Plant Cuttings

Do plants know math? Unwinding the story of plant spirals, from Leonardo da Vinci to now, by Stéphane Douady, Jacques Dumais, Christophe Golé & Nancy Pick, 2024. Princeton University Press.

Do plants know math? is the intriguing question* posed by Stéphane Douady, Jacques Dumais, Christophe Golé & Nancy Pick in their book of the same name [which tome is here appraised]. As a Botanist I was understandably eager to find out if plants actually ‘know math’**.

What plant maths* is considered?

At the start it needs to be said that Do plants know math? isn’t about whether plants know the full range of mathematics. Instead, its focus – as is clarified in the book’s sub-title, Unwinding the story of plant spirals, from Leonardo da Vinci to now – is about spirals [which view is reinforced within the text by the authors, “This book is mostly about spirals” (p. 5)]. More specifically, the tome is primarily concerned with the spirals related to the placement of leaves upon the stem of a plant, the phenomenon termed phyllotaxis. Somewhat irritatingly – considering the importance of the word phyllotaxis to the book book [it’s the book’s first word, “Phyllotaxis is an adventure in curiosity” (p. 1)] – the authors don’t tell us until page 2 that it comes from the Ancient Greek words for ‘leaf arrangement’***. Although the book considers the ways in which leaves are arranged around the stems of – mainly – angiosperms, it also looks at other spiral patterns such as those seen in the placement of scales of conifer cones and the patterning of florets of the flower heads of plants such as sunflowers. [Ed. – although why the cover of a book devoted to phyllotaxis is adorned with the fiddlehead of a fern that “follows an entirely different growth mechanism from Fibonacci phyllotaxis” (p. 20] is a mystery.]

Having defined what the book is about,

What do you get?

The bulk of the book is split between an Introduction [which you must read first] and 21 numbered Chapters. Additionally, there’s an Appendix, Notes, and an Index.

Recognising that phyllotaxis is quite a technical topic, the authors usefully present us with a section headed Phyllotaxis in 10 Terms in the Introduction. In that section they introduce, define and explain terms relevant to phyllotaxis – e.g. spirals [stated examples of which are scales on a pineapple, leaves of an artichoke, florets at the centre of a dahlia], parastichy numbers, Fibonacci sequence, divergence angle, generative spirals, lattices, meristems, and whorled phyllotaxis. That gives the reader a good idea of the technical nature of the book’s subject matter. The Introduction concludes with several practical activities (as do several of the chapters…) – such as identifying and drawing the spirals of the arrangement of florets on the disc of a dahlia (p. 13). Those activities are presumably intended to reinforce some of the key terms or ideas of a chapter, and would, I imagine, be useful class exercises for those teaching students about phyllotaxis and plant spirals. The tome therefore has pedagogic value beyond the educational value of the informational presented in the main text.

I have no wish to spoil the reader’s enjoyment of the book by providing summaries of each chapter, but the following information should help to give some insight into what the book covers, and how its subject matter is dealt with.

Chapter 1 The (So-Called) Fibonacci Sequence in History [a fascinating account of the history of the number sequence named after Fibonacci, which all began with reproduction of rabbits (p. 26), or, rather, Sanskrit poetry (p. 28)…]; Chapter 4 First Spirals in Dew [with the fascinating tale of Charles Bonnet who pondered the question of why the texture of a leaf is different on the top than on the bottom, and whether plants breathe, and whose work “spurred other scientists to the discovery of photosynthesis” (p. 53)]; Chapter 5 Biomathematics on a Watch Face [which documents the birth of ‘botanical biomath’ (p. 68) with the work of Karl Schimper, and Alexander Braun – whom Douady et al. describe as ‘frenemies’]; Chapter 7 Irrational Angles in a French Garden [with talk of upside-down rainbows, crystallography, and the two Bravais brothers – the younger of which, Auguste, died a ‘martyr to science’]; Chapter 8 A Glimpse of the Growing Tip [looking at the important role of microscope observations of phyllotaxis, and insights into the phenomenon by Wilhelm Hofmeister]; Chapter 11 Sunflowers on Turing’s Primitive Computer [the advent of computers greatly helped to speed up analysis of phyllotaxis]; Chapter 16 Leaf Bud Kirigami [looking at the non-Fibonacci topic of leaf-folding in buds, and containing an exercise to construct your own cut-and-fold maple leaf from paper with the Japanese technique of Kirigami (Hannah Bellis)]; Chapter 17 The Hormone That Makes Spirals [regardless of any mathematical interpretation of spiral patterns it is plant physiology that makes the leaves, so it’s good to see biology amongst the physics and maths of the book]; and Chapter 21 A Spiral Dinner (with Recipes) [yes, which gives recipes for creating a plant-based dining experience**** – and concocting a signature drink, the rum-coconut-and-pineapple-based phyllo colada.]

The Appendix provides more in-depth notes to topics in several of the chapters. One of the most interesting – for me – related to Leonardo da Vinci and his notebooks, and the recognition that “he was likely the first to observe that each ring in a tree trunk represents a year in the tree’s life” (p. 289). Which 15^th/16^th century insight provides the basis of the 20^th century tree-ring-dating technique of dendrochronology. Do plants know math? Isn’t just about spirals.

The approx. 17 pages of Notes primarily provide sources for statements made in the main text of the book (and which are indicated in-text by super-scripted numbers). Sources cited are arranged in Introduction/Chapter/Appendix order, and are principally split between books – many of the statements made are drawn from historical texts – and scientific articles (as befits the scientific nature of the book’s subject matter). However, several of the Notes are unsourced statements or further explanation of terms or concepts mentioned in-text. Having read the book, it seems that the majority of the notes regarding sources relate to text quoted by the authors – which is as it should be. However, this does mean that in several places within the text, sources are needed but missing. For example, no sources are cited for the first two paragraphs of Chapter 4 (pp. 53/54) re ‘Enlightenment scientist’ Charles Bonnet [and the 3^rd paragraph only has a source for the quote in the last sentence of that text]; the statement that “(More recently, scientists have established that some plants do absorb dew through their leaves, mainly species that live in the desert.)” (p. 57) is unsourced; and the 5 paragraphs re biographical details of Alan Turing, on pp. 161-163, are unsourced [the only source stated is for a quote in the last sentence of the 5^th paragraph].

The Index is a comparatively short 5 pages of 2-columned entries, from ‘acorn caps’ to ‘zigzag line of Van Iterson’.

Is it only about maths?

No, but there is quite a lot of maths in the book.

Although readers should be aware of how mathematical a book this is, I would encourage them not to be put off by that. Indeed, anticipating that sort of reaction, the authors provide readers with a ‘get out of jail free card’: “This book intends to introduce you to some math (as far as you dare go [my emphasis])” (p. 2). [Ed. – FYI, although page 134 was as far as I ‘dared go’ with trying to follow the maths, that did not prevent me enjoying the remaining half of the book.]

It’s much more than maths

Had the book been just about the maths related to phyllotaxis, it could be much, much shorter by ‘cutting to the chase’ (Paige Pfeifer) and simply answering the question posed in its title, in a mere handful of pages [Ed. – As it is, and although we are told the answer – that’s not until p. 280]. But, the authors have chosen not to follow that route, and the book is all the better for it. What we get instead is a well-written, lavishly illustrated, scholarly tome that’s enriched with the history of the ideas surrounding the discovery of spiral patterns in plants and the many attempts by many individuals over hundreds of years to understand the how and why of the phenomenon. As a result, you get a most satisfying book – albeit one with plenty of maths – that delivers a great read for the reader [Ed. – and one that’s not without some humour; yes, I here have in mind things like the title for Chapter 7 Irrational Angles in a French Garden… Vive l’entente cordiale.]

Over the course of its 287 or so pages of main text, Douady et al. cover topics such as plant anatomy, plant morphology, and plant physiology (and phyllotaxis, and ‘phytomaths’), with very human tales about such scientific luminaries as Leonardo da Vinci, Karl Schimper, the Bravais brothers, Willhelm Hofmeister, Simon Schwendener [who “was so poor that he had to choose between marriage and botany” (p. 129)], and Alan Turing, all of whom have contributed to unravelling plant spirals*****. [Ed. – And, as befits any biological book worth its salt, Charles Darwin gets an honorary mention. Not because he contributed to elucidation of the problem, but because he “joked that the topic [phyllotaxis] would drive him to a miserable death” (p. 265)]. And, almost as important as those who investigated the phenomenon, Douady et al. give us a fascinating insight into the course of scientific investigation. In other words, and as the authors advise, “This will be no dry scientific treatise, but instead a very human adventure. In this book, you will delve into the hearts and minds of scientists who have unlocked the secrets of “plant-mathematics” over the course of several centuries” (p. 1). There is therefore much more than maths in Do plants know math?

Do plants know math? is ‘quirky’, with its recipes, heartfelt personal biographies of the authors, practical activities, humour, and numerous questions for readers posed in the figure legends. It’s probably the only botanical text to use the term ‘frenemies’. And, amongst the many truthful statements are numerous ‘fibs’******.

Summary

I liked Do plants know math? by Stéphane Douady, Jacques Dumais, Christophe Golé & Nancy Pick. Although I struggled with the maths, you don’t have to be a maths whizz to appreciate the book. It’s a very worthwhile read for the historical perspective it provides on the development of our understanding of the formation of patterns in plants, the tales it tells about the scientists who contributed to its unravelling (which includes the authors), and insights it shares into the meandering course of scientific investigation. Whilst it may have a maths angle, this is very much a book about plants and people.

* As those of you who know me might have suspected, my knee-jerk reaction to seeing the question posed by Stéphane Douady et al’s book is to reply: No, but they might know maths. That response relates to the notable difference that exists in the short-hand English language word used by the peoples either side of the trans-Atlantic divide for the longer word mathematics. For more on the important battle between ‘math’ and ‘maths’, see here, here, here, here, Melissa, Mariusz Alza, Simon Kewin, and Candace Osmond.

** Not a ‘spoiler alert’, but to reassure you that this question is answered, on page 280 of the book.

*** The origin of use of the term phyllotaxis is explained by the authors: “His [Karl Schimper (Heinz Tobien)’s] wordsmithing left a lasting mark on plant mathematics, for, we believe, in 1830 he coined the word “phyllotaxis.” Based on the Ancient Greek words for “leaf arrangement”(p. 69). For more on phyllotaxis, see Irving Adler et al. (1997); Jan Traas (2013); Cris Kuhlemeier (2017); Okabe Takuya et al. (2019); Sören Strauss et al. (2020) [and the commentary thereon by Adrien Sicard (2020)].

**** Disappointingly, none of the vegetables were spiralised (Erin Alderson). But, the dessert did use phyllo pastry [Ed. – which I thought was a pun on filo pastry, and which would have been amusing. However, I now know that phyllo pastry is a legitimate alternative name for filo pastry (Caroline), especially in the USA, which is not quite so amusing. Indeed, as Douady et al. tell us in the book, “The name “phyllo dough” comes from the same Greek root as phyllotaxis: phyllo means “leaf.” In the case of the dough, the leaf is of course the thin sheet of pastry” (p. 286). And which reminds me that I really should have read Chapter 21 more carefully before making comical comments about this part of the meal(!)]

***** For more on plant spirals, Sandy Hetherinton’s Linnean Society’s fascinating lecture entitled “The Origin and Evolution of Botanical Spirals” is available on YouTube, and can be viewed here. Dealing with with Fibonacci spirals, non-Fibonacci spirals, and the development of fiddleheads in ferns, it is well worth a watch and a listen [Ed. – and nicely links to the book’s cover image… [comment added after, and in respopnse to, unironedman’s comment below.]]

***** One of the most unusual aspects of the book is that the chapters usually start with a 7-line poem, whose syllable count per line is in the Fibonacci sequence, 1, 1, 2, 3, 5, 8, and 13. As one of its many activities, readers are invited to try their hand at creating one. Furthermore, “In 2006, poet Gregory K. Pincus created a popular variation called a “fib,” with just six lines (20 syllables) altogether. Like all Fibonacci poems, fibs fall at the pleasing intersection of language and math” (p. 77).

REFERENCES

Irving Adler et al., 1997. A history of the study of phyllotaxis. Annals of Botany 80(3): 231-244; https://doi.org/10.1006/anbo.1997.0422

Cris Kuhlemeier, 2017. Phyllotaxis. Current Biology 27(17): R882-R887; https://doi.org/10.1016/j.cub.2017.05.069

Adrien Sicard, 2020. How bright is gold: is there a photosynthetic advantage to the golden angle? New Phytologist 225(1): 13-15; https://doi.org/10.1111/nph.16183

Sören Strauss et al., 2020. Phyllotaxis: is the golden angle optimal for light capture? New Phytologist 225(1): 499-510; https://doi.org/10.1111/nph.16040

Okabe Takuya et al., 2019. The unified rule of phyllotaxis explaining both spiral and non-spiral arrangements. JR Soc. Interface: 1620180850; http://doi.org/10.1098/rsif.2018.0850

Jan Traas, 2013. Phyllotaxis. Development 140(2): 249–253; https://doi.org/10.1242/dev.074740