# This Math Explains Why Cauliflowers Look Too Oddly Perfect

## Science

08 July, 2021 - 08:37pm 86 views

What you'll find is that what at first sight looks like an amorphous blob has a striking regularity.

If you take a good look, you will see that the many florets look alike and are composed of miniature versions of themselves. In math, we call this property self-similarity, which is a defining feature of abstract geometrical objects called fractals. But why do cauliflowers have this property?

Our new study, published in Science, has come up with an answer.

There are many examples of fractals in natures, such as ice crystals or branches on trees. In math, the number of copies of an initial pattern goes on infinitely. Cauliflowers present a high level of such self-similarity, involving seven or more copies of the "same" bud.

This is most conspicuous on the romanesco cauliflower (sometimes called romanesco broccoli, because of its color), one of the first images that will appear if you search "plant fractals" online.

What is striking about the romanesco is the very well-defined, pyramidal buds which accumulate along endless spirals. Though less immediately obvious, a similar arrangement is present in other cauliflowers too.

Spirals are found in many plants, it is the main pattern of plant organization – a topic which has been studied for well over 2,000 years. But although cauliflowers share spirals with most other plants, their self-similarity is unique.

Where does this special feature come from? And are the cauliflower spirals originating from the same mechanisms as those in other plants?

About 12 years ago, two of my colleagues in France, François Parcy and Christophe Godin, were starting to ask exactly these questions and invited me to join the effort. We spent many hours frantically dismantling florets, counting them, measuring angles between them, studying the literature on the molecular mechanisms underlying the growth of cauliflowers, and trying to create realistic computational models of these mysterious cabbages.

Most available data was on Arabidopsis thaliana, also known as the "thale cress" flowering plant. Though this is a weed, it is of paramount importance in modern plant biology because its genetics have been extensively studied for many years, including many variants. And it turns out to be related to all cabbages, belonging to the family known as brassicaceae.

Arabidopsis in fact has its own version of the cauliflower, arising from a simple mutation involving only one pair of similar genes (see image below). So the genetics of this mutant plant are very similar to the genetics of cauliflower.

If you spend some time observing the branches along the stem of, say, some weeds in your garden (which likely include close relatives of Arabidopsis), you will see how they are quite closely following each other, with the same angle between each successive pair. And if there are enough organs along this spiral, you will start seeing other spirals, going both clockwise and anticlockwise (see image on the right).

If you manage to count the spirals, they will typically be numbers somewhere along the Fibonacci sequence, where the next number in the sequence is found by adding up the two numbers before it. This gives 0, 1, 1, 2, 3, 5, 8, 13, etc.

On a typical cauliflower, expect to see five spirals going clockwise and eight anticlockwise, or vice-versa (see images below). But why? To understand how the geometry of plants develop over their lifetime, we need mathematics – but also microscopes.

We know now that for every plant, the main spiral is already formed at microscopic scales. This happens very early in its development. At this stage, it comprises spots, in which very specific genes are expressed (turned on). The genes expressed in a spot determine whether this spot will grow into a branch, a leaf, or a flower.

But the genes are actually interacting with each other, in complex "gene networks" – leading to specific genes being expressed in specific domains and at specific times. This is beyond simple intuition, and mathematical biologists therefore rely on differential equations to write models of these gene networks to predict their behavior.

To work out how cauliflowers grow into their peculiar shape after the first few leaves have formed, we built a model which included two main components. These were a description of the spiral formation that we see in large cauliflowers, and a model of the underlying gene network that we see in Arabidopsis. We then tried to match the two so we could work out which genetics led to cauliflower structure.

We found that four main genes are the crucial players: their initials are S, A, L, and T, which we obviously joked about.

The "A" is missing in Arabidopsis flowering plants that have mutated to become cauliflower-like, and is also a gene that drives spots to become flowers.

What makes cauliflower so special is that these spots at the growing tip try to turn into flowers for some time (up to several hours), but keep failing at it for lack of "A". Instead, they develop into stems, which turn into stems, etc – multiplying almost infinitely without growing leaves, which gives rise to near-identical cauliflower buds.

The time they spend trying is fundamental – getting this right in our model allowed us to reproduce cauliflowers and romanescos exactly on the computer. We confirmed this was right by altering the growth in a real-life Arabidopsis cauliflower mutant plant, effectively turning it into a shape much alike a miniature romanesco.

### What fractals, Fibonacci, and the golden ratio have to do with cauliflower

09 July, 2021 - 07:42am

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.

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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### Why do cauliflowers look so odd? We've cracked the maths behind their 'fractal' shape

09 July, 2021 - 07:42am

Assistant professor of Mathematics, University of Nottingham

University of Nottingham provides funding as a founding partner of The Conversation UK.

Have you ever stared at a cauliflower before preparing it and got lost in its stunningly beautiful pattern? Probably not, if you are in your right mind, but I reassure you it’s worth a try. What you’ll find is that what at first sight looks like an amorphous blob has a striking regularity.

If you take a good look, you will see that the many florets look alike and are composed of miniature versions of themselves. In maths, we call this property self-similarity, which is a defining feature of abstract geometrical objects called fractals. But why do cauliflowers have this property? Our new study, published in Science, has come up with an answer.

There are many examples of fractals in natures, such as ice crystals or branches on trees. In maths, the number of copies of an initial pattern goes on infinitely. Cauliflowers present a high level of such self-similarity, involving seven or more copies of the “same” bud.

This is most conspicuous on the romanesco cauliflower (sometimes called romanesco broccoli, because of its colour), one of the first images that will appear if you search “plant fractals” online. What is striking about the romanesco is the very well defined, pyramidal buds which accumulate along endless spirals. Though less immediately obvious, a similar arrangement is present in other cauliflowers too.

Spirals are found in many plants, it is the main pattern of plant organisation – a topic which has been studied for well over 2,000 years. But although cauliflowers share spirals with most other plants, their self-similarity is unique. Where does this special feature come from? And are the cauliflower spirals originating from the same mechanisms as those in other plants?

About 12 years ago, two of my colleagues in France, François Parcy and Christophe Godin, were starting to ask exactly these questions and invited me to join the effort. We spent many hours frantically dismantling florets, counting them, measuring angles between them, studying the literature on the molecular mechanisms underlying the growth of cauliflowers, and trying to create a realistic computational models of these mysterious cabbages.

Most available data was on Arabidopsis thaliana, also known as the “thale cress” flowering plant. Though this is a weed, it is of paramount importance in modern plant biology because its genetics have been extensively studied for many years, including many variants. And it turns out to be related to all cabbages, belonging to the family known as brassicaceae. Arabidopsis in fact has its own version of the cauliflower, arising from a simple mutation involving only one pair of similar genes (see image on the left). So the genetics of this mutant plant are very similar to the genetics of cauliflower.

If you spend some time observing the branches along the stem of, say, some weeds in your garden (which likely include close relatives of Arabidopsis), you will see how they are quite closely following each other, with the same angle between each successive pair. And if there are enough organs along this spiral, you will start seeing other spirals, going both clockwise and anticlockwise (see image on the right).

If you manage to count the spirals, they will typically be numbers somewhere along the Fibonacci sequence, where the next number in the sequence is found by adding up the two numbers before it. This gives 0, 1, 1, 2, 3, 5, 8, 13, etc. On a typical cauliflower, expect to see five spirals going clockwise and eight anticlockwise, or vice-versa (see images below). But why? To understand how the geometry of plants develop over their lifetime, we need mathematics – but also microscopes.

We know now that for every plant, the main spiral is already formed at microscopic scales. This happens very early in its development. At this stage, it comprises spots, in which very specific genes are expressed (turned on). The genes expressed in a spot determine whether this spot will grow into a branch, a leaf or a flower.

But the genes are actually interacting with each other, in complex “gene networks” – leading to specific genes being expressed in specific domains and at specific times. This is beyond simple intuition, and mathematical biologists therefore rely on differential equations to write models of these gene networks to predict their behaviour.

To work out how cauliflowers grow into their peculiar shape after the first few leaves have formed, we built a model which included two main components. These were a description of the spiral formation that we see in large cauliflowers, and a model of the underlying gene network that we see in Arabidopsis. We then tried to match the two so we could work out which genetics led to cauliflower structure.

We found that four main genes are the crucial players: their initials are S, A, L and T, which we obviously joked about. The “A” is missing in Arabidopsis flowering plants that have mutated to become cauliflower-like, and is also a gene that drives spots to become flowers.

What makes cauliflower so special is that these spots at the growing tip try to turn into flowers for some time (up to several hours), but keep failing at it for lack of “A”. Instead, they develop into stems, which turn into stems etc – multiplying almost infinitely without growing leafs, which gives rise to near-identical cauliflower buds.

The time they spend trying is fundamental – getting this right in our model allowed us to reproduce cauliflowers and romanescos exactly on the computer. We confirmed this was right by altering the growth in a real-life Arabidopsis cauliflower mutant plant, effectively turning it into a shape much alike a miniature romanesco.

It is amazing how complex nature is. The next time you have cauliflower for dinner, take a moment to admire it before you eat it.

Write an article and join a growing community of more than 129,900 academics and researchers from 4,090 institutions.

### Alien-like vegetables have mathematical “memory” — study

08 July, 2021 - 01:00pm

There’s more going on in a Romanesco than meets the eye.

The gentle, green conical towers sprouting from its surface — made themselves from even smaller conical towers — are mesmerizing. And the reason is more to do with math than flavor — the surface of Romanesco is a textbook example of a natural fractal.

These organic patterns appear throughout nature. But until now scientists have known little about how the self-similar patterns in sunflower heads, cauliflower, or the centers of daisies actually develop. The answer may be more human than you’d expect.

First, the team built a virtual network of the plant’s main regulatory genes including those involved in curd and flower development. They then investigated how these curds could be triggered in a plant with cauliflower-like structures, Arabidopsis thaliana.

While this delicate, white petaled flower has little resemblance to Romanesco in its wild form, the team found that modifying its genes in a 3D model could trigger remarkably similar curd formation.

“Self-similarity allows high curd compactness, which attracts humans for food,” Godin tells Inverse.

“Conspicuous fractal curds also attract humans by their fascinating beauty. We could say at least that cauliflowers and romanesco benefit from the symbiosis with humans that cultivate them and make them proliferate.”

Godin also says that this new work could lead to the “development of bio-inspired fractal mathematical constructions.”

“This provides new ways to think about fractals as growing structures with local distributed growth regulation,” Godin says.

Fractals in nature are a beautiful, albeit sometimes eery, reminder of the universe’s strange inner workings, and this research could offer an opportunity to pull back that curtain a little further.

The word fractal is derived from the Latin word fractus which means “broken” or “fractured” and is used to describe repeating patterns in nature, design, or human physiology. Branching trees, human lungs, and data management systems are some ways fractals show up in our daily life.

The Fibonacci Sequence is also another common way to describe a particular kind of fractal that grows larger and larger with new branches.

Francois Parcy, a senior author on the paper and plant physiology researcher at the Laboratoire Physiologie Cellulaire et Végétale in France, tells Inverse that a branching fireworks display is a good example. The Fibonacci Sequence is also what gives rise to the Golden Ratio — a curve that evokes both beauty and perfect order.

With the exponential growth of its curds, Romanesco falls into this mathematical ideal of beauty. Which Godin says isn’t necessarily beneficial for the plant — aside from how this pattern attracts human cultivators.

This human-selected gene variation allowed the team to determine that a form of “memory” was responsible for the plant's recursive and immature curd growth — the fractals that draw us to eat a Romanesco and feel somehow different from if we ate a standard white, fluffy cauliflower.

Parcy explains the process a little more to Inverse:

“A stem normally grows with leaves and you rarely have many stems piling up on each other with elongation,” Parcy says “Memory of the floral state generates stems with no leaves — flowers have no leaves — and free from the control of apical dominance.” This refers to a process that restricts bud growth beneath a new shoot.

“We have understood how a few perturbations can change a flowering shoot into cauliflowers or in Romanesco,” adds Parcy. “And how they can be changed drastically by human selection.”

“We understand well the Arabidopsis cauliflower but we know there must be more mutations to explain the compact cauliflower curd,” he says.

“We have not found the nature of all the mutations that are responsible for the curd shape or the Romanesco increased bud production,” he adds.

But, as they do here, there is a new avenue opening up for future research, in which labs can use genetic technology to potentially transform plants as we know them, and create new patterns using these organic blueprints.

“A recent paper provides hundreds of sequences for cabbage broccoli and cauliflower that could be used within our framework to go one step further,” he says.

“Maybe we could create cauliflowers from other plants!”

### Cauliflower and Chaos, Fractals in Every Floret

08 July, 2021 - 01:00pm

Scientists take a crack at recreating the hypnotic fractal spirals of the Romanesco cauliflower.

Monks once hoped to turn lead into gold through alchemy. But consider the cauliflower instead. It takes just two genes to transform the ordinary stems, stalks and flowers of the weedy, tasteless species Brassica oleracea into a formation as marvelous as this fractal, cloudlike vegetable.

This is the true alchemy, says Christophe Godin, a senior researcher at the National Institute for Research in Digital Science and Technology in Lyon, France.

Dr. Godin studies plant architecture by virtually modeling the development of the forms of different species in three dimensions. He wondered what genetic modification lurked behind cauliflower’s nested spirals and the logarithmic chartreuse fractals of Romanesco, a cauliflower cultivar that could almost be mistaken for a crystal.

“How is nature able to build such unexpected objects?” he asked. “What can be the rules behind this?”

Fifteen years ago, Dr. Godin met François Parcy, a plant biologist with the National Center for Scientific Research in Grenoble, France. In Dr. Parcy, Dr. Godin recognized a fellow fiend for fractal florets.

“There is no way you cannot notice it is such a gorgeous vegetable,” Dr. Parcy said, in reference to Romanesco.

Buoyed by a passion for Brassica, Dr. Godin and Dr. Parcy investigated the genetic mystery of the fractal geometry in both Romanesco and standard cauliflower, conjuring the plants in mathematical models and also growing them in real life. Their results, which suggest the fractals form in response to shifts in the networks of genes that govern floral development, are published Thursday in Science.

“It’s such a nice integration of genetics on one hand and rigorous modeling on the other,” said Michael Purugganan, a biologist at New York University who was not involved with the research. “They’re trying to show that by tweaking the rules of how genes interact you can get dramatic changes of a plant.”

In the early 2000s, Dr. Parcy believed he understood the cauliflower. He even taught classes on its flower development. “What is a cauliflower? How can it grow? Why does it look like this?” he said.

Cauliflowers, like brussels sprouts, stem from centuries of selective breeding of Brassica oleracea. Humans bred brussels sprouts for lateral buds and cauliflower for flower clusters. Cauliflowers, however, do not produce flower buds; their inflorescences, or flower-bearing shoots, never mature to produce flowers. Instead, cauliflower inflorescences generate replicas of themselves in a spiral, creating clusters of curds like plant-based cottage cheese.

As the two researchers discussed cauliflower, Dr. Godin suggested that if Dr. Parcy truly understood the plant, it should be easy to model the vegetable’s morphological development. As it turned out, it was not.

The two first confronted the curdled quagmire on the blackboard, sketching out various diagrams of genetic networks that could explain how the vegetable mutated into its current shape. Their muse was Arabidopsis thaliana, a well-studied weed in the same family as cauliflower and its many cousins.

If a cauliflower has a single cauliflower at the base of the plant, Arabidopsis has many cauliflower-like structures along its elongated stem. But what genes could refine these lesser cauliflowers into one grand, compact cauliflower? And if they identified those genes, could they warp these cauliflowers into the peaks that Romanescos form?

(Over the course of the research, Dr. Parcy also collected several specimens of Romanesco from his local farmer’s market, sequenced and dissected them. He and his colleagues then dined on the leftovers, most often raw with different dips, along with glasses of beer.)

Many initial models flopped, bearing little resemblance to cauliflowers. At first, the researchers believed the key to cauliflowers lay in the length of the stem. But when they programmed Arabidopsis with and without a short stem, they realized they did not need to reduce the stem size of the cauliflowers, either in the 3-D models or in real life.

And the cauliflowers they simulated and grew were simply not fractal enough. The patterns were visible only at two fractal scales, such as one spiral nested in another spiral. By contrast, a regular cauliflower often displays self-similarity in at least seven fractal scales, meaning a spiral nested in a spiral nested in a spiral nested in a spiral nested in a spiral nested in a spiral nested in, ultimately, another spiral.

So instead of focusing on the stem, they concentrated on the meristem, a region of plant tissue at the tip of each stem where actively dividing cells produce new growth. They hypothesized that making the meristem bigger would increase the number of shoots produced.

The only problem was that the researchers did not know what gene might control the meristem’s pace of shoot production.

One day, Eugenio Azpeitia, then a postdoctoral fellow in Dr. Godin’s lab, remembered a gene that was known to change the size of the central zone of the meristem. The three researchers enjoyed a brief moment of euphoria, and then waited patiently for months for their newly modified Arabidopsis to grow. When the shoots sprouted, they had cauliflowers with distinct conical tips.

“Very reminiscent of what occurs in the Romanesco,” Dr. Godin said proudly.

Normally, when a plant sprouts a flower, the flowering tip of the plant prevents more growth from the stem. A cauliflower curd is a bud that was designed to become a flower but never makes it all the way there, and instead makes a shoot. But the researchers’ experiments in the meristem found that because this shoot has passed through a transient floral stage, it is exposed to a gene that triggers its growth. “Because you have been a flower, you are free to grow and you can make a shoot,” Dr. Parcy said.

This process creates a chain reaction where the meristem is creating many shoots that, in turn, creates many more shoots, enacting the fractal geometry of a cauliflower.

“It’s not a normal stem,” Dr. Godin said. “It’s a stem without a leaf. A stem with no inhibition.”

“That’s the only way to make a cauliflower,” Dr. Parcy said.

The researchers say there are likely other mutations responsible for the spectacular shape of Romanesco. Ning Guo, a researcher at the Beijing Vegetable Research Center who is also studying the potential genetic mechanism behind the architecture of the cauliflower curd, says the paper has offered “a lot of inspiration.”

“The story is not yet finished,” Dr. Godin said, adding that he and Dr. Parcy will continue refining their cauliflower models. “But we know we are on the right track.”

But they are open, they say, to studying anything that flowers.

### Where does the shape of the Romanesco cauliflower come from?

08 July, 2021 - 01:00pm

by CNRS

This study shows that the brief incursion of buds into a flowering state profoundly affects their functioning and allows them, unlike normal stems, to grow without leaves and to multiply almost infinitely. The atypical shape of the Romanesco is explained by the fact that its stems produce buds more and more rapidly (whereas the production rate is constant in other cauliflowers). This acceleration gives each floret a pyramidal appearance, making the fractal aspect of the structure clear.

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### How Romanesco cauliflower forms its spiraling fractals

08 July, 2021 - 01:00pm

The swirling green cones that make up the head of Romanesco cauliflower also form a fractal pattern — one that repeats itself on multiple scales. Now, the genes that underlie this stunning structure have been identified, and the fractal pattern has been replicated in a common lab plant, Arabidopsis thaliana, researchers report in the July 9 Science.

“Romanesco is one of the most conspicuous fractal shapes that you can find in nature,” says Christophe Godin, a computer scientist with the National Institute for Research in Digital Science and Technology who is based at ENS de Lyon in France. “The question is, why is that so?” The answer has long eluded scientists.

Godin and his colleagues knew an Arabidopsis variant could produce small cauliflower-like structures. So the team manipulated the genes of A. thaliana in both computer simulations and growing experiments in the lab. Working with the extensively studied plant helped the researchers simplify their experiments and distill the essential fractal-spawning mechanism (SN: 6/15/21).

By altering three genes, the researchers grew a Romanesco-like head on A. thaliana. Two of those genetic tweaks hampered flower growth and triggered runaway shoot growth. In place of a flower, the plant grows a shoot, and on that shoot, it grows another shoot, and so on, says plant biologist François Parcy at CNRS in Paris. “It’s a chain reaction.”

The researchers then altered one other gene, which increased the growing area at the end of each shoot and provided space for spiraling conical fractals to form. “You don’t need to change the genetics much to get this form to appear,” says Parcy. The team’s next step, he says, “will be to manipulate these genes in cauliflower.”

E. Azpeitia, et al. Cauliflower fractal forms arise from perturbations of floral gene networks. Science. Vol. 373, July 9, 2021, p. 192. doi: 10.1126/science.abg5999.

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