What fractals, Fibonacci, and the golden ratio have to do with cauliflower - Ars Technica

Enlarge / Macro-photo of green Romanesco cauliflower showcases the vegetable's striking fractal pattern.

Cathy Scola / Getty Images

It's long been observed that many plants produce leaves, shoots, or flowers in spiral patterns. Cauliflower provides a unique example of this phenomenon, because those spirals repeat at several different size scales—a hallmark of fractal geometry. This self-similarity is particularly notable in the Romanesco variety because of the distinctive conical shape of its florets. Now, a team of French scientists from the CNRS has identified the underlying mechanism that gives rise to this unusual pattern, according to a new paper published in Science.

Fractal geometry is the mathematical offspring of chaos theory; a fractal is the pattern left behind in the wave of chaotic activity. That single geometric pattern repeats thousands of times at different magnifications (self-similarity). For that reason, fractals are often likened to Russian nesting dolls. Many fractal patterns exist only in mathematical theory, but over the last few decades, scientists have found there are fractal aspects to many irregular yet patterned shapes in nature, such the branchings of rivers and trees—or the strange self-similar repeating buds that make up the Romanesco cauliflower.

Each bud is made up of a series of smaller buds, although the pattern doesn't continue down to infinitely smaller size scales, so it's only an approximate fractal. The branched tips, called meristems, make up a logarithmic spiral, and the number of spirals on the head of Romanesco cauliflower is a Fibonacci number, which in turn is related to what's known as the "golden ratio."

The person most closely associated with the Fibonacci sequence is the 13th-century mathematician Leonardo Pisano; his nickname was "filius Bonacci" (son of Bonacci), which got shortened to Fibonacci. In his 1202 treatise, Book of Calculation, Fibonacci described the numerical sequence that now bears his name: 1, 2, 3, 5, 8, 13, 21... and on into infinity. Divide each number in the sequence into the one that follows, and the answer will be something close to 1.618, an irrational number known as phi, aka the golden ratio. And there is a special "golden" logarithmic spiral that grows outward by a factor of the golden ratio for every 90 degrees of rotation, of which a "Fibonacci spiral" is a close approximation.

Scientists have long puzzled over possible underlying mechanisms for this unusual patterning in the arrangement of leaves on a stem (phyllotaxis) of so many plants—including pine cones, daisies, dahlias, sunflowers, and cacti—dating all the way back to Leonardo da Vinci. Swiss naturalist Charles Bonnet (who coined the term "phyllotaxis") noted that these spirals exhibited either clockwise or counterclockwise golden ratios in 1754, while French brothers Auguste and Louis Bravais discovered in 1837 that the ratios of phyllotaxis spirals were related to the Fibonacci sequence.

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  Illustrations of different kinds of phyllotaxis spirals in nature.

  Eugenio Azpeitia et al., Science 2021

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  Simulated morphologies and light micrographs of various cauliflower structures

  Eugenio Azpeitia et al., Science 2021

In 1868, German botanist Wilhelm Hofmeister came up with a solid working model. He found that nascent leaves ("primordia") will form at the least crowded part of the meristem, and as the plant grows, each successive leaf will move outward radially, at a rate proportional to the stem's growth. The second leaf, for instance, will grow as far as possible from the first, and the third will grow at a distance farthest from both the first and the second leaves, and so on. It's not a hardcore law of nature or some kind of weird botanical magic: that Fibonacci spiral is simply the most efficient way of packing the leaves.

According to the authors of this latest paper, the spiral phyllotaxis of cauliflower is unusual because those spirals are conspicuously visible at several different size scales, particularly in the Romanesco variety. They maintain that cauliflowers are basically failed flowers. The whole process depends on those branched tips, or meristems, which are made up of undifferentiated cells that divide and develop into other organs arranged in a spiral pattern. In the case of cauliflower, these cells produce buds that would normally bloom into flowers. Those buds develop into stems instead, but unlike normal stems, they are able to grow without leaves and thereby produce even more buds that turn into stems instead of flowers.

This triggers a chain reaction, resulting in that trademark pattern of repeating stems upon stems that ultimately forms the edible white flesh known as the "curd." In the case of the Romanesco variety, its stems produce buds at an accelerating rate (instead of the constant rate typical of other forms of cauliflower). So its florets take on that distinctive pyramid-like shape that showcases the fractal patterns so beautifully.

The puzzle, per the authors, is how these gene regulatory networks that initially evolved to produce flowers were able to change so drastically. So co-author Eugenio Azpeitia and several colleagues combined in vivo experiments with 3D computational modeling of plant development to study the molecular underpinnings of how buds form in cauliflower (both edible cauliflower and the Romanesco variant).

Apparently, this is the result of self-selected mutations during the process of domestication, which over time drastically changed the shapes of these plants. The authors found that, while the meristems fail to form flowers, the meristems do experience a transient period where they're in a flower-like state, and that influences later steps in development. In the case of Romanesco cauliflower, the curd adopts a more conical shape instead of a round morphology. The end result is those fractal-like forms at several different size scales.

"These results reveal how fractal patterns can be generated through growth and developmental networks that alter identities and meristem dynamics," the authors concluded. "Our models now clarify the molecular and morphological changes over time by which meristems gain different identities to form the highly diverse and fascinating array of plant architectures found throughout nature and crops."

DOI: Science, 2021. 10.1126/science.abg5999  (About DOIs).

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