Hidden math link helps designers build fantastic shapes

Man presents small figure built of a framework of plastic rods and strings

Princeton researchers combined two disciplines to help designers create unique shapes. Photos by Aaron Nathans.

Termite mounds are remarkable structures that regulate temperature, balance airflow and maintain structural stability in some of Earth’s harshest climates. And like other irregular, disordered systems, they can be difficult to replicate with modern engineering techniques.

Now, researchers at Princeton’s engineering school have developed a system for designers to mimic irregular natural structures like termite mounds or human bones — not only their microstructural patterns, but their mechanical properties as well.

“We created a theory that is applicable to two distinct physical systems,” said Glaucio Paulino, the Margareta Engman Augustine Professor of Engineering at Princeton. “Knowing one such system can help to understand the other one better.”

In an article published March 19 in the Proceedings of the National Academy of Sciences, the researchers explain how they developed the method by combining two disciplines: origami, which studies how surfaces fold along creases; and tensegrity, which explores structures held together by compression and tension. Origami is commonly used to create objects that fold into compact shapes and expand to deploy in tasks such as space exploration. Tensegrity describes structures like the human skeleton, which holds its shape through a balanced distribution of stress among hard bones and soft tissues.

By exploring the mathematics that govern origami and tensegrity, the researchers learned that the systems’ underlying math rules are essentially the same. Although not obvious to non-mathematicians, the formula governing origami’s precise folds can be translated into the rules that govern tensegrity’s force distribution.

“It turns out that the same equation describes both engineering structures, origami and tensegrity,” said Xiangxin Dang, a postdoctoral researcher at Princeton and the article’s first author. “These two different types of structures are connected by math.”

Regular shapes, such as a cube or a sphere, are easy to design because they can be described by a small number of variables, Dang said. But irregular shapes, such as a termite mound or a complex section of bone, can demand many such variables to describe such disordered systems. This can make some designs impractical because these variables form large systems of equations demanding extensive analysis.

A paper cylinder on a table next to a framework model — A object folded with origami next to an object balanced with tensegrity.

“Without symmetry, the math appears far more complex,” Dang said. “But we found a way to bypass that complexity when a non-symmetric system inherits properties from a symmetric one.”

Using their new theory — called the invariant dual mechanics of tensegrity and origami — the researchers can start with a symmetric structure with known mechanical properties, such as stability or flexibility, and transform it into a non-symmetric form. The invariance (a math term for an element that does not change during an operation) allows them to determine the same properties for the new structure, without having to perform complex calculations on the new form.

The researchers said the application works for design. It can also work for optimization, in which engineers fine tune specific properties from a group of designs. Using the invariant duality, the engineers could easily try out new versions of stable or flexible structures without relying on trial and error, which would require complex calculations for each new shape. Instead, the engineers could start with a regular shape and adjust it as needed.

For example, consider an auto designer looking for an efficient autobody. Using older methods, the designer would have to repeatedly model the design and calculate the aerodynamics for each version. If a similar invariant method could be established, then the designer could start with a simple shape and tweak it to improve airflow.

Dang said early work on coupling the math behind force and motion was performed several decades ago as part of a branch of math called rigidity theory. But he said the work had not been pursued in a significant way. Researchers in Paulino’s lab, who often apply abstract math concepts to engineering applications, wanted to know if they could develop applications by interpreting the math through origami and tensegrity.

“We wanted to explore the problem in a way that could lead to engineering solutions,” Dang said.

Dang said the math described in the article can be applied to areas including robotics, which often involves irregular components, and metamaterials, in which the geometry of a material has a direct impact on its properties. The article, “Invariant dual mechanics of tensegrity and origami,” was published March 19 in the Proceedings of the National Academy of Sciences, https://www.pnas.org/doi/10.1073/pnas.2519138123. Authors are Xiangxin Dang and Glaucio Paulino, of Princeton. Support for the project was provided in part by the National Science Foundation, Princeton Materials Institute (PMI) and the Princeton Catalysis Initiative (PCI).