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While they are undeniably beautiful, roses aren’t among my favorite flowers—too clichéd and romantic. I much prefer an indestructible cactus, a carnivorous plant or a hipster monstera, which, of course, is a must-have in any Japandi-style apartment. But after hearing about the research findings of physicist Eran Sharon’s lab at the Hebrew University of Jerusalem, I’m reconsidering roses. From a mathematical perspective, the flowers are pretty cool—or at least their petals are.
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You can find many videos online explaining that the arrangement of a plant’s petals is related to the golden ratio or the Fibonacci sequence, indicating a deep connection between botany and mathematics. The reason isn’t that nature has a particular affinity for mathematics. Rather the most sunlight reaches the petals when they form an angle with one another that is irrational, meaning its measurements cannot be expressed as the ratio of two integers. The irrationality ensures that no petal is directly above another.
Such insights have fascinated mathematicians and physicists for decades, which is why they have taken a closer look at plant growth. How, for example, can the curly leaf shape of Lollo Rossa lettuce be explained? Why does a carrot bend when it’s cut lengthwise? Questions like these can be answered quite easily with a little math and physics. But the arrangement of rose blossoms has been a long-standing scientific puzzle.
Irregular Growth Leads to Tension
The shape of a leaf is influenced by many different factors, such as the speed of growth, the material and the environment it grows in. For example, if the edge of a leaf grows faster than its center, this can lead to a curled or wavy shape—as with Lollo lettuce. With carrots, on the other hand, the opposite is the case: the inside of the vegetable grows faster than its outside, but because of its firm structure, it usually retains a straight shape. If you quarter the vegetable lengthwise, the structure is weakened, and the insides of the pieces will bend outwards, making visible the pressure that has existed inside the carrot.
Rose petals, however, differ from other flower shapes. They are not simply wavy or curved but have almost triangular tips. Such irregular segments are distinct among flowers. And this raises questions: What distinguishes rose petals from the leaves, petals and shoots of other plants?
To find out, Sharon and his team dissected rose petals, devised theoretical models, created computer simulations and conducted experiments with rubber surfaces. They discovered that the shape of rose petals has a completely different geometric cause than that of other plants.
An Incompatible Geometry
The curling of flower petals can be explained mathematically: because of their differing growth rates, petals “want” to curl, but this is prevented by external factors—such as the tissue not being elastic enough. It’s like trying to flatten the surface of a sphere onto a plane—it’s simply impossible. This “Gauss incompatibility” leads to mechanical stresses that produce virtually all flower shapes.
Gauss incompatibility can only explain smooth and periodic patterns, however, as Sharon and his team were aware. But the shape of rose petals is neither periodic nor smooth: there are isolated locations where a point is formed. To understand this, the researchers first cut the petals into fine strips—once along the horizontal axis and once along the vertical axis. The team’s aim was to visualize the mechanical pressure acting on the petals, similar to cutting carrots to reveal their internal pressure.
The longitudinal strips of the rose petals did indeed curl up, suggesting that the petal preferentially curves in this direction. When cut along the horizontal axis, however, the petal segments remained unchanged. This means a petal possesses a kind of conical structure, which is compatible with a flat geometry and therefore shouldn’t create any internal tension. After all, a flat sheet of paper can easily be rolled into a cone or a cylinder. A rose petal can, in principle, be laid flat on a surface; this confirms that Gauss incompatibility isn’t what causes the unusual shape of the petal. But what causes it then?
Similar questions are addressed in mechanics and industrial manufacturing processes because it’s crucial to know whether a manufactured component exhibits internal stresses and could break. Besides the Gauss equations, which must be considered, there are other equations at play: the Mainardi-Codazzi-Peterson equations. These state, among other things, that the normal vector (the vector perpendicular to the surface) must be uniquely defined. And as Sharon’s team discovered, this isn’t the case with rose petals. They therefore exhibit a geometric incompatibility, albeit a different one than that of other flowers.
Imagine cutting a slice of a pie chart, a shape that roughly corresponds to a flower petal. Now the left side of the slice wants to curl up, and so does the right side—but in different ways. This creates a point in the circular segment where tension accumulates, which in turn leads to the formation of a pointed edge. This can even happen in several places, resulting in multiple points.
Sharon’s research group was able to confirm this mechanism, which had never been observed in nature otherwise, using computer simulations and experiments. Although the team cannot rule out the possibility of other natural processes exhibiting Mainardi-Codazzi-Peterson incompatibility, rose blossoms are currently the only known example.
And that’s what makes roses so unique. I’m still not sure if that’s enough to convince me of their beauty. But I’d never previously considered why flowers look the way they do, so I acknowledge that roses add some new insights into mathematics—and therefore appreciate them a little more.
And as physicist Richard Feynman pointed out in an interview in the early 1980s, looking at a flower with a scientist’s eyes raises “all kinds of interesting questions,” which, in his view, “only adds to the excitement, the mystery and the awe of a flower.”
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission. It was translated from the original German version with the assistance of artificial intelligence and reviewed by our editors.