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UK hobbyist stuns maths world with ‘amazing’ new shapes

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UK hobbyist stuns maths world with 'amazing' new shapes

David Smith, a retired print technician from the north of England, was pursuing his hobby of looking for interesting shapes when he stumbled onto one unlike any other in November. 

When Smith shared his shape with the world in March, excited fans printed it onto T-shirts, sewed it into quilts, crafted cookie cutters or used it to replace the hexagons on a soccer ball — some even made plans for tattoos.

The 13-sided polygon, which 64-year-old Smith called “the hat”, is the first single shape ever found that can completely cover an infinitely large flat surface without ever repeating the same pattern.

That makes it the first “einstein” — named after the German for “one stone” (ein stein), not the famed physicist — and solves a problem posed 60 years ago that some mathematicians had thought impossible.

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After stunning the mathematics world, Smith — a hobbyist with no training who told AFP that he wasn’t great at maths at school — then did it again.

While all agreed “the hat” was the first einstein, its mirror image was required one in seven times to ensure that a pattern never repeated.

But in a preprint study published online late last month, Smith and the three mathematicians who helped him confirm the discovery revealed a new shape — “the spectre.”

It requires no mirror image, making it an even purer einstein.

Craig Kaplan, a computer scientist at Canada’s Waterloo University, told AFP that it was “an amusing and almost ridiculous story — but wonderful”.

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He said that Smith, a retired print technician who lives in Yorkshire’s East Riding, emailed him “out of the blue” in November.

Smith had found something “which did not play by his normal expectations for how shapes behave”, Kaplan said.

If you slotted a bunch of these cardboard shapes together on a table, you could keep building outwards without them ever settling into a regular pattern.

Using computer programs, Kaplan and two other mathematicians showed that the shape continued to do this across an infinite plane, making it the first einstein, or “aperiodic monotile”.

When they published their first preprint in March, among those inspired was Yoshiaki Araki. The Japanese tiling enthusiast made art using the hat and another aperiodic shape created by the team called “the turtle”, sometimes using flipped versions.

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Smith was inspired back, and started playing around with ways to avoid needing to flip his hat.

Less than a week after their first paper came out, Smith emailed Kaplan a new shape.

Kaplan refused to believe it at first. “There’s no way it can be that easy,” he said.

But analysis confirmed that Tile (1,1) was a “non-reflective einstein”, Kaplan said.

Something still bugged them — while this tile could go on forever without repeating a pattern, this required an “artificial prohibition” against using a flipped shape, he said.

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So they added little notches or curves to the edges, ensuring that only the non-flipped version could be used, creating “the spectre”.

‘Hatfest’
Kaplan said both their papers had been submitted to peer-reviewed journals. But the world of mathematics did not wait to express its astonishment.

Marjorie Senechal, a mathematician at Smith College in the United States, told AFP the discoveries were “exciting, surprising and amazing”.

She said she expects the spectre and its relatives “will lead to a deeper understanding of order in nature and the nature of order.”

Doris Schattschneider, a mathematician at Moravian College in the US, said both shapes were “stunning”.

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Even Nobel-winning mathematician Roger Penrose, whose previous best effort had narrowed the number of aperiodic tiles down to two in the 1970s, had not been sure such a thing was possible, Schattschneider said.

Penrose, 91, will be among those celebrating the new shapes during the two-day “Hatfest” event at Oxford University next month.

All involved expressed amazement that the breakthrough was achieved by someone without training in maths.

“The answer fell out of the sky and into the hands of an amateur — and I mean that in the best possible way, a lover of the subject who explores it outside of professional practice,” Kaplan said.

“This is the kind of thing that ought not to happen, but very happily for the history of science does happen occasionally, where a flash brings us the answer all at once.”

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Estonia summons Russian embassy chief over GPS jamming

Estonia summons Russian embassy chief over GPS jamming

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Estonia summons Russian embassy chief over GPS jamming

Estonia’s foreign ministry has summoned the head of the Russian embassy in Tallinn to protest the jamming of GPS signals, the Baltic country said on Wednesday.

Estonia accuses neighbouring Russia of violating international regulations by disturbing GPS navigation in airspace above the Baltic states, echoing concerns from airlines that say they have contended with such interference for months.

The Estonian foreign ministry on Wednesday summoned Russia’s charge d’affaires to convey its message. Moscow and Tallinn last year expelled each other’s ambassadors as relations deteriorated in the wake of Moscow’s full-scale invasion of Ukraine in 2022.

“Jamming the GPS signal is an element of Russia’s hybrid activities, disrupting everyday life and threatening the security of allies,” Estonian Foreign Minister Margus Tsahkna said in a statement.

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Russia’s foreign ministry did not immediately respond to a request for comment.

GPS, short for Global Positioning System, is a network of satellites and receiving devices used for positioning, navigation and timing on Earth in everything from ships and planes to cars.

The jamming of GPS signals can be disruptive to commercial airliners but they can usually navigate by other means.

Finnish carrier Finnair last month said it would pause all flights to Tartu in Estonia to allow the local airport to upgrade its navigation system, which has so far relied only on GPS.

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Australia boosts funds to Taiwan-ally Tuvalu amid Pacific contest for influence with China

Australia boosts funds to Taiwan-ally Tuvalu amid Pacific contest for influence with China

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Australia boosts funds to Taiwan-ally Tuvalu amid Pacific contest for influence with China

Australia will quadruple its financial assistance to Tuvalu, a Pacific Island nation at risk from rising seas, to cement a landmark climate migration and security deal as China also courts small island states.

On a visit to Tuvalu, foreign minister Penny Wong said on Wednesday evening Australia has committed A$110 million ($72.27 million) in its national budget to Tuvalu.

The sum includes A$50 million ($32.85 million) to build the first undersea cable connecting the island’s residents to global telecommunications services, and A$19 million for a land reclamation project to fortify Tuvalu’s coastline from rising seas.

Another A$15 million will be spent on a national security coordination centre, as well as A$10 million in direct budget support.

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The funding is a significant boost on the A$17 million ($11.17 million) Australia provided to Tuvalu in 2023-24.

China’s ambitions for a greater security presence in the Pacific became an election issue in Tuvalu in January, as two leadership contenders said Tuvalu should consider switching ties from Taiwan to Beijing for more funds, and revise a new security pact with Australia.

Neighbour Nauru cut ties with Taiwan a fortnight before Tuvalu’s vote, after China built a port and promised more aid.

Tuvalu’s new prime minister Feleti Teo pledged to stick with Taiwan, and to ratify the Falepili Union signed with Australia in November. The treaty allows Tuvalu citizens to migrate to Australia for work or study, while recognising Tuvalu continues to exist despite the rising sea levels.

“Australia has provided a security guarantee to support Tuvalu in a humanitarian disaster, a pandemic or the event of attack,” Wong said in a speech on Wednesday evening.

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“It is also the first time in history that two nations have agreed in a legally binding instrument that statehood endures in the face of sea level rise,” she added.

The treaty allows Australia to vet Tuvalu’s deals with third countries in a broad range of security areas from ports to telecommunications.

On Thursday, Wong and Teo are expected to say that the security cooperation does not limit Tuvalu’s ability to enter into diplomatic agreements with other nations, according to an advance copy of a joint statement viewed by Reuters.

“We recognise that the people of Tuvalu deserve the choice to live, study and work elsewhere, as climate change impacts intensify at home,” the text of the statement reads.

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Probe launched after Boeing cargo plane lands in Istanbul without front landing gear

Probe launched after Boeing cargo plane lands in Istanbul without front landing gear

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Probe launched after Boeing cargo plane lands in Istanbul without front landing gear

A FedEx Airlines Boeing cargo plane landed at Istanbul Airport on Wednesday without its front landing gear, a Turkish Transport Ministry official said, adding there were no casualties and authorities had launched an investigation.

The aircraft, flying from Paris Charles de Gaulle Airport, informed the control tower in Istanbul that its landing gear had failed to open and touched down with guidance from the tower, managing to remain on the runway, a ministry statement said.

Airport rescue and fire fighting teams were scrambled before landing, but no one was injured. The ministry gave no reason for the failure.

The aircraft involved is a nearly 10-year-old Boeing 767 freighter, one of the most common cargo planes and based on the 767 passenger model dating back to the 1980s.

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An official from Turkey’s transport ministry said its teams were conducting examinations at the scene as part of the ongoing investigation, but did not provide further details.

Boeing referred queries to FedEX, which said in a statement it was coordinating with investigation authorities and would “provide additional information as it is available”.

Video footage obtained by Reuters showed sparks flying and some smoke as the front of the plane scraped along the runway before coming to a halt and being doused with firefighting foam. No fire appeared to have broken out.

The video showed the pilots holding the plane’s nose above the runway for several seconds after the main wheels had touched the ground, apparently executing the emergency drill for landing with a retracted nose gear that pilots train for, according to the SKYbrary aviation database.

In June last year, a small 22-year-old Boeing 717 flown by Delta Airlines made a similar smooth landing without a nose gear in Charlotte, North Carolina, in an incident later blamed on a fractured component.

The runway was temporarily closed to air traffic, but other runways at Istanbul airport were still operating normally, the airport operator IGA said.

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Manufacturers are not typically involved in the operation or maintenance of jets once they enter service, but Boeing has been under intense media and regulatory scrutiny following a series of incidents on its smaller 737.

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