Mathematics discovers that rose petals are found to contain a unique geometry

Nature and humans have different solutions for dealing with flat surfaces in a world that is three-dimensional. Among plants, the uneven growth of their leaves or their tissue causes the appearance of curvatures that relieve tension, without which they would break. An example is the wrinkling of lettuce at its edge.

Until now, the so-called planar plant morphogenesis was explained by the egregious theorem, a geometrical theory postulated by the mathematician Carl Friedrich Gauss two centuries ago. But roses follow their own geometric principle. A paper published in the latest issue of Science shows how their petals, that start out curved, end up as cutting edges through a mechanism never observed in the natural world until now.

The egregious theorem was not formulated to explain patterns observed in nature. Gauss formulated it as follows: “If a curved surface is developed on any other surface whatever, the measure of curvature in each point remains unchanged.”

This has many consequences in the realm of mathematics and physics. But there is one simple enough to grasp: the impossibility of faithfully capturing the globe on a flat surface, as the polar areas would appear exaggeratedly large. In such maps, both parallels and meridians are straight lines, when in reality they are curved and circular. This geometric frustration also occurs in the plant world and governs the tension between form and growth.

Michael Mose, from the Racah Institute of Physics at the Hebrew University of Jerusalem and co-author of the roses research, gives two examples to explain the departure point and what they have achieved. One example is that of the carrot. During its growth, the inner part of the carrot expands more than its outer layers, which generates internal tension. When cut into four pieces lengthwise, the pieces immediately curve outward, relaxing that tension. “The growth of the carrot caused a geometric incompatibility in this case, i.e., a preferred shape that cannot be achieved,” says Mose.

The other example is closer to the mystery of roses. If the edges of a sheet grow faster than the center, the distances between points on the sheet should tend toward a curved geometry. But the sheet maintains a uniform thickness, which prevents it from bending. “The result is frustration: the sheet tries to bend and stay flat at the same time, a contradiction. This is known as Gaussian incompatibility and explains the shape of almost all leaves and flowers,” he says.

However, roses are an exception to this rule. Unlike the smooth contours of other flowers which are usually curved, their petals are very particular. The youngest and inner ones are flat and curved. But as they grow unevenly there is a geometric frustration and what was curved becomes triangular at the edges.

“Their characteristic shapes, especially the sharp cusps at the edges, cannot be explained by geometrical principles known as Gaussian incompatibility,” says Mose.

Using computer models, artificial flowers and the cultivation of red baccara roses with their more than forty dark red petals, the researchers were able to confirm that they comply with their own unique geometrical principle.

The petal shape is governed by a type of geometrical frustration that is different from Gaussian; it comes from the violation of a series of equations known as Mainardi-Codazzi-Peterson (MCP). Also, from the mathematical realm of curved surface geometry, these equations describe how the bending of a surface must have a smooth transition from one point to another to avoid unnatural tears and folds in three-dimensional space.

“The rose is, to our knowledge, the only known natural system shaped by this form of incompatibility, but it may not be the only one,” says Mose.

The researchers demonstrate that rose petals grow in a simple, uniform, symmetrical way; nothing in their growth pattern suggests the final shape. “However, this growth causes MCP incompatibility, which generates internal stresses. These stresses, which are uniform, bend the petal into a shape that concentrates stresses and curvature at arbitrary points, shaping the petal edges into their iconic cusp shapes,” says Mose.

In a report also published by Science, Hong Kong City University mechanical engineering researchers Lishuai Jin and Qinghao Cui point out that it is not only genetics or environment that affect growth and shape, but also the limits imposed by geometry.

As for its implications, beyond the aesthetics of roses, they argue that many of the designs of today’s materials are based on Gaussian incompatibility — for example, the manufacture of tires.

And they conclude by highlighting the avenues opened up by a study of the geometric frustration of rose petals: “Exploiting the Mainardi-Codazzi-Peterson incompatibility could allow localized, programmable shape changes without requiring large-scale variations in surface distances. Also, combining the Gaussian and Minardi-Codazzi-Peterson incompatibilities could lead to as yet unobserved deformation behaviors.”

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