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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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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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Coca-Cola brews up sale of high street coffee giant Costa

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Coca-Cola brews up sale of high street coffee giant Costa

The Coca-Cola Company is brewing up a sale of Costa, Britain’s biggest high street coffee chain, more than six years after acquiring the business in a move aimed at helping it reduce its reliance on sugary soft drinks.

Sky News can exclusively reveal that Coca-Cola is working with bankers to hold exploratory talks about a sale of Costa.

Initial talks have already been held with a small number of potential bidders, including private equity firms, City sources said on Saturday.

Lazard, the investment bank, is understood to have been engaged by Coca-Cola to review options for the business and gauge interest from prospective buyers.

Indicative offers are said to be due in the early part of the autumn, although one source cautioned that Coca-Cola could yet decide not to proceed with a sale.

Costa trades from more than 2,000 stores in the UK, and well over 3,000 globally, according to the latest available figures.

It has been reported to have a global workforce numbering 35,000, although Coca-Cola did not respond to several attempts to establish the precise number of outlets currently in operation, or its employee numbers.

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This weekend, analysts said that a sale could crystallise a multibillion pound loss on the £3.9bn sum Coca-Cola agreed to pay to buy Costa from Whitbread, the London-listed owner of the Premier Inn hotel chain, in 2018.

One suggested that Costa might now command a price tag of just £2bn in a sale process.

The disposal proceeds would, in any case, not be material to the Atlanta-based company, which had a market capitalisation at Friday’s closing share price of $304.2bn (£224.9bn).

At the time of the acquisition, Coca-Cola’s chief executive, James Quincey, said: “Costa gives Coca-Cola new capabilities and expertise in coffee, and our system can create opportunities to grow the Costa brand worldwide.

“Hot beverages is one of the few segments of the total beverage landscape where Coca-Cola does not have a global brand.

“Costa gives us access to this market with a strong coffee platform.”

However, accounts filed at Companies House for Costa show that in 2023 – the last year for which standalone results are available – the coffee chain recorded revenues of £1.22bn.

While this represented a 9% increase on the previous year, it was below the £1.3bn recorded in 2018, the final year before Coca-Cola took control of the business.

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Coca-Cola has been grappling with the weak performance of Costa for some time, with Mr Quincey saying on an earnings call last month: “We’re in the mode of reflecting on what we’ve learned, thinking about how we might want to find new avenues to grow in the coffee category while continuing to run the Costa business successfully.”

“It’s still a lot of money we put down, and we wanted that money to work as hard as possible.”

Costa’s 2022 accounts referred to the financial pressures it faced from “the economic environment and inflationary pressures”, resulting in it launching “a restructuring programme to address the scale of overheads and invest for growth”.

Filings show that despite its lacklustre performance, Costa has paid more than £250m in dividends to its owner since the acquisition.

The deal was intended to provide Coca-Cola with a global platform in a growing area of the beverages market.

Costa trades in dozens of countries, including India, Japan, Mexico and Poland, and operates a network of thousands of coffee vending machines internationally under the Costa Express brand.

The chain was founded in 1971 by Italian brothers Sergio and Bruno Costa.

It was sold to Whitbread for £19m in 1995, when it traded from fewer than 40 stores.

The business is now one of Britain’s biggest private sector employers, and has become a ubiquitous presence on high streets across the country.

Its main rivals include Starbucks, Caffe Nero and Pret a Manger – the last of which is being prepared for a stake sale and possible public market flotation.

It has also faced growing competition from more upmarket chains such as Gail’s, the bakeries group, which has also been exploring a sale.

Coca-Cola communications executives in the US and UK did not respond to a series of emails and calls from Sky News seeking comment on its plans for Costa.

A Lazard spokesperson declined to comment.

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Israel is accused of allowing famine to fester in Gaza and global condemnation is deafening

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Israel is accused of allowing famine to fester in Gaza and global condemnation is deafening

Tom Fletcher, speaking on behalf of the United Nations, did not mince his words.

Gaza was suffering from famine, the evidence was irrefutable and Israel had not just obstructed aid but had also used hunger as a weapon of war.

His anger seeped through every sentence, just as desperation is laced through the report from the Integrated Food Security Phase Classification (IPC).

Gaza latest: UK calls out Israel for ‘manmade catastrophe’

Conditions are expected to worsen, it says, even though the Gaza Strip has been classified as a level 5 famine. There is no level 6.

A child attempts to access food from a charity kitchen in Khan Younis. Pic: Reuters
Image:
A child attempts to access food from a charity kitchen in Khan Younis. Pic: Reuters

But it took only moments for the Israeli government to respond in terms that were just as strident. The report dismissed as wholly inaccurate, based on biased, inaccurate data and influenced not by fact, but by the whims of Hamas.

COGAT, the Israeli agency that oversees humanitarian efforts in Gaza, claimed the IPC had ignored its data and presented a “one-sided report”, before claiming that “hundreds of truckloads of aid are still awaiting collection by the UN and international organisations”.

What is so striking is that there is no grey area between these two versions.

In one, Israel has obstructed the delivery of aid and allowed hunger to turn into famine; in the other, it is Hamas that has caused the crisis by stealing aid and exploiting hunger as a political tool to try to win global sympathy.

People in Beit Lahia take sacks of flour from an aid convoy en route to Gaza City. Pic: AP
Image:
People in Beit Lahia take sacks of flour from an aid convoy en route to Gaza City. Pic: AP

Journalists are not allowed to enter Gaza, so we are reliant on the work of colleagues who live there.

But the images are striking – emaciated people holding begging bowls, people scrambling towards aid drops or clambering over trucks carrying bags of flour. And all around them, shattered buildings.

Aid is continuing to be dropped by air, but humanitarian groups say it is not enough. Pic: Reuters
Image:
Aid is continuing to be dropped by air, but humanitarian groups say it is not enough. Pic: Reuters

We heard from a man in his 70s, who used to weigh 70kg, but who has lost almost half his body weight.

“Now, because of malnutrition, my weight has dropped to just 40,” Hassan Abu Seble said. “I suffered both a stroke and a heart attack. They had to put in a stent to help me recover, and I thank God that my organs are still functioning.”

The Israeli government, and many across the country, will maintain that Hamas bears the responsibility for everything that has happened to Gazans – that it was the attack on 7 October, 2023, that was the sole precipitant for the suffering, death and hunger that has followed.

But from around much of the rest of the world, the condemnation is deafening, accusing Israel of allowing famine to fester.

The body of a child is carried from the scene of an Israeli military strike in Gaza City. Pic: AP
Image:
The body of a child is carried from the scene of an Israeli military strike in Gaza City. Pic: AP

David Lammy, Britain’s foreign secretary, said the Israeli government had caused a “man-made famine” by blocking the distribution of aid, and described that as a “moral outrage”.

The question, as so often before, is what that rhetoric leads to. And, so long as the United States doesn’t join the chorus of disapproval, does widespread global disapproval mean anything?

There is also a question now of Gaza’s future.

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In the Jewish quarter of Jerusalem’s Old City, we found a large sign that says “Make Gaza Jewish Again”. It is a slogan, and a sentiment, that is supported by plenty.

“Yes, of course I agree,” says one man as he walks past, carrying a large pack of drinks. It turns out that he used to live in a Jewish settlement in Gaza until it was shut by the Israeli government two decades ago, but he has never stopped believing that Gaza is rightly Israel’s property.

“The people there now – they should leave. They could go to Jordan, Lebanon, Egypt. It is our land. And yes, I would like to go back there.”

He did not believe there was a famine. “They have lots of food,” he told me.

Another man, Avraham, was more conciliatory, but insisted there had never been a country like Israel “that is fighting a war against a country but is also sending in so much humanitarian aid for the people”.

Gaza City is now the focal point of so much. Famine is spreading from this heart just as troops prepare to encircle the city. A ceasefire could come, but so could a huge military assault. And all the while, the hunger will get worse.

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Plans for huge new Chinese embassy delayed by government

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Plans for huge new Chinese embassy delayed by government

Approval of a huge new Chinese embassy in London has been delayed by the government over redacted areas on the embassy’s plans.

Beijing hasn’t fully explained why there are blacked-out areas in its planning application after housing minister Angela Rayner demanded an explanation earlier this month.

The government has now delayed its decision over whether construction can go ahead from 9 September to 21 October, saying it needed more time to consider the application.

The Chinese embassy in London expressed “serious concern” over the delay and said host countries have an “international obligation” to support the construction of diplomatic buildings.

“The Chinese side urges the UK side to fulfil its obligation and approve the planning application without delay,” said the embassy in a statement.

Site of planned Chinese embassy
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Site of planned Chinese embassy

Royal Mint Court, the site of the proposed embassy. File pic: PA
Image:
Royal Mint Court, the site of the proposed embassy. File pic: PA

DP9, the planning consultancy working for the Chinese government, said its client felt it would be inappropriate to provide full internal layout plans.

It added that additional drawings provided an acceptable level of detail, after the government asked why several areas were blacked out.

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Protests have been held outside the proposed site. File pic: Feb 2025, PA
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Protests have been held outside the proposed site. File pic: Feb 2025, PA

“The Applicant considers the level of detail shown on the unredacted plans is sufficient to identify the main uses,” said DP9 in a letter to the government.

“In these circumstances, we consider it is neither necessary nor appropriate to provide additional more detailed internal layout plans or details.”

The embassy, which would be the largest in Europe, is planned for the 216-year-old site of the old Royal Mint Court next to the Tower of London.

However, opposition from local residents, lawmakers and pro-democracy campaigners means planning approval has been delayed for the past three years.

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Earlier this month, the embassy described claims that the building could have “secret facilities” used to harm Britain’s
national security as “despicable slandering”.

However, the executive director of the Inter-Parliamentary Alliance on China, which has ties to a network of politicians critical of the country, called the explanations “far from satisfactory”.

Luke de Pulford, who is a long-standing critic of the embassy plans, said the “assurances amount to ‘trust me bro'”.

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