Isaac Newton was baffled. He was already famous for discovering how gravity holds the universe together and for using that knowledge to predict the movements of celestial bodies, such as the moon’s path around the Earth. Now, by taking the sun’s gravitational tugs into account, he sought to improve his lunar predictions. Instead, it made them worse.

The setback, Newton’s friend Edmond Halley reported, “made his head ache, and kept him awake so often, that he would think of it no more.” Newton felt his defeat so keenly that he recalled it more than once in his old age.

Today it’s called the three-body problem. Famous in science and science fiction for orbital perturbations and chaotic phenomena, it’s recently become a concern of atomic experts and military planners. As Beijing rapidly expands its nuclear arsenal, they warn that the world of atomic superpowers is about to escalate to three from two. The outcome, they add, compared with the Moscow-Washington standoff, now 70 years old, could represent a dangerous new kind of unthinkable.

The looming era could encourage “states to resort to nuclear weapons in a crisis,” Andrew F. Krepinevich Jr., a senior fellow at the Center for a New American Security, recently warned. He cited the natural instabilities observed by physicists and astronomers as a portent.

Experts say the tripolar age could put human survival at risk. But they also cite a number of three-body lessons from nature — starting with Newton’s — that illuminate the issue and suggest possible ways forward. So far, however, no answer stands out. The world’s nuclear thinkers are finding the knotty topic to be as intractable as it was for Newton.

“We have a conceptual problem,” said Ernest J. Moniz, a physicist who as the secretary of energy in the Obama administration oversaw the U.S. nuclear arsenal. “We’ve got to change the traditional approach of equalizing weapons or strategic delivery systems, but how to do that is still unclear.”

France A. Córdova, an astrophysicist and past director of the National Science Foundation, said the study of three-body phenomena in the natural sciences could nonetheless help reveal the military risks. “Things are changing very rapidly,” she said. “Anything that helps in understanding that is great.”

Security-minded hawks want to expand the American arsenal in response to China’s nuclear rise and the threat of Beijing’s closing ranks with Moscow. Doves see a window for three-body downsizing. They want to break the problem into smaller and more manageable parts. For instance, they argue that Washington should deal with the two superpowers independently and seek diplomatic bonds that reinforce two-body stability.

Recently, the Biden administration called for a further simplification. Jake Sullivan, the national security adviser, argued that the American response should focus less on the quantity of the nation’s nuclear arms than on their quality. To deter attacks successfully, he said in a speech, the American military has no need for arms that “outnumber the combined total of our competitors.”

In everyday life, groups of twos and threes can seem inconsequential. Two friends joining another brings the total to three. It’s the sum of the parts — what scientists call a linear increase.

But in many aspects of nature, threes have an almost magical power to sow chaos, to become more than the sum of their parts. Scientists call them nonlinearities. In short, the interval from two to three can produce a counterintuitive jump in complexity, as Newton found to his dismay.

“Our intuitions fail us,” Michael Weisberg, a philosopher of science at the University of Pennsylvania, said of the three-body tumult. Steven Strogatz, an applied mathematician at Cornell University, agreed: “Threes are inherently problematic. Things get tricky.”

Atoms illustrate the complexity jump. Hydrogen, the simplest, has two main parts — a nucleus and a single circling electron. Physicists can predict with great accuracy the future states of the subatomic particle, said Michael S. Lubell, a professor of physics at the City College of New York.

But helium — the next larger atom — has two electrons. The interplay of those two particles with the element’s nucleus throws them into a complicated state beyond the comprehension of science. “There’s no exact solution,” Dr. Lubell said. “You can’t find out what’s happening to their behavior, their location or anything else. It doesn’t scale. Things get chaotic.”

Surprisingly, the jump in disorganization also shows up in the world’s oceans and atmosphere — in whirlpools and maelstroms, tornadoes and hurricanes. If two of the swirling bodies get close, they move ahead in straight lines or circle each other.

“With three, things immediately get more complicated,” said Michael J. Shelley, a specialist in fluid dynamics at New York University. “They can collapse into each other. It gets very disordered and unpredictable. There’s a huge difference.”

Notably, the jump also shows up in human life as groups of three cause social complexities to soar — markedly in young families. Two siblings have one relationship. But a third child results in seven kinds of ties among the siblings — three one-on-one relationships, three one-on-two relationships and one group relationship. Parents, by definition, are outnumbered, and bedlam can ensue.

In the cosmos, stars also come in chaotic threesomes. The celebrated science fiction novel “The Three-Body Problem,” by Liu Cixin, features three stars that whirl around one another in unruly orbits. As a result, the planet Trisolaris suffers cycles of blistering heat and icy cold that can reverse in minutes, producing an alien civilization obsessed with survival.

Clusters of three stars, however, turn out to be relatively rare in the universe because stragglers in wide orbits often get ejected or absorbed by passing star systems. “Roughly, for every two binaries, there’s one triple,” said Andrei A. Tokovinin, an astronomer at the Cerro Tololo Inter-American Observatory, which has headquarters in La Serena, Chile.

The Cold War — for all its terrors and crises — avoided nuclear war in part because its mature structures echoed the binary stability that astronomers see in the heavens and that young families see in the relatively simple play of two children.

The tense age began as the world’s first thermonuclear arms were tested by Washington in 1952 and Moscow in 1955‌. By nature, the weapons could produce blasts a thousand times more powerful than the Hiroshima bomb. The ensuing arms race fed the Cold War’s fear of mutual annihilation — ridiculed in “Dr. Strangelove,” the classic 1964 film.

Soon, the antagonists seized on force parity as a way to reduce the risk of conflict. Negotiated accords set Moscow and Washington on roughly equal footings meant to replace war with taut stalemates — as is the case with Russia and the United States today.

“We’re at a stable equality,” said William I. Newman, a professor of astrophysics at the University of California, Los Angeles, who aids the University of California’s management of the Los Alamos weapons lab. “Any departure from that will enhance the instability.”

The looming departure is Beijing’s plan to produce 1,500 nuclear warheads by 2035, as the Pentagon estimates. If achieved, the rise would represent a fivefold increase from the “minimum deterrent” that Beijing possessed for more than a half-century and would make it a nuclear peer of Moscow and Washington.

Dr. Newman calls the tripolar state “much less resilient” than the bipolar standoff. Even so, three-body theorists see a number of ways that the unthinkable might be avoided.

For instance, Dr. Krepinevich, in a Foreign Affairs article last year, argued that Moscow could fade into economic and strategic insignificance, leaving a strong Beijing and Washington to “navigate their way to a new bipolar equilibrium.” The armed revolt over the weekend in Russia drives home not only Moscow’s weakness but the threat of new instability in an atomic superpower.

On a different note, Siegfried S. Hecker, a former director of the Los Alamos weapons laboratory in New Mexico, argued that Washington should aim to deal with the rival superpowers as separate entities.

“I don’t see Russia and China getting together” on atomic strategies, he said. “I see it as two bipolars.” As the Ukraine war rages and Washington has little interaction with Moscow, Dr. Hecker added, now is a good time “to work with the Chinese” in building a two-body relationship.

The main worry of military planners is that Beijing will not only achieve weapons parity with Washington but also form a military pact with Moscow.

“We’re not seeing yet a full-fledged, really cemented, long-lasting, resilient geopolitical alliance,” Gen. Mark A. Milley, the outgoing chairman of the Joint Chiefs of Staff, told Foreign Affairs magazine last month. “Could that happen in the future? It could, and we need to be wary of that, and we need to do what we can to make sure that doesn’t happen.”

Ardent security analysts call for quick expansions of the American arsenal. In March, the Livermore weapons lab in California published a lengthy report arguing that the buildup should begin in early 2026 as New START, one of the last major arms-control agreements between Moscow and Washington, expires. Swarms of retired warheads, the report said, should then be redeployed on missiles, bombers and submarines.

Despite proposals for Washington to match the combined forces of Beijing and Moscow, analysts say bids for weapon parity are likely to fail. That’s because Washington’s rivals would see what’s coming and, in response, would most likely expand their own arsenals.

Parity will be “continuously sought but never achieved,” as Dr. Krepinevich put it last year in Foreign Affairs. Other analysts agree. Rather than weapon equivalence, they see endless arms races whose moves and countermoves could raise the risk of miscalculation and war.

Despite the prospect of complicated new threats and uncertainties from three atomic superpowers, Newton’s bane can still offer practical advice, said Melvin G. Deaile, director of the School of Advanced Nuclear Deterrence Studies at Maxwell Air Force Base in Montgomery, Ala.

Most basically, Dr. Deaile said, it argues that the armed forces of the United States should adopt a strategy of unrelenting flexibility given the tumult to come. “Static deterrence will not suffice,” he stated in an Air Force journal. “Deterrence will have to become agile.”

In an interview, Dr. Deaile expressed confidence that the existential threats of the tripolar world could be managed successfully.

Dr. Deaile said he drew inspiration from Carl von Clausewitz, the Prussian war theorist of the early 19th century. He said the theoretician was a pioneer in applying three-body logic to conflict management. In his military classic, “On War,” Clausewitz cited not only Newton by name but also a standard demonstration showing how an object suspended over three magnets makes unpredictable moves.

“Yes, the system is dynamic,” Dr. Deaile said of a tripolar world. “Yes, it constantly changes. But we have to realize that this problem is bounded and has some stability to it.” Despite the likelihood of clashes among three atomic superpowers, he said, “there are still pathways to maintain stability.”

A retired Air Force colonel, Dr. Deaile echoed the approach of Dr. Hecker, the former director of Los Alamos, the birthplace of the bomb.

Separately, each expert argued that keeping an uneasy peace among nuclear foes required them to talk, to share concerns and to take modest steps at confidence-building. “We have to keep the lines of communication open and interacting,” Dr. Deaile said.

After all, he added, “None of these nations want to wipe each other off the face of the earth.”

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