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Why does cauliflower look so strange? We have blasted the mathematics behind their ‘fractal’ form

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

If you look good, you will see that the many flowers are similar and consist of miniature versions of themselves. In mathematics, we call this property self-similarity, which is a defining feature of abstract geometric objects called fractals. But why does cauliflower have this property? Our new study, published in Science, has come up with an answer.

There are many examples of fractals in nature, such as ice crystals or branches on trees. In mathematics, the number of copies of an initial pattern continues indefinitely. Cauliflower presents a high level of such selfishness that involves seven or more copies of the “same”

; bud.

Green cauliflower in polygonal star shape with spiral patterns.
Romanesco cauliflower.
Kert / Shutterstock

This is most noticeable on Roman cauliflower (sometimes called romanesco broccoli because of its color), one of the first images that appear if you search for “plant fractals” online. What is striking about romanesco are the very well-defined, pyramid-shaped buds that accumulate along endless spirals. Although less immediately obvious, a similar arrangement is also present in other cauliflowers.

Spirals are found in many plants, it is the most important pattern of plant organization – a subject that has been studied for well over 2,000 years. But even though cauliflower shares spirals with most other plants, their own resemblance is unique. Where does this particular feature come from? And do cauliflower coils originate from the same mechanisms as those in other plants?

Scientific mystery

About 12 years ago, two of my colleagues in France, François Parcy and Christophe Godin, started asking exactly these questions and encouraged me to take part in the effort. We spent many hours furiously dismantling flowers, counting them, measuring angles between them, studying the literature on the molecular mechanisms underlying cauliflower growth, and trying to create a realistic calculation model of these mysterious cabbages.

A type of green cauliflower in many pointed star shapes with spiral patterns.
Arabidopsis cauliflower mutant.
BlueRidgeKitties, CC BY-NC

Most of the available data was on Arabidopsis thaliana, also known as the “thale cress” flowering plant. Although this is a weed, it is of utmost importance in modern plant biology because its genetics have been thoroughly researched for many years, including many varieties. And it turns out to be related to all the cabbages that belong to the family known as brassicaceae. Arabidopsis actually has its own version of the cauliflower that originates from a simple mutation that involves only a few similar genes (see image to the left). So the genetics of this mutant plant is very similar to the genetics of cauliflower.

Image of Arabidopsis thaliana.
Arabidopsis thaliana.
lehic / Shutterstock

If you spend some time observing branches along the trunk of e.g. Some weeds in your garden (which probably include close relatives to Arabidopsis), you will see how they follow quite closely with each other at the same angle between each subsequent pair. And if there are enough organs along this spiral, you will begin to see other spirals, both clockwise and counterclockwise (see image to the right).

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

Five spiral-shaped clock with similar flowers on a cauliflower
Five spiral-shaped clock with similar flowers on a cauliflower.
Etienne Farcot, Author provided
Wight against clockwise spirals highlighted on the same cauliflower as above.
Eight helically counterclockwise on the same cauliflower as above.
Etienne Farcot, Author provided

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

But the genes actually interact with each other in complex “gene networks” – which leads 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 find out how cauliflower grows to their peculiar shape after the first few leaves are formed, we built a model that contained two main components. These were a description of the spiral formation that we see in large cauliflowers, and a model for the underlying gene network that we see in Arabidopsis. We then tried to match the two so we could find out which genetics led to cauliflower structure.

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

What makes cauliflower so special is that these spots on the growth tip try to become flowers for some time (up to several hours), but continue to fail at the missing “A”. Instead, they develop into stems that turn into stems, etc. – multiply almost infinitely without growing leaves, giving rise to almost identical cauliflower buds.

The time they spend trying is basic – getting this right in our model allowed us to reproduce cauliflower and romanescos accurately on the computer. We confirmed that this was right by changing the growth in real life Arabidopsis cauliflower mutant plant that effectively transforms it into a shape resembling a miniature romanesco.

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

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