Apples: The mathematical analysis shows how the apple gets its shape

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According to a new mathematical study of the fruit, the apple gets its “dimple-like tip” and shape due to different growth rates between the mass and the stem.

Apples are relatively spherical, except for the dimples at the top, according to a team at Harvard University in Cambridge, Massachusetts, who set out to mathematically understand why the fruit has this unusual form factor.

They turned to a gel that can adjust its shape over time to mimic the growth of an apple and compared it to the growth of real apples from an orchard.

When combined with mathematical models, it was found that the underlying fruit anatomy, the way it grows at different rates, and mechanical instability all play a common role in the rise of the dimples, lower ridges, and the general shape of the fruit.

Apples are relatively spherical except for the dimple at the top, according to a team at Harvard University in Cambridge, Massachusetts, who set out to mathematically understand why the fruit has this unusual form factor

Mathematical models showed that the underlying fruit anatomy, the way it grows at different rates, and mechanical instability all play a common role in the rise of the dimples, lower ridges, and the general shape of the fruit

Mathematical models showed that the underlying fruit anatomy, the way it grows at different rates, and mechanical instability all play a common role in the rise of the dimples, lower ridges, and the general shape of the fruit

The evolution of the apple

Apples first developed in Central Asia from the wild ancestor Malus sieversii – which is still growing today.

They have been grown as a fruit in Asia and Europe for thousands of years and brought to North America by European colonists.

An apple tree grown from seed is usually very different from its parents and often produces different fruits than desired by the breeder.

For generations, apple varieties have been used by grafting them onto substrates.

These rhizomes allow the speed of growth and the resulting size of the tree to be controlled, which makes harvesting easier.

There are over 7,500 known apple varieties, several of which are bred for different tastes and uses – from cooking to eating.

Worldwide apple production was 86 million tons in 2018.

The research team began by collecting apples at various stages of growth from an orchard at Peterhouse College, University of Cambridge in the UK.

They then worked to map the growth of the dimple over time by measuring the various stages of growth.

Dr. Lakshminarayanan Mahadevan, the study’s lead author, had already developed a simple theory to explain the shape and growth of apples, but the project began to bear fruit when researchers were able to combine observations of real apples at different stages of growth.

“Biological forms are often organized by the presence of structures that act as focal points,” he said.

To understand the evolution of the shape of the apple and the tip, researchers turned to a long-standing mathematical theory known as singularity.

Singularity is used to describe a wide variety of different phenomena, from black holes to more mundane examples such as the patterns of light at the bottom of a swimming pool, the breaking up of droplets, and the propagation of cracks.

“These foci can sometimes take the form of singularities where deformations are localized,” Mahadevan said, adding, “a ubiquitous example is the tip of an apple, the inner depression where the stem meets the fruit.”

The team suggests that the “singularity” in this case is a slight change in the rate of growth around the stem compared to other parts of the apple – creating a dimple.

“The exciting thing about singularities is that they are universal,” said co-lead author of this article, Thomas Michaels of University College London.

“The apple tip has nothing to do with patterns of light in a swimming pool or a drop breaking off from a column of water, but it has the same shape as it.

“The concept of universality goes very deep and can be very useful because it connects singular phenomena that are observed in very different physical systems.”

Building on this, the team used numerical simulations to understand how the differential growth between the fruit rind and the core drives the formation of the bumps.

Top view and cross-section from simulations of apple growth showing the development over time of the compressive stresses around the central stem region, which are responsible for the formation of the multiple humps

Top view and cross-section from simulations of apple growth showing the development over time of the compressive stresses around the central stem region, which are responsible for the formation of the multiple humps

WHAT IS SINGULARITY THEORY?

A singularity is a point at which a given math object is undefined.

It’s also a point where the math object stops behaving well in a certain way.

This could be the point where there is a lack of distinctness or analytics – being different or not doing the expected results.

Singularities occur naturally in a variety of different areas of math and science, according to the Isaac Newton Institute.

“As a result, the singularity theory lies at the crossroads of the paths that connect the most important fields of application of mathematics with its most abstract parts.”

“In general, a singularity is a point where an equation explodes or degenerates,” explains WolframMathWorld.

“Singularities are extremely important in complex analysis because they characterize the possible behavior of analytical functions.”

They then confirmed the simulations with experiments that mimicked the growth of apples with gel that swelled over time.

The experiments showed that different growth rates between the mass of the apple and the stem region led to the dimple-like tip.

“It was particularly exciting to be able to control and reproduce the morphogenesis – or the shape growth – of individual cusps in the laboratory with simple material toolkits,” said Aditi Chakrabarti, co-author of the paper, about the gel.

“The variation in the geometry and composition of the gel mimetics showed how multiple bumps form.”

These changes and hump shapes are “observed in some apples and other stone fruits such as peaches, apricots, cherries and plums,” he added.

The team found that the underlying fruit anatomy, along with the mechanical instability, may play a common role in the formation of multiple bumps in all similar fruits.

“Morphogenesis, literally the origin of shape, is one of the big questions in biology,” Mahadevan said.

“The shape of the humble apple has enabled us to study some physical aspects of a biological singularity.

“Of course, we now need to understand the molecular and cellular mechanisms behind hump formation as we slowly approach a broader theory of biological form.”

The authors say future work will need to investigate the nature and dynamics of the molecular signals that trigger the inhibition of growth near the stem.

They also want to study the mechanisms, cell number, cell Size and cell shape up to tissue changes in the fruit.

The results were published in the journal Natural physics.

The ultimate egg quation! Scientists are developing a universal formula for the shape of each bird’s egg – a breakthrough that could shed light on how and why they evolved

Scientists came up with a “universal formula” to describe the shape of each bird’s egg, saying it could one day shed light on why and how eggs evolved.

The egg shape has long drawn the attention of mathematicians, engineers, and biologists as it is believed to be the “perfect shape” for its intended purpose, according to the team behind the new formula at the University of Kent at Canterbury.

The “perfect shape” claim is based on the fact that it is large enough to incubate an embryo, small enough to leave the body efficiently, and does not roll away after being laid.

The new formula for the egg shape is based on four parameters: egg length, maximum width, displacement of the vertical axis and the diameter at a quarter of the egg length.

It can be used in a variety of disciplines, particularly the food and poultry industries, and will lead to future research inspired by the egg, the team said.


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