For hundreds of years, mathematicians have been striving for improved approaches to calculate minimum surfaces. Recently, Computer Science and Engineering Department Assistant Professor Albert Chern and postdoctoral researcher Stephanie Wang added a new page to this book with their paper: "Computing Minimal Surfaces with Differential Forms," which was recently published by ACM Transactions on Graphics.

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"Other than modeling soap films, these shapes are the foundation for many microstructures, such as in tissues in biology," said Chern. "If you zoom into bone structures, you'll see these membranes that tend to form minimal surfaces. Basically, the surface tension on either side of the membrane is canceling itself out."

 

These measurements are also applicable in architecture. Chern mentions Munich's Olympic Park, which was inspired by simple surfaces. Finding the most efficient approach to discover minimum surfaces is a worthwhile proposition regardless of the application.

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Plateau's problem

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Plateau's issue refers to calculating a minimum surface within a boundary curve—the route around its edge—after Joseph Plateau, who was born in 1801 and enjoyed experimenting with soap bubbles. Our capacity to solve for minimum surfaces has gradually increased as new technologies have appeared. Most recently, mathematicians have adopted an ancient technique known as gradient descent, which was formerly considered cutting-edge.

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Oddly enough, Chern got into this area, to some degree, because of ants. "I was looking at optimal transport problems," says Chern. "Let's say you have an ant hill and a little ways away you have a pile of food. What is the optimal path for the ants to take to get that food and bring it back to their anthill?"

 

In reality, ants cannot perform math, but they have solved the problem by designing optimum pathways to food sources anyways. Without a doubt, this capacity originated as a survival technique to maximize nutritional advantage at the lowest metabolic cost. Chern found parallels between this optimal transport issue and minimum surfaces and, in a way, integrated them with each other.

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Chern's novel approach has the benefit of not caring about differences in a shape's topology. Previous methods, such as employing triangular mesh, need a variety of topology-based approaches. As a result, dealing with various surface topologies has become considerably simpler.

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"It's a completely different approach to looking at minimal surface problems—generalizing this interesting optimal transport problem," said Chern. "And then it suddenly becomes this big, well-known mathematical problem, and that gives it a very different flavor."

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References

staff, Science X. “A Brief History of Minimal Surfaces and the Ants That Love Them.” Phys.org, Phys.org, 17 Nov. 2021, https://phys.org/news/2021-11-history-minimal-surfaces-ants.html. 

 

Figure References

“Minimal Surfaces Examples.” Thor Architects, 5 Nov. 2020, https://www.thorarchitects.com/parametric-geometry/minimal-surfaces-1/.