deodands

Fibonacci Phyllotaxis

This entry describes the phenomenon of Fibonacci phyllotaxis. Many other authors have done this: a couple which are on line are Knott, who concentrates on the elementary mathematics of Fibonacci numbers and the informative and good looking interactive website at Smith. Also see the references for printed discussions.

Phyllotaxis means here the arrangement of structures, such as leaves or florets, in plants.

Here's an sunflower, where some florets have been removed to make it easier to see some spirals. The shallower spiral, or parastichy, at the bottom repeats all the way around. Counting the removed florets, there are 34 such spirals. The steeper spiral repeats 55 times. The astonishing thing is that in most of the sunflower head you examine, you will also find 34 and 55 spirals. Or if not that then 21 and 34 together, or maybe even 55 and 89.  And of course these numbers are adjacent members of the Fibonacci series: 1,1,2,3,5,8,13,21,34,55,89,... Perhaps the more astonishing thing biologically is that you don't find near approximations: if the spirals are clear you won't find 34 spirals one way and 57 the other. 

This phenomenon is not restricted to sunflowers. The example below shows the more common case where spirals appear to wind around the cylindrical stem of a plant, in this case a Euphorbia:

These spirals visually joining nearby points are called parastichies and the parastichy number for a spiral counts how many such spirals fit on the cylinder, or equivalently how many points around the plant but not on the spiral have to be skipped in the vertical direction between two points of the spiral.

Thus one prominent parastichy on the left hand slice of the specimen is a 5 parastichy because there are four other branching points spread around the cylinder between any two consecutive points on the spiral. The parastichy in the other direction is a 3 parastichy and the pair is called a (3,5) parastichy pair. Like the sunflower, this specimen exhibits a number of different parastichy pairs, each of which consists of two adjacent Fibonacci numbers. Yet more remarkable is that this property can be found in very many examples in many different species of plants.

Explaining this ubiquity is the problem of Fibonacci phyllotaxis. (A substantial recent book on Fibonacci phyllotaxis is edited by Jean and Barabé (1998)). The appearance of these numbers, or variants on them, turns out to be intimately related to the divergence angle, the difference in angle between successive points on the stem. If that angle in a cylindrical lattice is close to a simple function of the Golden Ratio, then Fibonacci numbers naturally appear. Moreover the Golden arrangement typically has the property of optimal packing. All of these relationships have been closely studied in the mathematical phyllotaxis and number theoretic literature (reviewed in Adler et al (1997) and Jean and Barabé (1998)), and each of them has been adduced at one time or another as the explanation for Fibonacci phyllotaxis, often with varying degrees of mysticism or arguments from evolutionary optimality attached.  According to Adler et al (1997) the first to explicitly recognise that Fibonacci numbers were involved in phyllotaxis were Schimper (1831) and Braun (1831).

Another excellent example of Fibonacci phyllotaxis, very well known to Turing, is the fir cone.