The other day I came across a book at barnes and noble that intrigued me. I discovered it while looking though the science section. It claimed that phi- the golden ratio- was essentially one of the most important numbers of all time. Phi, 1.6180339… , apparently appears in all sorts of weird places. I’ve done a little reseach myself and found it’s linked to the pentagram, the shell of some sea mollusk, and fibonnaci numbers. Unfortunately, I did not buy the book because it costs $25 and I’m a cheap college student. So,whats the straight dope on this number? Where else is it important, and why haven’t I heard of it before?
Never hear of Phi. I was thinking of Pi though, but judging by the depth of your searches I might need to do a little reading.
I am not all that fond of Phi myself, so I would be hesitant to call it an important number. Because Fibonnaci numbers show up in nature, Phi is sort of a natural number. It’s very good for making pretty patterns, but not all that useful for doing mathematics. Not compared with something like pi anyway. I believe that in the movie Pi they call it theta, but I could be remembering wrong.
Oh dear Cecil. Ryan, Hyyru; Hyyru, Ryan.
Or however you spell that guy’s name.
Anyone want Cherry Phi?
The Christmas (2002) edition of New Scientist ran a story about how much of what has been said about phi being the “The Divine Proportion” and “golden ratio” is just after the fact rationalisation. The stories about phi being used in the construction of the pyramids or the Parthenon, or the framing of the Mona Lisa’s face, are just BS.
Desmostylus got it right with the idea of the “golden ratio”. John Cleese made a TV programme recently that discussed this ratio and how it was applied to how we identify “beautiful people”. The idea of a “golden rectangle” (with the ratio of 1:1.6180339…) is well known in classical architecture/art. I am not convinced by the New Scientist assertion that this is just “after-the-fact” rationalisation. However, I did not see the programme myself. Without wishing to get in to a Great Debate, I personally believe that the “Golden Ratio” is fundamental to all things beautiful (Georgian architecture, Renaissance Art, conifer trees etc etc…
In case you’re wondering, RyanD004, there was(is?) a member here by the name of Hiryuu that has opened numerous Phi-related threads, as far as I can tell, nobody was ever really able to penetrate the meaning of his posts because he wouldn’t respond properly to questions and kept ranting on about how space is doughnut-shaped when viewed at a quantum level multiplied by love (or something quite like that).
For this reason, please expect some of the current membership to view threads about Phi with a somewhat jaundiced eye.
Anyway, I seem to remember someone saying that Phi appears a lot in nature because it is related to efficient packing algorithms and the like - if this is true then it makes sense that various evolutionary processes might have stumbled on it by chance (or even tended toward it, if the efficiency gradient is not terribly steep) and stayed there.
No one has mentioned this, so I thought I would throw it in. Phi (I’ve never heard it called that either, but let’s run with it) is not just some random pretty number, it comes from geometry, and it’s easy to derive.
Phi is the ratio of the sides of a rectangle, constructed so that if a strip is cut from one end of the rectangle to leave a square, the strip and the original rectangle are similar (the exact same shape and proportion, different size). If we let the short side of the original rectangle = 1, and the long side = (1 + x), the strip would have dimensions of x and 1. Since they are similar, you can solve for x by the proportion:
x/1 = 1/(1 + x)
This rearranges into the quadratic equation:
x^2 + x - 1 = 0
Use the quadratic formula to solve for x and you get x = (-1 + [the square root of]5)/2. You also get a negative solution, which turns out to be -(1 + x), for the value of x obtained above.
Then Phi = 1 + x, or the opposite of your negative solution.
You could take a golden rectangle and cut an infinite number of progressively smaller sqaures off of it, and you will always have a golden rectangle left over.
I assume most of the people responding to your original question take this ratio as a given, but if you bothered to ask about, I figured you might not know where it comes from.
Phi isn’t the most important number, but it is a devastatingly interesting one. See the book The Divine Proportion: A Study in Mathematical Beauty by H.E. Huntley, published by Dover:
There are other books on the topic, but Huntley’s is one of the best at showing how ubiquiotous it is, showing up in places you would never guess. On the other hand, I think some aspects of phi have been overrated. The claim (first made by Pertner, I think, in the 19th century) that it was the “most pleasing rectangle”, and therefore shows up everywhere is just wrong – virtually none of the commonly-used rectangles – postcards, TV screens, movie ratios (widescreen and old format), paperback books, notepads, etc. – are “golden ractangles”. If it really were the most pleasing shape, at least some of these would be.
One could make a better case for e, the base of natural logarithms, being more “important” that phi. In fact, though, these are just two of the many numbers that show up with surprising frequency in mathematics. There are many such numbers, and shapes and mathematical situations that are similarly prevalent. Have a look at D.G. Wells’ Dictionary of Curious and Interesting Numbers:
or his equally interesting Dictionary of Curious and Interesting Mathematics:
Most important? Sheesh, what about most popular, most likely to succeed, and most studious? Cutest couple?
Here’s the best fivesome: e[sup]i pi[/sup] + 1 = 0
And don’t forget the landmark equation:
Phee / Phi = Pho • Phum
( I think Newton discovered it while standing on a giant’s shoulders.)
Phi pops up a lot in Computer Science.
Look at it this way, 2 is of course absolutely crucial in CS. One common way 2 pops up is in recurrence relations where, e.g., a tree of depth n is built up of 2 trees of depth n-1 giving recurrences like:
f(n) = 2f(n-1).
Sometimes tough, something is built of 1 thing of depth n-1 and another of depth n-2 giving:
f(n) = f(n-1) + f(n-2)
which of course gives Fibonacci numbers and the closed form relies on Phi (and Phi-hat).
This “not quite doubling” can occur surprisingly often.
In my own research, it has come up in ~5 different times, each in a completely unrelated way. So I know log[sub]2[/sub] Phi to 5 places since that appears in resulting formulas a lot.
So it is fairly common and important in some applications. But it is hardly in the same class as 0, 1 and 2.
I’m pretty sure (mainly in case you’re up for searching old threads) that our Phi-obsessed friend from about a year ago was Hiyruu not Hiryuu.
[symbol]f[/symbol] has a couple of interesting properties. It’s a root of x[sup]2[/sup] = x + 1, and 1/x = x - 1 (same equation). But it’s not as important as some other constants.
Phi is interesting without being important. Some of its properties have been alluded to above. The ratio of successive Fibonacci numbers approaches it and it is the aspect ratio of the “golden triangle”. That aspect ratio is said (without any evidence, to be sure) to be the most pleasing. It is, in a certain very precise sense, the most difficult number to approximate using rational fractions. Closely related is the fact that all the numerators in its continued fraction expansion are 1. (Well, any rational added to phi will be just as hard.) But it doesn’t show up in any fundamental equation the way e and pi do and is really not important for either mathematics or physics. And several of the cited properties are more properly properties of the Fibonacci sequence, which really does show up in a remarkable number of places. And, oh yes, phi is very close to the ratio of the mile to the kilometer. Thus if you know that 55 and 89 are successive Fibonacci numbers, you can guess that 55 miles is around 89 km. Or if you see the sign that the Confederation Bridge is 13.8 km, you know it is around 8.5 miles.
For your edification and amusement, I’ve posted an .mp3 of a fifteen-minute BBC radio program about phi on my web server.
Right here, baby. (3.4Mb)
This is the third in Simon Singh’s excellent short series Five Numbers.
Dr. Singh has the rare gift of being able to write and speak about numbers in a way that is quite compelling and rewarding, even for those queer ducks who don’t ordinarily find them interesting.
(The “Phi” installment isn’t my favourite, as it happens – that honour goes to the premiere episode, of which Zero is the star – but it’s quite good.)
Taking a look at the BBC link, I see that they’ve made all five programs available in Real Audio format. If you have Realplayer, and you don’t, as I do, harbour the same deep and abiding hatred for it that Gollum reserves for the Baggins, please save me a little bandwidth and listen to the RealPlayer version here.
That’s kind of cool; I didn’t know that. Would you mind explaining a little more what is meant by this? I guess I don’t know what “rational fractions” are, but it seems easy to me to approximate it using a rational number - just take the ratio of two consecutvie large Fibonacci numbers.
You can indeed represent any real number to any degree of accuracy using rational numbers. One way of measuring how simple an approximation is, is to look at the size of denominator required to reach any particular level of accuracy. This is what we mean by saying that it is hard to approximate [symbol]f[/symbol] simply: you need fractions with large denominators to get good approximations.
This follows from consecutive Fibonacci numbers being “worst case” numbers in a sense for gcd’s. If you look at Euclid’s 2300 year old gcd algorithm, with division instead of repeated subtraction in the middle, the “worst” numbers for making it run the longest are cons. Fib. numbers. But the Real Reason I’m posting this is to note that Euclid’s gcd algorithm is the oldest known non-trivial algorithm. Euclid didn’t know Fibonacci numbers from a hole in the ground but things involving Phi go back a long ways.