Glossary

The H. of G. P.

A key staging post in creating an explanation of Fibonacci phyllotaxis is what Turing called the Hypothesis of Geometrical Phyllotaxis.

After creating a theory of lattices on cylinders, Turing went on to consider lattices of more variable geometry. This raises the question of what kinds of transformations of parastichy numbers are possible when a phyllotactic lattice is deformed. As it is deformed, the principal parastichies will in general remain unchanged. They will only change when a new lattice point from a different parastichy moves so as to become closer to the origin, but generically the point that does so must have previously been the third parastichy. Thus one of the two principal parastichy numbers, together with the third parastichy number, will become the new principal parastichy number. Turing showed (p72 of Turing (1992)) that if that third parastichy number never lies between first and second parastichy number then a Fibonacci property, once begun, would persist. This constraint he named the  Hypothesis of Geometrical Phyllotaxis.

The figure shows an initial lattice on the left-hand side with principal parastichy numbers 2,3 and 1. The lattice is then squeezed and stretched, and as this happens the principal parastichy numbers change. The 'hypothesis of geometrical phyllotaxis' (HGP), is satisfied in this example.

It was this result which was surely the cause of Hoskins' view, reported in Max Newman's 1955 Royal Society memoir, that Turing had shown that

if a phyllotactic system is Fibonacci in character then it must change, if at all, to another Fibonacci system (AMT A/7; section contributed by Hoskin to MHA Newman's obituary in Biographical Memoirs of the Royal Society AMT A/6)

But the truth, as Turing recognised, is that this relies on the HGP being true and the theory so far provides no reason why it should be. Indeed he wrote (p72 of Turing (1992))

the phyllotactic systems of botany do not arise in this way’ More

However this idea is still worthwhile: the key insight it embodies is that of continuous change. Phyllotactic lattices are not laid down all at once on an infinite cylinder: they are produced locally, node by node, and the resulting pattern is also deformed by growth.

We have evidence of Turing working out these ideas in 'Outline of the development of the daisy'.

Parastichy number

Here are four representations of the same idealised growing tip (green) on which primordia (brown) are forming, later to become stem branching points, or florets, say. The first image shows the physical arrangement of the primordia on the tip. The image below gives a view from above, while that to the right stretches the surface of the tip so that it has a constant diameter. The image below that unrolls the resulting cylinder into a plane. In each representation the same primordia are linked by one example parastichy (red). A parastichy which is obvious to the eye in one representation is not necessarily so obvious in another. In what follows we concentrate on the last, planar representation.

Given any of these lattice plots, the eye picks out obvious straight lines, which correspond to spirals on the original plant. These are called parastichies.

To understand where the parastichy number comes from, pick one point and number the rest in sequence up the cylinder:

What Turing defined as the principal or first parastichy was, loosely, the number of the closest point to 0. Here the first parastichy number is 3, because the nearest point to 0 is 3:

The second parastichy number is 2. Note how the parastichy lines go the opposite way around the cylinder:

The third parastichy number is 1. Note that 3-2=1: in general the third parastichy is always the sum or difference of the first two. Turing proves this theorem on p57 of Turing (1992).

Turing is one of several who have constructed theories of geometrical phyllotaxis: the static properties of lattices. His opinion was that the theory had been partially

expounded...by some previous writers but often in a rather unsatisfactory form, and with the emphasis misplaced (p62 of Turing (1992).)

but he wasn't the last to consider that his theory was the best.

The details of Turing's definition are related to but different from those of Jean (1994) primarily because Jean also ensures that the parastichies wind in opposite directions around the cylinder.

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.

Morphogenesis

The process by which living things develop organised structures.

For a mathematician, one canonical question in morphogenesis might be how a uniform sheet of embryonic skin breaks symmetry and develops zebra stripes. A biologist investigating the same problem might ask what genes are promoting and suppressing each other, and how the proteins they encode interact.