Mathematicians end decades-long quest to find elusive ‘vampire einstein’ shape

Published

2 years ago

June 7, 2023

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What has 14 sides, is full of curves, and can perfectly cover a surface with no gaps or overlaps? It’s not a riddle — it’s a “vampire einstein.”

In March, a retired printing technician named David Smith stumbled upon a remarkable discovery in the world of mathematics. He found a 13-sided shape that could completely tile a surface without ever repeating. The shape, nicknamed “the hat” for its vaguely fedora-like shape was the culmination of decades of hunting by mathematicians around the world.

Since 1961 mathematicians had wondered if such a shape could exist. At first, mathematicians found a set of 20,426 shapes that could tile together while creating a pattern that never repeats (in contrast to the tiles on a kitchen floor, which do create a repeating pattern). Eventually, mathematicians found a set of 104 shapes that could create such a never-repeating tiling.

The middle and right shapes are examples of ‘Spectres’ — 14-sided shapes that can be tiled infinitely without ever creating a repeating pattern. (Image credit: Smith et al.)

Then in the 1970’s physicist and Nobel prize winner Roger Penrose found a pair of shapes that together created anon-repeating tiling. And for decades since, mathematicians continued to wonder if the same trick could be done with only a single shape. That semi-mythical shape, known formally as an aperiodic monotile, came to be known as “the einstein,” which means “one stone” in German.

But for all the celebration around Smith’s discovery of an einstein tile, there was one small fly in the ointment. In order to create the non-repeating tiling, the “hat” had to work with its mirror image. Technically it’s the same shape, just flipped, but some argued that Smith hadn’t really found a true einstein.

Now, however, Smith and his colleagues have laid those objections to rest: they’ve found a shape that can tile a surface without repeating or being flipped. They described the new shape May 28 in a paper published to the preprint database arXiv, though it has not yet been peer reviewed.related stories—Mathematicians make rare breakthrough on notoriously tricky ‘Ramsey problem’

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The team named their shape the “Spectre,” an homage to vampires that can’t see their own reflections and thus don’t need a mirror.

“In plane tiling, it is completely standard that tiles may be reflected; nevertheless, some people were dissatisfied that the aperiodic hat monotile requires reflections to tile the plane,” co-author Joseph Samuel Meyers wrote on Mastodon. “In our new preprint, we present the Spectre, the first example of a vampire einstein: an aperiodic monotile that tiles the plane without reflections.”

To find the ghostly shape, the team started with the original “hat” shape and added an extra side to it. That new shape still required its mirror image to fully tile, but the researchers discovered that by transforming the 14-sided shape’s straight edges into curved ones, they could dispense with mirror images and work with just the one shape.

Technology

Amazon CEO Jassy says AI will lead to ‘fewer people doing some of the jobs’ that get automated

Published

28 mins ago

July 1, 2025

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Amazon CEO Jassy says AI will lead to 'fewer people doing some of the jobs' that get automated

AI will change the workforce, says Amazon CEO Andy Jassy

Amazon CEO Andy Jassy said the rapid rollout of generative artificial intelligence means the company will one day require fewer employees to do some of the work that computers can handle.

“Like with every technical transformation, there will be fewer people doing some of the jobs that the technology actually starts to automate,” Jassy told CNBC’s Jim Cramer in an interview on Monday. “But there’s going to be other jobs.”

Even as AI eliminates the need for some roles, Amazon will continue to hire more employees in AI, robotics and elsewhere, Jassy said.

Earlier this month, Jassy admitted that he expects the company’s workforce to decline in the next few years as Amazon embraces generative AI and AI-powered software agents. He told staffers in a memo that it will be “hard to know exactly where this nets out over time” but that the corporate workforce will shrink as Amazon wrings more efficiencies out of the technology.

It’s a message that’s making its way across the tech sector. Salesforce CEO Marc Benioff last week claimed AI is doing 30% to 50% of the work at his software vendor. Other companies such as Shopify and Microsoft have urged employees to adopt the technology in their daily work. The CEO of Klarna said in May that the online lender has managed to shrink its headcount by about 40%, in part due to investments in AI and natural attrition in its workforce.

Jassy said on Monday that AI will free employees from “rote work” and “make all our jobs more interesting,” while enabling staffers to invent better services more quickly than before.

Amazon and other tech companies have also been shrinking their workforces through rolling layoffs over the past several years. Amazon has cut more than 27,000 jobs since the start of 2022, and it’s announced smaller, more targeted layoffs in its retail and devices units in recent months.

Amazon shares are flat so far this year, underperforming the Nasdaq, which has gained 5.5%. The stock is about 10% below its record reached in February, while fellow megacaps Meta, Microsoft and Nvidia are all trading at or very near record highs.

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Politics

PM faces threat of major rebellion during key vote today

Published

28 mins ago

July 1, 2025

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PM faces threat of major rebellion during key vote today

Sir Keir Starmer continues to face the threat of a major rebellion during a key vote on welfare reforms later – despite making last-minute concessions to disgruntled Labour MPs.

Work and Pensions Secretary Liz Kendall has confirmed that all existing claimants of the personal independence payment (PIP), the main disability benefit, will be protected from changes to eligibility.

The combined value of the standard Universal Credit allowance and the health top-up will rise “at least in line with inflation” every year of this parliament.

And an additional £300m for employment support for sick and disabled people in 2026 has been announced, which will rise every year after.

Ms Kendall has also promised that a consultation into PIP – “co-produced” with disabled people – will be published next autumn.

She said the U-turn on welfare cuts will cost taxpayers about £2.5bn by 2030 – less than half the £4.8bn the government had expected to save with its initial proposals.

Modelling by Ms Kendall’s own department, released yesterday, suggested the proposals would push 150,000 more people into poverty by 2030, down from the 250,000 estimated under the original plan.

But after announcing the U-turns, Labour MPs were still publicly saying they could not back the plans as they do not go far enough to allay their concerns.

Disabilities minister Stephen Timms would not say he was “confident” the proposals would pass the Commons when asked on Sky News’ Politics Hub with Sophy Ridge.

“We’ve got a very strong package, I certainly hope it passes,” he replied.

A total of 86 charities united yesterday to call on MPs to reject the reforms, saying they will harm disabled people and calling it “a political choice”.

The likes of Oxfam, Child Action Poverty Group, Mind and Shelter said the bill has been brought to a vote without consulting disabled people and without any assessment “of its impact on health and employment outcomes”.

When asked to name “a single” disability organisation in favour of the reforms, Ms Kendall declined to do so.

Several Labour MPs indicated they would still vote against the changes, leaving the government in the dark over how big a rebellion it still may face.

Ms Kendall tried to allay their fears, telling MPs: “I believe we have a fair package, a package that protects existing claimants because they’ve come to rely on that support.”

Richard Burgon presented a petition to parliament yesterday evening against the cuts, signed by more than 77,000 people.

Several Labour MPs questioned why the vote was going ahead before the review into PIP is published – including Rachael Maskell, who said she could not “countenance sick and disabled people being denied support” and added: “It is a matter of conscience.”

Connor Naismith said the concessions “undoubtedly improve efforts to secure welfare reform which is fair”, but added: “Unfortunately, I do not believe these concessions yet go far enough.”

Nadia Whittome accused the government of “ignoring” disabled people and urged ministers to go “back to the drawing board”.

Ian Byrne told the Commons he will vote against the “cruel cuts” to disability benefits because the “so-called concessions go nowhere near far enough”.

The vote will take place this evening, with coverage on Sky News’ Politics Hub live blog and on TV.

World

Benjamin Netanyahu to meet Donald Trump next week amid calls for Gaza ceasefire

Published

28 mins ago

July 1, 2025

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Benjamin Netanyahu to meet Donald Trump next week amid calls for Gaza ceasefire

Israeli Prime Minister Benjamin Netanyahu will be meeting Donald Trump next Monday, according to US officials.

The visit on 7 July comes after Mr Trump suggested it was possible a ceasefire in Gaza could be reached within a week.

On Sunday, he wrote on social media: “MAKE THE DEAL IN GAZA. GET THE HOSTAGES BACK!!!”

At least 60 people killed across Gaza on Monday, in what turned out to be some of the heaviest attacks in weeks.

According to the Hamas-run health ministry, 56,500 people have been killed in the 20-month war.

The visit by Mr Netanyahu to Washington has not been formally announced and the officials who said it would be going ahead spoke on condition of anonymity.

An Israeli official in Washington also confirmed the meeting next Monday.

White House Press Secretary Karoline Leavitt said the administration was in constant communication with the Israeli government.

She said Mr Trump viewed ending the war in Gaza and returning remaining hostages held by Hamas as a top priority.

The war in Gaza broke out in retaliation for Hamas’ 7 October 2023 attacks on southern Israel that killed 1,200 people and saw a further 250 taken hostage.

An eight-week ceasefire was reached in the final days of Joe Biden’s US presidency, but Israel resumed the war in March after trying to get Hamas to accept new terms on next steps.

Talks between Israel and Hamas have stalled over whether the war should end as part of any ceasefire.