Turing's Biomathematics

local blip that changes the balance between missionary load. Take too few sub-beaches and or the larger of the two. The latter choice — in Appendix: Turing's missionaries and cannibals, it doesn’t get you can’t have a pattern; take too many and the this case going from a (1,2) not to a (1,3) but to Biomathematics smoothed out. For if there is a temporary local calculations become overwhelming to a human. a (2,3) — preserves the property that the pairs excess of missionaries, the short term will see So Turing compromised on 20 sub-beaches, of numbers involved will always be Fibonacci relatively more missionaries than cannibals multiplying his computing load by 20, and thus numbers, for then the next pair will be (3,5) and arrive from the rest of the island, so even more produced a Figure in which a random starter then (5,8) and so on. Turing called this the ‘H of Nobody who heard Turing talk about his theory conversion than normal occurs. This instability pattern evolves to a series of three stripes. This GP’: the Hypothesis of Geometrical Phyllotaxis — and he spoke to mathematicians, chemists, becomes magnified, stopping only later when the is one of four separate runs he reports under and it is this hypothesis he was using the Ferranti physiologists, zoologists and botanists — had cannibals belatedly turn up for the feast. This different conditions, and, he says, ‘some’ of these Mark I to test. He was right: the ‘H of GP’ does much idea of what his mathematics was. It effect relies on a balance of speed and time, so it results were with the aid of the computer. Given a hold for a wide family of mathematical models contained an idea of genius, simple enough to has a characteristic distance: for each local peak first chance on the Ferranti Mark I at the beginning of spot formation in growing tissue. But Turing explain to an applied mathematician in a few lines of missionaries there will be a trough a particular of September and the submission of his paper never managed to prove it in his lifetime and today. 343 But Turing never gave a lay account of distance away, which in turn induces a further on the 9th of November, he must have worked the mathematicians and physicists — not his mathematics, sWo here is mine. 344 peak so on all the way around the island. remarkably fast and accurately in programming botanists — who carried out that proof many When the report to the Academy is due, the the machine. 347 decades later did so in ignorance of Turing’s work. graph is no longer flat. The cannibals have Giving the arranged themselves into civilised groups. In a Year or Two: The framing of the ‘H of GP’ must have been the missionaries bicycles reason why Turing was confident in 1951 that he Returning from these Western fantasies to Turing's Fibonacci could get himself a good theory in ‘a year or two. 348 Imagine a tropical island, with an impenetrable Turing’s actual paper, he showed that, starting strategy mountainous interior and a lush ring of habitable from a blank canvas, he could generate a sea-fringed forest. This paradise is at first characteristic length-scale. Given that, spots The second remarkable attack Turing made on inhabited by a group of noble savages, whose and stripes and all things math-biological can developmental biology was to develop a theory of specific lack of civilisation is that they are emerge. Turing simply presents this as a model the appearance of Fibonacci structure. Turing had competent hunter-gatherers but occasional with no discussion of why it has been chosen over created a mechanism for making spots appear, cannibals. Despite the occasional feast, for the rivals. I have long puzzled about how Turing came and the spots were separated by a typical most part the population is constrained by the to discover the Turing Instability. He acknowledged length-scale, which depended on a complicated amount of food they can sustainably extract from Waddington’s idea of ‘evocators’ as an example of but tractable way on the numbers put into the their habitat. They move around, and when they his own morphogens, a word Turing invented for model. The number of spots one might see ‘happen to pass by’ each other enough births unnamed biochemicals that could react and developing in a narrow ring of tissue, as he used happen to balance the inevitable deaths. Because diffuse across his canvas, but did he deliberately in his 1952 paper, would depend not only on that the supply of plant food is evenly spread, so is the construct a system of equations for them that length-scale but the circumference of the ring. But population: even if a particularly big feast on one would have the right property or did he discover Figure 122. Turing’s ‘tree’ classifying possible now suppose the ring is no longer narrow, so that shifts between patterns side of the island requires a few extra sacrifices or it when analysing what he thought a plausible patterns can develop across as well along it, in a encourages the survivors to seize the moment model? The technology for this analysis is, in the way where the placement of new spots depends and pass by each other a little more, emigration jargon of applied mathematics, a linearization on where the older ones went. What Turing saw and birth and malnutrition eventually smooth out into Fourier modes, which had been around for was that this dynamic model could allow the the temporary local blip. Then the missionaries decades and well known to any mathematician. pattern to get more and more complicated as the arrive. A sub-mission of theirs is to do Science Turing was familiar from electronics with the inter-spot distance got smaller. What emerges is a and they plot the population density around the possibility that instability could be encoded in a lattice of points, gently stretched as the geometry 346 island. This shows an uninteresting flat graph; system, as was FC Williams. And in Turing’s of the underlying biological ring where spot- the savages are unable to organise themselves. role as an applications consultant he might have formation is happening changes. Lattices are A constant small re-supply of missionaries keeps had to think about how to design crystallographic regular objects, amenable to mathematics, and in arriving, but with guns, germs and steel they also algorithms in Fourier space. this case they can be usefully described by pairs convert a fraction of the population. Since it is a of integers; simple lattices at the beginning of condition of being a missionary to be celibate, the The majority of the mathematics in Turing’s paper development have small pairs like (0,1) and (1,1) population falls a little. But the underlying food is concerned with what happens on the thin outer and (1,2); more complicated lattices, evolving supply is uniform, and so wherever you go you ring of the island, keeping track of the number of from these as the lattices stretched to fit the new find the same number of cannibals and a few missionaries per mile as we go around the beach. geometry of the growing plan, might be described missionaries, in what the scientific reports back Some calculations, such as working out how fast as (1,3), or (2,3). And this is where Turing quickly Figure 123. Classifications of leaf-placement on to the sponsoring Academy call a uniform the bicycles need to go to generate instability, can isolated a problem to prove. Independently of the plant stems from Bonnet’s eighteenth century text distribution. be calculated by hand, perhaps with a few turns model assumptions, it turns out to be an on plant growth. In the fourth example, the leaves on a mechanical calculator: he had a Brunsviga unavoidable feature of geometry that a (1,2) are arranged in a spiral and if the bottom leaf is Now we can see Turing’s stroke of genius: he machine at home to do just this. But others are pattern can only evolve to either a (1,3) or a (2,3): labelled as leaf 0 and numbering is continued gave the missionaries bicycles. 345 Because the more complicated. To cope with this, the beach more generally that if there are two numbers, then upwards, then leaf 0 is closest to leaves 1 and 2: missionaries can move around more quickly, can to be broken up into little sub-beaches, each the sum of the two will appear in the more this is why pattern this can be described as a something new happens. Now, when there is a of which is approximately constant in its complicated lattice paired with either the smaller (1,2) spiral. 172 173 Almost all of the archivists I have needed help Lyn Lloyd Irvine retained her maiden name in her Acknowledgements from have provided it with grace. But I am Notes writing, but called herself Mrs Newman throughout particularly grateful to those who have taken the her life as a married woman. (Mr) Pat Blackett was trouble to find out about my enquiry and to offer perhaps referred to on occasion as Professor extra suggestions: these include Fiona Colbert, The standard, and best, biography of Alan Turing Patrick Maynard Stuart Blackett, Baron We like to identify with our heroes. My own first Elizabeth Ennion-Smith, Patricia McGuire, Kathryn is still Andrew Hodges’ extraordinary Blackett OM CH PRS.

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