James' World 2

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How few corners can a shape have and still tile the plane?” mathematician Gábor Domokos asked me over pizza. His deceptively simple question was about the geometry of tilings, also called tessellations—arrangements of shapes, called tiles or cells, that fill a surface with no gaps or overlaps. Humans have a preoccupation with tessellation that dates back at least to ancient Sumer, where tilings featured prominently in architecture and art. But in all the centuries that thinkers have tinkered with tiles, no one seems to have seriously pondered whether there’s some limit to how few vertices—sharp corners where lines meet—the tiles of a tessellation can have. Until Domokos. Chasing tiles with ever fewer corners eventually led him and his small team to discover an entirely new type of shape.

It was the summer of 2023 when Domokos and I sat at a wood picnic table at the Black Dog, a cozy spot for pizza and wine just a few blocks from the Budapest University of Technology and Economics, where Domokos is a professor. He reached across the table to grab a paper pizza menu and flip it over, revealing a blank underside, and gestured to me to grab a pen. The midsummer sky was taking on shades of orange and indigo as I filled the menu with triangles. Domokos watched expectantly. “You’re allowed to use curves,” he finally said. I started filling the page with circles, which of course can’t fill space on their own. But Domokos lit up. “Oh, that’s interesting!” he said. “Keep going, you can mix shapes. Just try to keep the average number of corners as low as possible.”

I kept going. My page of circles filled with increasingly desperate, squiggly forms. Domokos’s pizza Margherita had long since disappeared, but he wasn’t quite ready to leave. A quick glance at my crude drawing wasn’t enough to determine its average number of corners, let alone the minimum possible. But the right answer must have been something less than the triangle’s three—otherwise, the question would be boring.

That observation seemed to satisfy the mathematician, who revealed that the real answer is two. “That’s an easy question,” he said. “But what about 3D?”

“This is a tool that can reasonably describe, at least to me, a wide range of more physically relevant things than just polyhedrons stuck together.” —Chaim Goodman-Strauss, mathematician

Now, more than a year after that evening at the pizza shop, Domokos has the answer. Finding it was an exciting, frustrating challenge that ultimately led him and three colleagues to discover “soft cells,” shapes that can fit together to completely fill a flat surface or a three-dimensional space with as few corners as possible. In two dimensions, soft cells have two corners bridged by curves. But in 3D, these curvy, almost organic forms have no corners at all. Once the researchers identified the new shapes, they began to see them all over the place—in nature, art and architecture. The results have now been published in the journal PNAS Nexus.

Although soft cells hadn’t been categorized by mathematicians before—no one had noticed or named them in an academic paper—they abound in art and nature, from the architecture of Zaha Hadid, “Queen of the Curve,” to the forms of zebra stripes. Krisztina Regős, Domokos’s graduate student, found the first natural 3D soft cell tucked away in the chambers of the nautilus shell, an object that’s become iconic for showcasing the convergence of math and biology. “They were in front of our eyes the whole time,” Regős says. This connection to such a famous shape led Domokos to fear that his group would be scooped. He swore his collaborators to secrecy until their discovery was ready to be published. (It came out in September.) At the end of his lesson over pizza, he even took the paper menu, folded it up and pocketed it. Just to be safe.

In retrospect, it should have been obvious that soft cells exist, says mathematician Joseph O’Rourke of Smith College, who wasn’t involved in the study. But to think to ask such a question, “to even imagine that you can tile space with no vertices,” he says, is original. “I found that quite surprising and very clever.”

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There’s the play kitchen her 3-year-old son inherited from his cousins. There’s the “random stuff” her mother-in-law buys online, all of it plastic and made up of countless tiny pieces. There’s the kid-sized workbench — Randall got that from her local Buy Nothing group, where neighbors can offload used items (and pick up more).

The sheer volume of stuff her son has to play with is overwhelming, Randall told Vox. The day we talked, she and her family were having guests at their Pacific Northwest home, so she was attempting to declutter, “finding all the parts and putting food in the toy kitchen and putting the tools in the workbench.” But it was always a losing battle.

It’s a familiar refrain among parents: One reader told Vox recently that her family was “absolutely drowning in toys.” And while adults have been complaining about kids’ junk for generations (please see my father’s fruitless search for my brother’s one-inch-long toy wrench in Los Angeles International Airport circa 1992), many millennial and Gen X parents have the sense that something is different now — that kids have more toys than in past decades, and that they seem to arrive in ways Randall describes as “unintentional.”

Historical data on the average number of toys per kid is surprisingly hard to come by, but there is evidence that Americans’ toy glut is increasing — and it’s not just a problem for affluent households.

US toy sales jumped from $22.3 billion in 2019 to $26 billion in 2020, and then to $30.1 billion in 2021, as parents struggled to entertain their kids at home during the pandemic. Sales dipped slightly in 2023, perhaps because of inflation, but remain solidly above 2019 levels.

“I don’t think we’ll ever go back,” Juli Lennett, a vice president and industry adviser for toys at the market research firm Circana, told me.

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A Dutch citizen — her real name is Jacquelina de Jong — she has lived in more than a half-dozen countries and picks up languages with ease. But she is equally at home on the internet, where she has built a global fan base that is dominated by American men. At 16, with the consent of her parents, she was pulling in upward of $50,000 some months, she says, charging for access to her online posts and images.

When The New York Times began investigating the culture of underage girl influencers more than a year ago, Jacky Dejo — or simply Jacky, as she is widely known by her followers on the internet — quickly emerged as a prominent and enigmatic figure.

Still underage, she was posting salacious images of herself on Instagram and had her own photo-selling platform. Everyone in the ever-growing world of child influencers seemed to know about her — mothers who managed their underage daughters’ Instagram accounts, men who followed the girls on various platforms, and anti-child-exploitation crusaders who condemned all of it.

To better understand the alluring and sometimes perilous lifestyle so many of these girls aspire to achieve, The Times asked Jacky to share her story. Her experience is hardly that of a typical American girl for many reasons, including Jacky’s independence and international escapades. But it illustrates in rare detail the dangers faced by child influencers everywhere, and how adolescence for many girls is being molded by platforms that value — and monetize — attention from men who are sexually interested in minors.

Jacky agreed to talk, and her father said he and her mother had no objections. (Since 16, she has been legally emancipated under Dutch law, but her father in particular remains a regular presence in her life.) What followed were months of conversations in English on Telegram and Zoom, and a visit by a reporter and photographer to St. Maarten in the Caribbean, where she lives.

More than a decade online has made Jacky suspicious and cynical. She can be thoughtful and nuanced, but can also exhibit the

bravado and self-certainty of a teenager. She decries online child exploitation, blaming parents as much as the leering men, but proudly proclaims that she has turned the web’s ever-present male gaze to her advantage.

As she recounts her experience, it becomes apparent how much of it has been spent fending off pedophiles, outsmarting scammers and shedding the innocence of childhood.

“If a psycho blackmails a girl in a bikini,” she asserted in one somewhat contentious exchange about the risks of posting sexualized photos, “I don’t think the bikini is the problem here.”

It began harmlessly in 2012, when she was 6 years old and her mother and father launched a parent-run Facebook account to share her snowboarding prowess. By the time she was 8, they had added an Instagram account, where they highlighted free gear she received from brands like Adidas and Nitro Snowboards. They also posted photos of her surfing and skateboarding, two other favorite activities.

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“Jacky Dejo,” a child influencer turned social media entrepreneur, in St. Maarten. Credit…Martina Tuaty for The New York Times

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Monday marked the 102nd anniversary of the discovery of Tutankhamun’s tomb in the Valley of the Kings on November 4, 1922, one of the most spectacular discoveries in archaeology. On that fateful day, British archaeologist Howard Carter wrote in his diary, “I discovered the first traces of the entrance to the tomb (Tut-ank-Amon),” marking the discovery of the tomb of the Golden King.

Howard Carter had been excavating in the Valley of the Kings for a decade and was no stranger to the challenges of treasure hunting in such an ancient and looted place. Since 1907, he had been working with British nobleman Lord Carnarvon, who financed the excavations in the lands washed by the Nile. However, Lord Carnarvon began to doubt that his investment would yield results. In 1922, when Carter had been excavating in the valley for five years without significant results, Lord Carnarvon pressured him to terminate the work. Lord Carnarvon granted Carter a last season of work in the autumn of that year—his final opportunity.

By a stroke of luck, Carter and his team made an incredible discovery, having begun excavations just three days earlier. A member of their team, a water boy, accidentally stumbled upon a stone that turned out to be the first step of an ancient staircase. Intrigued, Carter ordered to excavate quickly, and gradually, the team unearthed a series of descending steps leading to a sealed door with hieroglyphic inscriptions. These seals indicated that it was a royal tomb, and Carter realized he was facing the find of his life—a historic event.

Despite the anxiety, Carter decided to stop before opening the tomb. He knew he had to wait for the arrival of Lord Carnarvon, who was in England and would want to witness the opening. So Carter ordered to cover the steps again and sent an urgent telegram to Lord Carnarvon, notifying him of the find. The wait lasted almost three weeks, certainly eternal for the archaeologist. Lord Carnarvon finally arrived in Egypt on November 23, accompanied by his daughter, Evelyn Herbert. The next day, Carter and Lord Carnarvon uncovered the staircase again and examined together the threshold of the tomb.

On November 26, Carter made a small opening in the door of the tomb and, with a candle, peered inside to see the interior. When Lord Carnarvon asked him if he saw anything, Carter replied with the phrase that would go down in history: “Yes, I see wonderful things,” he replied. Inside, the archaeologist glimpsed an amazing collection of objects that shone with the reflection of the light: chests, statues, gilded furniture, and other objects destined for the young pharaoh in his journey to the afterlife.

The tomb of King Tutankhamun is globally famous as the only royal tomb in the Valley of the Kings whose contents were discovered intact and relatively complete. On February 16, 1923, Howard Carter became the first person in over 3,000 years to set foot in the chamber containing Tutankhamun’s sarcophagus. The burial chamber was officially opened in mid-February 1923, after Carter and his sponsor, Lord Carnarvon, first contemplated the interior of a burial chamber that had remained closed for over 3,300 years.

Inside the tomb, sealed for over 3,300 years, they found more than 5,400 artifacts, including the ruler’s gold mask, chariots, a bed, jewelry, board games, food remains, and numerous figurines, many in perfect condition. Among the most dazzling items of the 18-year-old pharaoh is the mortuary mask, which exceeds six kilos of gold. They also found a leopard skin mantle, four game boards, six chariots, 30 jars of wine, and 46 bows.

The artifacts in Tutankhamun’s tomb reflect the lifestyle in the royal palace and include items he would have used in daily life, such as clothing, jewelry, cosmetics, incense, furniture, chairs, toys, vessels, weapons, and others. Among the treasures discovered were personal articles and weapons, revealing unknown aspects of his daily life and rituals, including an amazing collection of objects destined for his journey to the afterlife. A total of around 5,000 artifacts were discovered tightly packed inside the tomb, which, despite its immense wealth, was very modest in size and architectural design compared to other tombs in the Valley of the Kings. According to data from the Ministry of Tourism and Antiquities, Tutankhamun’s tomb is number 62 in the Valley of the Kings.

In December 1922, the first artifact was removed from the tomb, and cleaning of the antechamber began, which took seven weeks. The classification work extended for years, as it involved more than 5,000 unique pieces. Some of the most fascinating and meticulous moments in the exploration of Tutankhamun’s tomb was the revelation of treasures hidden among the layers of linen that wrapped his mummy. After years of excavation and cataloging the objects found in the burial chambers, Carter and his team faced the last challenge: unrolling the bandages that covered the pharaoh, a process that began in 1925.

With utmost care, the archaeologists and doctors proceeded to remove the layers of linen that had been placed in embalming ceremonies to protect the body in its journey to the afterlife. As they removed each layer, they discovered an impressive variety of jewelry and amulets carefully arranged among the bandages. A total of 143 pieces were hidden alongside the body of the pharaoh. Among them, gold diadems, intricate necklaces, bracelets of various metals and precious stones, and a series of amulets and talismans stood out, all with deep symbolic and religious meaning in ancient Egypt. These objects, besides beautifying the deceased, were believed to possess protective and magical powers that would help the pharaoh in his eternal life.

Some of the most notable findings included the royal diadem, which adorned the head of Tutankhamun and was made of gold, lapis lazuli, and other precious stones. This symbol of royalty identified him as pharaoh even in death. The archaeologists also discovered two daggers, one of iron and the other of gold, placed at the pharaoh’s waist. The iron dagger, forged with a rare material for the time and decorated with intricate motifs, is particularly famous for its composition, as recent studies suggested that this iron may have come from a meteorite.

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Tutankhamun © (photo credit: Jaroslav Moravcik. Via Shutterstock)

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I didn’t find math particularly exciting when I was in high school. To be honest, I only studied it when I went to university because it initially seemed quite easy to me. But in my very first math lecture as an undergraduate, I realized that everything I thought I knew about math was wrong. It was anything but easy. Mathematics, I soon discovered, can be really exciting—especially if you go beyond the realm of pure arithmetic.

In physics, the truly surprising content—concepts that go against your intuition about the universe—emerges around high school, when students can glimpse the strange quantum world and encounter Einstein’s general and special theories of relativity. School mathematics cannot keep up with these wonders. You learn elementary arithmetic operations, integration, and derivation, the basic handling of probabilities and vectors. If you’re lucky, ambitious teachers might show you a simple proof. And that’s it. So it’s no wonder that many pupils fail to develop a real passion for the subject.

Yet mathematics offers all sorts of surprises, such as the Banach-Tarski paradox, which states that you can double a sphere almost magically, or the fact that there are infinitely many different infinities. What really blew me away was discovering how deeply mathematics is interwoven with the strangest physical phenomena. It’s not necessarily quantum physics itself that gives rise to the incredible effects; no, the systems always follow the strict rules of mathematics. As chemist Peter Atkins put it in his 2003 book Galileo’s Finger, “Determining where mathematics ends and science begins is as difficult, and as pointless, as mapping the edge of a morning mist.”

Few examples illustrate the mixing of math and physics better than a discovery made by physicist Michael Berry. In 1984 Berry revealed a profound and largely unexpected geometric side to quantum mechanics. This geometry, Berry realized, gives quantum particles a kind of memory.

Nothing Should Actually Happen

At the time, Berry was investigating a very simple system: the quantum state of a particle, such as a neutron, in a changing environment. Neutrons have a quantum property called spin, which acts like a tiny magnet that the particles carry with them. This spin can either be oriented with the north pole facing upward or downward—so physicists speak of neutrons having “spin up” or “spin down.” The spin of a neutron is influenced by external magnetic fields.

Berry used mathematical means to investigate what would happen to the neutron if the direction of the magnetic field changed slowly. According to the so-called adiabatic theorem, which was introduced in the early 20th century, the quantum properties of the particle should not change as a result: its energy, momentum, mass, and spin remain the same.

If you slowly turn the direction of the magnetic field and then move it back in the original direction, this action should, in principle, not actually change anything. “That, at any rate, was the prevalent opinion among physicists for many years,” wrote Berry in an article in Scientific American in December 1988. But a “change on the phase of a wave function was overlooked.”

One of the strangest phenomena of quantum mechanics is wave-particle duality: quantum objects can be imagined as pointlike shapes, but they also exhibit wave behavior like water. A phase describes a displacement of the wave by a certain angle—for example, the cosine function is nothing other than a phase-shifted sine function.

As Berry recognized in his calculations, a slow change in the magnetic field causes the wave function of the neutron to rotate by a certain phase. This means that the wave function of the particle shows what happened in the past (in this case, the change in the magnetic field). Further, Berry recognized that this phase does not only occur in the special case of a particle in a magnetic field. Various situations in which a quantum system is slowly changed and then returned to its original conditions leave traces in the wave function.

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Ancient Book Found Under Nile River – Experts Turn Pale After Transcribing

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Ancient Book Found Under Nile River

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As memes go, it wasn’t particularly viral. But for a couple of hours on the morning of November 6, the term “darkest timeline” trended in Google searches, and several physicists posted musings on social media about whether we were actually in it. All the probabilities expressed in opinion polls and prediction markets had collapsed into a single definite outcome, and history went from “what might be” to “that just happened.” The two sides in this hyperpolarized U.S. presidential election had agreed on practically nothing—save for their shared belief that its outcome would be a fateful choice between two diverging trajectories for our world. That raises rather obvious (but perhaps pointless) questions: Could a “darkest

timeline” (or any other “timeline,” for that matter) be real? Somewhere out there in the great beyond, might there be a parallel world in which Kamala Harris electorally triumphed instead?

It turns out that, outside of fostering escapist sociopolitical fantasies and putting a scientific gloss on the genre of counterfactual history, the notion of alternate timelines is in fact something physicists take very seriously. The concept most famously appears in quantum mechanics, which predicts a multiplicity of outcomes—cats that are both alive and dead and all that. If a particle of light—a photon—strikes a mirror that is only partially silvered, the particle can, in a sense, both pass through and reflect off that surface—two mutually exclusive outcomes, known in physics parlance as a superposition. Only one of those possibilities will manifest itself when an observation is made, but until then, the particle juggles both possibilities simultaneously. That’s what the mathematics says—and what experiments confirm. For instance, you can create a superposition and then uncreate it by directing the light onto a second partially silvered mirror. That wouldn’t be possible unless both possibilities remained in play. Although this feature is usually framed in terms of subatomic particles, it is thought to be ubiquitous across all scales in the universe.

What supports the idea that these timelines are real, and not just imaginative fictions, is that they can “interfere” with one another, either enhancing or diminishing the probability of their occurrence. That is, something that might have happened but doesn’t has a measurable effect on what does, as if the former reaches from the shadowy realm of the possible into the world of the actual.

Consider the bomb detector that physicists Avshalom Elitzur and Lev Vaidman proposed in 1993 and that has since been demonstrated (fortunately not with real bombs): Perform the experiment with the partially silvered mirror but place a light-sensitive bomb along one of the two paths the photon can take. This blockage prevents you from uncreating the superposition to restore the traveling photon to its original state. It does so even if the bomb never goes off, indicating that the photon never touched it. The mere possibility that the photon could strike the bomb affects what happens. In theory, you could use this principle—known as counterfactual definiteness—to take x-ray images of cells without subjecting them to damaging radiation. In an emerging subject known as counterfactual quantum computing, a computer outputs a value even if you never press the “run” button.

One way to think about counterfactual definiteness is known as the many-worlds interpretation. A photon striking a mirror causes the cosmic timeline to branch, creating one world in which the particle passes through the mirror and one in which it reflects off that surface. Each of us is stuck inside our world and therefore sees only one outcome at a time, but the other is still there, visible to an inhabitant of the alternate world. All such worlds, taken together, constitute a “multiverse.”

Whether they agree with the many-worlds interpretation or not, physicists and philosophers certainly love to argue about it. Some admire its elegance; others grouse about conceptual difficulties, such as the slippery matter of what exactly constitutes a “world.” Quantum theory not only allows multiple worlds but also offers an infinity of ways to define them.

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An intriguing video of a stone house in a mountainside!

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Scientists Make Startling Discovery Of Ancient Stone House

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