You know how when you look down at the top of a pinecone

or the center of a daisy, you can discern two opposing sets of spirals. It turns out that they always reflect two

adjacent Fibonacci numbers. Fibonacci numbers are

formed by the recurrence f(0)=0, f(1)=1, f(n)=f(n-2) + f(n-1). The first few Fibonacci numbers are therefore 0, 1, 1, 2, 3, 5, 8, 13, 21. Though I don’t remember exactly the

derivations involved, the golden ratio is related to

the Fibonacci series. Why do plants show these patterns,

and why are they connected with the golden ratio?

I will place a WAG that fractal geometry is responsible.

I remember reading in an old mathematics textbook that the “golden ratio” was also found in the dimensions of the covers of paperback books–better marketing, because that shape is more pleasing to the human eye. Is that true?

I’ve never measured a paperback book, so I can’t say for sure, but it’s quite possible… It’s been known since the time of the ancient Greeks that rectangles in the Golden Ratio are most pleasing to the eye. For those having trouble remembering their high school algebra, the Golden Ratio, usually represented as phi, is defined as sqrt(5)/2 + 1/2 ~= 1.618034 . It has the interesting properties that phi[sup]2[/sup] = phi + 1, and 1/phi = phi - 1 . The connection with Fibonacci numbers is that, for large numbers, the ratio of two adjacent Fibonacci numbers approaches phi: 8/5 = 1.6, 13/8 = 1.625, 1597/987 = 1.6180344, 121393/75025 = 1.618034, and so on.

I never have found a satisfactory explanation of the sunflower thing, but it seems like it *should* be pretty simple. Does anyone know?

The golden mean is probably found in the proportions of many books, although some decide to be funky and do goofy proportions. 5:8, 21:34 (golden mean), 1:square root of 3, and 3:4 are common modern book proportions. 2:3 was used in medieval times but rarely now. My measurement of a standard paperback is 4.25" by 7", which is a proportion of .607:1 The golden mean is ~.618:1. So, pretty close. All the standard proprtions are not too far from the golden mean anyhow.

Ian Stewart’s *Nature’s Numbers: The Unreal Reality of Mathematics* has a pretty good discussion of this in Chapter 9. I’m too lazy to try to do justice to it here, but the two-cent version is that the so-called “golden number” and its derivative, the “golden angle”, arise naturally as a result of the dynamics of packing the greatest amount of stuff into a finite space. While Stewart confesses that there is a certain lack of logical rigor in some of the computer-model experiments that have been done to support this, he finds the arguments for the theory sufficiently strong.

I’ve always felt that the golden ratio was “oversold”. In the last century Pertner (I think) did a test, surveying people to see which proportioned rectangle they thought was “best”, and the histogram supposedly peaked at the golden rectangle. Ever since, eople have been claiming that the golden rectangle was somehow the most pleasing, and that it shows up everywhere. (Take a look, for instance, at the Walt Disney short “Donald in MatheMagicland” for a LOT of examples.)

The problem is, it ain’t true!

Despite what has been said earlier in this thread, paperback books are NOT made in the shape of a golden ratio rectangle. (measure and see). Nor are credit cards, billboards, movie screens (not widescreen or, earlier, the old film ratio), television screens or computer monitors, postcards, pads of paper, legal pads, doorways, or just about anything else rectangular in everyday use. Certainly if the Golden Rectangle were the most pleasing you would expect the advertising community – always anxious to win a few popularity points – would force billboards to this “optimum” shape. But they don’t.

This has been bugging me for a few years, and I’m surprised it hasn’t really been brought up by others. There has been an article recently in the Skeptical Inquirer about this, but that’s about it.

Back to the original topic – why do sunflowers and pinecones grow in Fibonacci sequence – you can find some reporting on the phenomenon in books on the Golden Ratio (LIke Huntley’s “The Divine Proportion”, published by Dover, or “The Curves of Life”, also published by Dover, although I don’t recall the author), but no reason is given. I recall seeing a recent argument that the curves are NOT, as has often been asserted, logarithmic spiral (which are related to the golden ratio), but are due to a different cause – only I can’t recall what that was. but the spirals, whatever their true shape, still seem to occur in Fibonacci numbers.

**phyllotaxis**

Read The Book of Numbers, pp.111-126, by Conway and Guy. It includes the section *Say, Bud, Where Do You Think You’re Going?"* Very readable. Hell, read the whole book.

The Fibbonacci Sequence (1,1,2,3,5…) is defined as the sequence that term n = term (n-1) + term(n-2), and where term 1 = 1 and term 2 = 1. The ratio of (term n)/(term(n-1)) converges on the Golden Ration as n approaches infinity.

However, the Fibonacci Sequence is not unique in this. Any series where term n = term (n-1) + term(n-2) will also do the same thing. The assignment of 1 to the first two terms of the Fibonacci Sequence is rather arbitrary. Although I haven’t done the research, I think it is resonable to believe that in nature there are things that emulate a series in which something is equal to the sum of the two things that proceed it, but not necessarily the Fibonacci Sequence.

Take this example. Let’s start a series with the first term of R^0 and the second term of R^1, where R = the golden ratio. By definition R^0 + R^1 = R^2, and R^1 + R^2 = R^3, and so on. This give us the series:

1, R, R^2, R^3,…

Each term is equal to the sum of two terms that proceed it, and the ratio of (term n)/(term(n-1)) is always exactly equal to the Golden Ratio. Way more elegant than the Fibonacci Sequence, and IMHO much more likely to be emulated by nature.