Plants grow linearly, producing leaves and flowers along newly formed stems. It’s a remarkably straightforward process, but challenging for the mind’s eye because the cell-producing tip (we call this the apical meristem) is a moving target. The meristem makes new cells that “grow up” (not a botanical term), constantly pushing the growing tip further along. Each season is a refrain.

The new stem and leaves are this year’s song. The tip and buds they produce will reprise the lines next season. (see also Primary Growth)

Under a microscope, an apical meristem is just a batch of super-condensed cells, but baked into the stem physiognomy are clues of remarkable orchestration that bud produces. A score exists, a set of plans, a formula the plant follows. Plans are concise, with canned instructions for different kinds of leaves and buds, like “typical details” in architectural drawings, or monotonous refrains. Internal dynamics determine how often one kind of structure is repeated and when to kick in instructions for different sorts of leaves and buds. The result is a chorus of patterns we can study, recognize, and even (usually) predict. Some plants are ballads, others are symphonies..

Like one of those endless Irish folk songs, botanists expect plants to be *iterative* (repeating, re-iterating patterns of growth), creating leaves and buds that recur along a spiral, even multiple spirals, wrapping loosely or tightly around the stem (the axis). Each kind of plant has its own repertory of patterns, its phyllotaxy (leaf = *phyllon*, arrangement = *taxis*)* that devolves from secrets of spatial organization determined in the growing tip.**

To a good extent, plant study becomes a numbers game. In fact, the regularity of growth creates a characteristic botanists call “ranking” – the number of positions (radii) you can count around a stem until a leaf position repeats. Examine the stem of an oak, leaves repeat in five positions (ranks) along the stem; willows do this in 13 positions. Or look at the growing tip of a stem many people think might be a root – a potato, and you’ll find the repeating spiral.

When it comes to numbers and and how we use them to analyze and predict things like plant structure, look to the past, look eastward, key-holing through the work of a scholar who lived in Pisa between 1170 and 1250. That scholar, a son (*filio*) of the Bonacci family, was the mathematician eventually known as Fibonacci. In great part, our reliance on the 1, 2, 3-numbering starts with his manuscript, *Liber abbaci* on the *modus Indorum*, which presents the Hindu-Arabic numeral system as a practical and useful technique people should adopt. Mario Livio quotes Fibonacci:

*“The nine Indian figures are: 9 8 7 6 5 4 3 2 1.With these nine figures, and with the sign 0, any number may be written..*“

Fibonacci was inspired; the ten-digit system became standard throughout Europe. In fact, by the time printing was established, the “Indian” system was integrated in European life to the extent that Fibonacci’s original texts were lost in time. With the secret totally out and no market in mystery to drive sales, *Liber abbaci* failed to make the list of earliest printed books, the *incunabula* (books from the cradle of printing). ***

Numbers, their descriptive, analytic, and predictive values are fundamental to objective science, and the system Fibonacci popularized is a major driver of discovery. Oddly, this is not how we remember Fibonacci today. His name surfaces in discussions because among the examples presented as *modus Indorum*, Fibonacci included a self-generating (recurrent) number series, a continuum populated automatically as each number is the sum of the two preceding (smaller) numbers, i.e. : 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …**** Though we call these Fibonacci numbers, they aren’t his. Fibonacci was using observations long-known to earlier mathematicians simply to make his point..

When graphed, the sequence creates an expanding spiral with proportions approaching another mathematical and artistic concept, the golden ratio (phi). We find the ratios and relationships have predictive value, and even align with our own sensibilities, reflecting something about packing and proportions. How do objects like spheres and soap bubbles pack efficiently, and what proportions are well-balanced? And plants, they are built of cells, which occupy space and have to fit together is some organic geometry. How does geometry work for them? Given solids and angles, proportion falls into place because things have to scale up to work properly. The geometric relates to natural patterns in our world, patterns we know intuitively, proportions that impact our sense of what is natural.

So golden ratios and self-generating schemes are real, though a bit irrational. Don’t go out checking every plant for lockstep compliance to numerology. Nature doesn’t fit in a box; you can count the ribs on Golden Barrel Cacti, and they hover around Fibonacci numbers, but the evidence is equivocal. There are many forces outside pure geometry driving the arrangement of leaves and buds, Still, there’s something lyrical about the idea, and it’s fun to ponder.

*“the phyllotaxis rules I have described cannot be taken as applying to all circumstances, like a law of nature. Rather, in the words of the famous Canadian mathematician Coxeter, they are “only a fascinatingly prevalent tendency.”* Livio, 2002

*Bonnet, Charles, 1754. *Recherches sur l’usage des feuilles dan les plantes: et sur quelques autres sujets relatifs à l’hi*, Gottingue, Leide. (HNT 476698)

** The Computable Plant – http://computableplant.ics.uci.edu/publications.html

***Late in the game however, a 19th century scholar, Italian prince Baldassarre Boncompagni-Ludovisi, studied extant manuscripts (only 12 exist today), publishing Fibonacci’s text in a two-volume set in 1857. A set of those volumes is held in Huntington collections, on deposit from the Italian Ministry of Culture (Rare Books 750821 & 701012, part of the Burndy holdings). The world has, again, forgotten the contributions of *Liber abbaci*, which is unfortunate. From the entry by Kurt Vogel, in the Dictionary of Scientific Biography, on Leonardo Fibonacci: *“In surveying Leonardo’s activity, one sees him decisively take the role of a pioneer in the revival of mathematics in the Christian West. Like no one before him, he gave fresh consideration to the ancient knowledge and independently furthered it. In arithmetic he showed superior ability in computations. Moreover, he offered material to his readers in a systematic way and ordered his examples from the easier to the more difficult.”*

****Yes, there are negative Fibonacci numbers, with a sequence like this: … −8, 5, −3, 2, −1, 1, 0, 1, 1, 2, 3, 5, 8… .

Mario Livio, 2002. *The Golden Ratio: The Story of Phi….*