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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’

—Centuries old ‘impossible math problem cracked using physics of Schrödinger’s cat

—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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UK

Parents of boy whose outbursts left them fearing for their lives say Adolescence ‘touched a nerve’

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Parents of boy whose outbursts left them fearing for their lives say Adolescence 'touched a nerve'

A teenager suddenly becomes violent, his anger towards women fuelled by online influencers, while his parents struggle to process what their son is capable of.

Does this sound familiar?

It’s the story of the hit drama Adolescence – but for Jess and Rob, it’s their life.

Their 14-year-old son Harry’s violence has escalated so rapidly he’s had to be taken into care. We’ve changed all their names to protect their identities.

Until the age of 12, Harry’s parents say he was a “wonderful” son. But they saw a change in his personality, which they believe was sparked by an incident when he was hit by a girl. Soon, he developed an online interest in masculine power and control.

Becky Johnson Adolescence  feature
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Harry’s personality changed after he was assaulted (this image shows an actor in a Sky News reconstruction)

“Harry became obsessed with being strong, and I think he developed a difficulty around certain female people because of the assault,” Jess says.

“He had to be in charge… in every setting,” Rob adds.

Then one night, he punched his mother, Jess. His parents called the police in the hope it would shock him out of doing it again. But, as time went on, the violence escalated.

“We probably must have called the police over 100 times,” Rob says.

One attack was so serious, Jess ended up in hospital. The violence spilled outside the home too as Harry assaulted neighbours and friends.

Then he threatened to stab a teacher.

“Every time we think it can’t get any worse, something else happens and it does get worse,” Rob says. “Unfortunately, him getting hold of a knife is quite likely to happen.”

They say Adolescence, which stars Stephen Graham, Ashley Walters and Owen Cooper, touched a nerve.

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Jess and Rob say they called the police 100 times (this image shows actors in a Sky News reconstruction)

“My worst fear is that he’s going to end up killing one of us,” says Jess. “If not us, then somebody else…”

It’s a shocking thought for any parent to have. As well as contacting police, the family have tried many times to get help from social services and the NHS for Harry’s deteriorating mental health.

“We’ve been told that we’re using too many resources and accessing too many services,” Rob says. “We tried for 18 months to get him more intensive therapeutic help. At every turn it was ‘no, no, no’.”

Read more:
Starmer backs campaign to show Adolescence in schools
Inside the online gangs where boys compete to be cruel

They have found help with an organisation called PEGS that supports parents who are victims of their own children’s violence.

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PEGS founder Michelle John says many families struggle to have their concerns taken seriously

Last year it was contacted by over 3,500 families, a 70% increase on 2023. Founder Michelle John says many families struggle to have their concerns taken seriously.

“What we’re hearing time and time again is that referrals are not being picked up because thresholds aren’t being met and perhaps the parent or caregiver isn’t a risk to the child,” she says. “Families are falling through gaps.”

In some parts of the country, local organisations are attempting to fill those gaps. Bright Star Boxing Academy in Shropshire has children referred by schools, social workers and even the police.

Joe Lockley, who runs the academy, says the problem is services that deal with youth violence are “inundated”.

“The biggest cause of the violent behaviour is mental health,” he says. “They lack that sense of belonging and control, and it’s quite easy to gain that from the wrong crowd and getting involved in violence.

“Social media is having a huge impact, especially around that young person’s identity.”

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Ethan at the Bright Star Boxing Academy

Ethan, 18, agrees. He joined the academy aged 14. By then he had already been arrested several times for getting into fights.

He believes bullying sparked anxiety and depression. “Someone could look at me, I’d be angry,” he says.

“Social media – that’s definitely a massive part. You’ve got so many people that are living this material life. They’ve got loads of money.

“My main thing was seeing people with amazing bodies – I felt I couldn’t reach that point and it made me self-conscious, which would add on to the anger which then turned to hatred towards other people.”

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Ethan says boxing has helped him turn his life around

Without the support of the boxing academy, he believes, he wouldn’t have been able to turn his life around.

“I would either be in prison or I would have done something a lot worse to myself,” he says.

“It’s just this massive mess in your head where you’ve got a million thoughts at once – you don’t know what to think or how to even speak sometimes,” he adds.

“All we need is someone that’s got the time for us… and the understanding that it’s a war in our heads.”

A government spokesperson told Sky News: “We have seen too many preventable tragedies caused by the failings of mental health services, and it’s unacceptable that young people have not been getting the care and treatment they need to keep them, their families and the wider public safe.

“We are working to ensure children and their families get that help. We are investing over £50m to fund specialist support in schools, launching a Young Futures hub in every community, and providing access to a specialist mental health professional in every school in England.”

Anyone feeling emotionally distressed or suicidal can call Samaritans for help on 116 123 or email jo@samaritans.org in the UK. In the US, call the Samaritans branch in your area or 1 (800) 273-TALK

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Technology

UK loses bid to keep Apple appeal against demand for iPhone ‘backdoor’ a secret

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UK loses bid to keep Apple appeal against demand for iPhone 'backdoor' a secret

Chief Executive of Apple, Tim Cook gives a thumb’s up during a tour the Apple Headquarters on December 12, 2024 in London, England. 

Chris Jackson | Getty Images

Apple has triumphed over an effort from the U.K. government to keep details secret of its appeal against an order to create a “backdoor” to iPhone users’ data.

The U.K.’s Investigatory Powers Tribunal on Monday published a ruling dismissing the government’s attempt to prevent details from a hearing on the appeal from being made public. The government had tried to keep the information secret on the grounds it posed risks to national security.

Judges Rabinder Singh and Judge Jeremy Johnson said in their ruling that the U.K. government’s request to keep details of the hearing private “would be the most fundamental interference with the principle of open justice.”

“It would have been a truly extraordinary step to conduct a hearing entirely in secret without any public revelation of the fact that a hearing was taking place,” they said.

Britain’s Home Office was not immediately available for comment when contacted by CNBC.

‘Backdoor’ to encrypted data

The ruling relates to an appeal made by Apple against a demand from the U.K. government to allow officials to access iPhone users’ encrypted data via a technical “backdoor.”

This backdoor would allow the government to access information secured by Apple’s Advanced Data Protection (ADP) system, which applies end-to-end encryption to a wide range of iCloud data.

Governments in the U.S., U.K. and EU have long expressed dissatisfaction with end-to-end encryption, arguing it enables criminals, terrorists and sex offenders to conceal illicit activity.

In the U.K., the Investigatory Powers Act of 2016 empowers the government to compel tech companies to weaken their encryption technologies through so-called “backdoors” — a heavily controversial policy for both the tech industry and privacy campaigners.

Apple — which is known for its pro-privacy stance — has pushed back on efforts to weaken its encryption tools, saying this would undermine its security and put users at risk.

As a result of the government’s order, Apple withdrew its ADP system for U.K. users in February. In a blog post at the time, the tech giant said it has “never built a backdoor or master key to any of our products or services and we never will.”

“We are deeply disappointed that our customers in the UK will no longer have the option to enable Advanced Data Protection (ADP), especially given the continuing rise of data breaches and other threats to customer privacy,” Apple said in the post.

“Apple remains committed to offering our users the highest level of security for their personal data and we are hopeful that we will be able to do so in the future in the United Kingdom.”

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Environment

Trump orders new review of U.S. Steel acquisition by Japan’s Nippon Steel

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Trump orders new review of U.S. Steel acquisition by Japan's Nippon Steel

A tugboat pushes a barge near the U.S. Steel Corp. Clairton Coke Works facility in Clairton, Pennsylvania, on Sept. 9, 2024.

Justin Merriman | Bloomberg | Getty Images

President Donald Trump on Monday ordered the planned acquisition of U.S. Steel to undergo a new review after the company’s pending purchase by Japan’s Nippon Steel was earlier blocked by President Joe Biden.

Trump directed the Committee on Foreign Investment in the United States to review the proposed deal again to assist “in determining whether further action in this matter may be appropriate,” according to a presidential action issued by the White House on Monday.

U.S. Steel shares spiked nearly 12% in reaction to the decision.

Trump gave the committee 45 days to submit a recommendation on whether measures proposed by U.S. Steel and Nippon “are sufficient to mitigate any national security risks.”

This is a developing story. Please check back for updates.

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