Hidden math link helps designers build fantastic shapes

A mathematical connection between origami and tensegrity allows design to quickly create irregular shapes that otherwise require intensive computation

Princeton University, Engineering School

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Princeton researchers combined two disciplines to help designers create unique shapes. 

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Credit: Aaron Nathans/Princeton University

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.

“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).

 

A object folded with origami next to an object balanced with tensegrity.

Credit

Aaron Nathans/Princeton University

Origami and tensegrity [VIDEO] |

Researchers at Princeton’s engineering school have developed a system for designers to mimic irregular natural structures like termite mounds or human bones.

Credit

Princeton Engineering

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Journal

Proceedings of the National Academy of Sciences

DOI

10.1073/pnas.2519138123 

Method of Research

Experimental study

Subject of Research

Not applicable

Article Title

Invariant dual mechanics of tensegrity and origami

 

Renewable energy is more cost effective than direct air capture at reducing carbon, new study finds

A new study in Communications Sustainability has found that the case for investing in direct air capture weakens substantially once it is directly compared against solar and wind.

Boston University School of Public Health

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May 4, 2026 — The case for investing in direct air capture weakens substantially once it is directly compared against solar and wind, according to a peer-reviewed analysis published today in Communications Sustainability. Across nearly every U.S. region and every year through 2050, an amount of money spent deploying wind or solar delivers more combined climate and public health benefit than if it is spent on direct air capture, even under extremely optimistic assumptions of the development of direct air capture.

Prior assessments of direct air capture, or DAC, have largely asked whether the technology removes more carbon than its operations emit, or whether the cost per ton clears a social-cost-of-carbon benchmark. Both tests implicitly compare DAC against doing nothing. The new study, led by researchers at PSE Healthy Energy with collaborators at Boston University School of Public Health and the Harvard T.H. Chan School of Public Health, instead compares DAC against the renewable energy the same dollars could fund. This is a stricter and, the researchers argue, more policy-relevant bar.

"Our study underscores that being carbon negative isn't enough to make direct air capture a good investment," noted Dr. Yannai Kashtan, lead author and Air Quality Scientist at PSE Healthy Energy.

The researchers modeled the health and climate benefit of cost-equivalent deployments of DAC, utility-scale solar, and onshore wind across 22 U.S. grid regions from 2020 through 2050. They tested four DAC scenarios anchored at today's commercial performance (about 5,500 kilowatt-hours and $1,000 per ton of CO₂ captured) at one end, and at the other an ambitious progress scenario in which DAC's energy use falls by more than two-thirds and its cost by half (1,500 kWh and $500 per ton). They also modeled a "breakthrough" (800 kWh and $100 per ton) at the extreme low end of published projections.

Even in the ambitious progress scenario, a dramatic technological advance well beyond anything DAC has demonstrated, renewables still delivered several-fold more climate and health benefit per dollar nationally. Only under the more aggressive breakthrough scenario did grid-connected DAC do the best nationally, and even then wind and solar continued to beat DAC across large portions of the country, including most of the Upper Midwest. Under today's commercial performance, grid-connected DAC produced more greenhouse gases and air pollution damage through 2050 than it offset.

"There's a rapidly growing variety of interventions out there to mitigate greenhouse gases, and potentially affect public health as well. Our research here shows the power of cost-effectiveness analysis to ensure that capital invested in climate mitigation has the most 'bang for the buck' for the climate, while having the fewest side effects," said Dr. Jonathan J. Buonocore, senior author and assistant professor of environmental health at Boston University School of Public Health and the Institute for Global Sustainability.

The new analysis also incorporated both climate and local health impacts, and underscored a reality that conventional carbon accounting misses. If DAC is connected to a grid powered even in part by fossil fuels, building DAC will generate new sulfur dioxide, nitrogen oxides, and fine particulate matter concentrated in the communities near the power plants supplying that electricity. Renewable deployment does the opposite, producing health benefits in every region and scenario modeled.

The analysis isn't an argument against DAC, the authors note. The technology may still help draw down legacy atmospheric CO₂ once ongoing emissions are largely abated. What the analysis offers is a sharper, opportunity-cost-based benchmark for when DAC deployment becomes worthwhile, substantially stricter than the carbon-neutrality and cost-parity tests the field has traditionally relied on. “If your sink is overflowing, turn off the tap before you begin mopping the floor,” said Kashtan.

This research was supported by the ClimateWorks Foundation.

The paper, "Direct air capture has substantial health and climate opportunity costs," was published in Communications Sustainability on May 4, 2026 (DOI: 10.1038/s44458-026-00068-0). The authors are Yannai Kashtan, Drew R. Michanowicz, and Seth B.C. Shonkoff of PSE Healthy Energy; and Joseph Pendleton, Brian Sousa, Mary D. Willis, and Jonathan J. Buonocore of the Boston University School of Public Health (Pendleton is now at the Harvard T.H. Chan School of Public Health).

** 

About Boston University School of Public Health 

Founded in 1976, Boston University School of Public Health is one of the top ten ranked schools of public health in the world. It offers master's- and doctoral-level education in public health. The faculty in six departments conduct policy-changing public health research around the world, with the mission of improving the health of populations—especially the disadvantaged, underserved, and vulnerable—locally and globally.

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Journal

Communications Sustainability

DOI

10.1038/s44458-026-00068-0 

Method of Research

Computational simulation/modeling

Subject of Research

Not applicable

Article Title

Direct air capture has substantial health and climate opportunity costs

Article Publication Date

4-May-2026

 

Nutrient imbalance may drive coral disease more than heat stress, new study suggests

University of Southampton

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Black Band Disease is caused by dark microbial mats moving across the coral surface.

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Credit: Credit M. Sweet/University of Derby

Key points:

• Abnormal ratios of seawater nutrients reduce a coral’s resistance to infection.
• Opportunistic microbes take over its microbiome and cause disease.
• Diseases can emerge from within the coral, not just externally.
• Heat stress remains important, but scientists think skewed nutrient ratios are key to the development of the so-called ‘Black Band Disease’.

Scientists led by the University of Southampton have revealed that an imbalance of nutrients in seawater can cause coral disease – possibly to a greater extent than that from heat stress of warming oceans.

New research conducted at Southampton’s Coral Reef Laboratory, and with colleagues at the University of Derby, shows disruption of the delicate nutrient balance of the sea can destabilise microbial communities that live in harmony with corals, triggering disease.

Devastating outbreaks include the common and highly destructive Black Band Disease (BBD) which moves across the coral surface, killing coral tissue, and leaving behind just bare skeleton.

Although not the only cause of BBD, the team found 88 percent of recorded incidents of the disease occurred in regions with highly imbalanced seawater nutrient ratios.

Findings of the new study are due for publication in the journal Nature Communications.

Corals are a group of unique animals, which live in symbiosis with photosynthetic microalgae. Together they form three-dimensional structures known as coral reefs, which about 25 percent of all marine biodiversity depends on.

Coral reefs in tropical seas are under mounting threat from climate change. Along with rising sea temperatures, coastal developments, overfishing and pollution from agriculture and industry also remain consistent threats to their future.

The hidden disease trigger

Led by Associate Professor Cecilia D’Angelo of the University of Southampton, the team demonstrated that skewed ratios of seawater nutrients, such as an imbalance of nitrogen and phosphorus, disrupt the delicate microbial communities living with corals. These microbes, both in and on corals, are collectively called the ‘coral microbiome’.

“When nutrients are out of balance, the interactions between members of the coral microbiome begin to break down,” says Dr D’Angelo. “This creates spaces for opportunistic microbes to take over and cause disease.”

Using controlled laboratory experiments in the Coral Reef Laboratory at the University of Southampton, the researchers demonstrated that imbalances of nitrate and phosphate can promote the formation of disease lesions similar to those of Black Band Disease. Importantly, the microbial communities that caused disease in the lab, not only visually resemble their real-life counterparts on reefs, they also contain related microbe species with an equally destructive effect on corals.

Microbial networks collapse under stress

Healthy corals rely on their complex microbial networks to help maintain stability and resist infection. The study found that nutrient imbalance in the seawater fragments these networks in the laboratory corals, reducing their connectivity and resilience.

As networks break down, opportunistic microbes, in particular dark-coloured, photosynthetic ones called ‘cyanobacteria’, rapidly increase in abundance. These form disease-causing ‘microbial mats’ – dense, web-like structures that cover the coral tissue. Secondary pathogens benefit from this and join the disease communities to intensify tissue damage and cause the mats to spread across the coral surface.

Dr Raphaela Gracie from the University of Southampton, a postdoctoral researcher in the team and first author on the paper, explains: “Strikingly, many of the microbes responsible for the disease were already present in healthy coral tissue before symptoms appeared—highlighting that this disease can emerge from within the organism itself, rather than from external infection.”

“Therefore, our research reframes a key coral disease as a micro-ecological imbalance as opposed to a simple pathogenic invasion,” adds Dr D’Angelo. “This follows similar principles as opportunistic diseases in humans, for instance fungal infections that follow on from the disturbance of the natural human microbiome by antibiotic treatments.”

Implications for coral reef conservation

To understand the broader relevance of their findings, the researchers analysed global records of BBD outbreaks between 2000 and 2023. They found that over 88 percent occurred in regions with highly imbalanced nutrient ratios, whereas only 16 percent were found in reefs that were recently exposed to heat stress.

“Our results show that a vast majority of BBD outbreaks occur in reefs exposed to chronic nutrient imbalance, indicating that water quality management could be a crucial tool for mitigating reef coral diseases in the future,” says Dr Raphaela Gracie.

However, warming oceans still remain a most severe concern for coral reef survival, as anomalously high temperatures can cause fatal coral bleaching, a major driver of coral reef decline. Furthermore, rising seawater temperatures may shift the nutrient balance in a way that promotes BBD.

Nutrient imbalances can also be caused by human activities including agricultural runoff and wastewater discharge. Reducing such disturbances and managing nutrient levels at a local scale could help to stabilise coral microbiomes and prevent disease.

Cecilia D’Angelo, whose work is supported by a Research Leadership Award from the Leverhulme Trust, concludes: “Our results show that it’s not just how much nutrients are in the water, but that the balance between nitrogen and phosphorus needs to be considered as well,” she continues, “Restoring this balance in areas affected by human activities has the potential to reduce disease risk at the local scale.”

Ends

Notes to Editors

1) The paper ‘Breakdown of microbial networks links nutrient stress and reef coral disease’ is due to be published in the journal Nature Communications, DOI: 10.1038/s41467-026-72175-4, at 10am British Summer Time on Tuesday 5 May 2026 and is strictly embargoed until this date and time. Once published, it can be viewed here: https://www.nature.com/articles/s41467-026-72175-4

2) Images of a coral reef, black band disease, the laboratory, and the lead researchers, can be downloaded here: https://safesend.soton.ac.uk/pickup?claimID=wQ5TwAARtb4tnJBv&claimPasscode=FRyAyzEPqrkygkcx&emailAddr=273643

3)For interviews or further info, please contact Peter Franklin, Media Manager, University of Southampton. press@soton.ac.uk +44 23 80593212, or contact Joerg Wiedenmann (Head of the Coral Reef Laboratory)  jw1w07@soton.ac.uk, or Cecilia D’Angelo c.dangelo@soton.ac.uk.

4) A video accompanying this press release can be viewed and embedded here: https://www.youtube.com/watch?v=nHag36Lc1WE&t=1s  

5) For more information about the University of Southampton’s Coral Reef Laboratory visit: https://www.southampton.ac.uk/oes/research/facilities/corals.page

6) More on Ocean and Earth Science at the University of Southampton can be found at: https://www.southampton.ac.uk/oes/index.page

7) The Leverhulme Trust is an independent charity that seeks to fund blue skies research and scholarship which has the potential to generate new ideas and research breakthroughs that benefit society. The Leverhulme Research Leadership Awards enable talented scholars to build a research team of sufficient scale to tackle a distinctive research problem. https://www.leverhulme.ac.uk/

8) About 25% of all marine biodiversity depends on coral reefs, the three-dimensional calcareous framework built by corals, a group of unique animals, which live in symbiosis with photosynthetic microalgae. Climate change, in particular increasing seawater temperatures, threatens to disrupt this productive association with devastating knock-on effects to ecosystem services provided by coral reefs such as food supply, coastal protection, attraction of tourists and access to biopharmaceuticals. Coral diseases contribute to the decline of reefs around the globe.

9) The University of Southampton drives original thinking, turns knowledge into action and impact, and creates solutions to the world’s challenges. We are among the top 100 institutions globally (QS World University Rankings 2026). Our academics are leaders in their fields, forging links with high-profile international businesses and organisations, and inspiring a 25,000-strong community of exceptional students, from over 135 countries worldwide. Through our high-quality education, the University helps students on a journey of discovery to realise their potential and join our global network of over 300,000 alumni. www.southampton.ac.uk

Black Band Disease under the fluorescence microscope.

Coral reefs sustain large parts of marine biodiversity.

Experimental aquarium of the Coral Reef Laboratory at the University of Southampton.

Credit

D'Angelo/Wiedenmann/University of Southampton

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Journal

Nature Communications

DOI

10.1038/s41467-026-72175-4 

Method of Research

Observational study

Subject of Research

Animals

Article Title

Breakdown of microbial networks links nutrient stress and reef coral disease

Article Publication Date

5-May-2026

WORD OF THE DAY

Relamination is described, a mechanism that has been shaping continents for billions of years

The study redefines the evolution of continents and provides a framework for interpreting the geochemistry of major mountain ranges

Spanish National Research Council (CSIC)

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An international team led by researchers from the National Museum of Natural Sciences (MNCN-CSIC) has identified a key mechanism that has shaped Earth’s continents over billions of years. This mechanism is the deep re-lamination of subducted continental crust, a process that explains the origin of certain magmas and offers a new perspective on continental evolution from the Archean (between 3.8 and 2.5 billion years ago) to recent times. The study, published today in the journal Nature Geoscience, combines numerical geodynamic modeling and high-pressure experiments to unravel how fragments of continental crust can give rise to hybrid magmas that fuel major magmatic events following continental collisions, generating new crust.

During continental collisions, one plate sinks beneath another—a process known as subduction. This study demonstrates that the less dense crust breaks away from the subducted plate and rises again, becoming integrated into the lithospheric mantle of the overlying plate in a process called relamination. The relaminated material mixes mechanically with the mantle, creating a hybrid reservoir from which characteristic magmas emerge—known as post-collisional magmas—that form large granitic batholiths such as the Sierra de Gredos and Guadarrama in Central Spain. These processes, in which two continental masses collide, are what produce mountain ranges or orogens. “What we have learned from this research is why these magmas are found in both modern orogens and in formations dating back to the Archean,” explains Daniel Gómez Frutos, a researcher at the MNCN who is currently working at the University of Portsmouth. “Our models show that relamination is a recurring process in continental collisions. Fragments of subducted crust do not disappear; they are reincorporated into the continent and leave a very clear chemical signature in the magmas produced millions of years later.new crust”.

A model for simulating continental movements

The team supplemented the numerical models with high-pressure, high-temperature melting experiments, in which they mixed varying proportions of peridotite (mantle) and continental crust, reproducing the physical interaction described by the simulations. “The results were conclusive: the experimental melts reproduce the geochemical signature of post-collisional magmas, primarily granitic; features that cannot be explained by simple mantle melting or by the differentiation of basaltic magmas,” explains MNCN researcher Antonio Castro. An example of this type of magma is sanukitoids, a variety of granitic rocks with a high magnesium content found in environments at the margins of continental tectonic plates that have undergone collision processes. “The experiments make it clear that without the deep incorporation of continental crust, the magmas we observe in mountain ranges around the world cannot form.

The key lies in this intimate and solid mixing between re-laminated crust and mantle peridotite,” says Castro. Relamination is an ancient driver of continental growth. The process described is currently active and has been fundamental since the dawn of plate tectonics. The presence of Archaean sanukitoids, which are very similar to modern post-collisional magmas, supports the idea that relamination was already occurring more than 2.5 billion years ago. In addition, the team has compiled global isotopic data for 87Sr (strontium) and 143Nd (neodymium) that demonstrate that post-collisional magmas retain the “memory” of the previously subducted crust. “Relamination not only explains where these magmas come from, but also offers a way to trace what type of crust was subducted in each past collision,” notes Taras Gerya of ETH Zurich. “It is a direct window into the fate of those continents that, apparently, disappear upon colliding with others,” he continues.

The work redefines our understanding of continental evolution and offers a mechanism that explains the reincorporation of crust, contributing to its expansion and rejuvenation. “Relamination allows us to understand why we find Archean and Proterozoic magmas in certain places, something we had previously been unable to explain,” concludes Gómez Frutos.

The full paper includes high-resolution 2D numerical geodynamic simulations, piston-cylinder melting experiments, detailed geochemical analyses, and an extensive global database of post-collisional rocks. The study, led by Daniel Gómez-Frutos together with Antonio Castro, Attila Balázs, and Taras Gerya, was conducted by researchers from the MNCN-CSIC, ETH Zurich, and the University of Portsmouth.

CSIC Comunicación

comunicacion@csic.es  

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Journal

Nature Geoscience

Method of Research

News article

Subject of Research

Not applicable

Article Title

Continental evolution influenced by relamination of deeply subducted continental crust

Article Publication Date

5-May-2026