## Sunflowers, Alan Turing and the Fibonacci sequence

The seeds in sunflower seed-heads are organized in two sets of spirals ("parastichies"), one counterclockwise and one clockwise; the numbers of spiral arms in one direction and in the other are usually two consecutive Fibonacci numbers, that is, numbers from the series 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... where each number is the sum of the two preceding. The genesis of this arrangement, which is typical of plants in the family Compositae (daisies, asters, etc.), is still not completely understood. Recently Jonathan Swinton and Erinma Ochu organized a "citizen science experiment" in Britain to plant, grow and harvest a large number of sunflowers so a census could be made to determine how often the numbers of spirals have this exact form, and to record the most common deviations from the pattern. The results of the experiment were published in Royal Society Open Science, May 18, 2016.

Table 1. Specimens of sunflower heads are grouped and counted by the larger of the two numbers of spirals they bear. Here the vertical bars are truncated at #= 30; the full table shows, approximately, #(21)=40, #(34)=172, #(55)=240, #(89)=105. The Fibonacci specimens overwhelmingly outnumber the others; the most common variants are Fibonacci $\times 2$: 2, 4, 6, 10, 16, 26, etc., the Lucas series: 1, 3, 4, 7, 11, 18, etc. (same definition as Fibonacci, but starting with 1, 3), and Fibonacci $-1$: 1, 2, 4, 7, 12, 10, etc. All the images in this item are reproduced from Swinton et al., are © 2016 The Authors, and published by the Royal Society under the terms of the Creative Commons Attribution License.

The report included images of the standard configuration and some of the variants, with a few parastichies marked for reference:

A Fibonacci configuration (89,55) and a double-Fibonacci (68,42).

A flower head exhibiting two numbers (76, 47) from the Lucas sequence, and an anomalous example: "While the relatively unambiguous anticlockwise 56 count might be interpreted as close to 55, the equally unambiguous clockwise 77 count is far from Fibonacci."

As the authors remark in their conclusion, along with verifying the prevalence of Fibonacci and Fibonacci-like configurations, they have conducted "a systematic survey of cases where Fibonacci structure, defined strictly or loosely, did not appear. Although not common, such cases do exist and should shed light on the underlying developmental mechanisms."

Where does Alan Turing fit into this picture? It turns out that he spent some time with this phenomenon in the 1950s, and formulated a conjecture ("the Hypothesis of Geometrical Phyllotaxis"), unpublished until very recently, describing how a smooth change during development should lead to a transition from "from an adjacent pair of a Fibonacci-type sequence ... to the next higher pair in the sequence: this fact has provided a powerful generic explanation for the appearance of Fibonacci structure." More immediately, the citizen science project that this article presents was organized in 2012 by the Museum of Science and Industry (Manchester, UK) to mark the centenary of Turing's birth.

## Compute at your own risk

Here's how the story appeared in the New York Times, by Elaine Glisak on May 18, 2016: "On May 5, aboard a plane about to fly from Philadelphia to Syracuse, a passenger, apparently alarmed by math equations her seatmate was working on, caused its departure to be delayed when she suggested he might be a terrorist. The man turned out to be Guido Menzio, a University of Pennsylvania economics professor working on a differential equation."

An earlier account ran in the Philadelphia Inquirer (by Julie Shaw, May 7): "Passenger thinks Penn prof doing math is 'terrorist;' flight delayed." We learn that Menzio "has dark, curly hair and wears glasses, according to his Penn faculty web photo" and that he was on his way to give a talk at Queen's University in Kingston, Ontario, on "The (Q, S, s) Pricing Rule." Shaw refers to a social media post by Menzio: after the woman sitting next to him had passed a note to a flight attendant, the plane returned to the gate and the woman left her seat. "He was then asked by the pilot to get off the plane, and at that point, he was 'met by some FBI looking man-in-black,' he wrote in his post. After first being asked about the woman who had been sitting next to him, Menzio said he was then told that the woman 'thought I was a terrorist because I was writing strange things on a pad of paper. I laugh. I bring them back to the plane. I showed them my math.' Apparently, his math is pretty high level and technical."

The possibility that Menzio may have alarmed his seat-mate by having a dark complexion is mentioned gently in La Repubblica - Torino's headline: "Mediterranean face and strange notes: Torinese economist suspected of terrorism in the US" (May 7, Paolo Griseri) and more extravagantly by Catherine Rampell in her Rampage column in the Washington Post, May 7: "Ivy League economist ethnically profiled, interrogated for doing math on American Airlines flight" with the subtitle "Move over, Clock Boy. Another swarthy-looking nerd is alarming the authorities."

## "Fresh solutions" to the 4-body problem

The article "Central configurations of four bodies with an axis of symmetry" by Bálint Érdi and Zalán Czirják of Eötvös University in Budapest (Celest Mech Dyn Astr (2016) 12:33-70) was picked up by Douglas P. Hamilton in Nature "News & Views," May 4, 2016. "Describing the motion of three or more bodies under the influence of gravity is one of the toughest problems in astronomy. The report of solutions to a large subclass of the four-body problem is truly remarkable." The subclass is that of central configurations (those for which the solutions are invariant under rotation and the resulting forces on all bodies are directed towards the center of mass of the system). Hamilton: "Although the set of all possible central configurations of four bodies remains unknown, Érdi and Czirják have taken a large stride forward by solving all of those in which two of the bodies lie along an axis of symmetry." Hamilton illustrates the three subclasses of configurations that Érdi and Czirják determined. Here is the convex one:

Masses $m_1$ and $m_2$ are on the $x$-axis; the other two masses are equal (to $m$) and are positioned symmetrically with respect to that axis; center of mass is at the origin. In the convex case the line $mm$ intersects the axis between $m_1$ and $m_2$. Image adapted from Hamilton.

As Hamilton observes, the authors simplified their calculation by using the angles $\alpha$ and $\beta$ to locate the $m$-masses. In fact, using angles allowed Érdi and Czirják to apply the trigonometric identity $\sqrt{\tan^2\theta + 1} = 1/\cos\theta$ to great advantage in eliminating a troublesome radical from their computations.

## Math $-$ a piece of cake?

It's that and more (pie, custard, mayonnaise, lasagne, ...) in an article by Natalie Angier above the fold on page 1 of of the New York Times Science section, May 3, 2016. Angier initiates us into the math-culinary world of Eugenia Cheng, a mathematician currently at the School of the Art Institute of Chicago, "where she teaches math to art students, lectures widely and continues her research ... on the side." She works in "a rarefied field called category theory, which is so abstract that 'even some pure mathematicians think it goes too far,' Dr. Cheng said. At the same time, Dr. Cheng is winning fame as a math popularizer, convinced that the pleasures of math can be conveyed to the legions of numbers-averse humanities majors still recovering from high school algebra." How does this work? "Dr. Cheng adopts a literal approach to making math more appetizing. 'Math is about taking ingredients, putting them together, seeing what you can make out of them, and then deciding whether it's tasty or not,' she said. Every chapter in How to Bake $\pi$ [her book; Basic Books, 2015] offers recipes for desserts and other dishes that encapsulate mathematical themes." Some examples from the article:

• [Cake vs. custard] If you're making a cake, you can throw together the flour, sugar, butter and eggs however you please, and the cake will come out fine $-$ that's an associative process. Not so for preparing custard. You must first combine the sugar and egg yolks and whisk them into a froth before you pour in the cream. Blend the ingredients in a different order, she said, "and you end up with a runny mess."
• [Mayonnaise vs. hollandaise] To demonstrate how math seeks to identify underlying similarities across a broad set of problems, for example, Dr. Cheng starts with a recipe that can be readily tweaked to make mayonnaise instead of hollandaise sauce. "Books might tell you that hollandaise sauce needs to be done differently," she writes, "but I ignore them to make my life simpler. Math is also there to make things simpler, by finding things that look the same if you ignore some small detail."
• [Lasagne] Her recipe for lasagna illuminates the importance of context to math. Dr. Cheng lists among the basic ingredients "fresh lasagna noodles," and then points out that another cookbook might deem the noodles not truly basic and instead describe their preparation from scratch. So, too, do numbers change their character and degree of basicness depending on context. The number 5, for example, when viewed among the natural, or counting, numbers is one of those elemental creatures: a prime number, divisible only by 1 and itself. But in the context of the so-called rational numbers, which include fractions, 5 loses its prime identity and gains versatility, able to be divided into ever tinier slivers, like a cake at a dieters' convention.
Angier consulted veteran mathematics explainers and popularizers; they are very positive. John Baez: "She's trying to explain math to everybody, with or without pre-existing expertise, and I think she's doing wonderfully." Stephen Strogatz: "She conveys the spirit of inventiveness and creativity in math that all mathematicians feel but do a very poor job communicating when teaching math."

Tony Phillips
Stony Brook University
tony at math.sunysb.edu