I first came across the online mathematical aid called GeoGebra some years ago and noticed it more recently at reference 1. I am still working away at the paper at reference 2 and I have been playing with the GeoGebra program at reference 4, intended as a toy demonstration of some of the possibilities of the spiral growth of plants, of which it is a very common – but not universal – feature. There is a lot of spiral growth about and, in this program, it is modelled on a disc, rather in the way of a disc of the sunflower.
A lot of what follows is encapsulated in Figure 2 of said reference 2. A report on work in progress and I shall acknowledge some of the various other material which I have consulted along the way in a report to come. In the meantime, I just thank Boris Rozin for drawing my attention to the importance of the human visual system here. To adapt a common phrase, it takes two to spiral.
The model
We have a circular generating zone of radius 1 (aka meristem). Our nodes (aka primordia) are generated on the periphery of that zone at intervals of time of T (aka plastochron) and angle α (aka divergence angle), as measured from the centre of the zone, from the last one.
Noting in passing that building biology to support constant time and constant angle – if that is where we end up – might well prove difficult – despite all the effort which has been put into taking biological clocks apart over the last few decades.
The units of α are such that α = 0 ≡ 1 ≡ 360° ≡ 2π radians ≡ one complete revolution about the generating zone. Where ‘≡’ means equivalent to, slightly weaker than equals.
All this is to say that each successive node is generated at time T after its predecessor and at clockwise angle α from that predecessor. The start point is due east, that is to say, middle right.
Generated nodes move radially outwards with a velocity of 1.
I dare say the model works by stepping through T units of time the requisite number of times, generating a new node and moving the rest radially outwards, at each step. A reasonable approximation.
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| Figure A |
The model displays graphically a nominated number of generated nodes given nominated values for T and α. A simple example is snapped above, with α = 4/5, with the key point being that α is a rational for which 5 is the minimum denominator. That is to say that there is no way of expressing α as an integral ratio with denominator less than 5, but plenty of ways of expressing it with denominator more than 5. The numbers 4 and 5 are relatively prime. All this means that we get five straight arms. It is the order of their generation which is the function of four.
For a given number of points, as T gets smaller, the lines shrink and the red dots get closer together, at the limit collapsing to a point on the periphery of the green generating zone.
Note that the five arms are well spaced out around the generating zone. They are not going to interfere with each other, visually or otherwise. On result of which is that we see the arms but we do not see the generating spiral, the spiral which joins up the nodes in order of generation, which would be whizzing round and round the generating zone.
Note also that biology does not do limits. They might be convenient mathematically, but they don’t fly at the level of cells. The real world is granular, not smooth all over.
Replacing arms
In this model, generated nodes are points, as in geometry. Unlike the generating zone, they have position but no extent. However, on the screen or page, they do have extent and may be thought of as being approximated by small circles, which has some effect on their collective perception, as for example when they run together into a thick straight line.
In the left-hand panel above, the 25 radial lines which are the result of that rational value of α = 24/25 are clearly visible. In the right-hand panel however, with its much larger value of T, those lines appear to have been replaced by a single spiral, a consequence of the successive points on this spiral being a lot closer together than successive points on one of the radiating arms. It’s all in the eyes of the beholder.
Note that this anti-clockwise spiral is not the generating spiral, although the nodes are in the right order. That second spiral is clockwise and goes round nearly once for every successive pair of nodes on the first. There is a lot more of it.
Bending arms
Another trick is to bend the arms of the left-hand panel, rather than replacing them with the spiral of the right-hand panel.
Here we have the same value of T as in Figure A above, but a slightly bigger value of α = 0.803, rather than 4/5. So rather than coming back to its starting point after 5 iterations, it overshoots and generation is slightly offset.
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| Figure B |
Here, in the left-hand panel, a slightly small value of α = 0.797, in which case the arms bend the other way. While if we go for a small value of T, as in the right-hand panel, we don’t change the structure, we just pull the spiral arms in a bit tighter. A big value does the other thing.
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| Figure C |
Jumping ahead, things get even more interesting when we use a value close to 4/5. Spirals popping up all over the place, rather, indeed, as in the sunflower. Notice also how different things look when we toggle between points and blobs, between left and right.
In building slightly more elaborate models, the idea of contact between nodes is important, with the spirals that are going to stand out being the ones where successive nodes are in tangential contact.
So what is going on? How did we jump from Figure B to Figure C? I offer a couple of tasters in what follows.
Taster 1
In the snap above, in which ‘G’ for growth takes the place of ‘T’ for time, a collage of material lifted from reference 2, we have some hints about what is to come – and the curious connection to Fibonacci numbers and other mathematical matters, connections which have fascinated all kinds of people for a long time.
Row A is more on how appearances can change with G (or T).
Row B is the case when the divergence angle α is the golden angle, a simple function of the golden number, with pairs of families of spirals emerging along, as it were, the rational convergents of that angle – irrational, so we never actually get there with a rational approximation.
The golden angle is 137.51° or α = 0.381966. The golden number has the very special – unique - expansion as a continued fraction snapped above.
Row C illustrates the sensitivity of this simple model to small changes in α.
Taster 2
Another angle is to take more interest in the size of nodes and in their regular arrangements, assuming for the moment that they are all the same size and do not grow. As far as seeing spirals and lines is concerned, touching nodes is good, as illustrated at A above. We can draw plenty of straight lines on A linking regular series of nodes, and one such is shown, but those nodes will not be in contact and the eye is unlikely to see the lines without more help.
Touching in two directions gives us two families of lines or spirals: with B being a square lattice and C being a rhombic lattice. Not all that much freedom as to the angle – if one is going to maintain the two directions of contact. Whereas at D1 we have a triangular lattice where there is very little freedom at all; all one can do is rotate the whole thing, as at D2, which might not interest a mathematician, but might interest a botanist.
The Snipping Tool (from Microsoft) having rather shrunk D1 along the way to D2. But that is accidental. And fits what is left of the page.
The D lattices offer the greatest packing density, which may be relevant.
That apart, all this becomes relevant when we move from looking at our meristem from above, as it were, as at A above, to the unrolled cylinder views at C and D. The top of the rectangle is the perimeter of the generating zone and the two sides are identified. The bottom is where the oldest nodes are to be found. The numbering is newest zero – which means that all the numbers change at every step. The small black rhombus highlights the rhombic nature of the lattice. Spirals at A – or at Figure C above – have become the rather more tractable lines at D. Taken from Figure 3 of reference 2.
Suggestion box
Given that the image can be sensitive to small changes of α, it would be good if one could set it directly, rather than using a slide bar, with which it is sometimes difficult to get the particular value one wants. I dare say if one knew GeoGebra, it would be easy enough to do this for oneself – but I don’t – and don’t care to invest the time to learn, clever product though it is.
T also, although the image is not so sensitive to small changes, so this is less important.
And more nodes. This perhaps a bit greedy for a central resource, assuming that the GeoGebra code does not actually run on the target computer, out in the sticks, as it were.
Conclusions
I dare say I would have got to this point a good deal faster, should this paper have then been available, half a century ago.
Nevertheless, it all remains an interesting and intriguing business.
References
Reference 1: https://psmv5.blogspot.com/2024/06/trolley-696.html.
Reference 2: Phyllotaxis as geometric canalization during plant development - Christophe Godin, Christophe Golé, Stéphane Douady – 2020. Turned up by Google when looking for something else.
Reference 3: www.geogebra.org.
Reference 4: www.geogebra.org/m/q5ysr7bv#material/feycx5yb.
Reference 5: https://psmv5.blogspot.com/2024/06/corn-on-cob.html. Another line of inquiry.
