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A brief illustrated guide to ‘scissors congruence’ − an ancient geometric idea that’s still fueling cutting-edge mathematical research

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A brief illustrated guide to ‘scissors congruence’ − an ancient geometric idea that’s still fueling cutting-edge mathematical research

While scissors congruence accurately captures the modern algebraic notion of 2D area, things get more complicated in higher dimensions.
Maxine Calle, CC BY-ND

Maxine Calle, University of Pennsylvania and Mona Merling, University of Pennsylvania

In math class, you probably learned how to compute the area of lots of different shapes by memorizing algebraic formulas. Remember “base x height” for rectangles and “½ base x height” for triangles? Or “𝜋 x radius²” for circles?

But if you were in math class in ancient Greece, you might have learned something very different. Ancient Greek mathematicians, such as Euclid, thought of area as something geometric, not algebraic. Euclid’s geometric perspective, recorded in his foundational work “Elements,” has influenced research programs across centuries – even the work of mathematicians today, like the two of us.

A gif of a blue blob bouncing between a triangle, circle and square, along with formulas for their areas.
Math today treats area algebraically, using formulas for simple shapes and calculus for more complicated regions.
Maxine Calle, CC BY-ND

Modern mathematicians refer to Euclid’s concept of “having equal area” as “being scissors congruent.” This idea, based on cutting up shapes and pasting them back together in different ways, has inspired interesting mathematics beyond just computing areas of triangles and squares. The story of scissors congruence demonstrates how classical problems in geometry can find new life in the strange world of abstract modern math.

Euclid’s notion of area

Today, people think of the area of a shape as a single number that can be computed using algebraic formulas or calculus. So what does it mean to think of area as something geometric the way the ancient Greeks did?

Imagine you’re back in math class and you have a pair of scissors, some tape and a piece of construction paper. Your teacher instructs you to make a new flat, two-dimensional shape using all of the construction paper and only straight-line cuts. Using your scissors, you cut the paper into a bunch of pieces. You start moving these pieces around – maybe you rotate them or flip them over – and you tape them back together to form a new shape.

A drawing of a student doodling on some paper. Above them, a pentagon is cut into 3 triangles, and those triangles are reassembled into a new polygon.
What kind of shapes can you make from a pentagon using only scissors and tape?
Maxine Calle, CC BY-ND

Using your algebraic formulas for area, you could check that the area of your new shape is equal to the original area of the construction paper. No matter how a 2D shape is cut up – as long as all the pieces are taped back together without overlap – the area of the old and the new shape will always be equal.

A blue pentagon with a central eyeball is cut up into pieces.
You’re not allowed to make curved cuts or tape overlapping pieces.
Maxine Calle, CC BY-ND

For Euclid, area is the measurement that is preserved by this geometric “cutting-and-pasting.” He would say that the new shape you made is “equal” to the original piece of construction paper – mathematicians today would say the two are “scissors congruent.”

What can your new shape look like? Because you’re only allowed to make straight-line cuts, it has to be a polygon, meaning none of the sides can be curved.

A gif of a pentagon splitting up into five triangles
When a polygon is cut up into pieces, its area doesn’t change.
Maxine Calle, CC BY-ND

Could you have made any possible polygon with the same area as your original piece of paper? The answer, amazingly, is yes – there’s even a step-by-step guide from the 1800s that tells you exactly how to do it.

In other words, for polygons, Euclid’s notion of area is exactly the same as the modern one. In fact, you may have even used Euclid’s idea of area before in computations without knowing it.

For example, you can use scissors congruence to compute the area of a pentagon. Since area is preserved if you cut the pentagon up into smaller triangles, you can instead find the area of these triangles (using “½ base x height”) and add them up to get the answer.

A pentagon, the same pentagon split up into five triangles, and the five separate triangles
It can be easier to compute the area of a pentagon by chopping it up into smaller triangles.
Maxine Calle, CC BY-ND

Hilbert’s third problem

Perhaps the most infamous appearance of scissors congruence is on the famous German mathematician David Hilbert’s list of problems, which consisted of some of the most important mathematical questions of the 1900s. Of the original 23 problems Hilbert proposed, some have been solved, some have been shown to be unsolvable and others are still unresolved. The third problem on the list, and the first to be resolved, is about scissors congruence.

Instead of two-dimensional polygons, Hilbert asked about their three-dimensional cousins: polyhedra. Euclid’s notion of scissors congruence was known to be an accurate description of two-dimensional area, but could it be a good notion of three-dimensional volume?

An illustration of the five platonic solids
There are just five Platonic solids – polyhedra, whose faces are all the same polygon.
Maxine Calle, CC BY-ND

The answer came within a year, provided by one of Hilbert’s students, Max Dehn. Dehn’s solution to the problem was very different from the two-dimensional case. He showed that when polyhedra are cut up, volume is not the only thing that is preserved. There is another preserved measurement, now called the Dehn invariant, which is constructed from the lengths of edges and the angles between the faces of the polyhedron.

If two polyhedra are scissors congruent, then they have to have the same Dehn invariant. So, if Dehn could find two polyhedra with the same volume but different values of this invariant, that would prove the answer to Hilbert’s third problem is no – scissors congruence doesn’t precisely capture 3D volume.

A tetrahedron splits into pieces. In its thought bubble is a cube, also splitting into pieces.
There is no way to cut a tetrahedron into pieces and glue them back together to make a cube with the same volume.
Maxine Calle, CC BY-ND

This is exactly what Dehn did, showing that the invariants associated to a cube and tetrahedron with the same volume are different. This means that there’s no possible way to cut up a tetrahedron into a finite number of pieces and reassemble them back into a cube with the same volume.

Are volume and the Dehn invariant all we need to know? If two polyhedra have the same volume and the same Dehn invariant, does that tell us they’re scissors congruent? It took mathematicians another 60 years to answer this question. In 1965, Jean-Pierre Sydler confirmed that the answer is yes, closing this chapter on scissors congruence.

Strange shapes and stranger connections

But the story doesn’t end there. Mathematics is full of shapes living in higher dimensions – like 4D, 100D, 3,485D or any dimension you can imagine – which are impossible to visualize. An active new research area called generalized scissors congruence seeks to uncover whether Hilbert’s question about scissors congruence can also be stated – and maybe even solved – for these strange shapes.

An illustration of various shapes lining a landscape, with arrows pointing between them. Some resemble waves.
Mathematicians study other strange kinds of geometries, like hyperbolic and spherical geometry.
Maxine Calle, CC BY-ND

However, what it means for two things to be scissors congruent is now far more complicated. While Hilbert and Dehn cared about things like volume and angles, other mathematicians could exchange these physical traits for something far less tangible.

A recent research program pioneered by mathematicians Jonathan Campbell and Inna Zakharevich proposes a unifying framework for generalized scissors congruence. This framework is built using a very abstract, seemingly unrelated mathematical toolkit called algebraic K-theory.

A triangle jumps into a machine and splits into three triangles.
Algebraic K-theory was developed in the late 20th century in the field of abstract math known as algebraic topology.
Maxine Calle, CC BY-ND

The big idea of K-theory is that mathematical objects can be understood by how they decompose into fundamental building blocks – much like molecules are broken up into atoms. With a little bit of adjustment, mathematicians can harness the machinery of K-theory and apply it to generalized scissors congruence problems.

This use of K-theory reimagines the problem of scissors congruence and opens the doors for future research. But at the end of the day, scissors congruence is a concrete idea that you don’t need fancy math to understand – just some patience, creativity, a pair of scissors and a lot of tape.The Conversation

Maxine Calle, Ph.D. Candidate in Mathematics, University of Pennsylvania and Mona Merling, Assistant Professor of Mathematics, University of Pennsylvania

This article is republished from The Conversation under a Creative Commons license. Read the original article.

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Measles can ravage the immune system and brain, causing long-term damage – a virologist explains

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theconversation.com – Peter Kasson, Professor of Chemistry and Biomedical Engineering, Georgia Institute of Technology – 2025-03-31 07:16:00

Measles infections send 1 in 5 people to the hospital.
wildpixel/ iStock via Getty Images Plus

Peter Kasson, Georgia Institute of Technology

The measles outbreak that began in west Texas in late January 2025 continues to grow, with 400 confirmed cases in Texas and more than 50 in New Mexico and Oklahoma as of March 28.

Public health experts believe the numbers are much higher, however, and some worry about a bigger resurgence of the disease in the U.S. In the past two weeks, health officials have identified potential measles exposures in association with planes, trains and automobiles, including at Washington Dulles International Airport and on an Amtrak train from New York City to Washington, D.C. – as well as at health care facilities where the infected people sought medical attention.

Measles infections can be extremely serious. So far in 2025, 14% of the people who got measles had to be hospitalized. Last year, that number was 40%. Measles can damage the lungs and immune system, and also inflict permanent brain damage. Three in 1,000 people who get the disease die. But because measles vaccination programs in the U.S. over the past 60 years have been highly successful, few Americans under 50 have experienced measles directly, making it easy to think of the infection as a mere childhood rash with fever.

As a biologist who studies how viruses infect and kill cells and tissues, I believe it is important for people to understand how dangerous a measles infection can be.

Underappreciated acute effects

Measles is one of the most contagious diseases on the planet. One person who has it will infect nine out of 10 people nearby if those people are unvaccinated. A two-dose regimen of the vaccine, however, is 97% effective at preventing measles.

When the measles virus infects a person, it binds to specific proteins on the surface of cells. It then inserts its genome and replicates, destroying the cells in the process. This first happens in the upper respiratory tract and the lungs, where the virus can damage the person’s ability to breathe well. In both places, the virus also infects immune cells that carry it to the lymph nodes, and from there, throughout the body.

YouTube video
Measles can wipe out immune cells’ ability to recognize pathogens.

What generally lands people with measles in the hospital is the disease’s effects on the lungs. As the virus destroys lung cells, patients can develop viral pneumonia, which is characterized by severe coughing and difficulty breathing. Measles pneumonia afflicts about 1 in 20 children who get measles and is the most common cause of death from measles in young children.

The virus can directly invade the nervous system and also damage it by causing inflammation. Measles can cause acute brain damage in two different ways: a direct infection of the brain that occurs in roughly 1 in 1,000 people, or inflammation of the brain two to 30 days after infection that occurs with the same frequency. Children who survive these events can have permanent brain damage and impairments such as blindness and hearing loss.

Yearslong consequences of infection

An especially alarming but still poorly understood effect of measles infection is that it can reduce the immune system’s ability to recognize pathogens it has previously encountered. Researchers had long suspected that children who get the measles vaccine also tend to have better immunity to other diseases, but they were not sure why. A study published in 2019 found that having a measles infection destroyed between 11% and 75% of their antibodies, leaving them vulnerable to many of the infections to which they previously had immunity. This effect, called immune amnesia, lasts until people are reinfected or revaccinated against each disease their immune system forgot.

Occasionally, the virus can lie undetected in the brain of a person who recovered from measles and reactivate typically seven to 10 years later. This condition, called subacute sclerosing panencephalitis, is a progressive dementia that is almost always fatal. It occurs in about 1 in 25,000 people who get measles but is about five times more common in babies infected with measles before age 1.

Researchers long thought that such infections were caused by a special strain of measles, but more recent research suggests that the measles virus can acquire mutations that enable it to infect the brain during the course of the original infection.

There is still much to learn about the measles virus. For example, researchers are exploring antibody therapies to treat severe measles. However, even if such treatments work, the best way to prevent the serious effects of measles is to avoid infection by getting vaccinated.The Conversation

Peter Kasson, Professor of Chemistry and Biomedical Engineering, Georgia Institute of Technology

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Supreme Court considers whether states may prevent people covered by Medicaid from choosing Planned Parenthood as their health care provider

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theconversation.com – Naomi Cahn, Professor of Law, University of Virginia – 2025-04-02 17:04:00

Planned Parenthood clinics, like this one in Los Angeles, are located across the United States.
Patrick T. Fallon/AFP via Getty Images

Naomi Cahn, University of Virginia and Sonia Suter, George Washington University

Having the freedom to choose your own health care provider is something many Americans take for granted. But the Supreme Court is weighing whether people who rely on Medicaid for their health insurance have that right, and if they do – is it enforceable by law?

That’s the key question at the heart of a case, Medina v. Planned Parenthood South Atlantic, that began during President Donald Trump’s first term in office.

“There’s a right, and the right is the right to choose your doctor,” said Justice Elena Kagan on April 2, 2025, during oral arguments on the case. John J. Bursch, the Alliance Defending Freedom lawyer who is representing South Carolina Director of Health and Human Services Eunice Medina, countered that none of the words in the underlying statute had what he called a “rights-creating pedigree.”

As law professors who teach courses about health and poverty law as well as reproductive justice, we think this case could affect access to health care for 72 million Americans, including low-income people and their children and people with disabilities.

Excluding Planned Parenthood

The case started with Julie Edwards, who is enrolled in Medicaid and lives in South Carolina. After she struggled to get contraceptive services, she was able to receive care from a Planned Parenthood South Atlantic clinic in Columbia, South Carolina.

Planned Parenthood, an array of nonprofits with roots that date back more than a century, is among the nation’s top providers of reproductive services. It operates two clinics in South Carolina, where Medicaid patients can get physical exams, cancer screenings, contraception and other services. It also provides same-day appointments and keeps long hours.

In July 2018, however, South Carolina Gov. Henry McMaster issued an executive order that barred health care providers in South Carolina that offer abortions from reimbursement through Medicaid.

That meant Planned Parenthood, a longtime target of conservatives’ ire, would no longer be reimbursed for any type of care for Medicaid patients, preventing Edwards from transferring all her gynecological care to that office as she had hoped to do.

Planned Parenthood and Edwards sued South Carolina, claiming that the state was violating the federal Medicare and Medicaid Act, which Congress passed in 1965, by not letting Edwards obtain care from the provider of her choice.

A ‘free-choice-of-provider’ requirement

Medicaid operates as a partnership between the federal government and the states. Congress passed the law that led to its creation based on its power under the Constitution’s spending clause, which allows Congress to subject federal funds to certain requirements.

Two years later, due to concerns that states were restricting which providers Medicaid recipients could choose, Congress added a “free-choice-of-provider” requirement to the program. It states that people enrolled in Medicaid “may obtain such assistance from any institution, agency, community pharmacy, or person, qualified to perform the service or services required.”

This provision is at the core of this case. At issue is whether a civil rights statute provides a right for Medicaid beneficiaries to sue a state when their federal rights have been violated. Known as Section 1983, it was enacted in 1871.

Bursch, backed by the Trump administration, argued before the court that the absence of words like “right” in the Medicaid provision that requires states to provide a free choice of provider means that neither Edwards nor Planned Parenthood has the authority to file a lawsuit to enforce this aspect of the Medicaid statute.

Nicole A. Saharsky, Planned Parenthood’s lawyer, argued that the creation of a right shouldn’t depend on “some kind of magic words test.” Instead, she said it was clear that the Medicaid statute created “a right to choose their own doctor” because “it’s mandatory” that the state provide this option to everyone with health insurance through Medicaid.

She also emphasized that Congress wanted to protect “an intensely personal right” to be able “to choose your doctor, the person that you see when you’re at your most vulnerable, facing … some of the most significant … challenges to your life and your health.”

Restricting Medicaid funds

Through a federal law known as the Hyde Amendment, Medicaid cannot reimburse health care providers for the cost of abortions, with a few exceptions: when a patient’s life is at risk or her pregnancy is due to rape or incest. Some states do cover abortion when their laws allow it, without using any federal funds.

Therefore, Planned Parenthood only gets federal Medicaid funds for abortions in those limited circumstances.

McMaster explained that he removed “abortion clinics,” including Planned Parenthood, from the South Carolina Medicaid Program because he didn’t want state funds to indirectly subsidize abortions.

South Carolina “decided that Planned Parenthood was unqualified for many reasons, chiefly because they’re the nation’s largest abortion provider,” Bursch told the Supreme Court.

But only 3% of Planned Parenthood’s services nationwide last year were related to abortion. Its most common service is testing for sexually transmitted diseases. Across the nation, Planned Parenthood provides health care to more than 2 million patients per year, most of whom have low incomes.

Man with gray hair in a suit and red tie speaks at a podium.
South Carolina Gov. Henry McMaster speaks to a crowd during an election night party on Nov. 3, 2020, in Columbia.
Photo by Sean Rayford/Getty Images

Section 1983

Because the Medicaid statute itself does not allow an individual to sue, Edwards and Planned Parenthood are relying on Section 1983.

Lower courts have repeatedly upheld that the Medicaid statute provides Edwards with the right to obtain Medicaid-funded health care at her local Planned Parenthood clinic.

And the Supreme Court has long recognized that Section 1983 protects an individual’s ability to sue when their rights under a federal statute have been violated.

In 2023, for example, the court found such a right under the Medicaid Nursing Home Reform Act. The court held that Section 1983 confers the right to sue when a statute’s provisions “unambiguously confer individual federal rights.”

Consequences beyond South Carolina

The court’s decision in the Medina case on whether Medicaid patients can choose their own health care provider could have consequences far beyond South Carolina. Arkansas, Missouri and Texas have already barred Planned Parenthood from getting reimbursed by Medicaid for any kind of health care. More states could follow suit.

In addition, given Planned Parenthood’s role in providing expansive contraceptive care, disqualifying it from Medicaid could harm access to health care and increase the already-high unintended pregnancy rate in America.

The ramifications, likewise, could extend beyond the finances of Planned Parenthood.

If the court rules in South Carolina’s favor, states could also try to exclude providers based on other characteristics, such as whether their employees belong to unions or if they provide their patients with gender-affirming care, further restricting patients’ choices.

Or, as Kagan observed, states could go the opposite direction and exclude providers that don’t provide abortions and so forth. What’s really at stake, she said, is whether a patient is “entitled to see” the provider they choose regardless of what their state happens to “think about contraception or abortion or gender transition treatment.”

If the Supreme Court rules that Edwards does have a right to get health care at a Planned Parenthood clinic, the controversy would not be over. The lower courts would then have to decide whether South Carolina appropriately removed Planned Parenthood from Medicaid as an “unqualified provider.”

And if the Supreme Court rules in favor of South Carolina, then Planned Parenthood could still sue South Carolina over its decision to find them to be unqualified.The Conversation

Naomi Cahn, Professor of Law, University of Virginia and Sonia Suter, Professor of Law, George Washington University

This article is republished from The Conversation under a Creative Commons license. Read the original article.

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Feeling FOMO for something that’s not even fun? It’s not the event you’re missing, it’s the bonding

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theconversation.com – Jacqueline Rifkin, Assistant Professor of Marketing, Cornell University – 2025-04-02 07:48:00

They had so much fun without me.
Milko/E+ via Getty Images

Jacqueline Rifkin, Cornell University; Barbara Kahn, University of Pennsylvania, and Cindy Chan, University of Toronto

Imagine you’ve planned the trip of a lifetime for your animal-loving family: a cruise to Antarctica with the unique opportunity to view penguins, whales and other rare wildlife. Your adventure-loving kids can kayak through fjords, plunge into icy water and camp under the Antarctic sky.

But rather than being ecstatic, as you anticipated, your kids whine about skipping an after-school scout meeting at a neighbor’s house. Missing this ordinary weekly event triggers such intense FOMO – “fear of missing out” – for them that they don’t want to go on your amazing expedition.

If this kind of debacle sounds familiar to you – or at least if you find it perplexing – you’re not alone. The three of us are marketing professors and social psychologists who focus on how consumers make decisions and how this shapes well-being. We’ve been studying FOMO for over a decade and recently published our work in the Journal of Personality and Social Psychology. Over the years, we’ve learned what really drives intense feelings of FOMO – which explains why a run-of-the-mill meeting might feel more crucial than an over-the-top vacation.

FOMO’s real trigger

People use the term FOMO in many different ways. In our research, we focus on a very specific type of FOMO: the kind that occurs when people miss out on events that involve valued social connections.

With this kind of FOMO, we found that the pain of missing out is not related to missing the actual event or opportunity – although that could be there as well. The FOMO we study happens when people miss the chance to bond with friends, co-workers or teammates they care about.

So, the critical part of FOMO is missing out on interactions with people you value. FOMO about a group dinner at a restaurant isn’t really about the food and great lighting. Nor is FOMO about a concert just about the band’s performance. Instead, it’s about the lost opportunity to connect and make memories with people who are important to you.

Why is this upsetting? Imagine the scenario where all your best friends go out to dinner without you. They bond and make lasting memories with each other – and you’re not there for any of it.

If they get closer to each other, where does that leave you? What happens to your social relationships and your sense of belonging? Do you become a less important friend? Less worthy of future invites? Or even kicked out of the group altogether? The anxiety of FOMO can begin to spiral.

People with what psychologists call an anxious attachment style chronically fear rejection and isolation from others. Because FOMO involves anxiety about future social belonging, it may not come as a surprise that people who are naturally more anxious about their friendships tend to get more intense FOMO. When we asked people in one of our studies to scroll social media until they encountered something social they missed, we found that the more anxiously attached a participant was, the more intense FOMO they experienced.

cheerful group laughing together around an outdoor dining table
They’ll always remember that summer cookout – and you weren’t there.
Maskot/DigitalVision via Getty Images

Not just missing Coachella

Getting FOMO for an amazing event you can’t attend makes sense. But if FOMO is less about the event itself and more about the social bonding, what happens when you miss something that’s not really fun at all?

We find that people anticipate FOMO even for unenjoyable missed events. As long as there is some form of missed social bonding, feelings of FOMO emerge. One of our studies found that people anticipated more FOMO from missing an un-fun event that their friends would be at, than a fun event without their friends.

For better or for worse, sad and stressful events can often be emotionally bonding: Going to a funeral to support a friend, cleaning up the mess after a party, or even white-knuckling through a harrowing initiation ceremony can all offer opportunities to forge stronger connections with one another. Stressful contexts like these can be fertile grounds for FOMO.

How to fend off FOMO

Popular discussions about the negative consequences of FOMO tend to focus on the FOMO people feel from compulsively scrolling on social media and seeing what they missed out on. Consequently, much of the suggested advice on how to mitigate FOMO centers on turning off phones or taking a vacation from social media.

Those recommendations may be tough for many people to execute. Plus, they address the symptoms of FOMO, not the cause.

Our finding that the core of FOMO is anxiety about missed social relationships yields a simpler strategy to combat it: Reminding yourself of the last time you connected with close friends may provide a sense of security that staves off feelings of FOMO.

In an experiment testing multiple interventions, we asked 788 study participants to look through their social media feeds until they encountered a post of a missed social event. We asked about 200 of these participants to immediately rate how much FOMO they were feeling. They averaged a 3.2 on a 1-to-7 scale.

Another group of about 200 participants also scrolled through their social media feeds until they encountered a post of a missed social event. But before indicating how much FOMO they were feeling, we asked them to think back to a prior experience socializing and bonding with their friends. Encouragingly, this reflection exercise seemed to curtail FOMO. Their average FOMO rating was 2.7 out of 7, a significant drop.

group of older women on a neighborhood walk with a leashed dog
Reminding yourself about other good times with your pals can help keep FOMO at bay.
AJ_Watt/E+ via Getty Images

With the remaining participants, we tested other strategies for mitigating FOMO – thinking about the next time they might see their friends or imagining what they’d say to a FOMO-suffering friend – but the simple reflection exercise was by far the most promising.

So, reminding yourself of the meaningful relationships you already have and reaffirming your social belonging in the moment may help combat the rush of anxiety that is characteristic of FOMO.

And missing out on social bonding experiences doesn’t have to be anxiety-provoking. In fact, in our activity-packed, hectic lives, missing some “must-attend” events may be a welcome relief – especially if you remind yourself that your social belonging is not in jeopardy. Cue a recent wave of counter-FOMO programming called JOMO, or “Joy of Missing Out.”

To quote Stuart Smalley, the fictional self-help guru of 1990s “Saturday Night Live,” reminding yourself that “I’m good enough, I’m smart enough, and doggone it, people like me!” might be just the trick to mitigate FOMO.The Conversation

Jacqueline Rifkin, Assistant Professor of Marketing, Cornell University; Barbara Kahn, Patty and Jay H. Baker Professor of Marketing, University of Pennsylvania, and Cindy Chan, Assistant Professor of Marketing, University of Toronto

This article is republished from The Conversation under a Creative Commons license. Read the original article.

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