Saturday 18 December 2010

Dan Brown part 2: φ

Just as I recently announced I have written an article about one of the many things Dan Brown could and should have gotten right in his books but did not. Those things are legion; and people more patient than me have already had their shot at it. For example there is a really excellent (and really long) article in Danny’s (don’t know his surname, I’m afraid) blog No Loss for Words titled Dan Brown is a fraud: A list of errors in Angels and Demons, which deals with this one book alone, and to which I, some years ago, had the honor of contributing a little thing as well.

Despite this abundance of material I have decided to confine myself to a single passage in the book The Da Vinci Code, which deals with a mathematical fact, which naturally interests me.

For your enjoyment here is the passage in its whole atrocious glory:

He felt himself suddenly reeling back to Harvard, standing in front of his “Symbolism in Art” class, writing his favorite number on the chalkboard.
1.618
Langdon turned to face his sea of eager students. “Who can tell me what this number is?”
A long-legged math major in back raised his hand. “That’s the number PHI.” He pronounced it fee.
“Nice job, Stettner,” Langdon said. “Everyone, meet PHI.”
“Not to be confused with PI,” Stettner added, grinning. “As we mathematicians like to say: PHI is one H of a lot cooler than PI!”
Langdon laughed, but nobody else seemed to get the joke.
Stettner slumped.
“This number PHI,” Langdon continued, “one-point-six-one-eight, is a very important number in art. Who can tell me why?”
Stettner tried to redeem himself. “Because it’s so pretty?”
Everyone laughed.

And I laughed too, a kind of sad, hollow laugh. Who would have thought it possible to get so much wrong in so little a piece of text? We are indeed witnessing a master at work. (Please excuse my bitterness, this thing happens to touch the field of my profession—who wouldn’t be touchy there?)

Most of which can be said (complained) about the matter has been said by the world’s most intelligent human being, the uncomparable Cecil Adams, but I’d like to add a few points of my own.

First of all, we are talking about a Greek letter here, φ. Am I too demanding in assuming that it should be possible, in the 21st century, to print some Greek letters in a book instead of writing it as “phi”? I mean, obviously this blog can do it, why not Doubleday Group publishers? Alright, minor point. By the way, if anything it should be “phi”, not “PHI”. “PHI” is obviously uppercase, which in Greek is Φ. The Golden Section (which is the thing meant here), however, is always a lowercase φ.

Secondly, the main point of Cecil Adams, φ is not 1.618. Just like 1/3 is not 0.333. φ is an irrational number, which have a unending, nonrepeating sequence of decimals after the comma. You just cannot write them that way. How to do it, then? Well, simply write

φ=

which is the definition of φ. If you like to have the decimal expansion, please at least write it with dots: φ=1.618… But this thing that Brown (or Langdon, depending on how you look at it) wrote down, simply is not φ.

But this has been said before. Now I come to that which has not.

It is this: There are much too few letter in the alphabet for mathematics. There are 26, each lowercase and uppercase. Letters from other latin script variants are not used (e.g. ä, ç, ñ). And then there are the greek letters, where those that look like latin letters are excluded (an uppercase α, for example, looks just like an A). There are even a couple of hebrew letters that are used. Still it is too little. Therefore all letters are reused all the time and each one is used to name at least half a dozen things from various areas of mathematics. φ too. The most important are as angle parameters in functions, and the Euler φ function.

The point I’m trying to make here is that the Golden Section is not an important mathematical concept, and even less a frequently encountered one. I must have written several thousand φs since I began studying mathematics, and at most 20 of them denoted the Golden Section. If you ask a mathematician what φ is, in 99 of 100 cases (or more) the answer will not be Brown’s. The concept of Brown is called the Golden Section or Golden Ratio. If we name it with a letter, it is traditionally φ, but most of the time φ denotes something else.

And finally:

“Not to be confused with PI,” Stettner added, grinning. “As we mathematicians like to say: PHI is one H of a lot cooler than PI!”

If that guy is a mathematician he must be very drunk. Or on drugs. Or both. Because what he say is utter, utter bosh.

How can anyone confuse φ with PI (π)? Once a tutor told us not to confuse “angles” with “angels”, but that was a joke. A singularly lame one, but still. Anyway, “As we mathematicians like to say: PHI is one H of a lot cooler than PI!”. No, we mathematicians don’t say things like that.

Dan Brown, as he has showed on other occasions, has a rather weird conception of the sense of humor of scientists. There are indeed mathematicians’ jokes and physicists’ jokes, which other people don’t find funny at all (neither do mathematicians, to be honest). Its an interesting phenomenon and I may write a short piece about it some time. [Here it is.] Anyway, these jokes are in no way like these things Brown puts into our mouths. Not a bit. It’s a completely different style.

And finally, no mathematician would say that φ is cooler than π because it isn’t. It simply isn’t. φ does have a couple of beautiful properties, mathematically, but they are rather trivial. There is nothing a high school student could not understand. For example, that there is a relationship between φ and the Fibonacci numbers is completely correct, but it is no difficult or deep thing. I have written a PDF document that shows how it is done and you will probably not need more than 10 minutes to read and understand all of it (it’s also in Wikipedia).

π, on the other hand, is a monster that has baffled mathematicians for millenia. It drove the ancient Greeks crazy. There’s still many unsolved question left that involve it. It pops up in unexpected places. It is the reason you cannot square the circle.

(Here is my first article about Dan Brown.)

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