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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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National Guard will begin carrying firearms in Washington DC, official says

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National Guard will begin carrying firearms in Washington DC, official says

National Guard troops deployed to Washington DC in an effort to mitigate crime will begin carrying firearms, an official has said.

Defence secretary Pete Hegseth ordered the authorisation of roughly 2,000 National Guard troops to begin carrying weapons.

The majority of the guard members will carry M17 pistols, their service-issued weapons, while a small number will be armed with M4 rifles, reports Sky’s US partner organisation, NBC News.

The troops are authorised to use their weapons for self-protection.

A White House official told NBC News that despite being armed, as of Saturday night, the National Guard troops in DC are not making arrests, and will continue to work on protecting federal assets.

The troops were largely deployed from outside the state and were framed by President Trump as a concerted effort to tackle crime and homelessness in the nation’s capital.

Such deployments are not common, and are typically used in response natural disasters or civil unrest.

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Democrats have bashed the deployment as partisan in nature, accusing Mr Trump of trying to exert his presidential authority through scare tactics and said his primary targets have been cities with black leadership.

Armed members of the South Carolina National Guard patrol outside of Union Station. Pic: AP
Image:
Armed members of the South Carolina National Guard patrol outside of Union Station. Pic: AP

Pentagon plans to deploy US army to Chicago

Yesterday it was reported that the Pentagon was drafting plans to deploy the US army in Chicago, the largest city in the state.

The governor of Illinois then accused Mr Trump of “attempting to manufacture a crisis” and “abusing his power to distract from the pain he is causing working families”.

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Officials familiar with the proposals told the Washington Post that several options were being weighed up by the US defence department, including mobilising thousands of National Guard troops in Chicago as early as September.

Mr Trump had told reporters on Friday that “Chicago is a mess”, before attacking the city’s mayor, Brandon Johnson, and hinting “we’ll straighten that one out probably next”.

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Flesh-eating screwworm parasite detected in person in US for first time

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Flesh-eating screwworm parasite detected in person in US for first time

A case of the flesh-eating screwworm parasite has been detected in a person in the United States for the first time.

The parasitic flies eat cattle and other warm-blooded animals alive, with an outbreak beginning in Central America and southern Mexico late last year.

It is ultimately fatal if left untreated.

The case in the US was identified in a person from Maryland who had travelled from Guatemala.

Beth Thompson, South Dakota’s state veterinarian, told Reuters on Sunday that she was notified of the case within the
last week.

A Maryland state government official also confirmed the case.

The person was treated and prevention measures were implemented, Reuters reports.

The Centres for Disease Control and Prevention (CDC) and the Maryland Department of Health did not immediately respond to requests for comment.

What is screwworm?

The female screwworm fly lays eggs in the wounds of warm-blooded animals and once hatched, hundreds of screwworm larvae use their sharp mouths to burrow through living flesh.

It can be devastating in cattle and wildlife, and has also been known to infect humans.

Treatment is onerous, and involves removing hundreds of larvae and thoroughly disinfecting wounds. They are largely survivable if treated early enough.

The confirmed case is likely to rattle the beef and cattle futures market, which has seen record-high prices because of tight supplies.

The US typically imports more than a million cattle from Mexico each year to process into beef. The screwworm outbreak could cost Texas – the biggest cattle-producing state – $1.8bn (£1.3bn) in livestock deaths, labour costs and medication
expenses.

A view shows a calf after being sprayed with a disinfectant spray to prevent screwworm. Pic: Reuters
Image:
A view shows a calf after being sprayed with a disinfectant spray to prevent screwworm. Pic: Reuters

The United States Department of Agriculture (USDA) has set traps and sent mounted officers along the border, but it has faced criticism from some cattle producers and market analysts for not acting faster to pursue increased fly production via a sterile fly facility.

What is a sterile fly facility?

The case also comes just one week after the US agriculture secretary, Brooke Rollins, travelled to Texas to announce plans to build a sterile fly facility there in a bid to combat the pest. Ms Rollins had pledged repeatedly to keep screwworm out of the country.

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A sterile fly facility produces a large number of male flies and sterilises them – these males are then released to mate with wild female insects, which collapses the wild population over time. This method eradicated screwworm from the US in the 1960s.

Mexico has also taken efforts to limit the spread of the pest, which can kill livestock within weeks if not treated. It had started to build a $51m sterile fly production facility.

The USDA has previously said 500 million flies would need to be released weekly to push the fly back to the Darien Gap, the stretch of rainforest between Panama and Colombia.

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UK

Urgent letter to home secretary over violence against women and girls strategy – as it omits child abuse

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Urgent letter to home secretary over violence against women and girls strategy - as it omits child abuse

Ten child protection organisations have written an urgent letter to the home secretary expressing concern about the omission of child sexual abuse from the government’s violence against women and girls strategy, following a Sky News report. 

Groups including the NSPCC, Barnardo’s and The Children’s Society wrote to Yvette Cooper to say that violence against women and girls (VAWG) and child sexual abuse are “inherently and deeply connected”, suggesting any “serious strategy” to address VAWG needs to focus on child sexual abuse and exploitation.

The letter comes after Sky News revealed an internal Home Office document, titled Our draft definition of VAWG, which said that child sexual abuse and exploitation is not “explicitly within the scope” of their strategy, due to be published in September.

Poppy Eyre when she was four years old
Image:
Poppy Eyre when she was four years old

Responding to Sky News’ original report, Poppy Eyre, who was sexually abused and raped by her grandfather when she was four, said: “VAWG is – violence against women and girls. If you take child sexual abuse out of it, where are the girls?”

The Centre of Expertise on Child Sexual Abuse, which is funded by the Home Office and a signatory to the letter, estimates 500,000 children in England and Wales are sexually abused every year.

The NSPCC “welcome” the government’s pledge to halve VAWG in a decade, but is “worried that if they are going to fulfil this commitment, the strategy absolutely has to include clear deliverable objectives to combat child sexual abuse and exploitation too”, the head of policy, Anna Edmundson, told Sky News.

Poppy is a survivor of child sexual abuse
Image:
Poppy is a survivor of child sexual abuse

She warned the government “will miss a golden opportunity” and the needs of thousands of girls will be “overlooked” if child sexual abuse and exploitation is not “at the heart of its flagship strategy”.

The government insists the VAWG programme will include action to tackle child sexual abuse, but says it also wants to create a distinctive plan to “ensure those crimes get the specialist response they demand”.

“My message to the government is that if you’re going to make child sexual abuse a separate thing, we need it now,” Poppy told Sky News.

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Rape Crisis, which is one of the largest organisations providing support to women in England and Wales, shares these concerns.

It wants plans to tackle child sexual abuse to be part of the strategy, and not to sit outside it.

“If a violence against women and girls strategy doesn’t include sexual violence towards girls, then it runs the risk of being a strategy for addressing some violence towards some females, but not all,” chief executive Ciara Bergman said.

A Home Office spokesperson said the government is “working tirelessly to tackle the appalling crimes of violence against women and girls and child sexual exploitation and abuse, as part of our Safer Streets mission”.

“We are already investing in new programmes and introducing landmark laws to overhaul the policing and criminal justice response to these crimes, as well as acting on the recommendations of Baroness Casey’s review into group-based Child Sexual Exploitation, and the Independent Inquiry into Child Sexual Abuse,” they added.

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