Ever wondered how your middle school algebra lessons could possibly apply to real life? It turns
out that math is ever-present in the world around us, and your garden may be the first place to start
looking. Studies of phyllotaxis, the arrangement of leaves on plants, have led to discoveries that traits
like divergence angles of leaves from stems can form fascinatingly exact patterns [1]. One of the most
prevalent patterns in plants is the well-known Fibonacci sequence (1, 1, 2, 3, 5, 8, …), which is also found
in many other iterations within the natural world [1]. The resulting spirals align with primordium plant
growth and are evident in the center of a sunflower, the aerial view of a radial succulent, and the
polygons of Brassica oleracea in the image above [2]. Aside from the Fibonacci spiral, common formulas
that plant branches follow include distichous (alternating), decussate (paired at right angles), and
tricussate (whorled trios of leaves) patterns [3].
But why would plants need to follow this math? There is substantial evolutionary justification
for this visually stunning geometry; plants are largely immobile creatures who adapt to their
environments primarily through their own growth (or lack thereof). As such, a plant’s best chance of
survival requires using growth forms to maximize a leaf’s exposure to sunlight and protection from the
elements – and then doing the same for every other leaf [4]. Leaves must grow in close proximity to
conserve space and resources on the plant, but current hypotheses state that they also biochemically
repel other leaves from growing too near them in order to fully utilize their environment [3]. This level
of complexity requires the absolute precision that is found in the mathematical formulas we use to
describe these natural phenomena.
While it’s true that certain patterns are far more recognizable than others – the golden ratio and
Fibonacci sequence are rarely inconspicuous – that doesn’t mean that plants with seemingly irregular or
random branching patterns don’t follow discernable equations. For instance, the species Orixa japonica
has the eponymous orixate arrangement, which was long thought to be mathematically inexplicable [3].
Its leaves grow asymmetrically rather than in a spiral or radial form and occur in a repeating series of
four different angles from consecutive leaves. Surprisingly, this highly specific pattern occurs in other
plant species across the evolutionary tree, indicating that there must be some mathematically driven
mechanism causing it. Researchers only came up with an appropriate model for its manifestation in the
past few years using computer simulations, and in doing so revised a previous phyllotaxis formula to
better describe several other plants [3].
Despite the longstanding fascination of the relationship between mathematical concepts and
the botanical world, concepts of it being leaf-inhibitory based remain in speculation and there is no
unifying theory for determining formulas or the mechanisms behind this [2]. There is undoubtedly a
great deal of complexity in this phenomenon, and yet its amazing visual effects can be appreciated by
even the most untrained eye in the simplest plant in a garden.
References:
Shipman, P.D.; Newell, A.C. Phyllotactic Patterns on Plants. Physical Review Letters 2004, 92, 1-4.
Newell, A.C.; Shipman, P.D. Plants and Fibonacci. Journal of Statistical Physics 2005, 121 937-968.
Yonekura, T.; Iwamoto, A.; Fujita, H.; Sugiyama, M. Mathematical model studies of the comprehensive generation of major and minor phyllotactic patterns in plants with a predominant focus on orixate phyllotaxis. PLoS Computational Biology 2019, 15, e1007044.
Burakoff, M. Decoding the Mathematical Secrets of Plants’ Stunning Leaf Patterns. Smithsonian Magazine 2019. https://getpocket.com/explore/item/decoding-the-mathematical-secrets-of-plants-stunning-leaf-patterns?utm_source=pocket-newtab (accessed September 1, 2020).
Image source: Hesselink, A. romanesco; 2005.