clean up for dinner and that's where math makes a lot of headway. Setting up the formal systems, abstraction, finding counterxeamples, tweaking everything so its just right.

    I prove less and less as years go by and more "reasonable discussions" What might hold up for a proof 300-400 years ago perhaps, but with more pictures and conversation.

    just my two sense.

  12. OK OK my bad. Apologies. I said that poorly. (Does that make me the "apologetic mathematician"? Har har.) A more honest sentence would be "I am insecure about not having a math degree and intimidated by people who do." But people like unapologetic and Calculus Dave regularly do a fantastic job in comments around here and elsewhere bringing rigor into the discussion. But even when done with skill, you have to admit, the formal, purely abstract, axiomatic structure can turn people off.

    I see it in students all the time - if they can 'see' a relationship, if it makes sense and they can reconcile it with their observations of the world around them, they are much more able to incorporate it into their problem solving arsenal. I run into trouble in Algebra 2 with things like the log laws and the angle/sum difference formulas. Sure I can show them an algebraic proof, and they agree with every step, but it's hard for them to intuitively agree with why sin(A+B) has to = sinAcosB + cosAsinB. Couple that with the fact that we have so much crammed into that course that there is not time to give kids lots of examples and direct experience with the relationships... and then we lament and wonder why the general public hates and fears math.

  13. A student recently coined the phrase "Pushing the 'I Believe' Button" to describe that necessity for getting through Algebra 2.

  14. The geometry book I have has the annoying tendency of creating drawings that 'tempt' you. Triangles with 89.9 degree angles, lines that are not quite parallel (they probably intersect about six miles away) and so on. I get the intention, but to a kid: "its close enough." which is true. If they mis-apply a postulate or a theorem on such a diagram they won't be that far off.

  15. Kate: you're a mathematician. I only have a master's and I call myself one. What makes a mathematician? A love of mathematics, an appreciation for all the conversation we're having right now.

    Mathematicians come in all varieties, but my favorite type are kids.

  16. Kevin - Which book is that? That does sound annoying. Good observation with "Know your audience".

  17. I put a sign up in my classroom (and in my office) that says 'Real mathematicians ask why."

    I made it when I had a beginning algebra class of older students years ago who always asked why. I told them they might be novices, but they were real mathematicians because they were trying to make sense out of it.

  18. Forget about "high school math classes", man this stuff works with middle school kids. Start with definition 1 in Book 1: A point is that which has no part and walk away. Kids will gnaw on that for hours...especially when they realize that as soon as you draw a point on the board it isn't a point anymore. With some of my advanced kids, it seems like these philosophical discussions is what keeps them coming back. It is the carrot I dangle to get them to show their stinkin' work.

    Nice post Kate. It has had me thinking all weekend.

  19. sorry this is a little off topic, but could you show me how to measure a hip roof without being up on the roof? I would have the width of the eaves, the width of the roof ridge, and the height of the ridge (not the sloped height, but just the verticl height). Hopefully this makes sense.

  20. Well, that's true, that there's still something to chew on there. But a modern reading of Euclid sort of discards the definitions. A point is whatever you say it is. A point is a pair of numbers. Or a point is a line in the projective plane.

    The point (har) is that the Elements have some basic terms, and some basic relations between those basic terms, that can be filled with whatever you want, as long as certain statements about those terms and relations (the postulates) are true. Then everything else in the book (the propositions) holds true, as interpreted in terms of how you filled in the basic pieces.

  21. Using imagination in math can be tricky. For example, to establish the infinitude of primes, you imagine the largest prime and use it to construct a larger one, thus proving that you only imagined that you imagined it.

  22. to Kate: "Geometry" by Larson, Boswell, Kanold, and Stiff

  23. Unapologetic, your point, “it's conceivable that spacetime and everything in it is actually not continuous, and that the string manages to jump past being exactly 1cm long,” is well-taken. I was working up an answer based on the belief that an inch is defined in terms of a physical bar, but upon a bit of research, I learned I’m hopelessly behind the times – in 1958 it was defined in terms of a meter, and in 1983 a meter was defined to be the distance travelled by light in 1/299,792,458th of a second. The Wikipedia article continues on to say that the standards board neglected to distinguish between a quantum vacuum and free space. Just in case anyone thought filaments and electrons were too picky.
    Kate, to your question, “If you can have a 1" string and not know you do...can you really have a 1" string? If you don't know about it? :-),” I would answer, “Sure.” Can you have your trousers 33” long and not know about it? Would you say I really don’t have trousers on?:-) Your argument reminds me of the joke, “If a man says something in a forest, and there’s no woman to hear him, is he still wrong?” On a bit more serious note, I think a fundamental assumption of science is that there is an underlying reality which exists independently of its apprehension, Bishop Berkeley notwithstanding.
    As to the game of thinking up unintended solutions to problems, I will only be able to give you a bit of flavor – it was several years ago and my memory can be described as pre-Alzheimer’s at best. The problem that started it all was intended for younger kids, and went something like this: “In a parade, two giraffes, three elephants, a lion, and two hippos pass by. How far back in the parade is Simba?” I mentioned that ‘simba’ is the Swahili word for lion, just in case anyone had been on Mars and missed 'The Lion King.' Right away someone pointed out Simba might be one of the giraffes or any of the other animals. Others had the animals walking side by side in various configurations. It was suggested that the parade did not start with the animals. It was also suggested that the animals weren’t listed in the same order as that in which they passed by. And so on.
    I presented Kate’s inch/pi inches problem at the dinner table last Sunday. My wife went for inch yes pi inches no. Much to my delight, two of those fifth graders, now eighth graders, argued against my wife’s position with an enthusiasm usually reserved for questions like which shoes go with which skirt.

  24. I am delighted that your family was discussing this at the dinner table. Awesome! Yesterday I heard on NPR a cartoonist being interviewed who said he painted his dining room with that chalkboard know, if I had kids, I would totally do that. Except for math. Not cartoons. Well maybe cartoons sometimes.

    I wouldn't say you weren't wearing trousers, just that you wouldn't know they were 33" long. I does have a 'tree falls in the woods' quality.

  25. > Now saw this cube three times...
    > Question: did you see sawdust?

    If it said "cut this cube", I wouldn't see sawdust. That's the math puzzle convention I'm used to. Sawing can take out 1/16th of an inch, so it might be a clue that we're looking at a problem involving critical points, boundary conditions, etc. I think I found myself visualizing a saw, _tentatively_ with sawdust.


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