Look carefully! Mathematicians have invented a new 13-sided shape that can be tiled infinitely without ever repeating a pattern. They call it “the einstein.”
For decades, mathematicians wondered if it was possible to find a single special shape that could perfectly tile a surface, without leaving any gaps or causing any overlaps, with the pattern never repeating. Of course, this is trivial to do with a pattern that repeats — just look at a bathroom or kitchen floor, which is probably made up of simple rectangular tiles. If you were to pick up your floor and move it (called a “translation” in mathematics), you could find a position where the floor looks exactly the same as before, proving that it’s a repeating pattern.
In 1961, mathematician Hao Wang conjectured that aperiodic tilings, or tilings that never become a repeating pattern, were impossible. But his own student, Robert Berger, outwitted him, finding a set of 20,426 shapes that, when carefully arranged, never repeated. He then slimmed that down to a set of 104 tiles. That means that if you were to buy a set of those tiles, you could arrange them on your kitchen floor and never find a repeating pattern.
In the 1970s, Nobel prize-winning physicist Roger Penrose found a set of only two tiles that could be arranged together in a nonrepeating pattern, now known as a Penrose tiling.
Here we see the first four iterations of the H metatile and its supertiles. (Image credit: Smith el at. (2023))
Since then, mathematicians around the world have searched for the aperiodic tiling holy grail, called “the einstein.” The word doesn’t come from the famous Albert but from the German translation of his last name: one stone. Could a single tile — one “stone” — fill a two-dimensional space without ever repeating the pattern it creates?
The answer was just discovered by David Smith, a retired printing technician from East Yorkshire, England. How did he come across this remarkable solution? “I’m always messing about and experimenting with shapes,” Smith told The New York Times (opens in new tab) . “It’s always nice to get hands-on. It can be quite meditative.”
Smith and his co-authors dubbed the new shape “the hat,” mostly because it vaguely resembles a fedora. Although mathematicians have known about the shape, which has 13 sides, they had never considered it a candidate for aperiodic tiling.
“In a certain sense, it has been sitting there all this time, waiting for somebody to find it,” Marjorie Senechal (opens in new tab) , a mathematician at Smith College who was not part of the study, told The Times.Related stories—Mathematicians make rare breakthrough on notoriously tricky ‘Ramsey problem’
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Smith worked closely with two computer scientists and another mathematician to develop two proofs showing that “the hat” is an aperiodic monotile — an einstein. One proof relied on building larger and larger hierarchical sets of the tiles, showing how the pattern never repeats as the surface area grows. The other proof relied on the team’s discovery that there wasn’t just one of these tiles, but an infinite set of related shapes that could all do the trick. The team’s paper is available on the preprint server arXiv (opens in new tab) but has not yet been peer-reviewed, and the proofs have not yet been scrutinized.
These kinds of aperiodic tilings are more than mathematical curiosities. For one, they serve as a springboard for works of art, like the Penrose tiling found at the Salesforce Transit Center (opens in new tab) in San Francisco, and reveal that some medieval Islamic mosaics employed similar nonrepeating patterns.
Aperiodic tilings also help physicists and chemists understand the structure and behavior of quasicrystals, structures in which the atoms are ordered but do not have a repeating pattern.
If you know a teacher, or you are a teacher, you’ll understand that many in the profession consider their jobs one of the best in the world, while also acknowledging it can be one of the most stressful.
Teaching in a war zone takes it to another level on both fronts.
But imagine teaching in a war zone in say the London Underground or the Paris or New York metros?
Well, that’s exactly what is happening in Kharkiv, Ukraine’s second largest city – a city attacked by Russia on a constant basis.
In 2023, the authorities banned children from being taught in regular school buildings here because of the threat they were under from missile strikes.
So Kharkiv city and its education department started working on a plan.
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That plan has concluded with the construction of six schools in six metro stations, teaching 4,800 children daily whose families haven’t fled the war.
Intrigued by this new educational experiment, as it’s still considered, we were told to meet at the University metro station in the centre of the city and wait to be taken to the school.
Soviet-era metro stations, and particularly the platforms, are universally grand and ornate, although Ukraine’s are slightly more modest.
Pedestrian walkways though are always crammed with florists, newspaper stands, mobile phone shops and cafes, and Kharkiv’s are no different.
What is different though is that next to the main platform entrance, an adjacent passageway has been sealed off by a white wall with three doors built into it.
We were ushered inside, past a guard and into a concourse with two desks.
In front of us, we could see a wide staircase leading to a long corridor. Above us, large, shiny silver metal tubes stretched into the distance, pumping in air from above ground.
Lining the corridor, we could see a series of white interconnecting cabins, all brightly lit, and from them, we could hear the sounds of children laughing, music, and teachers conducting class in this strange subterranean school.
Perhaps the only school in the world with one corridor.
A door to one of the classes opened, and in pairs, a class of six and seven-year-olds emerged – their teacher at the front and an assistant at the rear.
They’re heading to the bathroom for a toilet break, the staff must keep an eye on them.
If any of the children were to make it out of the school, they could easily get lost in the underground system.
The pupils are spread across seven classrooms with a mix of yellow and green desks and chairs, and walls decorated with bright pictures like rainbows, sunshine and sunflowers, to try to create a cheery atmosphere.
“At first, the children were bewildered by studying here, it felt unfamiliar to them, but interacting with each other and with their teachers helped them to adapt,” Iryna Tarasenko from Kharkiv’s education department told me.
Iryna is showing me around the school. They feel it’s their way of contributing to the war effort.
She said: “In the summer of 2023, the Defence Council didn’t permit children to study in regular school buildings, so, we had to find a solution. This is our frontline – our educational frontline.”
There are of course health concerns for both the teachers and the pupils spending so much time underground, so medical examinations are done on a regular basis.
It is an opportunity for doctors and behavioural specialists to monitor the effects of this war on them.
Dmytro Mitelyov is the neurologist on duty monitoring the children’s behaviour, checking for any signs of stress – physical and mental.
He gently asks each child how they’re feeling, and asks if they have any aches and pains.
“These are children who haven’t left Kharkiv since the war began, they live in a constant state of stress, multifaceted stress caused by a range of factors. They are exposed to things they see and hear, like alarms, explosions, and they feel it all,” Dmytro told me.
“When these children grow older, all the trauma they’ve endured during their formative years can, unfortunately, leave a lasting and serious imprint on their mental health. They haven’t had a healthy, carefree childhood – the kind where they can learn, play, and feel safe as children normally would.”
As the first group’s school day ends, children from one of Kharkiv’s most bombed districts, wrap up warm before being led to the surface to catch a school bus to their old school – which was hit by a missile.
It’s still where they meet their parents for normal pick-up.
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We jump on the bus with them for the journey past destroyed government buildings, shopping centres and restaurants.
When the air raid suddenly sounds, I look around to see how the children and teachers on this bus will react.
They barely flinch it’s so common.
“People have, to some extent, adapted,” one of the headmistresses, Olena Nikolienko, said.
“If we see missiles coming, heaven forbid, we’ll stop near a designated shelter, following our evacuation plan, and that’s where the children will go for cover.”
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Remember, these children are safe underground at school, up here they are not.
During the bus ride, 10-year-old Alisa began quietly sobbing. Her teacher told me her grandfather was killed four months ago on the frontline, and since then she has been prone to tears.
Alisa’s teacher comforted her and told her everything would be ok.
It feels like everyone here has to carry their own sadness.
Everyone.
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0:18
‘Ukraine war will end sooner under Trump’
Parents, standing in the first snowfall of the season, greet the children as they get off the bus.
The kids run off the bus, and start throwing snowballs at each other, enjoying this precious moment.
Joe Biden has met with Xi Jinping for the last time as US president, where the Chinese leader said he is “ready to work” with Donald Trump.
Speaking at the Asia-Pacific Economic Cooperation forum in Lima, Peru, Mr Biden said the US and China’s relationship should be about “competition, not conflict”.
“We haven’t always agreed, but our conversations have always been candid and always been frank,” he said.
“We’ve never kidded one another. These conversations prevent miscalculations, and they ensure the competition between our two countries will not veer into conflict.
“We’ve been level with one another. I think that’s vital.”
But despite Mr Trump’s proposed measures, Mr Xi said his country’s goal “of a stable, healthy and sustainable China-US relationship remains unchanged”.
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“Our commitment to mutual respect, peaceful coexistence and… cooperation as principles for handling China-US relations remains unchanged,” he added.
The Chinese president then said the country is “ready to work with the new US administration to maintain communication, expand cooperation and manage differences, so as to strive for a steady transition”.
Neither Mr Xi nor Mr Biden responded to a question about whether there were concerns about Mr Trump’s proposed tariffs.
The president-elect has also named several China hawks to his transition team, such as Senator Marco Rubio as secretary of state and Representative Mike Waltz as national security adviser.
Former heavyweight champion Mike Tyson says the health scare in June, which forced the postponement of his boxing match with Jake Paul, almost cost him his life.
“I almost died in June,” Tyson wrote on X, adding that he “had 8 blood transfusions. Lost half my blood and 25lbs in hospital”.
The 58-year-old, who lost to the YouTuber-turned-boxer in a unanimous points decision in Texas, tweeted that the situation was far worse than people had realised.
Following Tyson’s recovery, after reportedly suffering an ulcer flareup, he reflected on the result of the fight.
“This is one of those situations when you lost but still won… no regrets to get in the ring one last time,” he said.
That contrasts with Tyson’s comments after the match in which he refused to confirm whether it would be his last fight.
“It depends on the situation,” Tyson said before suggesting a fight with Paul’s older brother, Logan Paul, who was standing near him in the ring.
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Critics argued that the revised match – which involved fewer and shortened rounds, as well as heavier gloves – fell short of entertaining.
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Paul beats Tyson by unanimous decision
Some claimed it appeared more like a glorified sparring session.
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It was Tyson’s first sanctioned pro bout since 2005, when he lost to Kevin McBride and then went into retirement – only briefly reappearing for an exhibition match in 2020 against Roy Jones Jr.
Paul has plenty to prove
Paul, 27, still has plenty to prove. The social media influencer wants to be a championship fighter and compete for a championship belt within two years. “I think it could happen in the next 24 months,” Paul said.
“I truly, truly believe in my skills and my ability and my power. And the cruiserweight division is seemingly open for the taking on that timeline,” he added.
The only professional match he has lost so far has been to Tommy Fury, the less-accomplished brother of former heavyweight champion Tyson Fury.
But there are questions about when Paul will fight a contender in his prime, as opposed to former champions or mixed martial artists.
Intriguingly, in the days before his fight with Tyson, Paul mentioned super middleweight champion Canelo Alvarez, considered one of the best boxers in the world.
According to reports, the Texas bout earned Paul $40m (£31.7m) versus $20m (£15.8m) for Tyson.
Netflix said 60 million households worldwide viewed the contest on its streaming platform, and nearly 50 million tuned in to watch the undercard which saw Ireland’s Katie Taylor beat Puerto Rico’s Amanda Serrano.
Even so there were some technical glitches. More than 90,000 users reported problems on Netflix at its peak, according to the website Downdetector, which tracks outages.