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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.

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.

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.”

“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.

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.”

Einstein turned this insight into an equation that described the jittering mathematically. His Brownian motion paper is widely recognized as the first incontrovertible proof that atoms and molecules really exist—and it still serves as the basis for some stock market forecasts.

He was trying to explain an odd fact that was first noticed by English botanist Robert Brown in 1827. Brown looked through his microscope and saw that the dust grains in a droplet of water were jittering around aimlessly. This Brownian motion, as it was first dubbed, had nothing to do with the grains being alive, so what kept them moving?

If you’ve been to a conference or played with a cat, chances are you’ve seen a laser pointer in action. In the nearly six decades since physicists demonstrated the first laboratory prototype of a laser in 1960, the devices have come to occupy almost every niche imaginable, from barcode readers to systems for hair removal.

So Einstein made an inspired guess: Maybe photons like to march in step, so that the presence of a bunch of them going in the same direction will increase the probability of a high-energy atom emitting another photon in that direction. He called this process stimulated emission, and when he included it in his equations, his calculations fit the observations perfectly

A laser is just a gadget for harnessing this phenomenon

When scientists first saw the data suggesting that they’d captured a gravitational wave, they thought the results seemed to good to be true. Past claims of gravitational waves have proven unreliable, and there are many possible sources of error.

Gravitational waves confirmedAstrophysicists have announced the discovery of gravitational waves, ripples that travel at the speed of light through the fabric of space-time. A 1916 theory of Albert Einstein’s predicted their existence. .oembed-asset-photo-image { width: 100%; }

There were many new phenomena that were discovered and that came to be understood in the centuries that followed Newton’s era. Take electricity and magnetism, for example. Until the 19th century, we didn’t really know what electricity or magnetism were, or how they worked. Isaac Newton certainly didn’t have a clue.

To many physicists around the turn of the 20th century, the state of physics seemed very settled. The Newtonian worldview had been very successful, and for a very long time.

In 1905, however, a revolution in physics did come. And perhaps even more surprising than the revolution itself was where that revolution came from. In 1905, Albert Einstein was not working as a professor at some prestigious university. He was not famous, or even well-known among other physicists.

Things didn’t stay this way for long, however. In 1905, Einstein wrote not one or two, but four absolutely groundbreaking papers. Any one of these four papers would have made him a star within the field of physics, and would have certainly secured him a position of prominence in the history of science.

It seems that having so many breakthroughs of this magnitude in such a short period of time had never happened before, and has never happened since. In the first of Einstein’s 1905 papers, he proposed that light doesn’t only behave like a wave, but that it is also made up of individual pieces or particles.

But Einstein’s paper provided concrete empirical evidence that atoms were, in fact, real and tangible objects. He was even able to use these arguments to make a pretty good estimate for the size and mass of atoms and molecules. It was a huge step forward.

The equations that physicists use to describe the propagation of light waves—what are known as Maxwell’s equations—predict that light should move through space at a speed of about 670 million miles per hour. And more interestingly, these equations don’t make any reference to any medium that the light waves propagate through.

Although no experiment had ever detected this aether, they argued that it must fill virtually all of space. After all, they argued, the light from a distant star could only reach us if there was a continuous path filled with aether, extending all the way from the star to us.

ventually, though, physicists discovered that there was no aether. It would be Einstein who would come up with an equation to explain this conundrum.

Tyson praises science as an “exercise in finding what is true” that’s based on peer-reviewed experimentation backed by other experiments and counter-experiments that gives birth to an “emergent truth.” He points out that science is “not something to toy with.” “You can’t say, ‘I chose not to believe in E=mc2,’” he says, referring to physicist Albert Einstein’s corroborated theory of special relativity. “You don’t have that option. It is true, whether or not you believe in it.”

Tyson warns that every minute someone is in denial of a scientific truth delays the “political solution that should have been established years ago.”  “Recognize what science is, and allow to be what it can and should be: In the service of civilization,” he says. “It’s in our hands.”

In 1972, using duct tape and spare parts in the basement on the campus of the University of California, Berkeley, Dr. Clauser and a graduate student, Stuart Freedman, who died in 2012, endeavored to perform Bell’s experiment to measure quantum entanglement. In a series of experiments, he fired thousands of light particles, or photons, in opposite directions to measure a property known as polarization, which could have only two values — up or down. The result for each detector was always a series of seemingly random ups and downs. But when the two detectors’ results were compared, the ups and downs matched in ways that neither “classical physics” nor Einstein’s laws could explain. Something weird was afoot in the universe. Entanglement seemed to be real.

in 2002, Dr. Clauser admitted that he himself had expected quantum mechanics to be wrong and Einstein to be right. “Obviously, we got the ‘wrong’ result. I had no choice but to report what we saw, you know, ‘Here’s the result.’ But it contradicts what I believed in my gut has to be true.” He added, “I hoped we would overthrow quantum mechanics. Everyone else thought, ‘John, you’re totally nuts.’”

the correlations only showed up after the measurements of the individual particles, when the physicists compared their results after the fact. Entanglement seemed real, but it could not be used to communicate information faster than the speed of light.

In 1982, Dr. Aspect and his team at the University of Paris tried to outfox Dr. Clauser’s loophole by switching the direction along which the photons’ polarizations were measured every 10 nanoseconds, while the photons were already in the air and too fast for them to communicate with each other. He too, was expecting Einstein to be right.

Quantum predictions held true, but there were still more possible loopholes in the Bell experiment that Dr. Clauser had identified

For example, the polarization directions in Dr. Aspect’s experiment had been changed in a regular and thus theoretically predictable fashion that could be sensed by the photons or detectors.

Anton Zeilinger

added even more randomness to the Bell experiment, using random number generators to change the direction of the polarization measurements while the entangled particles were in flight.

Once again, quantum mechanics beat Einstein by an overwhelming margin, closing the “locality” loophole.

as scientists have done more experiments with entangled particles, entanglement is accepted as one of main features of quantum mechanics and is being put to work in cryptology, quantum computing and an upcoming “quantum internet

One of its first successes in cryptology is messages sent using entangled pairs, which can send cryptographic keys in a secure manner — any eavesdropping will destroy the entanglement, alerting the receiver that something is wrong.

, with quantum mechanics, just because we can use it, doesn’t mean our ape brains understand it. The pioneering quantum physicist Niels Bohr once said that anyone who didn’t think quantum mechanics was outrageous hadn’t understood what was being said.

In his interview with A.I.P., Dr. Clauser said, “I confess even to this day that I still don’t understand quantum mechanics, and I’m not even sure I really know how to use it all that well. And a lot of this has to do with the fact that I still don’t understand it.”

General relativity describes gravity not as a force, as the physicist Isaac Newton thought of it, but rather as a curvature of space and time due to the mass of objects

The reason Earth orbits the sun is not because the sun attracts Earth, but instead because the sun warps space-time, he said

Einstein noted that the public recognition of his accomplishment had a political slant. “Today I am described in Germany as a ‘German servant,’ and in England as a ‘Swiss Jew,’ ” he said. “Should it ever be my fate to be represented as a bête noire, I should, on the contrary, become a ‘Swiss Jew’ for the Germans and a ‘German servant’ for the English.”

After World World II, a new generation of physicists in the United States began to focus on relativity from their perch within the “military-industrial complex.” Here, political exigencies accelerated a deeper appreciation of Einstein’s theory, in unanticipated ways.

With GPS, the warping of time that Einstein imagined assumed operational significance. (Later, GPS was opened to the commercial market, and now billions of people rely on general relativity to find their place in the world, every single day.)

One common strategy for thinking about this is to suggest that what we used to call the whole universe is just a small part of everything there is, and that we live in a kind of bubble universe, a small region of something much larger

Newton realized there had to be some force holding the moon in its orbit around the earth, to keep it from wandering off, and he knew also there was a force that was pulling the apple down to the earth. And so what suddenly struck him was that those could be one and the same thing, the same force

That was a physical discovery, a physical discovery of momentous importance, as important as anything you could ever imagine because it knit together the terrestrial realm and the celestial realm into one common physical picture. It was also a philosophical discovery in the sense that philosophy is interested in the fundamental natures of things.

There are other ideas, for instance that maybe there might be special sorts of laws, or special sorts of explanatory principles, that would apply uniquely to the initial state of the universe.

The basic philosophical question, going back to Plato, is "What is x?" What is virtue? What is justice? What is matter? What is time? You can ask that about dark energy - what is it? And it's a perfectly good question.

right now there are just way too many freely adjustable parameters in physics. Everybody agrees about that. There seem to be many things we call constants of nature that you could imagine setting at different values, and most physicists think there shouldn't be that many, that many of them are related to one another. Physicists think that at the end of the day there should be one complete equation to describe all physics, because any two physical systems interact and physics has to tell them what to do. And physicists generally like to have only a few constants, or parameters of nature. This is what Einstein meant when he famously said he wanted to understand what kind of choices God had --using his metaphor-- how free his choices were in creating the universe, which is just asking how many freely adjustable parameters there are. Physicists tend to prefer theories that reduce that number

You have others saying that time is just an illusion, that there isn't really a direction of time, and so forth. I myself think that all of the reasons that lead people to say things like that have very little merit, and that people have just been misled, largely by mistaking the mathematics they use to describe reality for reality itself. If you think that mathematical objects are not in time, and mathematical objects don't change -- which is perfectly true -- and then you're always using mathematical objects to describe the world, you could easily fall into the idea that the world itself doesn't change, because your representations of it don't.

physicists for almost a hundred years have been dissuaded from trying to think about fundamental questions. I think most physicists would quite rightly say "I don't have the tools to answer a question like 'what is time?' - I have the tools to solve a differential equation." The asking of fundamental physical questions is just not part of the training of a physicist anymore.

The question remains as to how often, after life evolves, you'll have intelligent life capable of making technology. What people haven't seemed to notice is that on earth, of all the billions of species that have evolved, only one has developed intelligence to the level of producing technology. Which means that kind of intelligence is really not very useful. It's not actually, in the general case, of much evolutionary value. We tend to think, because we love to think of ourselves, human beings, as the top of the evolutionary ladder, that the intelligence we have, that makes us human beings, is the thing that all of evolution is striving toward. But what we know is that that's not true. Obviously it doesn't matter that much if you're a beetle, that you be really smart. If it were, evolution would have produced much more intelligent beetles. We have no empirical data to suggest that there's a high probability that evolution on another planet would lead to technological intelligence.

A classic example of this endemic bias at work is illustrated through Einstein. He was disturbed by the implications of an expanding universe. For thousands of years it was assumed — outside of some theological circles — that matter was eternal. The notion that it came into being at a discreet point in time naturally implied that something had caused it and quite possibly that that something had done it on purpose. Not willing to accept this new information, Einstein added a now famous “fudge factor” to his equations to maintain the solid state universe that he was comfortable with — something he would later describe as “the greatest blunder of my career.”

If there is great resistance to notions of design and causality in science, it is exponentially greater when it comes to theology.

If a “fact” cannot be understood, fitted into a conceptual framework that we have reason to believe in, or confirmed independently some other way, it risks becoming what journalists like to call a “permanent exclusive” — wrong.

it is as if two magic coins, flipped at different corners of the cosmos, always came up heads or tails together. (The spooky action takes place only in the context of simultaneous measurement. The particles share states, but they don’t send signals.)

fashion, temperament, zeitgeist, and sheer tenacity affected the debate, along with evidence and argument.

The certainty that spooky action at a distance takes place, Musser says, challenges the very notion of “locality,” our intuitive sense that some stuff happens only here, and some stuff over there. What’s happening isn’t really spooky action at a distance; it’s spooky distance, revealed through an action.

Why, then, did Einstein’s question get excluded for so long from reputable theoretical physics? The reasons, unfolding through generations of physicists, have several notable social aspects,

What started out as a reductio ad absurdum became proof that the cosmos is in certain ways absurd. What began as a bug became a feature and is now a fact.

“If poetry is emotion recollected in tranquility, then science is tranquility recollected in emotion.” The seemingly neutral order of the natural world becomes the sounding board for every passionate feeling the physicist possesses.

Musser explains that the big issue was settled mainly by being pushed aside. Generational imperatives trumped evidentiary ones. The things that made Einstein the lovable genius of popular imagination were also the things that made him an easy object of condescension. The hot younger theorists patronized him,

There was never a decisive debate, never a hallowed crucial experiment, never even a winning argument to settle the case, with one physicist admitting, “Most physicists (including me) accept that Bohr won the debate, although like most physicists I am hard pressed to put into words just how it was done.”

Arguing about non-locality went out of fashion, in this account, almost the way “Rock Around the Clock” displaced Sinatra from the top of the charts.

The same pattern of avoidance and talking-past and taking on the temper of the times turns up in the contemporary science that has returned to the possibility of non-locality.

the revival of “non-locality” as a topic in physics may be due to our finding the metaphor of non-locality ever more palatable: “Modern communications technology may not technically be non-local but it sure feels that it is.”

Living among distant connections, where what happens in Bangalore happens in Boston, we are more receptive to the idea of such a strange order in the universe.

The “indeterminacy” of the atom was, for younger European physicists, “a lesson of modernity, an antidote to a misplaced Enlightenment trust in reason, which German intellectuals in the 1920’s widely held responsible for their country’s defeat in the First World War.” The tonal and temperamental difference between the scientists was as great as the evidence they called on.

Science isn’t a slot machine, where you drop in facts and get out truths. But it is a special kind of social activity, one where lots of different human traits—obstinacy, curiosity, resentment of authority, sheer cussedness, and a grudging readiness to submit pet notions to popular scrutiny—end by producing reliable knowledge

What was magic became mathematical and then mundane. “Magical” explanations, like spooky action, are constantly being revived and rebuffed, until, at last, they are reinterpreted and accepted. Instead of a neat line between science and magic, then, we see a jumpy, shifting boundary that keeps getting redrawn

Real-world demarcations between science and magic, Musser’s story suggests, are like Bugs’s: made on the move and as much a trap as a teaching aid.

In the past several decades, certainly, the old lines between the history of astrology and astronomy, and between alchemy and chemistry, have been blurred; historians of the scientific revolution no longer insist on a clean break between science and earlier forms of magic.

Where once logical criteria between science and non-science (or pseudo-science) were sought and taken seriously—Karl Popper’s criterion of “falsifiability” was perhaps the most famous, insisting that a sound theory could, in principle, be proved wrong by one test or another—many historians and philosophers of science have come to think that this is a naïve view of how the scientific enterprise actually works.

They see a muddle of coercion, old magical ideas, occasional experiment, hushed-up failures—all coming together in a social practice that gets results but rarely follows a definable logic.

Yet the old notion of a scientific revolution that was really a revolution is regaining some credibility.

David Wootton, in his new, encyclopedic history, “The Invention of Science” (Harper), recognizes the blurred lines between magic and science but insists that the revolution lay in the public nature of the new approach.

What killed alchemy was the insistence that experiments must be openly reported in publications which presented a clear account of what had happened, and they must then be replicated, preferably before independent witnesses.

Wootton, while making little of Popper’s criterion of falsifiability, makes it up to him by borrowing a criterion from his political philosophy. Scientific societies are open societies. One day the lunar tides are occult, the next day they are science, and what changes is the way in which we choose to talk about them.

Wootton also insists, against the grain of contemporary academia, that single observed facts, what he calls “killer facts,” really did polish off antique authorities

once we agree that the facts are facts, they can do amazing work. Traditional Ptolemaic astronomy, in place for more than a millennium, was destroyed by what Galileo discovered about the phases of Venus. That killer fact “serves as a single, solid, and strong argument to establish its revolution around the Sun, such that no room whatsoever remains for doubt,” Galileo wrote, and Wootton adds, “No one was so foolish as to dispute these claims.

everal things flow from Wootton’s view. One is that “group think” in the sciences is often true think. Science has always been made in a cloud of social networks.

There has been much talk in the pop-sci world of “memes”—ideas that somehow manage to replicate themselves in our heads. But perhaps the real memes are not ideas or tunes or artifacts but ways of making them—habits of mind rather than products of mind

science, then, a club like any other, with fetishes and fashions, with schemers, dreamers, and blackballed applicants? Is there a real demarcation to be made between science and every other kind of social activity

The claim that basic research is valuable because it leads to applied technology may be true but perhaps is not at the heart of the social use of the enterprise. The way scientists do think makes us aware of how we can think

then there is the mishandling of time. The physicist Lee Smolin's recent book, Time Reborn, links the crisis in physics with its failure to acknowledge the fundamental reality of time. Physics is predisposed to lose time because its mathematical gaze freezes change. Tensed time, the difference between a remembered or regretted past and an anticipated or feared future, is particularly elusive. This worried Einstein: in a famous conversation, he mourned the fact that the present tense, "now", lay "just outside of the realm of science".

Recent attempts to explain how the universe came out of nothing, which rely on questionable notions such as spontaneous fluctuations in a quantum vacuum, the notion of gravity as negative energy, and the inexplicable free gift of the laws of nature waiting in the wings for the moment of creation, reveal conceptual confusion beneath mathematical sophistication. They demonstrate the urgent need for a radical re-examination of the invisible frameworks within which scientific investigations are conducted.

we should reflect on how a scientific image of the world that relies on up to 10 dimensions of space and rests on ideas, such as fundamental particles, that have neither identity nor location, connects with our everyday experience. This should open up larger questions, such as the extent to which mathematical portraits capture the reality of our world – and what we mean by "reality".

Love can make you less vulnerable to pain.

your heart beats as fast as your partner’s so they’re at the same rate. How romantic.

holding hands with the person you love may alleviate pain.

men adjust their walking speed to match their romantic partner’s pace

brain activated by feelings of intense love are the same areas that drugs use to reduce pain,”

people in a committed relationship who have been actively thinking about their partner actually avert their eyes from attractive members of the opposite sex unknowingly

men are more willing to take unnecessary risks for a romantic partner.

It makes your pupils grow.

pupil dilation correlates with intense emotional states

What happened to me demands explanation. Modern physics tells us that the universe is a unity—that it is undivided. Though we seem to live in a world of separation and difference, physics tells us that beneath the surface, every object and event in the universe is completely woven up with every other object and event. There is no true separation. Before my experience these ideas were abstractions. Today they are realities. Not only is the universe defined by unity, it is also—I now know—defined by love. The universe as I experienced it in my coma is—I have come to see with both shock and joy—the same one that both Einstein and Jesus were speaking of in their (very) different ways.

Today many believe that the living spiritual truths of religion have lost their power, and that science, not faith, is the road to truth. Before my experience I strongly suspected that this was the case myself. But I now understand that such a view is far too simple. The plain fact is that the materialist picture of the body and brain as the producers, rather than the vehicles, of human consciousness is doomed. In its place a new view of mind and body will emerge, and in fact is emerging already. This view is scientific and spiritual in equal measure and will value what the greatest scientists of history themselves always valued above all: truth.