The so-called Golden Ratio

June 13, 2004 9:47 PM

A good article on the so-called Golden Ratio.
posted by stbalbach (19 comments total)

Essential to this article is an intuitive understanding of the concept of irrational numbers, which I'll be happy to explain if anyone is interested. By the way, "rational" and "irrational" in this mathematical usage derive from Euclid's use in the context of ratios, not in the sense of "reasonable" and "unreasonable". Of course, the Greek roots of all these words are the same. (And, interestingly, the Pythagoreans might have thought of irrational numbers as being unreasonable, and that's why they kept their existence as their greatest secret.)
posted by Ethereal Bligh at 10:13 PM on June 13, 2004

Thanks stbalbach. Also lots of good links in this old thread.
posted by vacapinta at 11:32 PM on June 13, 2004

16:9 is my golden ratio.



I got nothing.
posted by cinematique at 12:49 AM on June 14, 2004

EB, you pique my interest. Can it really be said that they kept them a secret? To me, irrational numbers merely ("merely") frame the actual inadequacy of number mathematics in facing the world; they demonstrate something geometers could do that mathematicians couldn't.

But my grasp of mathematical theory fails at that point (and perhaps before). I for one would appreciate if you did explain, at least to the extent you feel feasible without annoying the others...

Aside: I find the obsession with the Golden Mean much more interesting than the phenomenon, itself...
posted by lodurr at 3:47 AM on June 14, 2004

great link stbalbach. I'm also interested in hearing more about irrational numbers EB, got some nice links to share?
posted by dabitch at 5:48 AM on June 14, 2004

Can it really be said that they kept them a secret? To me, irrational numbers merely ("merely") frame the actual inadequacy of number mathematics in facing the world; they demonstrate something geometers could do that mathematicians couldn't.

are you saying, people could figure them out themselves? The story with the pythagoreans is that they believed that mathematics was sacred and revealed truths about the universe of a divine nature; the existence of irrational numbers was a danger to this belief. We're used to them, but really it is difficult to comprehend how a number can have no ratio to something, be indistinct in some sense. We've basically accepted this through calculus - a number always nearing a point but never reaching it. But the greeks thought of numbers as concrete. The pythagoreans wrote their numbers as little dots. The highest form of math known to the greeks was geometry, the study of the ratios of parts of different shapes.
posted by mdn at 5:59 AM on June 14, 2004

that's why they kept their existence as their greatest secret

Maybe, but we're in territory where history merges into legend. Some commentators said that the Pythagoreans hushed up the discovery of irrationals; at MathWorld's Irrational Number page there's a good story, largely put about by Euclid, about how they were so fazed that they chucked the discoverer overboard. Other versions say they celebrated with a massive sacrifice. Anyway, it's good to see a site that debunks phi-mysticism.
posted by raygirvan at 7:40 AM on June 14, 2004

"the golden ratio" has been the name of my LJ for quite a while now. prolly since the last mefi post, heh.
posted by taumeson at 7:50 AM on June 14, 2004

A striking example of how strange irrational numbers really are is pi. If a perfect circle could somehow actually exist, either its diameter or its circumference would have to be irrational. Think about what it would mean to have a diameter which is an irrational number.
posted by callmejay at 8:32 AM on June 14, 2004

By the way, "rational" and "irrational" in this mathematical usage derive from Euclid's use in the context of ratios, not in the sense of "reasonable" and "unreasonable".

:::lightbulb goes on:::

Thanks!
posted by rushmc at 9:18 AM on June 14, 2004

:::lighbulb goes dim:::

Something I'm not grasping, here.

I know this isn't math class, but bear with me for a minute.

If "irrational" as applied to irrational numbers is 'derived from use in the context of ratios', wouldn't Pi be a rational number?

That is, since Pi can only be accurately expressed as a ratio, and never as a number, wouldn't that make it rational?
posted by lodurr at 10:00 AM on June 14, 2004

Pi cannot be expressed as a ratio of rational numbers.
posted by callmejay at 10:10 AM on June 14, 2004

lodur: rationals are defined as numbers that can be expressed exactly as a ratio a/b, where a and b are whole numbers (e.g. 22/7 or 355/113). Irrationals, such as pi, can't.

Many geometrical processes throw up lengths that can't be expressed in this form in the unit system used as 'input' for the drawing. It's just a property of our decimal system. For instance, if you draw a square whose side = 1 inch, the length of the diagonal is a classic irrational, SQRT(2).
posted by raygirvan at 11:42 AM on June 14, 2004

"Pi cannot be expressed as a ratio of rational numbers."—callmejay

If it could, it'd be rational...of course.

In the Euclidean sense, any qualitatively similar things can be put in ratio with each other, including a rational and irrational length. (Because both are lengths.) The diameter and the circumference of the circle are incommensurable to each other, but it's no big deal to put them in a ratio to each other, diameter:circumference. That relationship is definite, in just these terms there's nothing odd about it. It's only when you try to express that relationship using numbers that things get weird.

Incommensurability, irrational numbers mini-explanation to follow in a bit. I have enough time to be brief.

On preview: "It's just a property of our decimal system." Nope. You're conflating irrationals with rational numbers that cannot be fully expressed in a given number base. They're not the same things. But the way that school math is often taught makes it easy to think the two are the same.
posted by Ethereal Bligh at 11:50 AM on June 14, 2004

Nope. You're conflating irrationals with rational numbers that cannot be fully expressed in a given number base.

Yes, but a handwaving explanation seemed in order rather than wading into rigorous definitions way above the level being discussed. MathWorld is happy to introduce irrationals, in part, in terms of the decimal expansion properties.
posted by raygirvan at 12:41 PM on June 14, 2004

Hmm. I think that's a very, very bad approach to the concept. It's very misleading. A rational number with a never-ending decimal expansion is far, far more the same sort of thing (well, the point is that it's exactly the same sort of thing) as a number with a limited decimal expansion than it is an irrational. An irrational is qualitatively a completely different beast than, say, a never-ending decimal expansion. And a rational number that is a never-ending decimal expansion can always be expressed as a limited expansion in some other number base (and the other way around). An irrational can never be expressed in a limited expansion regardless of the number base (well, if the number base itself isn't irrational).
posted by Ethereal Bligh at 12:53 PM on June 14, 2004

Um, the end of the third sentence should have been "...a never-ending decimal expansion of a rational number" (the last four words are important and I left them out).
posted by Ethereal Bligh at 12:56 PM on June 14, 2004

By way of an aside, a small (and not at all scientific) survey I carried out myself a few months ago revealed that all architects polled knew about the GR, and all believed that other architects used the GR in their work, but none of them had ever used it themselves.

I have.

Make whatever inference you wish.

That Mr. Devlin should stick to mathematics…?
posted by Dick Paris at 3:07 PM on June 14, 2004

Okay. Frankly, I didn't (as had been hoped) spend some time preparing something on this. I'm going to have to wing it.

Some background: doing Euclid in college, and then building through western mathematics as it historically developed (also, science) really made me come to strongly feel that many deeply interesting things, and some very important "get your hands on it and really understand it at some intuitive level" stuff is being short-circuited by the contemporary pedagogy which presents this stuff in its modern context. The thing is, the whole history of western rational thought is increasing abstraction of terms and ideas slowly but inexorably moving away from their original definitions...often to places that would be unthinkable and paradoxical from the perspective where they started. It's important to understand that (in my opinion) these later conceptions are just as "true", arguably more true, of course. But you get this mix of anachronistic terms, ideas that made perfect sense in their original incarnation being reincarnated into something (apparently) completely different...and this can be very misleading to anyone that doesn't have a perspective on it. In a way, this is part of the reason, I am guessing, that the "new math" pedagogy came to be popular in the sixties. The idea was to deeply formalize and abstract things at a very high level and avoid a lot of the historical cruft that is likely to mislead students. However, many people feel that this doesn't really work. One reason might be that people (at least children) aren't built to understand things as purely formal systems. They still want things rooted in common sense.

And we arrive at the Greeks. The most important thing to understand about the Greeks were that they were a radical combination of idealism, abstraction, and practicality. I really don't think that their way of thinking is easily accessible to us, since we tend to see things, for example, as dualisms where they see unity. Anyway. My point is that their approach to math is essentially the same as a really, really, really bright child would take in trying to understand the world around her. They thought about geometric shapes and relationships, but they only thought about geometry because, to them, geometry was real. Looking around them, they saw geometry. As opposed to say, numbers.

So. We're thinking about a square. Simple enough, right? you've got a square, and you draw a diagonal in it. Now, the sides of that square are a certain length, and the diagonal is a certain length. It's also pretty obvious that the relationship between the length of the side of a square and the length of its diagonal is always exactly the same. As a ratio, it's side:diagonal. (Note: ratios aren't fractions! Or are they? You get there from here.)

Now in the real world, when we have lengths of things that we want to compare, and determine their relationships, we measure them, right? Presented with, say, a square table with a diagonal drawn across it, what would that bright child do? She'd first copy that length of the side, and lay it along the diagonal. Hmm, okay, the diagonal is longer. With a little fiddling around, she'd discover that it's not twice as long, however. So now what to do?

What that child would do, what most of us would do, is figure that there's some way we can cut up the length of that side of the square so that it accurately measures the diagonal. So we cut the side into, say, eights (by repeating halfing, maybe). Nope, still won't quite measure the diagonal. We try some more divisions. Nope.

Now, in the real world, because things are fuzzy and our ability to cut things up and measure them is inexact, we'd likely come up with some division of the length of the side that is able to, it seems, measure the length of the diagonal. But before we reached that point, we or our very, very bright child might have thought, "huh, there must be a way to calculate this, to come up with the answer exactly, in abstraction". So we get out our self-taught geometry notes and try to figure out how to divide that line up so we can measure the length of the diagonal.

And what we discover, to our amazement, is that it's not possible. There is no division—a bazillion bazilion tiny little bits, even—that we could divide the length of the side of the square so that we use those little bits to exactly measure the length of the diagonal. No matter what we do, that last little bit is going to be just slightly too short or too long. As you approach an infinite division of the length of that side of the square, you'll still find that the bit is just a little too small or big to exactly measure the diagonal.

Now, we could have gone the other way. We could have tried to divide the diagonal into parts in order to measure the side. The same thing would have happened. Ultimately, what we discover is that, in terms of our commonsensical understanding of "measurement" it's simply not possible to measure the diagonal of a square with its side. Or vice-versa.

Isn't this truly contrary to common sense? It (supposedly) deeply offended the sense of the Pythagoreans, and that's why they kept it as their secret (or mystery, as you prefer).

Strictly speaking, in this sense, you can't really say that one of those lengths is "rational" and the other "irrational" intrinsically. Incommensurability means that one can't measure the other. However if you take the side of the square as your number unit, "one", then you can't say what the length of the diagonal is in those units in the way that we conventionally (commonsensically) expect we could. Or, conversely, if you make the diagonal the length "one", your "unit", then you can't state the length of the side in those conventional units. So, either way, you have the number "one", and other numbers like it (including, as should be clear, numbers with what are effectively partial units, like one-half, one-third, etc.), and you have things like these other values, whatever the diagonal of the square is, or the circumference of the circle (if you make the diameter your unit) that you can't say what those other numbers are, exactly. That's not to say that you can't talk about the relationship between the side of a square and its diagonal in other ways. It's a definite, known ratio. If an "irrational" number isn't exactly a number the way our commonsense tells it should be, that doesn't mean that it's not exactly what it is, or that what it is isn't knowable. We know what it is. It's just not what we expect. Frankly, we expected everything to be expressable as what we think of as "numbers". But, alas, here's a very important thing we cannot.

This is already too long, so I'll stop there.
posted by Ethereal Bligh at 10:01 PM on June 14, 2004