How Apples Get Their Shape

Physicists say a universal theory that describes everything from light reflecting in tea cups to black holes can explain why apples have a dip at the top.
Apples growing on a tree
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Jessica Orwig, Contributor

(Inside Science) -- Next time you’re about to bite into an apple, slice it open first and inspect its cross-section. If you look in the right spot, you’ll observe that the stem cavity -- where the surface dips down to meet the stem -- is so sharply sloped it nearly becomes a vertical line. Here the curvature, the local change in slope, is what mathematicians would call “singular.” Singularities show up in a large range of physical systems, from light reflecting in tea cups to black holes that warp space-time.

And it turns out, the theory developed for analyzing singularities can also be applied to describe how an apple gets its stem cavity.

Mathematicians and physicists have been using singularity theory for decades, said Thomas Michaels, a biophysicist at University College London. Michaels was co-lead author on a paper published this October in Nature Physics that is the first to mathematically model the apple cavity's shape -- what they refer to as the apple “cusp” in the study. Michaels said the idea came out of a casual conversation with lead author L. Mahadevan, an applied mathematician and physicist of Harvard University, who’d been thinking about the problem for some time. And it eventually led to a lot of apple picking, slicing, and tracing, Michaels said.

Mahadevan and Michaels had predicted early on that singularity theory could describe the apple cusp's shape. But in order to test their model, they needed to know when the cusp formed and how. That's where the apple picking came in.

Michaels collected and studied 100 apples at various stages of growth over a four-week window and found that the cusp does not fully form until the last half of development. In fact, if you look at the cross-section of a very young apple, it's practically a circle. 

Cross sections of bigger and bigger apples

The growth trajectory of an apple shows the formation of the near-singular cusp at the stalk.

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Reproduced with permission from "The cusp of an apple," by Aditi Chakrabarti, Thomas C. T. Michaels, Sifan Yin, Eric Sun & L. Mahadevan, Nature Physics volume 17, pages 1125–1129 (2021)

The cusp forms because as the apple matures, its core and stem grow at a slower rate than its cortex, the fleshy meat that constitutes the bulk of the apple. In fact, at the peak of its growth, the cortex grows five times faster than the core, causing it to rapidly expand around the stem, which leads to the apple's iconic cusp. This differential growth rate is also likely why apples have a cusp but other fruits like oranges, which grow differently, do not, Michaels said.

Armed with their differential growth model, the team first tested it with computer simulations run by Sifan Yin, a visiting student from Tsinghua University, who confirmed that, yes, you could get cusp-like shapes from a differentially growing sphere. 

After that, the team then reproduced the apple's shape from a polymer-based gel that co-author and Harvard postdoc Aditi Chakrabarti developed for the study. And while Michaels said the study was relatively straightforward from the get-go, there was one factor he hadn't expected: multiple cusps. You can see multiple cusps (or bumps at the top) in certain types of apples, like Red Delicious. Tomatoes are another great example of this shape.

Multiple "cusps" on top of tomato fruit

Experiments with a swelling gel show the formation of a cusp in the first image. The number of cusps changes as the diameter of the stalk gets biggers (subsequent images.) 

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Reproduced with permission from "The cusp of an apple," by Aditi Chakrabarti, Thomas C. T. Michaels, Sifan Yin, Eric Sun & L. Mahadevan, Nature Physics volume 17, pages 1125–1129 (2021)

"The biggest surprise to me was the multiple cusp development," Michaels said. "It appeared by chance," out of the gel models when Chakrabarti made the stalks slightly larger. "We realized that [multiple cusps] must be part of the story but it took us a while to understand the mechanism and fine-tune our model to account for them."

Michaels said the study was purely based on curiosity. But there may be promise for it from an agricultural standpoint, according to Peter Hirst, a professor of horticulture at Purdue University in Indiana.

"What is interesting is that sometimes when fruit grows, the skin can crack and that leads to huge economic problems," Hirst said. In particular, for apples, those cracks can occur in and around the cusp because the skin of the fruit can't expand fast enough, stressing it to the point of breakage.

"And so by being able to describe changes in shape and size in mathematical terms, I think one of the next steps will be to describe those stresses and to be able to predict and maybe even prevent some of these cracks and damages from occurring," Hirst said.

Michaels said that he has no immediate plans to follow up on this study, as his main focus is on physical models to better understand diseases like Alzheimer's. But there's always the potential for the future.

"In the general area of science, you never know what's going to be useful until later on, and things that were once deemed to be of only scientific interest become a huge practical interest," Hirst said. "So you can never tell where these things lead sometimes." 

Author Bio & Story Archive

Jessica Orwig is a freelance writer for Inside Science focusing on physics. She majored in astrophysics at Ohio State and earned her masters in science and tech journalism from Texas A&M. When she’s not at her computer you can find her biking, hiking, painting, or playing the piano.