# Talk:Phyllotaxis

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## Fibonacci numbers and phyllotaxis

I put some discssuion of the appearance of Fibonacci numbers and the golden ratio in phyllotaxis at Talk:Fibonacci number/Phyllotaxis. Some of it might be useful here once this article is fleshed out a little more. -- Dominus 15:40, 11 Mar 2004 (UTC)

I'm new but having read Talk:Fibonacci number/Phyllotaxis, I don't know why the arrangement of leaves on a stalk isn't just one more example of those things that are .618... of another or 1.618 times the other.

There are going to be a lot of appearances of φ in the world. Some of these are coincidences, where it seems that there is no particular reason why it should be 1.6 rather than 1.7 or 1.5 or 23. For example, is your navel really 1/φ of the way up your body? Not so far as anyone can tell. Maybe it is about three-fifths of the way up, but there's no reason to identify that 3/5 with anything related to φ.
Other appearances of φ, however, are not coincidences; there will be some reason why they appear, and when the number that is actually measured differs from φ, we can view that as a deviation. The distribution of leaves on the stem is one of these, because there's a simple mechanism that leads to favorable leaf placement. Leaf placement is more favorable for certain angles than for other angles, and the optimal leaf placement is achieved when the angle is 360/φ. So it seems clear that the appearance of φ here is not a coincidence, because plants that distribute their leaves with the 360/φ angle will tend to outcompete plants that distribute their leaves in some other way, and the closer the plant can get to the uniform 360/φ angle, the more successful it will tend to be. -- Dominus 14:34, 13 Jun 2005 (UTC)

"In modern times, researchers such as Snow and Snow have continued these lines of inquiry."

Who are "Snow and Snow"? I think that the article ought to cite this. That way, people could read more about the research being done on this topic. InAJar 16:39, 28 February 2007 (UTC)

Yes, it's annoying when you see that sort of thing. I've requested the person who added it provide a citation. They're still active, so it might happen. Richard001 02:26, 9 September 2007 (UTC)
Well... if the author meant Richard and Mary Snow's papers, like "Experiments on phyllotaxis. I. The effect of isolating a primordium" (Philosophical Transactions of the Royal Society of London), which was published in... 1932, then I'll have to redefine my notion of "modern times".
If anyone's interested in more recent research, I can recommend reviews like Didier Reinhardt's "Phyllotaxis - a new chapter in an old tale about beauty and magic numbers" (Curr.Op.Pl.Biol. 2005) or Cris Kuhlemeier's "Phyllotaxis" (Tr.Pl.Sc. 2007). One of the key research papers being e.g. D. Reinhardt's "Regulation of phyllotaxis by polar auxin transport" (Nature, 2003). All of them focus more on the biochemical regulation of phyllotaxis, though...

I. ADLER*, D. BARABE and R. V. JEAN in "A History of the Study of Phyllotaxis", Annals of Botany 80: 231-244, 1997 gives a great account of Snow and Snow's contributions. By modern times, they mean taking an experimental approach to discovering the rules that govern appearance of new primordia, in combination with an attempt at mathematization of the process, which formalizes the inquiry. For example:

"Some readers may find that the mathematical aspects of the history of the study of phyllotaxis are given more importance than the botanical aspects. It is rather that over the years botanists (such as Church, 1904; van Iterson, 1907; Richards, 1951) found that in order to improve our understanding of phyllotaxis it was necessary to elaborate mathematical models based on botanical hypotheses (such as those put forward by the botanists Hofmeister, 1868; Schwendener, 1878; Snow and Snow, 1962). Over the years these pioneers have promoted a global scientific approach where mathematics were finally permitted to take their place in the concert of the disciplines concerned by the challenges of phyllotaxis study...

Mary and Robert Snow initiated an experimental phase of the study of phyllotaxis. They studied the effect of isolating a leaf primordium of Lupinus albus. They concluded (Snow and Snow, 1931) that the position at which a new leaf primordium is initiated is influenced by the pre-existing leaf primordia adjacent to the site of initiation. They, and others after them, showed that the phyllotaxis of a growing plant can be altered by surgical or chemical intervention."

In Douady and Couder's "Phyllotaxis as a Dynamical Self Organizing Process Part II: The Spontaneous Formation of a Periodicity and the Coexistence of Spiral and Whorled Patterns", J. Theor. Biol., 1996, they state:

"The conditions for the appearance of a new primordium put forward by Snow + Snow (1952) are shown to form the rules of a dynamical iterative system more general than that based on Hofmeister's hypotheses (1868)...The transition between a whorled type of phyllotaxy and a spiral one can result from botanical experiments of the type initiated by Snow & Snow (1935). They showed that by breaking the symmetry of an apex with a diagonal cut a usually decussate plant (Epilobium hirsutum) could form spiral patterns. Revisiting the drawing of Snow & Snow, Richards (1951) showed that the formation of these spiral modes was obtained even though the plastochrone ratio was unchanged. This experiment was a demonstration of the fact that both the whorled modes and the spiral ones could result from the same dynamics with differences in the initial conditions only."

In summary, they were early contributors who helped construct the practice of the "most venerable" mathematical biology, which allowed us to predict some surprising results very precisely. (best, gahnett) — Preceding unsigned comment added by Gahnett (talkcontribs) 05:11, 27 July 2014 (UTC)

## Phi reference wrong

Here's what the article says:

A repeating spiral can be represented by a fraction describing the angle of windings leaf per leaf. Alternate leaves will have an angle of 1/2 of a full rotation. In beech and hazel the angle is 1/3, in oak and apricot it is 2/5, in poplar and pear it is 3/8, and in willow and almond the angle is 5/13.[2] The numerator and denominator normally consist of a Fibonacci number and its second successor. The number of leaves is sometimes called rank, in the case of simple Fibonacci ratios, because the leaves line up in vertical rows. With larger Fibonacci pairs, the ratio approaches phi and the pattern becomes complex and non-repeating.

If we try a few examples, we can see that the ratio does not approach phi. For example, 13/5 = 2.6, while phi is about 1.618. Or we can try fib(30)/fib(28) = 832040/317811 = 2.6180339887543225. Clearly this is not converging on phi.

If you want numbers that approximate phi, you can try a Fibonacci number and its successor. For example, fib(30)/fib(29) = 832040 / 514229 = 1.61803398874820362134. Therefore, the article appears to be incorrectly claiming that dividing a Fibonacci number into that number's second successor will approximate phi. In fact, to approximate phi you should divide a Fibonacci number into its successor, not its second successor. 13:09, 17 October 2008 (UTC)~~ —Preceding unsigned comment added by 220.253.81.139 (talk)

The 'rule' produces convergence to phi, not the actual numbers - see e.g. Ian Stewart and Douady and Couder's work - implies that the underlying dynamics of the system are involved in formation of a phi-compatible pattern (and convergence to phi)79.70.229.86 (talk) 22:36, 17 March 2012 (UTC)lofthouse79.70.229.86 (talk) 22:36, 17 March 2012 (UTC)

lofthouse, his point was that it DOESN'T converge to phi, when you use a Fibonacci number and its SECOND successor. And he's absolutely right. Uporządnicki (talk) 12:53, 11 May 2012 (UTC)
Just a quick look-and-think suggests to me that the sequence might be converging on the square of phi. That comes to about 2.617924. Uporządnicki (talk) 13:58, 10 May 2012 (UTC)
Without going into general details, the examples given necessarily converge to the square of phi, because the numerators and denominators given are alternate Fibonacci numbers (1,1,2,3,5,8,13...) Note that phi^(n+1)=phi^(n-1)+phi^(n), so that phi^2=phi^0 + phi^1 (2.61803... = 1 + 1.61803...). But without explanatory details concerning the underlying mechanisms, that amounts to numerology. Maybe someone with access to some of the more recent publications could supply some helpful material? JonRichfield (talk) 09:12, 11 May 2012 (UTC)

This page provides a good description of the patterns but little as to why or how they occur. Auxin plays a part but if anyone could add more that would be great. Smartse (talk) 14:32, 12 March 2009 (UTC)

Should also state that the pattern is not confined to plants - see e.g. <http://arxiv.org/abs/physics/0411169> - it appears in animal cells too.79.70.229.86 (talk) 22:39, 17 March 2012 (UTC)twl79.70.229.86 (talk) 22:39, 17 March 2012 (UTC)

## Larger Fibonacci pairs--really non-repeating?

Under the section Repeating Spiral, discussing successive plant leaves along a branch spiraling until a leaf finally lines up with the first leaf in consideration, and the pattern repeats, it says "With larger Fibonacci pairs, the pattern becomes complex and non-repeating."

I'm not an expert--neither a mathematician or a botanist. But it seems to me that that might be true only in practical terms, not mathematical. When I think it through, it seems to me that mathematically, with ANY Fibonacci pair "m/n", after "n" leaves, the spiral will get back to its starting point--having gone around the branch "m" times. So it seems to me that if "[w]ith larger Fibonacci pairs, the pattern becomes complex and non-repeating," this would only be due to the practical point that with larger Fibonacci pairs, you'll never find a branch with enough successive leaves.

If I'm wrong, I'd be grateful if someone could explain it to this layperson. But if I'm right, I think the statement in the article could do with clarification. Uporządnicki (talk) 13:28, 10 May 2012 (UTC)

Without going back to the source you quote, what you say is quite correct, and the quoted text is wrong, or at least a misstatement. M/N is necessarily rational. However, if you take the sequence 1,1,2,3,5,8,13,21,34,55... and successively divide each number by its predecessor, you will find that they converge rather quickly on (1+Sqrt(5))/2, (roughly 1.61803...) which is demonstrably necessarily irrational. The convergence goes successively from below then above, 1,2,1.5, 1.666..., 1.6, etc. It does not reach the actual value at any particular Fibonacci number, but you can in principle find any digit in the expansion by dividing large enough Fibonacci numbers. In lazy or lay speech, one may say that it reaches phi at infinity (which is only arguably meaningful), or one might say that it converges on phi, which is correct, because there is no point either larger than phi or smaller, that does not have a larger error than an infinite number of such divisions. To speak of "larger pairs" and non-repetition etc, is simply wrong. However, as you correctly reflect, the plant doesn't know its maths all that well, and reality is only a rough approximation to the maths of the abstract Fibonacci structure. JonRichfield (talk) 17:58, 11 May 2012 (UTC)

## Repeating spiral section

The article says that Alternate leaves will have an angle of 1/2 of a full rotation. But this assertion is not qualified at all. If it applies to a specific example, that is not mentioned. If this is a general rule, it is not explained further, and the examples that follow (which are specific) give different rotation values. Aboctok (talk) 23:48, 22 June 2012 (UTC)

Good catch! I've changed it to "Alternate distichous leaves will have an angle of 1/2 of a full rotation." Sminthopsis84 (talk) 01:18, 23 June 2012 (UTC)

## Phyllotactic Architecture

Phyllotaxy Towers are kind of Phyllotactic Architecture. — Preceding unsigned comment added by 2.187.98.165 (talk) 11:15, 4 November 2012 (UTC)

This is listed in the See also section. Where that was recently added is a section about plants, not clearly written, but probably intended to refer to "plant architecture", a sub-discipline of botany. Sminthopsis84 (talk) 14:55, 4 November 2012 (UTC)

This image and this can be used as an example for Phyllotaxy Towers.--2.187.113.14 (talk) 22:06, 17 November 2012 (UTC)

The article on P. Towers was recently deleted at AfD. I believe the references there would be useful in a new article on either Biomimetic architecture or Phyllotactic architecture, to cover (actual) domes as well as (putative) towers. There would then be a place for links in either direction with Phyllotaxis. There could also be a place for a properly-cited paragraph here about the application of phyllotaxis to the architecture of buildings. Chiswick Chap (talk) 08:24, 18 November 2012 (UTC)