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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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—Two mathematicians just solved a decades-old math riddle — and possibly the meaning of life

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.

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Counter-terrorism police investigating after two women injured in Leeds

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Counter-terrorism police investigating after two women injured in Leeds

Counter-terrorism police are investigating after an incident involving a crossbow and a firearm left two women injured in Leeds.

Police were called to Otley Road at 2.47pm on Saturday to reports of a “serious incident involving a man seen with weapons”, West Yorkshire Police said.

Officers arrived at the scene to find two women injured – and a 38-year-old man with a self-inflicted injury. All three were taken to hospital, with the man held under arrest, but their injuries are not believed to be life-threatening.

“Two weapons have been recovered from the scene, which were a crossbow and a firearm,” Counter Terrorism Policing North East said in a statement.

The incident happened on the ‘Otley Run’ pub crawl, with one venue saying it was closed for the evening due to “unforeseen circumstances”.

Officers guard one of the crime scenes in Leeds
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Officers guard one of the crime scenes

Officers inside the cordon in Leeds
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Officers inside the cordon in Leeds

Counter Terrorism Policing’s statement added: “Due to the circumstances surrounding the incident, Counter Terrorism Policing North East have taken responsibility for leading the investigation with the support of West Yorkshire Police.

“Extensive enquiries continue to establish the full circumstances and explore any potential motivation.”

Home Secretary Yvette Cooper described it as a “serious violent incident” and said she was being kept updated by police.

“Thank you to the police and emergency services for their swift response,” she said. “My thoughts are with the victims and all those affected by this attack.”

This breaking news story is being updated and more details will be published shortly.

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Wrexham promoted for third season in a row under Ryan Reynolds and Rob McElhenney

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Wrexham promoted for third season in a row under Ryan Reynolds and Rob McElhenney

Wrexham AFC have been promoted for the third season in a row.

The North Wales-based side has gone from the National League to the Championship in just three seasons, under its Hollywood owners Ryan Reynolds and Rob McElhenney.

Wrexham were second in the table and had a run of eight games unbeaten ahead of their match against Charlton Athletic on Saturday, which they won 3-0.

Wrexham's James McClean lifts the trophy after the Sky Bet League One match at SToK Racecourse, Wrexham. Picture date: Saturday April 26, 2025.
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Wrexham’s James McClean lifts the League One trophy. Pic: PA

Wrexham's Dan Scarr celebrates with the fans on the pitch after Wrexham won promotion to the Sky Bet Championship after the Sky Bet League One match at SToK Racecourse, Wrexham. Picture date: Saturday April 26, 2025. PA Photo. See PA story SOCCER Wrexham. Photo credit should read: Martin Rickett/PA Wire...RESTRICTIONS: EDITORIAL USE ONLY No use with unauthorised audio, video, data, fixture lists, club/league logos or "live" services. Online in-match use limited to 120 images, no video emulation. No use in betting, games or single club/league/player publications.
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Wrexham’s Dan Scarr celebrates with the fans on the pitch after Wrexham won promotion to the Championship. Pic: PA

It is the first time any club has been promoted for three consecutive seasons within the top five tiers of English football.

The third oldest association football club in the world, Wrexham AFC was bought by Reynolds and McElhenney in 2020, and has since been the subject of a Disney+ documentary, Welcome To Wrexham.

Reynolds, wearing a Wrexham sweatshirt, and McElhenney were pictured celebrating each goal, and after the game, as the fans came onto the pitch at the SToK Cae Ras (Racecourse Ground) to celebrate the victory with the players.

Wrexham co-owners Rob McElhenney (left) and Ryan Reynolds (right) and Ryan's wife Blake Lively before the Sky Bet League One match at SToK Racecourse, Wrexham. Picture date: Saturday April 26, 2025.
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Wrexham co-owners Rob McElhenney (L) and Ryan Reynolds and Ryan’s wife Blake Lively, before the match. Pic: PA

Both stars came onto the pitch after the supporters returned to the stands.

More on Ryan Reynolds

Speaking to Sky Sports, McElhenney praised those behind the scenes, referring to “so many that don’t get the credit they deserve, people who aren’t talked about”.

Reynolds said bringing success back to the club “seemed like an impossible dream” when they arrived in North Wales in 2020.

Wrexham's Sam Smith celebrates in front of the fans on the pitch after Wrexham won promotion to the Sky Bet Championship after the Sky Bet League One match at SToK Racecourse, Wrexham. Picture date: Saturday April 26, 2025. PA Photo. See PA story SOCCER Wrexham. Photo credit should read: Martin Rickett/PA Wire...RESTRICTIONS: EDITORIAL USE ONLY No use with unauthorised audio, video, data, fixture lists, club/league logos or "live" services. Online in-match use limited to 120 images, no video emulation. No use in betting, games or single club/league/player publications.
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Wrexham’s Sam Smith celebrates in front of the fans after Wrexham won promotion to the Championship. Pic: PA

He put the three promotions down to “the coaching staff, the greatest dressing room” and an “all for one, one for all” attitude throughout the club, adding he was “speechless with their commitment and their emotion”.

As for the mouth-watering prospect of another promotion to the promised land of the Premier League, the pair agreed it was “for tomorrow”, before ending the interview with a joint mic-drop.

Veteran striker Steven Fletcher said, “as soon as I came to this club, I knew it was something special. We want to go again. We’ll reset in the summer, take a break and go again”.

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Direct action group Just Stop Oil holds final protest, claiming it has been ‘successful’

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Direct action group Just Stop Oil holds final protest, claiming it has been 'successful'

“It has been a success.”

Just Stop Oil (JSO) insists it’s been “successful” – as its members ceremoniously hang up their orange high-vis vests during a march in central London.

Since the group formed three years ago, it’s drawn attention and criticism for its colourful, controversial protests, which ranged from disrupting sporting events to throwing soup on Vincent van Gogh’s Sunflowers, and climbing on gantries over the M25. It sprayed orange paint over Stonehenge, and cost police forces tens of millions of pounds.

Those days are now behind it; to the relief of many.

Read more
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As a few hundred activists marched through London on Saturday, blocking roads as they went; taxi drivers blared their horns and football fans shouted abuse from the pavement.

The PA News Agency filmed the moment a white minivan seemed to drive towards a group of protesters blocking the road.

Protesters shouted “I’m being pushed back!” to police, while the driver could be heard shouting “What about my right to get home?” to the officers gathered.

But JSO never set out to be popular. And it believes its tactics – though hated – have been successful; thanks to the new Labour government’s commitment to not issue new oil or gas exploration licences.

That’s why, it says, its ceasing direct action.

JSO hangs up its high vis jackets in central London on Saturday
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JSO hangs up its high-vis jackets in central London on Saturday

A washing line of high-vis jackets signifies JSO's disbanding
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A washing line of high-vis jackets signifies JSO’s disbanding

“This moment marks the success of the JSO campaign – our demand was to end new oil and gas licences and that is now government policy.

“As a result of which four billion barrels of oil are being kept under the North Sea. The campaign has reached a natural end.”

Dr Oscar Berglund, senior lecturer in international public and social policy, disagrees that JSO is disappearing because it’s been “successful”.

He told Sky News policing strength and public perception might have more to do with it.

“They have very low levels of popularity. About 17% of the British population are kind of broadly supportive of what Just Stop Oil do. And that’s too low to recruit.

“It’s difficult to recruit members to something that is that unpopular, and then that a lot of people for good reason I think have kind of stopped believing in that kind of disruption as a means to achieve meaningful change.”

Group triggers specific new protest laws

One thing it did change is the law.

Policing commentator Graham Wettone tells us: “Obstruction of the highway, obstruction of rail networks for example, these are specific offences now.

“It’s given the police more tactics, more methods, more offences they can consider, even stopping and searching somebody who may have something to either lock themselves on or glue themselves to something.”

A JSO activist holds a picture of an imprisoned colleague
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A JSO activist holds a picture of an imprisoned colleague

Emma Smart was held in prison for her activism with both Insulate Britain and Just Stop Oil.

“The high-vis might be going away,” she tells me, “but we aren’t.”

“These people aren’t going anywhere, we are still committed, dedicated, terrified by the failings of this government and governments around the world.”

JSO activists throw orange paint at van Gogh's sunflowers
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JSO activists throw orange paint at van Gogh’s sunflowers

Orange smoke set off by JSO protesters at Stonehenge
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Orange smoke set off by JSO protesters at Stonehenge

She hopes for a time of reflection before it returns in a new form but says the need for climate activism is stronger than ever.

She also believes that while most people dislike JSO tactics, it still raises awareness of the cause and might even push people to more moderate campaign groups.

Just Stop Oil came behind other, similarly controversial climate campaign groups like Insulate Britain and Extinction Rebellion, and as it says goodbye, its disruptive methods have been seized upon by other organisations like the Pro-Palestinian Youth Justice.

The infamous Just Stop Oil orange vests might be going away, but the individual activists, their cause and campaign tactics feel here to stay.

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