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A group of mathematicians and physicists has figured out how apples get their distinctive shapes

Apples are one of the world's oldest and most well-known fruits, having been around for thousands of years. But have you ever given much thought to the shape of an apple? Apples are relatively spherical in shape, with the exception of the distinctive dimple at the top of the fruit where the stem grows.

What causes apples to grow in such a distinctive shape?

An international team of mathematicians and physicists has now used observations, laboratory experiments, theory, and computation to better understand the growth and shape of an apple's cusp, according to the journal Nature.

The paper has been published in Nature Physics.

As L Mahadevan, the Lola England de Valpine Professor of Applied Mathematics, of Organismic and Evolutionary Biology, and of Physics at Harvard's John A. Paulson School of Engineering and Applied Sciences (SEAS), explained in a statement, "Biological shapes are frequently organized by the presence of structures that serve as focal points." "These focal points can sometimes take the form of singularities, in which deformations are concentrated in a specific area. "The cusp of an apple, which is the inward dimple where the stalk meets the fruit, is a common example of this phenomenon."

Even though Mahadevan had already devised a simple theory to explain the form and growth of apples, the project only started bearing fruit when the researchers were able to link observations of real apples at different growth stages with gel experiments to simulate the growth, in addition to theory and computations, and the results were published.

A collection of apples at various growth stages was collected by the researchers from an orchard at Peterhouse College at the University of Cambridge in the United Kingdom (the alma mater of another famous apple lover, Sir Isaac Newton).

These apples were used to create a time-lapse video showing the progression of the dimple, or cusp, as the team dubbed it.

Singularity theory, a mathematical theory that has been around for a long time, was used to help the researchers understand the evolution of the apple's shape, and specifically the cusp of the apple. Various phenomena, ranging from black holes to more mundane examples such as light patterns at the bottom of a swimming pool, droplet breakup, and crack propagation, are described by singularity theory. Black holes are one of the most well-known examples of singularity theory, but there are many others.

"One of the most fascinating aspects of singularities is that they are universal. Even though it has nothing in common with light patterns in a swimming pool or droplets breaking off from a column of water, the apple cusp has the same shape as both," said Thomas Michaels, a former postdoctoral fellow at SEAS and co-lead author of the paper who is now at University College London. "It's amazing how similar the shapes are," Michaels said. "The concept of universality is extremely complex, but it can be extremely useful because it connects singular phenomena observed in very different physical systems," says the author.

The researchers used numerical simulation to gain an understanding of how differential growth between the fruit cortex and the core contributes to the formation of the cusp, based on their theoretical framework. Later, they conducted experiments that mimicked the growth of apples by using gel that swelled over time, which confirmed the results of the simulations. The results of the experiments revealed that the dimple-like cusp was caused by the different rates of growth between the bulk of the apple and the stalk region.

In particular, Aditi Chakrabarti, a postdoctoral fellow at SEAS and co-author of the paper, said, "Being able to control and replay morphogenesis of singular cusps in the laboratory with simple material toolkits was particularly exciting." When the geometry and composition of the gel mimics were varied, it was demonstrated that multiple cusps formed, as seen in some apples and other drupes, such as peaches, apricots, cherries, and plums.

The researchers discovered that the underlying anatomy of fruits, combined with mechanical instability, may play a joint role in the development of multiple cusps in fruits.

"Morphogenesis, literally the origin of shape, is one of the great unsolved mysteries in biology," Mahadevan explained. " "The shape of the humble apple has provided us with an opportunity to investigate some physical aspects of a biological singularity." For the moment, however, we must focus on understanding the molecular and cellular mechanisms that underlie its formation, as we progress slowly toward a more comprehensive theory of biological shape."

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