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There are many reasons why Alan Mathison Turing (1912-1954) is well known: coming up with the concept of a simple finite-rule-following computer now called a Turing machine; helping to break the Enigma code used by the Germans in World War II; devising the Turing Test for determining whether a computer is intelligent; and committing suicide after undergoing estrogen “therapy” following his conviction for homosexual behavior. Indeed, a 2012 worldwide petition seeking to earn Turing a royal pardon for his crime finally achieved its desired aim on Christmas Eve, 2013. Less well known is Turing’s interest in *morphogenesis,* the study of how living things grow and form. In particular, Turing was working on a paper describing a mathematical model for plant *phyllotaxis* (leaf arrangement) and its connection to Fibonacci numbers when he died.

While searching the internet, one of my students came across the *Turing Archive* of King’s College, Cambridge, and found a document headed as follows [page 1]:

MORPHOGEN THEORY OF PHYLLOTAXIS

Part I

__Geometrical and descriptive phyllotaxis__

This paper is actually a draft, written after Turing’s death by N. E. Hoskin and B. Richards, based on Turing’s original typescript and lecture notes, and was intended to be the first part of a longer paper submitted to the *Philosophical Transactions of the Royal Society of London*. (Indeed, in a 1952 article for the *Transactions*, Turing mentioned that he was working on such a paper.) However, the entire completed paper did not appear in print until 1992, when it was published in *Morphogenesis,* one of four volumes published by Elsevier that were devoted to Turing’s “mature scientific writings” – as his friend and executor P. N. Furbank wrote in the preface – and even included some unpublished work. It has recently appeared again in *Alan Turing: His Work and Impact,* also published by Elsevier, in 2013. Despite the document not having been written by Turing himself, and despite heavy editing by the actual authors, there is agreement that the ideas contained in it were Turing’s and so I will refer to it as his paper from here on. For more detail on the document itself, see Jonathan Swinton’s article “Watching the Daisies Grow.”

In the draft version, Turing modeled a portion of a pine branch as a cylinder, and studied the lattice of points where the leaves grew out of the branch. Probably due to its draft nature, the document contains no figures, not even Figure 1, to which it refers in the passage below [page 2].

The pattern is remarkably regular and is seen to have the following properties.

1) If the cylinder is rotated and at the same time shifted along its length in such way as to make a leaf A move into the position previously occupied by a leaf B, every other leaf also moves into a position previously occupied by a leaf. This may be called the __congruence property__.

2) All the leaves lie at equal intervals along a helix. On the specimen in Fig. 1 the pitch of the helix is about 0.046 cm and the successive leaves differ in angular position by about 137°.

Turing pointed out that these properties are not independent; clearly any specimen with property 2 also has property 1, but that there are examples in nature of plants having property 1 but not property 2. Using cylindrical coordinates \((\theta, z)\) on a cylinder of fixed radius, he discussed the idea of *parastichy* – the helix mentioned above that joins one leaf to the next (and which is very apparent in pineapples and pine cones). He both counted the number of leaves and measured the central angle between adjacent leaves on each parastichy. Turing’s Table 1 demonstrates the connection of the Fibonacci numbers to an angle measuring approximately 137.5°.

Table 1 |
||

DIVERGENCE | ANGLES | |

Fraction of \(2\pi\) | Degrees, min., sec. | Degrees |

\(2\pi\,\dot\,\frac{1}{2}\) | \(180^{\circ}\) | |

\(2\pi\,\dot\,\frac{1}{3}\) | \(120^{\circ}\) | |

\(2\pi\,\dot\,\frac{2}{5}\) | \(144^{\circ}\) | |

\(2\pi\,\dot\,\frac{3}{8}\) | \(135^{\circ}\) | |

\(2\pi\,\dot\,\frac{5}{13}\) | \(138^{\circ}\,27^{\prime}\,41.5^{\prime\prime}\) | \(138.46154\) |

\(2\pi\,\dot\,\frac{8}{21}\) | \(137^{\circ}\,\,8^{\prime}\,34.3^{\prime\prime}\) | \(137.14286\) |

\(2\pi\,\dot\,\frac{13}{34}\) | \(137^{\circ}\,38^{\prime}\,49.4^{\prime\prime}\) | \(137.64706\) |

\(2\pi\,\dot\,\frac{21}{55}\) | \(137^{\circ}\,27^{\prime}\,16.4^{\prime\prime}\) | \(137.45454\) |

\(2\pi\,\dot\,\frac{34}{89}\) | \(137^{\circ}\,31^{\prime}\,41.1^{\prime\prime}\) | \(137.52809\) |

\(2\pi\,\dot\,\frac{55}{144}\) | \(137^{\circ}\,30^{\prime}\,00.0^{\prime\prime}\) | \(137.50000\) |

\(2\pi\,\dot\,\frac{89}{233}\) | \(137^{\circ}\,30^{\prime}\,38.6^{\prime\prime}\) | \(137.51073\) |

Limiting value | \(137^{\circ}\,30^{\prime}\,27.9^{\prime\prime}\) | \(137.5078\) |

*Turing’s Table 1 demonstrates the connection of the Fibonacci numbers to the
observed difference in angular position of approximately 137.5°.*

Thus, if we denote the Fibonacci numbers as \(f_1,\) \(f_2,\) \(f_3,\) etc., then

\[137.5078^{\circ}\approx 360^{\circ}\left({\lim_{n\rightarrow\infty}}\,{\frac{f_n}{f_{n+2}}}\right).\]

The following illustration drawn by Turing himself, labeled as the “daisy ring diagram,” illustrates a similar relationship between Fibonacci numbers and differences in position. It is not explicitly linked to the document we consider here.

*Turing's “daisy ring diagram” provides another example of the Fibonacci numbers in plant life.
(Source: Alan M. Turing Digital Archive, Archives Centre, King's College, Cambridge University,
UK. Image AMT/K/3, page 3. Copyright © P. N. Furbank. Reproduced with permission.)*

If we order the points based on their distance to the center, then the central angle between consecutive points is about 137.5°. For more on Fibonacci phyllotaxis in general, and on this diagram in particular, see the articles by Swinton.

In the draft typescript, Turing discussed other Fibonacci-like patterns that occur in the parastichy numbers of different plant species. The paper continues, covering material beyond the scope of this article, and so I will move on to discussing the student’s work and refer the interested reader to the original document itself.

In her paper, the student gave brief biographies of both Fibonacci and Turing and described the latter’s interest in phyllotaxis and its numerical intricacies. In addition to explaining much of the content, she also used diagrams to explain cylindrical coordinates and the notion of parastichy. Moreover, she gave a short history of the study of phyllotaxis, including how its mathematical modeling was rejected outright by some botanists, all the way from the late 1800’s up until the time when Turing was writing this paper. She even mentioned the continuation of this research after Turing’s death, citing work of Roger Jean and the development of the so-called “fundamental theorem of phyllotaxis.” It was clear that the student thoroughly enjoyed this assignment.

While the student wrote a very engaging paper overall, there were certainly some areas where her paper could have been improved. First, though interesting, the document she used was not written by Turing himself. Instead, she could have used Turing’s own unpublished notes, which are quite similar to the edited document. She also neither extended nor applied the mathematics beyond its initial scope. As an example, she could have calculated some other limits involving Fibonacci numbers, such as finding \[360^{\circ}\left({\lim_{n\rightarrow\infty}}\,{\frac{f_n}{f_{n+k}}}\right)\]

for different values of \(k,\) and possibly found examples of some of these angles occurring in nature. Or, she could have compared Turing’s original typescript to the typescript of Hoskin and Richards, to see what the latter pair had changed. So while she found and interacted with a historical document of personal interest that contained Turing’s ideas, she did not attain the level of engagement that I was aiming to foster via the assignment. Other students in the “report” group also wrote interesting, novel papers that were fun to read and well written, but came up short in the area of engagement. On the contrary, the next example of student work, involving a paper of Euler, exemplified the kind of engagement I wanted the assignment to elicit.

Christopher Goff (University of the Pacific), "How to Improve a Math History Assignment - Turing and Plant Phyllotaxis," *Loci* (July 2014)