Fibonacci phyllotaxis: Turing's problem
Fibonacci phyllotaxis
(This section 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.)
The basic observation is a simple one, going back at least as far as Goethe, and a simple and particularly relevant example comes from examining the spirals in which florets are arranged in the sunflower. Here's an example where some florets have been removed to make it easier to see the 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.
The process of structure formation in plants is called phyllotaxis, and explaining this observation is the problem of Fibonacci phyllotaxis. Another appearance of the patterns comes from the arrangement of side branches on the main stems of many plants. Two model examples are shown below:
Here, 'obvious' spirals have been drawn through adjacent branching points. The parastichy number for a spiral counts how many such spirals fit onto the cylinder, or equivalently how many points around the cylinder 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 of the points joined by the spiral. The parastichy in the other direction is a 3 parastichy and the pair is called a (3,5) parastichy pair. The sunflower above had a (34,55) pair around the outside.
Another example of Fibonacci phyllotaxis, very well known to Turing, was the fir cone (Figure 3).
The appearance of these numbers, or variants on them, is 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; 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. Turing thought of the problem in terms of explaining the Fibonacci numbers of the parastichies, and it is this approach I concentrate on here. According to Adler et al (1997) the first to explicitly recognise that Fibonacci numbers were involved were Schimper (1830) and Braun (1831).
Turing's interest
When and where did Turing's interest in this problem come from? We know that at school Turing was well acquainted with D'Arcy Thomson's classic On Growth and Form that discusses it; decades later, Turing is recorded as discussing daisies and fir-cones during off-duty periods at Bletchley Park (Hodges p207). We have little concrete idea of his thinking on the subject until 1951. When Turing returned to Cambridge for a year in 1947-1948 he attended the undergraduate physiology lectures of Lord Adrian, and Hodges has plausibly speculated that his prime interest by now was the possibility of a logical description of the nervous system. Indeed it was in a correspondence with a leading physiologist on the needs of a physiological theory of the brain that he continued:
'...my mathematical theory of embryology...is
yielding to treatment, and it will so far as I can see, give satisfactory
explanations of
(i) gastrulation
(ii) polygonally symmetrical structures, e.g. starfish, flowers
(iii) leaf arrangements, in particular the way the Fibonacci series
(0,1,1,2,3,5,8,13,...) comes to be involved
(iv) colour patterns on some animals, e.g. stripes, spots and
dappling
(v) pattern on nearly spherical structures such as some Radiolara...' more
Whatever the original trigger, these were strong claims and it
is the aim of this website to examine why Turing felt able to
make them and claim (iii) in particular. In the same month he
also wrote in a letter that
'Our new machine is to start arriving on Monday. I am hoping to do something about 'chemical embryology'. In particular I think I can account for the appearance of Fibonacci numbers in connection with fir-cones' more
He certainly could do something about chemical embryology. By November of that year he had submitted a paper to Philosophical Transactions. Read more about this paper
In 1952 he wrote that he had
'Had quite a jolly time lecturing on fir cones' more
in Cambridge, and in some time between then and 1954 he drafted parts of a paper on The Morphogen Theory of Phyllotaxis. This work was left incomplete, and indeed Robin Gandy wrote, after Turing's death, that
'When I was staying with Alan the weekend before Whitsun he also told me more or less where the computations had got to; but since his methods were so individual, he was unmethodical, I imagine it will be almost impossible for anyone to go on with the programme where he left off.' more
In fact Nick Hoskin did manage to make some progress with preparing
the work for publication, and Bernard Richards provided a third
section based on the MSc thesis he started under Turing. But the
resulting typescript was not published until
Saunders edited that volume of the Collected Works in
1992.
Jonathan Swinton 1/10/2002