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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.
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.
- 'It can be that easy' -
Craig Kaplan, a computer scientist at Canada's Waterloo University, told AFP that it was "an amusing and almost ridiculous story -- but wonderful".
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.
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.
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".
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."
© 2023 AFP
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As of Wednesday afternoon air quality in New York was clocking in as the worst in the world, according to an international monitor, as levels of pollution hung at hazardous levels well into the night.
Even as the apocalyptic skies gave residents the illusion of living in a sepia filter, festival organizers took a wait-and-see approach in consultation with local officials.
"We are a go," they announced by Thursday evening, as the smoke began to dissipate due to the shifting direction of the winds over the Canadian province of Quebec, where the fires are raging.
It was welcome news for thousands of revelers planning to attend the festival in New York's Queens borough, among them Simrya Anand who traveled from Boston to see artists including headliners Lizzo, Odesza and Kendrick Lamar.
"I was really worried about the weather," the 20-year-old told AFP. "I was thinking about, like, wearing a mask here but thankfully it looks like things have cleared up."
"But I wasn't considering not coming," she added with a smile.
Hamza Hussein, 25, just wrapped a graduate degree at New York University and was looking forward to attending his first Gov Ball ever, in particular the set of famed rapper Pusha T.
He and his friend were concerned about the potentially "poisonous" air -- but "we predicted it was going to go on because it's really hard to rebook all these artists."
Breathing easy
By Friday evening as Diplo took the stage, the air quality level had dropped to a refreshing 38 on the 500-point scale -- earlier in the week it had jumped past a dangerous 400 -- and festival-goers along with artists were able to breathe easy once more.
It was a relief for New Yorkers Andy and Bonnie Goolcharan, both in their early 50s, who said they had been ready to skip.
"We weren't going to come," Andy Goolcharan told AFP. "We thought it would be canceled... but it worked out."
And unlike many of their fellow attendees in their 20s, the couple said if both the smog and the festival had persisted, they would have stayed home.
More than 111 million people in the United States had been under air quality alerts as of Thursday due to the fires. The wildfire smoke from Canada was also detected several thousand miles away in Norway.
The mayors of New York, Montreal, Toronto, Washington and Philadelphia issued a joint statement Friday saying "this alarming episode serves as a stark reminder of the harmful impacts that the climate crisis is having on cities around the world.
The three-day Governors Ball music festival is set to continue through Sunday, and along with the headliners will feature Lil Nas X, Haim, Lil Uzi Vert, Rina Sawayama, Omar Apollo and Ice Spice.
© 2023 AFP
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‘We are not going to stand for it’: McCarthy vows to use Jim Jordan’s committee to target the AG
June 10, 2023
The Republican Speaker of the House, Kevin McCarthy, barely hours after the U.S. Dept. of Justice unsealed a 49-page, 37-felony count criminal indictment charging Donald Trump with violations of seven federal laws, decided to double-down on his defense of the ex-president by threatening to target the Attorney General of the United States and declaring House Republicans "are not going to stand for" the criminal prosecution of the ex-president.
McCarthy went on Fox News Friday afternoon, saying "this judgment is wrong by this DOJ. That they treated President Trump differently than they treat others. And it didn't have to be this way. This is going to disrupt this nation because it goes to the core of equal justice for all – which is not being seen today and we are not going to stand for it."
McCarthy, a California Republican who cobbled together a tenuous pact with far-right extremists to win his speakership on the 15th try, is incorrect on the facts.
The Dept. of Justice does not pass judgment, the courts – in this case a jury, does. The Dept. of Justice did not treat Trump "differently," except to give him multiple opportunities over an approximately two-year period to return national secrets he allegedly unlawfully removed, retained, and refused to return, even after being served with a subpoena and a search warrant.
What McCarthy does not do is claim Trump's actions were legal or reasonable, because the damning indictment makes clear they are not.
Later, McCarthy took to Twitter to effectively declare he would target the Attorney General of the United States, Merrick Garland, who – for nearly a quarter century – served as a federal appeals court judge and chief judge before being nominated to serve at Main Justice.
(Garland was nominated to the U.S. Supreme Court in 2016 but then-Senate Majority Leader Mitch McConnell refused to allow the confirmation to move process forward.)
"Many officials, from Secretary Hillary Clinton to then-Senator Joe Biden, handled classified info after their time in office & were never charged," tweeted the Speaker, not just wrongly, but grossly and dishonestly characterizing the allegations against Trump.
"Now Biden's leading political opponent is indicted—a double standard that must be investigated," he again dishonestly declared.
President Joe Biden had nothing to do with the decision of the Special Counsel to ask a Florida grand jury for an indictment. Nor was the President even told before Trump was indicted – like every American, President Biden learned of the Trump indictment through news reports. Attorney General Garland did not sign off on the decision to ask a grand jury for an indictment.
McCarthy, meanwhile, vowed House Judiciary Committee Chairman Jim Jordan and the House Republicans "will get answers."
"Merrick Garland: the American people elected us to conduct oversight of you. We will fulfill that obligation," he declared.
McCarthy made those remarks atop a Friday letter from Jordan to Garland that begins: "The Biden Department of Justice is reportedly about to indict a former president and President Biden's chief rival in the upcoming presidential election."
"According to reports, the Department will indict President Donald Trump, despite declining to indict former Secretary of State Hillary Clinton for her mishandling of classified information and failing to indict President Biden for his mishandling of classified information." (The letter does not mention former Vice President Mike Pence, who is not being charged for his mishandling of classified information."
On Thursday a defiant and angry McCarthy, after Trump was indicted, wrote: "Today is indeed a dark day for the United States of America."
"It is unconscionable for a President to indict the leading candidate opposing him," he said, which is egregiously false – Biden did not indict Trump, nor did his Attorney General or even Special Counsel; a grand jury of Florida citizens did.
"Joe Biden kept classified documents for decades," McCarthy charged, which is a legitimate claim and there is a current federal investigation underway. The difference is Biden did not take the documents, did not know they were among his papers, and immediately upon learning they were, contacted the National Archives to arrange their return.
Donald Trump, we now know, according to the indictment, packed some of the boxes himself, not only refused to return the documents but hid them from the Dept. of Justice and National Archives, lied about them, and kept them at times in public areas of his Florida resort and residence.
"I, and every American who believes in the rule of law," McCarthy wrong declared, "stand with President Trump against this grave injustice. House Republicans will hold this brazen weaponization of power accountable."
In response to McCarthy's remarks, U.S. Rep. Eric Swalwell (D-CA) posted a photo from the DOJ's indictment of Trump.
"These are the secrets that protect our troops. And Kevin McCarthy thinks it’s perfectly OK that Donald Trump stole and stored them like this," he charged.
Watch the video and see Rep. Swalwell's tweet above or at this link.
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