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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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Business

Sir Keir Starmer pledges to protect UK companies from Trump tariff ‘storm’

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Sir Keir Starmer pledges to protect UK companies from Trump tariff 'storm'

Sir Keir Starmer has said his government stands ready to use industrial policy to “shelter British business from the storm” after Donald Trump’s new 10% tariff kicked in.

The UK was among a number of countries hit with the lowest import duty rate following the president’s announcement on 2 April – which he called ‘Liberation Day’, while other nations, such as Vietnam, Cambodia and China face much higher US levies.

But a global trade war will hurt the UK’s open economy.

The prime minister said “these new times demand a new mentality”, after the 10% tax on British imports into America came into force on Saturday. A 25% US levy on all foreign car imports was introduced on Thursday.

It comes as Jaguar Land Rover announced it would “pause” shipments to the US for a month, as firms grapple with the new taxes.

On Saturday, the car manufacturer said it was working to “address the new trading terms” and was looking to “develop our mid to longer-term plans”.

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Jobs fears as Jaguar halts shipments

Referring to the tariffs, Sir Keir said “the immediate priority is to keep calm and fight for the best deal”.

Writing in The Sunday Telegraph, he said that in the coming days “we will turbocharge plans that will improve our domestic competitiveness”, adding: “We stand ready to use industrial policy to help shelter British business from the storm.”

It is believed a number of announcements could be made soon as ministers look to encourage growth.

NI contribution rate for employers goes up

From Sunday, the rate of employer NICs (national insurance contributions) increased from 13.8% to 15%.

At the same time, firms will also pay more because the government lowered the salary threshold at which companies start paying NICs from £9,100 to £5,000.

Also, the FTSE 100 of leading UK companies had its worst day of trading since the start of the pandemic on Friday, with banks among some of the firms to suffer the sharpest losses.

Sir Keir said: “This week, the government will do everything necessary to protect Britain’s national interest. Because when global economic sands are shifting, our laser focus on delivering for Britain will not. And these new times demand a new mentality.”

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Trump defiant despite markets

UK spared highest tariff rates

Some of the highest rates have been applied to “worst offender” countries including some in Southeast Asia. Imports from Cambodia will be subject to a 49% tariff, while those from Vietnam will face a 46% rate. Chinese goods will be hit with a 34% tariff.

Imports from France will have a 20% tariff, the rate which has been set for European Union nations. These will come into effect on 9 April.

Read more:
Red wall on Wall Street – but Trump undeterred
How will UK respond to Trump’s tariffs?

Sir Keir has been speaking to foreign leaders on the phone over the weekend, including French President Emmanuel Macron, Italian Prime Minister Giorgia Meloni and Australian Prime Minister Anthony Albanese, to discuss the tariff changes.

A Downing Street spokesperson said of the conversation between Sir Keir and Mr Macron: “They agreed that a trade war was in nobody’s interests but nothing should be off the table and that it was important to keep business updated on developments.

“The prime minister and president also shared their concerns about the global economic and security impact, particularly in Southeast Asia.”

👉 Follow Trump 100 on your podcast app 👈

Trump’s warning

Mr Trump has warned Americans the tariffs “won’t be easy”, but urged them to “hang tough”.

In a post on his Truth Social platform, he said: “We are bringing back jobs and businesses like never before.

“Already, more than FIVE TRILLION DOLLARS OF INVESTMENT, and rising fast!

“THIS IS AN ECONOMIC REVOLUTION, AND WE WILL WIN. HANG TOUGH, it won’t be easy, but the end result will be historic.”

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Politics

Sir Keir Starmer pledges to protect UK companies from Trump tariff ‘storm’

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Sir Keir Starmer pledges to protect UK companies from Trump tariff 'storm'

Sir Keir Starmer has said his government stands ready to use industrial policy to “shelter British business from the storm” after Donald Trump’s new 10% tariff kicked in.

The UK was among a number of countries hit with the lowest import duty rate following the president’s announcement on 2 April – which he called ‘Liberation Day’, while other nations, such as Vietnam, Cambodia and China face much higher US levies.

But a global trade war will hurt the UK’s open economy.

The prime minister said “these new times demand a new mentality”, after the 10% tax on British imports into America came into force on Saturday. A 25% US levy on all foreign car imports was introduced on Thursday.

It comes as Jaguar Land Rover announced it would “pause” shipments to the US for a month, as firms grapple with the new taxes.

On Saturday, the car manufacturer said it was working to “address the new trading terms” and was looking to “develop our mid to longer-term plans”.

Please use Chrome browser for a more accessible video player

Jobs fears as Jaguar halts shipments

Referring to the tariffs, Sir Keir said “the immediate priority is to keep calm and fight for the best deal”.

Writing in The Sunday Telegraph, he said that in the coming days “we will turbocharge plans that will improve our domestic competitiveness”, adding: “We stand ready to use industrial policy to help shelter British business from the storm.”

It is believed a number of announcements could be made soon as ministers look to encourage growth.

NI contribution rate for employers goes up

From Sunday, the rate of employer NICs (national insurance contributions) increased from 13.8% to 15%.

At the same time, firms will also pay more because the government lowered the salary threshold at which companies start paying NICs from £9,100 to £5,000.

Also, the FTSE 100 of leading UK companies had its worst day of trading since the start of the pandemic on Friday, with banks among some of the firms to suffer the sharpest losses.

Sir Keir said: “This week, the government will do everything necessary to protect Britain’s national interest. Because when global economic sands are shifting, our laser focus on delivering for Britain will not. And these new times demand a new mentality.”

Please use Chrome browser for a more accessible video player

Trump defiant despite markets

UK spared highest tariff rates

Some of the highest rates have been applied to “worst offender” countries including some in Southeast Asia. Imports from Cambodia will be subject to a 49% tariff, while those from Vietnam will face a 46% rate. Chinese goods will be hit with a 34% tariff.

Imports from France will have a 20% tariff, the rate which has been set for European Union nations. These will come into effect on 9 April.

Read more:
Red wall on Wall Street – but Trump undeterred
How will UK respond to Trump’s tariffs?

Sir Keir has been speaking to foreign leaders on the phone over the weekend, including French President Emmanuel Macron, Italian Prime Minister Giorgia Meloni and Australian Prime Minister Anthony Albanese, to discuss the tariff changes.

A Downing Street spokesperson said of the conversation between Sir Keir and Mr Macron: “They agreed that a trade war was in nobody’s interests but nothing should be off the table and that it was important to keep business updated on developments.

“The prime minister and president also shared their concerns about the global economic and security impact, particularly in Southeast Asia.”

👉 Follow Trump 100 on your podcast app 👈

Trump’s warning

Mr Trump has warned Americans the tariffs “won’t be easy”, but urged them to “hang tough”.

In a post on his Truth Social platform, he said: “We are bringing back jobs and businesses like never before.

“Already, more than FIVE TRILLION DOLLARS OF INVESTMENT, and rising fast!

“THIS IS AN ECONOMIC REVOLUTION, AND WE WILL WIN. HANG TOUGH, it won’t be easy, but the end result will be historic.”

Continue Reading

World

IDF says it mistakenly identified Gaza aid workers as threat – after video of deadly attack emerges

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IDF says it mistakenly identified Gaza aid workers as threat - after video of deadly attack emerges

The IDF says it mistakenly identified a convoy of aid workers as a threat – following the emergence of a video which proved their ambulances were clearly marked when Israeli troops opened fire on them.

The bodies of 15 aid workers – including eight medics working for the Palestine Red Crescent Society (PRCS) – were found in a “mass grave” after the incident, according to the head of the UN Office for Coordination of Humanitarian Affairs Jonathan Whittall.

The Israeli military originally claimed an investigation found the vehicles did not have any headlights or emergency signals and were therefore targeted as they looked “suspicious”.

But video footage obtained by the PRCS, and verified by Sky News, showed the ambulances and a fire vehicle clearly marked with flashing red lights.

In a briefing from the IDF, it said the ambulances arrived in the Tel Sultan neighbourhood in Rafah shortly after a Hamas police vehicle drove through.

Palestinians mourn medics, who came under Israeli fire while on a rescue mission, after their bodies were recovered, according to the Red Crescent, at Nasser hospital in Khan Younis in the southern Gaza Strip March 31, 2025. REUTERS/Hatem Khaled/File Photo
Image:
Palestinians mourning the medics after their bodies were recovered. Pic: Reuters

An IDF surveillance aircraft was watching the movement of the ambulances and notified troops on the ground. The IDF said it will not be releasing that footage.

When the ambulances arrived, the soldiers opened fire, thinking the medics were a threat, according to the IDF.

The soldiers were surprised by the convoy stopping on the road and several people getting out quickly and running, the IDF claimed, adding the soldiers were unaware the suspects were in fact unarmed medics.

An Israeli military official would not say how far away troops were when they fired on the vehicles.

The IDF acknowledged that its statement claiming that the ambulances had their lights off was incorrect, and was based on the testimony from the soldiers in the incident.

The newly emerged video footage showed that the ambulances were clearly identifiable and had their lights on, the IDF said.

The IDF added that there will be a re-investigation to look into this discrepancy.

Analysis: Video undermines Israel’s account of aid worker deaths

The clip is filmed through a vehicle windscreen - with three red light vehicles visible in front
Image:
The clip is filmed through a vehicle windscreen – with three red light vehicles visible in front

Addressing the fact the aid workers’ bodies were buried in a mass grave, the IDF said in its briefing this is an approved and regular practice to prevent wild dogs and other animals from eating the corpses.

The IDF could not explain why the ambulances were also buried.

The IDF said six of the 15 people killed were linked to Hamas, but revealed no detail to support the claim.

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Bodies of aid workers found in Gaza

The newly emerged footage of the incident was discovered on a phone belonging to one of the workers who was killed, PRCS president Dr Younis Al Khatib said.

“His phone was found with his body and he recorded the whole event,” he said. “His last words before being shot, ‘Forgive me, mom. I just wanted to help people. I wanted to save lives’.”

Sky News used an aftermath video and satellite imagery to verify the location and timing of the newly emerged footage of the incident.

More from Sky News:
Israeli troops expand Gaza ‘security zone’
What happened to the ceasefire?

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Aid worker attacks increasing

It was filmed on 23 March north of Rafah and shows a convoy of marked ambulances and a fire-fighting vehicle travelling south along a road towards the city centre. All the vehicles visible in the convoy have their flashing lights on.

The footage was filmed early in the morning, with a satellite image seen by Sky News taken at 9.48am local time on the same day showing a group of vehicles bunched together off the road.

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