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 They described the new shape May 28 in a paper published to the preprint database arXiv, though it has not yet been peer reviewed.related stories—Mathematicians make rare breakthrough on notoriously tricky ‘Ramsey problem’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.
EastEnders actor Shane Ritchie and comedian Paul Chuckle were among the stars at singer Linda Nolan’s funeral in Blackpool on Saturday.
Nolan died at the age of 65 last month having been diagnosed with breast cancer in 2005.
Her pink, sparkly coffin was carried into St Paul’s Church in her home city by her sons and others – while her sisters wore pink breast cancer broaches and earrings – opting for pink handbags to contrast with their black outfits.
Sisters Linda, Coleen, Bernie, Maureen, Anne, and Denise formed The Nolans in the 1970s and had hits including I’m In The Mood For Dancing and Gotta Pull Myself Together.
Linda Nolan also had a career in musical theatre, starring in Blood Brothers, Prisoner Cell Block H, and Pump Boys And Dinettes.
In her later years, she appeared on Celebrity Big Brother and wrote a column for the Daily Mirror newspaper.
A framed photo of her was placed outside the church and There You’ll Be by Faith Hill was played as mourners arrived.
In her eulogy, Denise Nolan-Anderson said: “She really loved going to premieres and opening nights, having her beautiful hair and make-up done, and always was the life and soul of any big occasion. She would have loved all the fuss today.”
She also said Linda showed “courage in the face of adversity”, adding: “It’s time to rest now Linda. The battle is over, you are free.”
Speaking before the service, Paul Elliott, better known as Paul Chuckle of the Chuckle Brothers, said he last saw her in 2024, adding: “She was just a fun, bubbly person. The world’s a darker place without her.”
Shane Richie was previously married to Coleen Nolan. Other stars who paid their respects at the funeral included Charlotte Dawson, singer Lisa Maffia, and comedian Tommy Cannon.
The family statement announcing her death read: “At around 10.20am on Wednesday 15th January, she passed peacefully, with her loving siblings by her bedside, ensuring she was embraced with love and comfort during her final moments, aged 65.”
After her initial diagnosis, Nolan helped to raise £20m for charities Breast Cancer Now and the Irish Cancer Society.
She had a mastectomy and was given the all-clear in 2011 – but was diagnosed with secondary breast cancer that spread to her liver in 2020 and her brain in 2023.
The singer was born in Dublin but made Blackpool her home – marrying her husband of more than two decades, the late Brian Hudson, there.
The family are raising money for Trinity Hospice in her memory – another organisation Nolan supported.
A medical plane carrying a young patient and her mother has crashed in a heavily populated area of Philadelphia – killing all six people on board.
The Jet Rescue Air Ambulance was carrying the girl and her mother from Northeast Philadelphia Airport to Springfield-Branson National Airport in Missouri – on their way home to Tijuana, Mexico, the company said.
It crashed about 30 seconds after taking off, erupting in a fireball and engulfing several homes in flames.
The plane came down near the Roosevelt Mall in the Rhawnhurst area of Pennsylvania at 6.30pm local time on Friday (11.30pm UK time). Four members of crew were also on board, the Federal Aviation Administration (FAA) confirmed.
The aircraft had climbed to 1,600ft (487m) and travelled around three miles (5km) from the airport – before it came down.
Jet Rescue Air Ambulance, which is based in Mexico, said: “At this time we cannot confirm any survivors.”
Shai Gold, who works in corporate strategy for the company, said it was “terribly shocked by this tragic turn of events”.
“This was a very seasoned crew,” she said. “We are a leading air ambulance company, we fly 600 to 700 times a year. We don’t really know what happened.”
Fire engines and other emergency vehicles were sent to the scene, with the blaze appearing to have been put out around two hours after the crash.
There were reports of people injured on the ground – but no details were confirmed.
A doorbell camera captured the plane falling from the sky in a streak of white and exploding as it hit the ground.
The owner of the camera, Jim Quinn, said: “All we heard was a loud roar and didn’t know where it was coming from. We just turned around and saw the big plume.”
The moment of the crash was also seen on dashcam footage of a car driving nearby.
Michael Schiavone, 37, was at home in a nearby neighbourhood when he heard a loud bang and his house shook. He described it as a “mini earthquake” and when he checked his home security camera, it looked like a missile came down.
Philadelphia Governor Josh Shapiro said: “We want to offer our thoughts and serious prayers for those that are grieving at this moment.”
President Donald Trump wrote on social media that he was “so sad” to see the crash.
“More innocent souls are lost. Our people are totally engaged. First responders are already being given credit for doing a great job.”
The FAA said the National Transportation Safety Board would lead an investigation into the crash. The NTSB said an investigator has arrived at the scene – and more officials would be deployed on Saturday.
Second US plane crash in a week
The crash comes days after a plane collided with a US Army helicopter as it came into land at an airport in Washington DC – killing all 64 people on board the plane – and three on the helicopter.
The aircraft had requested to land on the shorter runway at Reagan Washington National Airport – before it collided with the helicopter, and fell into the Potomac River at around 8.47pm local time (1.47am UK time) on Wednesday.
At a news conference on Friday, authorities said they have recovered 48 bodies so far – of which they have identified 28.
Head of Washington DC’s emergency medical services John Donnelly said he expects all the victims’ bodies to be found – but it would require the plane debris to be removed from the river.
Crews are currently working at the shoreline and more personnel were being deployed late on Friday, he said.
Kraken will delist Tether’s USDT and four other stablecoins in Europe to comply with MiCA regulations, following a phased approach to minimize market disruptions.