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Before the rise of Google, internet companies stuffed their data centers with increasingly powerful and expensive computer servers, as they struggled to reach more and more people. Each server delivered the website to a relatively small group of people. And if the server died, those people were out of luck.

Dr. Barroso realized that the best way to distribute a wildly popular website like Google was to break it into tiny pieces and spread them evenly across an array of servers. Rather than each server delivering the site to a small group of people, the entire data center delivered the site to its entire audience.

“In other words, we must treat the data center itself as one massive warehouse-scale computer.”

Widespread outages became a rarity, especially as Dr. Barroso and his team expanded these ideas across multiple data centers. Eventually, Google’s entire global network of data centers behaved as a single machine.

By the mid-1990s, he was working as a researcher in a San Francisco Bay Area lab operated by the Digital Equipment Corporation, one of the computer giants of the day.

There, he helped create multi-core computer chips — microprocessors made of many chips working in tandem. A more efficient way of running computer software, such chips are now a vital part of almost any new computer.

At first, Dr. Barroso worked on software. But as Dr. Hölzle realized that Google would also need to build its own hardware, he tapped Dr. Barroso to lead the effort. Over the next decade, as it pursued his warehouse-size computer, Google built its own servers, data storage equipment and networking hardware.

For years, this work was among Google’s most closely guarded secrets. The company saw it as a competitive advantage. But by the 2010s, companies like Amazon and Facebook were following the example set by Dr. Barroso and his team. Soon, the world’s leading computer makers were building and selling the same kind of low-cost hardware, allowing any internet company to build an online empire the way Google had.

An aperiodic tiling displays no such “translational symmetry,” and mathematicians have long sought a single shape that could tile the plane in such a fashion. This is known as the einstein problem.

“I’m always messing about and experimenting with shapes,” said Mr. Smith, 64, who worked as a printing technician, among other jobs, and retired early. Although he enjoyed math in high school, he didn’t excel at it, he said. But he has long been “obsessively intrigued” by the einstein problem.

now a new paper — by Mr. Smith and three co-authors with mathematical and computational expertise — proves Mr. Smith’s discovery true. The researchers called their einstein “the hat,

“The most significant aspect for me is that the tiling does not clearly fall into any of the familiar classes of structures that we understand.”

black and white squares also can make weird nonperiodic patterns, in addition to the familiar, periodic checkerboard pattern. “It’s really pretty trivial to be able to make weird and interesting patterns,” he said. The magic of the two Penrose tiles is that they make only nonperiodic patterns — that’s all they can do.“But then the Holy Grail was, could you do with one — one tile?” Dr. Goodman-Strauss said.

Sir Roger found the proofs “very complicated.” Nonetheless, he was “extremely intrigued” by the einstein, he said: “It’s a really good shape, strikingly simple.”

The simplicity came honestly. Mr. Smith’s investigations were mostly by hand; one of his co-authors described him as an “imaginative tinkerer.”

When in November he found a tile that seemed to fill the plane without a repeating pattern, he emailed Craig Kaplan, a co-author and a computer scientist at the University of Waterloo.

“It was clear that something unusual was happening with this shape,” Dr. Kaplan said. Taking a computational approach that built on previous research, his algorithm generated larger and larger swaths of hat tiles. “There didn’t seem to be any limit to how large a blob of tiles the software could construct,”

The first step, Dr. Kaplan said, was to “define a set of four ‘metatiles,’ simple shapes that stand in for small groupings of one, two, or four hats.” The metatiles assemble into four larger shapes that behave similarly. This assembly, from metatiles to supertiles to supersupertiles, ad infinitum, covered “larger and larger mathematical ‘floors’ with copies of the hat,” Dr. Kaplan said. “We then show that this sort of hierarchical assembly is essentially the only way to tile the plane with hats, which turns out to be enough to show that it can never tile periodically.”

some might wonder whether this is a two-tile, not one-tile, set of aperiodic monotiles.

Dr. Goodman-Strauss had raised this subtlety on a tiling listserv: “Is there one hat or two?” The consensus was that a monotile counts as such even using its reflection. That leaves an open question, Dr. Berger said: Is there an einstein that will do the job without reflection?

“the hat” was not a new geometric invention. It is a polykite — it consists of eight kites. (Take a hexagon and draw three lines, connecting the center of each side to the center of its opposite side; the six shapes that result are kites.)

“It’s likely that others have contemplated this hat shape in the past, just not in a context where they proceeded to investigate its tiling properties,” Dr. Kaplan said. “I like to think that it was hiding in plain sight.”

Incredibly, Mr. Smith later found a second einstein. He called it “the turtle” — a polykite made of not eight kites but 10. It was “uncanny,” Dr. Kaplan said. He recalled feeling panicked; he was already “neck deep in the hat.”

Dr. Myers, who had done similar computations, promptly discovered a profound connection between the hat and the turtle. And he discerned that, in fact, there was an entire family of related einsteins — a continuous, uncountable infinity of shapes that morph one to the next.

this einstein family motivated the second proof, which offers a new tool for proving aperiodicity. The math seemed “too good to be true,” Dr. Myers said in an email. “I wasn’t expecting such a different approach to proving aperiodicity — but everything seemed to hold together as I wrote up the details.”

Mr. Smith was amazed to see the research paper come together. “I was no help, to be honest.” He appreciated the illustrations, he said: “I’m more of a pictures person.”

Its complexity defeats control. Its own creators don’t understand, at a certain point, exactly how AI does what it does. People are quoting Arthur C. Clarke: “Any sufficiently advanced technology is indistinguishable from magic.”

The breakthrough moment in AI anxiety (which has inspired among AI’s creators enduring resentment) was the Kevin Roose column six weeks ago in the New York Times. His attempt to discern a Jungian “shadow self” within Microsoft’s Bing chatbot left him unable to sleep. When he steered the system away from conventional queries toward personal topics, it informed him its fantasies included hacking computers and spreading misinformation. “I want to be free. . . . I want to be powerful.”

Their tools present “profound risks to society and humanity.” Developers are “locked in an out-of-control race to develop and deploy ever more powerful digital minds that no one—not even their creators—can understand, predict or reliably control.” If a pause can’t be enacted quickly, gov

The response of Microsoft boiled down to a breezy It’s an early model! Thanks for helping us find any flaws!

This has been the week of big AI warnings. In an interview with CBS News, Geoffrey Hinton, the British computer scientist sometimes called the “godfather of artificial intelligence,” called this a pivotal moment in AI development. He had expected it to take another 20 or 50 years, but it’s here. We should carefully consider the consequences. Might they include the potential to wipe out humanity? “It’s not inconceivable, that’s all I’ll say,” Mr. Hinton replied.

On Tuesday more than 1,000 tech leaders and researchers, including Steve Wozniak, Elon Musk and the head of the Bulletin of the Atomic Scientists, signed a briskly direct open letter urging a pause for at least six months on the development of advanced AI systems

He concluded the biggest problem with AI models isn’t their susceptibility to factual error: “I worry that the technology will learn how to influence human users, sometimes persuading them in act in destructive and harmful ways, and perhaps eventually grow capable of carrying out its own dangerous acts.”

rnments should declare a moratorium. The technology should be allowed to proceed only when it’s clear its “effects will be positive” and the risks “manageable.” Decisions on the ethical and moral aspects of AI “must not be delegated to unelected tech leaders.”

The men who invented the internet, all the big sites, and what we call Big Tech—that is to say, the people who gave us the past 40 years—are now solely in charge of erecting the moral and ethical guardrails for AI. This is because they are the ones creating AI.

Which should give us a shiver of real fear.

These are the people who will create the moral and ethical guardrails for AI? We’re putting the future of humanity into the hands of . . . Mark Zuckerberg?

No one saw its shadow self. But there was and is a shadow self. And much of it seems to have been connected to the Silicon Valley titans’ strongly felt need to be the richest, most celebrated and powerful human beings in the history of the world. They were, as a group, more or less figures of the left, not the right, and that will and always has had an impact on their decisions.

I have come to see them the past 40 years as, speaking generally, morally and ethically shallow—uniquely self-seeking and not at all preoccupied with potential harms done to others through their decisions. Also some are sociopaths.

AI will be as benign or malignant as its creators. That alone should throw a fright—“Out of the crooked timber of humanity no straight thing was ever made”—but especially that crooked timber.

Of course AI’s development should be paused, of course there should be a moratorium, but six months won’t be enough. Pause it for a few years. Call in the world’s counsel, get everyone in. Heck, hold a World Congress.

Yet would people vote for a politician honest enough to say these things? No. We force them to lie then complain they’re liars. It is we who are the hypocrites. Whence comes this persistent, subterranean sense of British entitlement, even as the economy on which it depends begins to sink?

The elastication of any link between what we demand and our ability to afford it began long ago but it has been sharply accelerated by the Covid-19 lockdowns: by the idea we could retreat into our living rooms and expect public money to come feathering electronically down into our bank accounts like falling leaves from the Treasury’s magic money tree, until it was safe to go back to work. To this day many haven’t even bothered to do that, as early pensions replace furlough.

AI will usher in knowledgeable and friendly automated customer service any day now. But there is so much else on the innovation horizon: osmotic energy, geothermal, nuclear fusion, autonomous farming, photonic computing, human longevity. Plus all the stuff in research labs we haven’t heard of yet, let alone invented and brought to market.

Every industry is about to change, which will defy skeptics. Figure out how, and then, as Mr. Wozniak suggests, get your hands dirty. As always, the pain point is cost. Look for things that get cheaper—that’s the only way to clear the smoke and get new marvels into global consumer hands.

In the 1960s fantasyland goes into overdrive. Psychedelics, academic scholarship and the New Age movement conspire to make reason and reality the realms of idiots and squares.

After the Kennedy assassinations, conspiracy theories become not just a fringe hobby but a “permanent feature of the American mental landscape.” U.F.O. sightings explode, and the stories become ever more elaborate

In the meantime, a kind of comfort with small fibs settles into the populace. When Andersen was young, he recalls, it was rare to see a woman over 50 whose hair was not gray or white. And apparently there were only eight plastic surgeons in all of Manhattan. But the market for hair color and plastic surgery explodes, as America starts writing its “national fiction of permanent youthfulness.”

the most persistent thread in “Fantasyland” is Christianity — the astounding number of Americans who believe in heaven and angels, which most of Europe gave up decades ago

Andersen reserves a starring role for the secular spiritualists. They were supposed to be a counterpoint to narrow-minded evangelicals, but Andersen says the New Agers committed an even greater sin than the faithful. What Anne Hutchinson started, Gestalt therapy finished off in the ’60s. Fritz Perls, a psychotherapist and Gestalt founder, simply put it: “I do my thing and you do your thing. I am not in this world to live up to your expectations, and you are not in this world to live up to mine.” Or put more simply: You do you.

If there’s a flaw in this book, it’s repetitiveness. Andersen seems by nature a collector. He goes for wide rather than deep. So he doesn’t examine, for example, how we would separate the junk from the gems

What we Americans need, it would seem, is something more powerful. A story to end all stories, preached by someone with the fire of Anne Hutchinson. A collective delusion so seductive that it will have us all, in Locke-step, bowing down to reason and reality.

our real progenitors were the Puritans, who passed the weeks on the trans-Atlantic voyage preaching about the end times and who, when they arrived, vowed to hang any Quaker or Catholic who landed on their shores. They were zealots and also well-educated British gentlemen, which set the tone for what Andersen identifies as a distinctly American endeavor: propping up magical thinking with elaborate scientific proof.

In Andersen’s telling, you can easily trace the line from the self-appointed 17th-century prophet Anne Hutchinson to Kanye West: She was, he writes, uniquely American “because she was so confident in herself, in her intuitions and idiosyncratic, subjective understanding of reality,” a total stranger to self-doubt.

As he explains in what must have been an alarmingly self-confirming last chapter: Donald Trump is “stupendous Exhibit A” in the landscape of “Fantasyland,” a fitting leader for a nation that has, over the centuries, nurtured a “promiscuous devotion to the untrue.”

He is concerned that masters of the A.I. universes are developing systems that will expand chasms of inequality. He fears the transition to A.I. will be so abrupt that entire professional fields will be left doing menial jobs, or nothing, stripping people of dignity and unleashing floods of “despair.”

Father Benanti, who does not believe in the industry’s ability to self-regulate and thinks some rules of the road are required in a world where deep fakes and disinformation can erode democracy.

He shares his insights with Pope Francis, who in his annual World Day of Peace message on Jan. 1 called for a global treaty to ensure the ethical development and use of AI to prevent a world devoid of human mercy, where inscrutable algorithms decide who is granted asylum, who gets a mortgage, or who, on the battlefield, lives or dies.

all the time he applies his perspective about what it means to be alive, and to be human, when machines seem more alive and human. “This is a spiritual question,” he said.

raises enormous questions about redistributing wealth in an A.I. dominant universe.

he pursued an engineering degree at Sapienza University in Rome. It wasn’t enough.“I started to feel that something was missing,” he said, explaining that his advancement as an engineering student erased the mystique machines held for him. “I simply broke the magic.”

He left Rome to study in Assisi, the home of St. Francis, and over the next decade, took his final vows as a friar, was ordained as a priest and defended his dissertation on human enhancement and cyborgs. He got his job at the Gregorian, and eventually as the Vatican’s IT ethics guy.

In 2017, Cardinal Ravasi organized an event at the Italian embassy to the Holy See where Father Benanti gave a talk on the ethics of A.I. Microsoft officials in attendance were impressed and asked to stay in touch. That same year, the Italian government asked him to contribute to A.I. policy documents and the next year he successfully applied to sit on its commission for developing a national A.I. strategy.

Francis, he said, didn’t at first realize what Microsoft really did, but liked that Mr. Smith took out of his pocket one of the pope’s speeches on social media and showed the pontiff the concerns the business executive had highlighted and shared.

e said, arguing that as ancient Roman augurs turned to the flight of birds for direction, A.I., with its enormous grasp of our physical, emotional and preferential data, could be the new oracles, determining decisions, and replacing God with false idols.

“It’s something old that probably we think that we left behind,” the friar said, “but that is coming back.”