Trump’s Ban Leaves Refugees in Civil Liberties Limbo

At Terminal Four of the latest York’s John F. Kennedy airport Saturday afternoon, a huge selection of protesters chanted within the snowfall all day, pushed from the chaotic lobby by police. Within the terminal, volunteer attorneys and immigrant advocates scoured the crowd, asking one family after another when they were awaiting a family member or relative who had didn’t appear. Deeper inside the airport’s bureaucratic bowels, 11 foreign people waited within the appropriate no-man’s-land for the Customs and Border Protection workplace.

These were separated in one another and detained for over 16 hours, according to the brief updates CBP supplied immigrant liberties teams, and had no use of solicitors. They’d been forbidden from employing their electronics, legal advocates feared, even as their phones and social media reports were searched for evidence of wrongdoing. Also their names stayed mostly unknown.

“We’re not getting any informative data on who they are,” said Murad Awawdeh, an activist with the ny Immigration coalition who was on the scene. “But we’re attempting to continue the pressure right here right now to make sure these people move out. And we wish to make sure that Donald Trump sees that individuals won’t mean this. This might be outrageous. Exactly How are we closing our borders toward many vulnerable people in the world?”

Witness day one of President Donald Trump’s Muslim ban.

Human Effect

Over at JFK’s terminal one, a 26-year-old guy had spent the majority of the day trying, and a failure, to talk to his aunt, a Yemeni resident whoever trip from Jeddah, Saudi Arabia had landed at 11:30 that early morning. CBP had detained her upon arrival. Mohammed, who declined to offer his final name, stated that their aunt is 68 yrs . old, and suffers from both diabetes and high blood pressure. She had traveled toward United States to move in with her son, a citizen, and possessed a family visa.

Now, Mohammed states, CBP plans to deport the lady at 8pm Saturday night, and won’t allow him see her, talk to the lady, or share information regarding her condition. She’s had no access to a lawyer, and Mohammed worries for her well-being.

“It’s insane,” Mohammed stated. “She’s not coming right here to accomplish anything bad. She’s sick. We simply desired the girl in the future live the American fantasy.”

She ended up being one among the not known amount of travelers subject to the sweeping, yet vaguely worded, executive order that institutes “extreme vetting” processes for refugees from seven Muslim countries. It pubs refugees from Iran, Iraq, Libya, Somalia, Sudan, and Yemen from entering the united states of america for the next 120 times. Additionally produces a 90-day ban on brand new visas for immigrants from those countries, and bans Syrian refugees indefinitely.

But whilst it purports to mainly affect newcomers, the very first hours of its implementation unveiled the professional order to be much wider than that. It impacts both “immigrants and nonimmigrants,” meaning it offers even green-card holders who’ve been residing in america for decades, but who may presently be traveling abroad. Based on the White House press pool report, one senior official confirmed that green-card holders who are currently away from usa require a case-by-case waiver to come back. The Wall Street Journal in addition has verified that also people in america with dual citizenship in any of these countries may also be barred entry.

Over night, reports from immigrants worldwide started flowing in, according to Mana Yegani, a Houston-based attorney who represents Iranian immigrants. She along with other solicitors in her network began compiling them in a Bing Doc created by the American Immigration Bar Association. There were reports from Amsterdam and Frankfurt, where airlines had been refusing to board you aren’t an Iranian passport on US-bound flights. In Istanbul, a household of nine, including eight visa holders and another green card holder, was escorted off their plane.

“Up until 8:30 pm everyone was coming through with visas and green cards, no issue,” Yegani states. But once the text of this professional purchase was released, she claims, “We saw catastrophe occur.”

No Due Procedure

Reports of legal visa holders being detained delivered shockwaves through civil freedom and immigrant liberties groups alike. Saturday morning, the United states Civil Liberties Union filed a petition for writ of habeas corpus on behalf of Iraqi refugees Hameed Khalid Darweesh and Haider Sameer Abdulkhaleq Alshawi, who have been one of the JFK detainees. It contends that since they alongside asylum seekers take US soil, the us government is necessary beneath the Immigration and Nationality Act to at the least grant them an asylum hearing.

“We don’t think there’s a legitimate argument your administrator purchase can bypass the asylum legislation,” says Lee Gelernt, legal counsel with the ACLU. He acknowledges the situation may be different for folks who are not currently in america.

The ACLU’s suit additionally doesn’t address the constitutionality associated with the executive order it self. It might be rendered moot, or in other words, if the Trump management were to purchase all of the present detainees released. But Gelernt says, “The ACLU is prepared to go to court to challenge the exec purchase more generally speaking.” On Monday, the Council on American-Islamic Relations may also announce its constitutional challenge against the order.

The type of the immigration system makes this an arduous legal battle to fight, especially for those currently in CBP custody. “It’s all shrouded in privacy. You’re detained all night and questioned outside the existence of an attorney,” says Jackie Esposito, an immigration attorney plus one of the JFK protest’s organizers. “It’s one of the major flaws of our immigration system that there’s no due procedure.”

“This actually system that trafficked in too little transparency and human devastation,” states Daniel Altschuler, managing director of immigrant rights group Make the Road nyc, whom attended the JFK rally. “The actions regarding the president yesterday is only going to make that worse.”

The ACLU will concentrate on the professional order’s disproportionate affect Muslims over Christians. The White home maintains this will be distinctive from the Muslim ban President Trump promised regarding campaign path, in training, it amounts to properly that, Gelernt says. “It’s favoring one religion over another,” he claims. “That’s antithetical toward basics of this constitution.”

In addition operates counter up to a 1965 law which states that “no individual shall get any choice or priority or be discriminated against in issuance of an immigrant visa because of the person’s battle, sex, nationality, place of delivery, or host to residence.”

Voices of Protest

Meanwhile, the full effect continues to be to be seen, but some prominent oppositional sounds have actually answered quickly. Google CEO Sundar Pichai urged foreign employees to return towards US at the earliest opportunity in a memo acquired by Bloomberg. “It’s painful to begin to see the individual cost of this administrator purchase on our colleagues,” Pichai wrote. Around 187 Google employees might be rejected re-entry to the US. Apple CEO Tim Cook said within an email to workers that it’s “not an insurance policy we help.”

Other people, including Reza Zadeh, a Stanford professor and CEO regarding the machine learning startup Matroid, took to Twitter to state concern over unique futures in this country.

although Republican politicians have actually remained mainly silent, many Democrats have actually provided full-throated censure. “This is ill-conceived, misguided policy, also it undermines the collaboration we need to have with Muslim countries in fighting ISIL,” claims New York Congresswoman Nydia Velazquez1, who was simply attempting to negotiate the release of detainees at JFK. “But it’s additionally arbitrary, without clear guidelines, and it’s producing confusion for not only the families, but also for the [customs] agents.”

Saturday marked the start of what’s going to undoubtedly be described as a long battle over the constitutionality for the order and civil liberties of each individual who’s been refused entry. But as their fellow detainees waited inside, no doubt worried and wondering about their fates, Darweesh, 1st refugee to be released, had been steadfast in his belief that America is still welcome place for people like him.

“Thank you to definitely the individuals whom stumbled on support me,” he said. “This may be the soul of America, this is actually the land of freedom.”

Revision: A federal judge the Eastern District Court of New York issued a short-term remain on Trump’s executive purchase tonight, based on Dale Ho, Director of ACLU’s Voting Rights venture. The stay, which arrived in response up to a habeas corpus petition filed by the ACLU, implies that anyone currently in custody in the usa beneath the order can not be deported.

Correction on 1/28/17 at 9:30 pm ET: a youthful form of this tale misspelled Congresswoman Nydia Velazquez’s name.

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3 Critical concerns Tom cost Didn’t response on health insurance and Human solutions Hearing

Republican congressman and orthopedic surgeon Tom cost, President-elect Trump’s nominee for secretary of health insurance and Human solutions, invested just as much time today protecting their character as he did protecting his eyesight of medical for all Americans. Absolutely nothing he said precluded the Senate committee conducting their hearing from recommending which he obtain the work, although several key questions remain—and must be raised ahead of the Senate verifies him.

Democrats on Senate wellness, Education, Labor and Pensions committee had been especially enthusiastic about Price’s aspire to privatize Medicare and change the reasonably priced Care Act with something less comprehensive, and so they harshly criticized him for drafting legislation that preferred businesses he’d purchased.

Such concerns are of paramount importance, because, if verified, the six-term Georgia Republican will oversee vast social programs and lead the meals and Drug management, the Centers for infection Control and Prevention, the nationwide Institutes of Health, alongside agencies. The Department of Health and Human Services, having a budget of $1 trillion, may be the world’s largest supply of capital for medical research, plus it sets the course for how a national health care system navigates the increasingly digitized world of personal health data.

Price could have broad oversight over Food And Drug Administration policies regulating drugs, medical products, and diagnostic tests. That made lawmakers particularly interested in allegations, reported by CNN, the brand new York circumstances, and the Wall Street Journal, that Price traded a lot more than $300,000 worth of stocks in organizations that endured to benefit from legislation he supported or drafted. Three Democratic senators attempted unsuccessfully to postpone Price’s hearing pending an inquiry into whether he violated home ethics rules; when that work failed, they questioned the candidate directly.

An important question to be sure, but listed here are similarly essential concerns to consider when Price faces his formal confirmation hearing on Jan. 24.

Exactly how if the government make certain that every United states has access to affordable medical insurance?

The biggest concern facing Republicans wanting to repeal the Affordable Care Act is how to cover the approximately 20 million individuals now insured in legislation. So far, no one’s offered reveal plan.

Price has some ideas. He’s got introduced the Empowering people very first Act every year since the ACA took effect this year. It might change present income-based subsidies with age-based tax credits, which helps those who are young and healthy although not the elderly or those with pre-existing conditions. Their legislation, which never gained traction, could have rolled back the ACA’s expansion of Medicaid, an application that covers above 12 million low-income Americans.

Asked today where he stands on the government’s part in providing use of affordable healthcare, cost stated it is imperative that all Americans “have the chance to gain access” to insurance policy. He didn’t say how which may take place. Pressed on Trump’s promise to provide “insurance for everybody,” cost said, “i will be dedicated to ensuring every American has the protection they want.” Once again, he offered no particulars.

What Americans want is always to pay less for the protection they have. A Kaiser Family Foundation poll released last month discovered 67 percent of Us citizens think about reducing healthcare costs the utmost effective concern. Just 20 percent want to see Obamacare end straight away. If Republicans intend to repeal the Affordable Care Act, they’re want to something to change it. Neglecting to offer an instant alternative could result in as much as 32 million individuals losing coverage by 2026, according to a report the Congressional Budget workplace released yesterday.

Price stated the Trump administration could provide precisely its plan since March. Congress must press him to spell out what sort of replacement would make insurance more affordable without drastically increasing government spending. During the hearing, a few senators pointed out a 2015 Congressional Budget workplace report that discovered repealing the ACA could improve the federal deficit by $353 billion. Senators must press cost on these numbers, as it may end up being the public’s best chance to realize the financial repercussions of this GOP’s want to repeal and replace Obamacare.

How can you plan to alter the national Medicare programs, which can be changed lacking any ACA repeal, while protecting the low-income senior?

Medicare, which covers more than 55 million Us americans, has for 50 years guaranteed in full healthcare to seniors. Polls consistently show people rank it second only to personal safety in value. President-elect Trump has promised not to touch it, but Price—who has long favored privatizing the program—said final month that the Republican-controlled Congress could privatize Medicare this season.

The various tools are certainly set up. Despite the risk of a Democratic filibuster in the Senate, Republican leaders can use a parliamentary maneuver (that spending plan reconciliation loophole again) to radically remake Medicare having easy majority vote both in houses of Congress. Trump has stated their opposition to that plan, but anything can be done within political environment.

Price recently drafted a spending plan that will have cut paying for Medicare by $449 billion throughout the next 10 years. When Senator Elizabeth Warren asked him how that squared with Trump’s pledge never to to touch Medicare or Medicaid, Price said money was the wrong metric. “I believe the metric ought to be the care toward patients.”

While that non-answer is cause enough to think that cuts towards the program are in store, Congress should see whether Price intends to tailor his proposals to align with Trump’s promises.

How will your anti-abortion views effect your oversight associated with Food and Drug management and National Institutes of wellness?

Trump’s views on abortion and reproductive liberties are fluid, but their vice president, Mike Pence, among others in his administration staunchly oppose abortion. Cost includes a long record opposing a woman’s to select and enjoys a “100 percent” rating through the nationwide directly to lifestyle. In 2005, he co-sponsored a bill that could have defined human being life as starting at conception, which would have prohibited many abortions.

Cost has also supported a nationwide ban on abortions after 20 months, alleged “conscience clauses,” that allow physicians dictate the reproductive health care they offer considering their individual and religious opinions, as well as the right of insurance providers to reject coverage for contraceptive on those moral grounds. He also completely supports defunding Planned Parenthood.

While the nation’s top health official, Price will make and enforce policy regarding fetal muscle and embryonic stem mobile research.

Researchers utilize tissue from aborted fetuses and embryonic stem cells to examine very early illness development and explore experimental treatments for a number of disorders. President Obama advanced such work by reducing restrictions enacted by President Bush, and last year the National Institutes of Health supplied about $260 million to get results because area. The ongoing future of such research faces an uncertain future underneath the Trump management.

Though some lawmakers asked Price to spell out exactly how their proposition to change the low-cost Care Act might affect females, no-one asked him about abortion or stem cell research. That’s another question that must definitely be asked—and answered—before Price may be confirmed.

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It’s Time to Stand Up for the Climate—and for Civilization

During his campaign for president, Donald Trump promised to end action on climate change and kill the climate treaty adopted in 2015 in Paris. To truly understand why that’s such a big deal—perhaps the biggest deal ever—you need to think about a few things.

Yes, you need to think about the oft-repeated but nonetheless true and alarming statistics: 2014 was the hottest year ever recorded till 2015 snatched the crown—till 2016 obliterated the record. Last summer featured some of the hottest days ever reliably recorded on this planet: 128 degrees Fahrenheit in cities like Basra, Iraq—right at the edge of human endurance. Global sea ice has been at a record low in recent months.

But you need to think about more than that.

Think about the slow, difficult, centuries-long march of science that got us to the point where we could understand our peril. Think of Joseph Fourier in the 1820s, realizing that gases could trap heat in the atmosphere; John Tyndall in the middle of that century, figuring out that carbon dioxide is one of those gases; and the valiant Svante Arrhenius in the 1890s, calculating by hand how the global temperature rises in lockstep with carbon dioxide levels. Think of Hans Suess and Roger Revelle in the 1950s, fumbling toward an understanding that the oceans would not absorb excess CO2—the first modern realization that CO2 must be accumulating in the atmosphere and hence, as Revelle put it, “human beings are now carrying out a large-scale geophysical experiment of a kind that could not have happened in the past nor be reproduced in the future.” Think of Charles Keeling in 1958, installing the first real CO2 monitor on the side of Mauna Loa and for the first time watching the CO2 level steadily rise. Think of the scientists who built on that work, using satellites and ocean buoy sensors to erect a scaffolding of observations; think of the theorists who used that data and the new power of supercomputers to build models that by the 1980s had made it clear we faced great danger. Think of the men and women who educated those scientists and who built the institutions in which they were educated and who organized the learned societies that supported them. And think of the forums—like the UN and its Intergovernmental Panel on Climate Change—that brought them together from across the planet to combine their knowledge.

The Paris accord would limit the global temperature rise to 2 degrees Celsius—unless the incoming administration dismantles it.The Paris accord would limit the global temperature rise to 2 degrees Celsius—unless the incoming administration dismantles it.Jonathan Raa/Pacific Press/LightRocket/Getty Images

All this, taken together, is one part of what we call civilization.

Now think of the men and women of the diplomatic corps, who over generations have learned to build bridges across nations, to sometimes reconcile disputes short of war. The Paris accord was a triumph for them, not because it solved the problem (it didn’t, not even close) but because it existed at all. Somehow 195 nations—rich and poor, those with oil beneath their sand and those that have to import it—managed to agree that we should limit the rise in temperature to 2 degrees Celsius this century and set up an intricate architecture to at least begin the process. That too is an aspect of what we call civilization.

None of this should be taken for granted. The building blocks of our common home—science and diplomacy and also civility—are hard-won, and history would indicate that they can fade fast. In fact, we now seem likely to start tossing them away based on nothing but the politically useful whim that climate change is a hoax. When Trump announced on the campaign trail that he would “cancel” the Paris agreement, it represented an assault on civilization as surely as announcing that he would jail his political opponent represented an assault on democracy. He’s backed down from the latter plan and, under pressure, said he now has an “open mind” about Paris—though his chief of staff clarified that his “default position” is that climate change is bunk. In any event, he has packed his transition team and cabinet with a small band of climate deniers who have blocked action for years. Already they’ve announced their intention to end NASA’s climate research, which has been a bulwark of the scientific edifice. If they have their way, there will be no more satellites carefully measuring the mass of ice sheets so we can track their melt, no more creative and fascinating “missions to planet Earth” that the space agency has run so successfully. We seem intent on blinding ourselves, on ripping out the smoke detectors even as the house begins to burn.

Trump’s team can’t, by themselves, change everything. Engineers and entrepreneurs have done their jobs magnificently over the past decade, as the price of a solar panel has fallen 80 percent. Because of that work, the potential for rapid change is finally at hand. Denmark generated nearly half its power from wind in 2015, and not because it cornered the world’s supply of breeze. Given the new economics of renewable energy, progress will continue. But the climate question has never been about progress per se; we know that eventually we’ll move to the sun and wind. The issue has always been about pace, and now Trump will add serious friction, quite likely shifting the trajectory of our path enough that we will never catch up with the physics of climate change. Other assaults on civilization and reason eventually wore themselves out—fascism, communism, imperialism. But there’s no way to wait out climate change, because this test has a timer on it. Melt enough ice caps and you live on a very different planet. Either we solve this soon or we don’t solve it. And if we don’t, then the cascading crises that follow (massive storms, waterlogged cities, floods of migrants) will batter our societies in new ways that we are ill prepared to handle, as the xenophobia of this election season showed.

Which is why we need to rise to the occasion. Not only in our day jobs but in our roles as citizens—of city, state, country, planet. Engineers should, by all means, keep developing the next generation of batteries; but that work is merely necessary now, not sufficient. We must not watch idly as Trump takes a hammer to the mechanisms of our civilization, mechanisms that can’t be rebuilt in the time we have. We need to resist in all the nonviolent ways that we’ve learned over the past century and in new ones that the moment suggests. There will be marches and divestment campaigns, pressure to be put on city halls and statehouses. We will not lack for opportunity. If many join in, then civilization will not just endure but will emerge stronger for the testing, able to face our problems with renewed vigor. At best, it’s going to be a very close call.

Bill McKibben is the Schumann Distinguished Scholar in Environmental Studies at Middlebury College and founder of global grassroots climate campaign 350.org.

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Uber’s Mildly Helpful Data Tool Could Help Cities Fix Streets

Uber has a rocky history with city governments—to put it averagely. Due to the fact ridesharing giant has spread its solutions throughout the world, it offers jumped into battles over regulations that could curtail its activities. The newest battlefield is nyc, where Uber is refusing Mayor Bill de Blasio’s need so it give the city data on when and in which it falls down every passenger.

Now, Uber is making one thing of a peace offering. The organization is introducing a new solution that could assist towns master their traffic. it is called Uber motion, and it uses info on the vast amounts of rides Uber has completed. It’s free, open to anybody who would like to use it—today that’s limited by choose planning agencies and researchers—and lets users track vehicle travel times between any two points in a city whenever you want of time. It really seems pretty helpful.

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Built by way of a group around 10 designers in the last nine months, motion now provides information for Manila, Sydney, and Washington, DC—with dozens more cities in the future before it launches toward public in mid-February, and ultimately it’ll consist of data for every town that Uber, returning to very early 2016. Areas where Uber doesn’t offer enough trips to create dependable and anonymized data are greyed down.

“We don’t manage streets. We don’t plan infrastructure,” claims Andrew Salzberg, Uber’s chief of transport policy. “So why have these items bottled up when it can offer enormous value on metropolitan areas we’re employed in?”

It’s true that localities frequently don’t have the resources to obtain that information themselves. Physical sensors are high priced, probe cars can’t be every where at once, and information from business Inrix, which arises from commercial automobiles, will adhere to major thoroughfares. But if municipal authorities had the numbers, they could be capable spot islands in which transportation times are particularly rough, to discover spikes in travel times because of lack of infrastructure or just about any problem. But … not a lot of else. “Beyond that, I’m not sure if it is such a game changer,” claims Kevin Heaslip, a transport planner at Virginia Tech.

What planners actually want to understand, Heaslip says, is where people begin and end most of their trips. Understanding drive patterns provides a better notion of where you can concentrate resources, whether it’s improving roadways or accumulating public transportation. The US Department of Transportation tries to get that information using its nationwide domestic Travel Survey (those selected to participate get a type within the mail, plus $20 thank-you), nevertheless the ubiquity of Uber’s data would be a massive enhancement, Heaslip states. “That will be extremely helpful.” However it’s additionally helpful for Uber to hang onto the commercial advantage that comes with keeping that valuable data proprietary—so don’t hold your breathing.

Uber’s perhaps not the only business sharing the data its solutions generate. Through its “Connected people” system, Waze works together urban centers all over the world, exchanging user driving info for real-time and advance notice of construction and road closures to hold its maps. Cycling software Strava peddles data to towns and cities eager to know in which their residents are riding.

They’re part of a growing trend which personal organizations match their gobs of information with public agencies’ regulatory powers. Uber could be prepared to fight, but business can get easier once the neighborhood authorities are glad to really have the company around.

Salzberg claims he’s considering incorporating more capability to Movement while the task moves forward. Simply don’t expect Uber to give NYC—or anybody else—the info they really would like.

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How to Build Beautiful 3-D Fractals Out of the Simplest Equations

If you came across an animal in the wild and wanted to learn more about it, there are a few things you might do: You might watch what it eats, poke it to see how it reacts, and even dissect it if you got the chance.

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Original story reprinted with permission from Quanta Magazine, an editorially independent division of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences


Mathematicians are not so different from naturalists. Rather than studying organisms, they study equations and shapes using their own techniques. They twist and stretch mathematical objects, translate them into new mathematical languages, and apply them to new problems. As they find new ways to look at familiar things, the possibilities for insight multiply.

That’s the promise of a new idea from two mathematicians: Laura DeMarco, a professor at Northwestern University, and Kathryn Lindsey, a postdoctoral fellow at the University of Chicago. They begin with a plain old polynomial equation, the kind grudgingly familiar to any high school math student: f(x) = x2 – 1. Instead of graphing it or finding its roots, they take the unprecedented step of transforming it into a 3-D object.

With polynomials, “everything is defined in the two-dimensional plane,” Lindsey said. “There isn’t a natural place a third dimension would come into it until you start thinking about these shapes Laura and I are building.”

The 3-D shapes that they build look strange, with broad plains, subtle bends and a zigzag seam that hints at how the objects were formed. DeMarco and Lindsey introduce the shapes in a forthcoming paper in the Arnold Mathematical Journal, a new publication from the Institute for Mathematical Sciences at Stony Brook University. The paper presents what little is known about the objects, such as how they’re constructed and the measurements of their curvature. DeMarco and Lindsey also explain what they believe is a promising new method of inquiry: Using the shapes built from polynomial equations, they hope to come to understand more about the underlying equations—which is what mathematicians really care about.

Breaking Out of Two Dimensions

In mathematics, several motivating factors can spur new research. One is the quest to solve an open problem, such as the Riemann hypothesis. Another is the desire to build mathematical tools that can be used to do something else. A third—the one behind DeMarco and Lindsey’s work—is the equivalent of finding an unidentified species in the wild: One just wants to understand what it is. “These are fascinating and beautiful things that arise very naturally in our subject and should be understood!” DeMarco said by email, referring to the shapes.

Laura DeMarco, a professor at Northwestern University.Laura DeMarco, a professor at Northwestern University.Courtesy of Laura DeMarco

“It’s sort of been in the air for a couple of decades, but they’re the first people to try to do something with it,” said Curtis McMullen, a mathematician at Harvard University who won the Fields Medal, math’s highest honor, in 1988. McMullen and DeMarco started talking about these shapes in the early 2000s, while she was doing graduate work with him at Harvard. DeMarco then went off to do pioneering work applying techniques from dynamical systems to questions in number theory, for which she will receive the Satter Prize—awarded to a leading female researcher—from the American Mathematical Society on January 5.

Meanwhile, in 2010 William Thurston, the late Cornell University mathematician and Fields Medal winner, heard about the shapes from McMullen. Thurston suspected that it might be possible to take flat shapes computed from polynomials and bend them to create 3-D objects. To explore this idea, he and Lindsey, who was then a graduate student at Cornell, constructed the 3-D objects from construction paper, tape and a precision cutting device that Thurston had on hand from an earlier project. The result wouldn’t have been out of place at an elementary school arts and crafts fair, and Lindsey admits she was kind of mystified by the whole thing.

“I never understood why we were doing this, what the point was and what was going on in his mind that made him think this was really important,” said Lindsey. “Then unfortunately when he died, I couldn’t ask him anymore. There was this brilliant guy who suggested something and said he thought it was an important, neat thing, so it’s natural to wonder ‘What is it? What’s going on here?’”

In 2014 DeMarco and Lindsey decided to see if they could unwind the mathematical significance of the shapes.

A Fractal Link to Entropy

To get a 3-D shape from an ordinary polynomial takes a little doing. The first step is to run the polynomial dynamically—that is, to iterate it by feeding each output back into the polynomial as the next input. One of two things will happen: either the values will grow infinitely in size, or they’ll settle into a stable, bounded pattern. To keep track of which starting values lead to which of those two outcomes, mathematicians construct the Julia set of a polynomial. The Julia set is the boundary between starting values that go off to infinity and values that remain bounded below a given value. This boundary line—which differs for every polynomial—can be plotted on the complex plane, where it assumes all manner of highly intricate, swirling, symmetric fractal designs.

JuliaSet_450_double.pngLucy Reading-Ikkanda/Quanta Magazine

If you shade the region bounded by the Julia set, you get the filled Julia set. If you use scissors and cut out the filled Julia set, you get the first piece of the surface of the eventual 3-D shape. To get the second, DeMarco and Lindsey wrote an algorithm. That algorithm analyzes features of the original polynomial, like its degree (the highest number that appears as an exponent) and its coefficients, and outputs another fractal shape that DeMarco and Lindsey call the “planar cap.”

“The Julia set is the base, like the southern hemisphere, and the cap is like the top half,” DeMarco said. “If you glue them together you get a shape that’s polyhedral.”

The algorithm was Thurston’s idea. When he suggested it to Lindsey in 2010, she wrote a rough version of the program. She and DeMarco improved on the algorithm in their work together and “proved it does what we think it does,” Lindsey said. That is, for every filled Julia set, the algorithm generates the correct complementary piece.

The filled Julia set and the planar cap are the raw material for constructing a 3-D shape, but by themselves they don’t give a sense of what the completed shape will look like. This creates a challenge. When presented with the six faces of a cube laid flat, one could intuitively know how to fold them to make the correct 3-D shape. But, with a less familiar two-dimensional surface, you’d be hard-pressed to anticipate the shape of the resulting 3-D object.

“There’s no general mathematical theory that tells you what the shape will be if you start with different types of polygons,” Lindsey said.

Mathematicians have precise ways of defining what makes a shape a shape. One is to know its curvature. Any 3-D object without holes has a total curvature of exactly 4π; it’s a fixed value in the same way any circular object has exactly 360 degrees of angle. The shape—or geometry—of a 3-D object is completely determined by the way that fixed amount of curvature is distributed, combined with information about distances between points. In a sphere, the curvature is distributed evenly over the entire surface; in a cube, it’s concentrated in equal amounts at the eight evenly spaced vertices.

A unique attribute of Julia sets allows DeMarco and Lindsey to know the curvature of the shapes they’re building. All Julia sets have what’s known as a “measure of maximal entropy,” or MME. The MME is a complicated concept, but there is an intuitive (if slightly incomplete) way to think about it. First, picture a two-dimensional filled Julia set on the plane. Then picture a point on the same plane but very far outside the Julia set’s boundary (infinitely far, in fact). From that distant location the point is going to take a random walk across two-dimensional space, meandering until it strikes the Julia set. Wherever it first strikes the Julia set is where it comes to rest.

The MME is a way of quantifying the fact that the meandering point is more likely to strike certain parts of the Julia set than others. For example, the meandering point is more likely to strike a spike in the Julia set that juts out into the plane than it is to intersect with a crevice tucked into a region of the set. The more likely the meandering point is to hit a point on the Julia set, the higher the MME is at that point.

In their paper, DeMarco and Lindsey demonstrated that the 3-D objects they build from Julia sets have a curvature distribution that’s exactly proportional to the MME. That is, if there’s a 25 percent chance the meandering point will hit a particular place on the Julia set first, then 25 percent of the curvature should also be concentrated at that point when the Julia set is joined with the planar cap and folded into a 3-D shape.

“If it was really easy for the meandering point to hit some area on our Julia set we’d want to have a lot of curvature at the corresponding point on the 3-D object,” Lindsey said. “And if it was harder to hit some area on our Julia set, we’d want the corresponding area in the 3-D object to be kind of flat.”

This is useful information, but it doesn’t get you as far as you’d think. If given a two-dimensional polygon, and told exactly how its curvature should be distributed, there’s still no mathematical way to identify exactly where you need to fold the polygon to end up with the right 3-D shape. Because of this, there’s no way to completely anticipate what that 3-D shape will look like.

“We know how sharp and pointy the shape has to be, in an abstract, theoretical sense, and we know how far apart the crinkly regions are, again in an abstract, theoretical sense, but we have no idea how to visualize it in three dimensions,” DeMarco explained in an email.

She and Lindsey have evidence of the existence of a 3-D shape, and evidence of some of that shape’s properties, but no ability yet to see the shape. They are in a position similar to that of astronomers who detect an unexplained stellar wobble that hints at the existence of an exoplanet: The astronomers know there has to be something else out there and they can estimate its mass. Yet the object itself remains just out of view.

A Folding Strategy

Thus far, DeMarco and Lindsey have established basic details of the 3-D shape: They know that one 3-D object exists for every polynomial (by way of its Julia set), and they know the object has a curvature exactly given by the measure of maximal entropy. Everything else has yet to be figured out.

In particular, they’d like to develop a mathematical understanding of the “bending laminations,” or lines along which a flat surface can be folded to create a 3-D object. The question occurred early on to Thurston, too, who wrote to McMullen in 2010, “I wonder how hard it is to compute or characterize the pair of bending laminations, for the inside and the outside, and what they might tell us about the geometry of the Julia set.”

Kathryn Lindsey, a mathematician at the University of Chicago.Kathryn Lindsey, a mathematician at the University of Chicago.Courtesy of Kathryn Lindsey

In this, DeMarco and Lindsey’s work is heavily influenced by the mid 20th-century mathematician Aleksandr Aleksandrov. Aleksandrov established that there is only one unique way of folding a given polygon to get a 3-D object. He lamented that it seemed impossible to mathematically calculate the correct folding lines. Today, the best strategy is often to make a best guess about where to fold the polygon—and then to get out scissors and tape to see if the estimate is right.

“Kathryn and I spent hours cutting out examples and gluing them ourselves,” DeMarco said.

DeMarco and Lindsey are currently trying to describe the folding lines on their particular class of 3-D objects, and they think they have a promising strategy. “Our working conjecture is that the folding lines, the bending laminations, can be completely described in terms of certain dynamical properties,” DeMarco said. Put another way, they hope that by iterating the underlying polynomial in the right way, they’ll be able to identify the set of points along which the folding line occurs.

From there, possibilities for exploration are numerous. If you know the folding lines associated to the polynomial f(x) = x2– 1, you might then ask what happens to the folding lines if you change the coefficients and consider f(x) = x2 – 1.1. Do the folding lines of the two polynomials differ a little, a lot or not at all?

“Certain polynomials might have similar bending laminations, and that would tell us all these polynomials have something in common, even if on the surface they don’t look like they have anything in common,” Lindsey said.

It’s a bit early to think about all of this, however. DeMarco and Lindsey have found a systematic way to think about polynomials in 3-D terms, but whether that perspective will answer important questions about those polynomials is unclear.

“I would even characterize it as being sort of playful at this stage,” McMullen said, adding, “In a way that’s how some of the best mathematical research proceeds—you don’t know what something is going to be good for, but it seems to be a feature of the mathematical landscape.”

Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.

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