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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 They described the new shape May 28 in a paper published to the preprint database arXiv, though it has not yet been peer reviewed.related stories—Mathematicians make rare breakthrough on notoriously tricky ‘Ramsey problem’

—Centuries old ‘impossible math problem cracked using physics of Schrödinger’s cat

—Two mathematicians just solved a decades-old math riddle — and possibly the meaning of life

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.

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Convicted killer jailed after turning up at Cheryl Tweedy’s home for fourth time

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Convicted killer jailed after turning up at Cheryl Tweedy's home for fourth time

A convicted killer who turned up at Cheryl Tweedy’s home for a fourth time has been jailed.

Daniel Bannister, 50, was sentenced to 12 months after admitting a single charge of breaching a restraining order.

He was also given a new restraining order, which warns him against contacting the former Girls Aloud singer.

“You are causing her anxiety,” Judge Alan Blake told him.

“She does not wish any contact with you. You have shown defiance to the court order. You need to draw a line under that behaviour.”

Bannister turned up at Tweedy’s rural home for the fourth time on 19 June.

Reading Crown Court heard he arrived in a taxi just before 10pm and rang the intercom twice before peering over the gate.

Bannister believed the singer had invited him to her home over Microsoft Teams, the court was told.

Daniel Bannister. Pic: Thames Valley Police
Image:
Daniel Bannister. Pic: Thames Valley Police

Tweedy said she was “stunned” when Bannister visited her home yet again and had been forced to hire security.

“Each time he returns the worry of his intentions intensifies,” she said in a victim impact statement.

“I’m worried, nervous and on edge every time I open my gate. No person should have to feel this way.

“Daniel has made my young child scared,” she added.

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Bannister was initially jailed for four months in September last year – and handed a three-year restraining order.

But he breached it by turning up at Tweedy’s home in December.

In March, he was jailed for 16 weeks at Wycombe Magistrates’ Court for repeatedly going to Tweedy’s Buckinghamshire home while under the restraining order.

During that appearance, the court heard that Tweedy “immediately panicked” and was “terrified” when she saw him outside her home, fearing for the safety of her eight-year-old son Bear.

Bannister killed Rajendra Patel, 48, at a south London YMCA shelter in 2012 and pleaded guilty to manslaughter.

Mr Patel died from an injury to his leg, a court heard.

Tweedy’s former partner Liam Payne died last year in Buenos Aires, Argentina, after falling from his third-floor hotel balcony.

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Sooners QB Mateer will miss time for hand injury

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Sooners QB Mateer will miss time for hand injury

Oklahoma quarterback and Heisman frontrunner John Mateer will undergo surgery on his right hand, the school announced Tuesday.

Mateer suffered the injury to his throwing hand during the first quarter of Saturday’s 24-17 win over Auburn, with sources telling ESPN’s Pete Thamel that Mateer has a broken bone in the hand.

While Oklahoma coach Brent Venables didn’t provide a specific timetable for Mateer’s return, sources told Thamel that the estimated timeline after surgery is expected to be about a month, as the surgery is considered straight-forward.

Mateer is scheduled to undergo the surgery Wednesday in Los Angeles.

“After consulting with medical experts, it became clear that surgery is the best option for John and his short- and long-term future,” Venables said. “He’s extremely disappointed he will miss some game action but is eager to correct the issue and move forward. As he is with everything, we know he will be aggressive with his rehabilitation and work to return to the field as quickly as possible.”

Mateer’s injury is a massive blow to the seventh-ranked Sooners, who are off to a 4-0 start with wins over Michigan and Auburn.

Mateer, who transferred in from Washington State during the offseason, has been the catalyst behind Oklahoma’s stark turnaround after a 6-7 finish in 2024.

He’s completing 67.4% of his passes for 1,215 and six touchdowns. He’s also the Sooners’ leading rusher with 190 yards and five more touchdowns. Mateer ranks second nationally with 351.3 yards of total offense per game.

Despite suffering the injury early against Auburn, he finished the game and passed for 271 yards.

With his hot start, Mateer had emerged as the favorite to win the Heisman at +750, according to ESPN BET Sportsbook. Indiana QB Fernando Mendoza (+650) is the new Heisman favorite at ESPN BET Sportsbook.

Venables said that sophomore Michael Hawkins Jr. will start Oklahoma’s next game against Kent State on Oct. 4 Hawkins started four games for the Sooners last year. He passed for 783 yards and three touchdowns.

Oklahoma faces rival Texas on Oct. 11.

After a road trip to South Carolina, the Sooners close out the regular season with five straight games against ranked opponents: No. 13 Ole Miss, at No. 15 Tennessee, at No. 18 Alabama, No. 20 Missouri and No. 4 LSU.

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MLB playoff pressure rankings: Which World Series contenders must win now — or else?

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MLB playoff pressure rankings: Which World Series contenders must win now -- or else?

Which playoff team most needs to win the World Series?

This is a question we try to answer around this time every year. What builds the pressure to win right now? The answer is a little different for every team, and the force of that pressure changes with each passing season. Teams age. Free agents leave and arrive. Playoff disappointments pile up. Playoff absences chafe.

The more success a team has without winning it all, the more the pressure builds up. Not until it wins it all does that pressure finally release, resetting the valve, and fans of that team can relax. Only the Dodgers’ faithful are in a state of pure release — because L.A. won just last year.

Let’s take a look at how the current contenders rate on the pressure scale.


The Pressure Index formula

The original incarnation of our system was based on an old Bill James method for calculating “pressure points.” Last year, we tweaked our methodology a little to add measurable narrative-based factors to the numbers-based historical context, and that worked pretty well, so we’ve carried that over for the coming 2025 MLB playoffs.

The revised Pressure Index considers the following factors, ranked in order by the weight they carry in the final calculation:

1. Drought pressure: This is all about flags, both World Series titles and pennants. The championship part of this factor counts twice as much as the pennant factor. The current leader is the Cleveland Guardians, owners of baseball’s longest title drought at 76 years. Teams coming off a title — e.g., the Dodgers — have no drought pressure. The New York Yankees, as the defending American League champs, get a little release from the drought factor for ending their pennant-less streak, but the pressure won’t fully dissipate until the Bombers win it all.

2. Knock-knock pressure: There is a whole different pressure for a franchise that lingers well below .500 year after year than there is for a franchise that consistently plays winning baseball and yet can’t seem to get anywhere in October. Looking at the most recent 25-year window, the knock-knock factor counts winning seasons weighted from most recent to most distant. The Yankees, with their active streak of 33 straight winning seasons, have the most knock-knock points. The 2025 season standings were included in this calculation.

3. Flickering star pressure: Using AXE ratings, we calculate the average AXE of a team’s pending free agents. Players with a club option are not included in this tally. Using the average adds to the pressure of clubs such as the Philadelphia Phillies, who have star players (Kyle Schwarber, J.T. Realmuto and Ranger Suarez) about to hit free agency.

4. Exodus pressure: Another free agent factor: Using AXE ratings again, we tally the total AXE points for a team’s pending free agents. Again, players with a club option are not included. Losing a star player hurts — but so does having a lot of holes to fill.

5. Father Time pressure: Time comes for us all. The older a team is, the shorter its window of opportunity for elite contention. This factor is based on average team ages, hitters and pitchers combined.


The rankings

Note: This includes every team that currently has at least a 5% chance at making the postseason, per our daily simulations.

Pressure Index: 109.5

Last pennant: 1982

Last World Series title: Never

The Brewers topped our leaderboard a year ago and went on to drop a tough wild-card series at American Family Field to the Mets. Then, despite preseason forecasts that marked them as a fringe playoff hopeful, they went out and put up what might well end up as the most successful regular season in franchise history. Based on what we’ve seen to date, the Brewers have never been in a better position to win their first World Series. They’ve had great regular seasons before — perhaps not as great as this one — but the only thing that will quench the ever-thirsty fans in Wisconsin is the city’s first MLB championship since the 1957 Milwaukee Braves.


Pressure Index: 108.0

Last pennant: 1998

Last World Series title: Never

The Brewers have gone longer without a pennant and have also piled up more good seasons than San Diego, or else the Pads might have landed in the top spot. There isn’t a factor for trade deadline aggression, but if there were, the Padres might have overtaken the Brewers for that reason, too. Because many of the players moved at the deadline tend to be headed for free agency, that’s a pretty good proxy for the kind of internally generated pressure that goes with an aggressive deadline approach. The Padres have the third-most exodus points and are fifth in the flickering star column. Among those who might hit the market are Dylan Cease, Luis Arraez and Michael King (mutual option) and Robert Suarez (player option). Given that list and A.J. Preller’s frenetic deadline behavior, the time to win for San Diego is very much right now.


Pressure Index: 107.8

Last pennant: Never

Last World Series title: Never

Speaking of deadline activity, the Mariners’ pickups of Josh Naylor and Eugenio Suarez signaled a clear win-now approach. Both will be free agents after the season. Beyond that, as the only active team that has never won a pennant, the pressure has been building in Seattle since the day the Mariners debuted in 1977. Finally, though Cal Raleigh‘s history-making breakout season doesn’t factor into the Pressure Index, you could argue that in a sense it should because we can’t expect to see him at this level after 2025. That’s not a knock on him — he should remain an All-Star-caliber backstop. But few have ever reached the pinnacle Raleigh is at in 2025, and those who have gotten there have tended not to stay there.


Pressure Index: 106.5

Last pennant: 2016

Last World Series title: 1948

A late-season surge, combined with the Tigers’ collapse, has resoundingly returned the Guardians to this portion of our rankings. As has been the case the past couple of years, the Guardians’ pressure is almost entirely generated by their decades-long title drought. There are so many ways to contextualize it, but here’s a fun one: The last time Cleveland won the World Series, one of its top relief pitchers was Satchel Paige. Because winning a pennant relieves the drought points column a little, the Guardians’ 2016 pennant is recent enough to keep them behind the top three on this list in that area. However, only two teams have more knock-knock points than the consistently solid Guardians — the Yankees and Cardinals.


Pressure Index: 105.1

Last pennant: 1993

Last World Series title: 1993

The Blue Jays’ rise from fifth place to first place in the AL East has ratcheted up the tension for a franchise that hasn’t seen the World Series in more than three decades despite a number of strong teams during that span. The Jays also have some big-name free agents in Bo Bichette, Max Scherzer, Chris Bassitt and Shane Bieber, who has a player option. Imagine what the pressure might feel like if Vladimir Guerrero Jr. had not inked an extension back in April.


Pressure Index: 104.6

Last pennant: 2015

Last World Series title: 1986

If we based these ratings on media scrutiny and payroll, the Mets, Yankees and Dodgers would never fall out of the top three slots. For the Mets, their real-world pressure can’t be captured by an algorithm like this, mostly because of a combination of preseason expectation fueled by the addition of Juan Soto in the offseason, and the downward-pointing trajectory of what once looked like a banner season. If the Mets capture the NL’s last postseason spot — not even close to a sure thing at this point — the consternation generated by their lackluster second half can be addressed with a deep playoff run. But that doesn’t feel too likely at the moment.


Pressure Index: 104.5

Last pennant: 2024

Last World Series title: 2009

The defending AL champs! Good enough? Didn’t think so. The World Series returned to the Bronx last fall, but watching the Dodgers celebrate a title at Yankee Stadium didn’t do much to turn down the temperature on fan expectation. There was also the Soto factor, given his possible departure, which of course came to pass. Still, the Yankees restocked in free agency, added aggressively at the deadline and added even more points to their MLB-leading knock-knock total with a 33rd straight winning season. But let’s face it: This is the Yankees, and the only thing that will ease the scrutiny is a championship.


Pressure Index: 102.8

Last pennant: 2012

Last World Series title: 1984

Last season’s playoff berth felt like found money for the Tigers. Not so this time around, as Detroit has led the AL Central for virtually the entire season. Its lead over Cleveland in the division, once in double figures, has all but disappeared, and what looked like a certain return to the playoffs is now in doubt. A four-decade title drought feeds the Tigers’ Pressure Index, as does a fairly splashy free agent class that includes Kyle Finnegan, Jack Flaherty (player option), Charlie Morton, Alex Cobb, Chris Paddack, Tommy Kahnle and Gleyber Torres. Still, even if Detroit’s late-season swoon culminates in an early playoff exit, Tigers fans can take solace in a young roster core and a loaded minor league system. This is only the beginning of the adventure.


Pressure Index: 101.4

Last pennant: 2022

Last World Series title: 2008

Drought points are the most heavily weighted factor in the system, and the Phillies’ 2008 title and 2022 pennant are recent enough to keep their score low in that category. But they still land in the top 10 of the Pressure Index because of their looming free agent class (led by Schwarber and Ranger Suarez) and a veteran roster that ranks second in overall average team age. Under GM Dave Dombrowski, the Phillies have been building with the near-term window at the forefront of their planning for a few years now. At some point, and soon, it needs to pay off with their third World Series crown in 143 years of existence.


Pressure Index: 101.2

Last pennant: 1990

Last World Series title: 1990

The Reds’ Mets-fueled playoff odds resurgence lands them here in the contenders’ group, where Cincinnati’s ever-growing drought in both the championship and pennant columns looms large. This isn’t in the calculation, but you might also consider the Reds’ dynamic young rotation as a soft factor. The way Hunter Greene and Brady Singer have thrown lately, and the way Nick Lodolo and Andrew Abbott have pitched for much of the season, you’d love to see what the Reds might do in a playoff context. And although these are all pretty young pitchers — and the Reds have more in the pipeline — pitching is a mercurial thing. You want to take advantage of it while it’s going good.


Pressure Index: 99.9

Last pennant: 2016

Last World Series title: 2016

Coming to you live from Chicago: We can report that the euphoria over the Cubs’ drought-ending championship in 2016 is largely yesterday’s news. (No, it won’t be forgotten, but it’s been nine years, after all.) The Cubs entered the season as the NL Central favorite, and although it’s been a strong campaign on the North Side, they are looking at a wild-card berth. That’s progress — the Cubs had missed the playoffs four years running — and expectations remain high. A midseason offensive slump was a cause for concern, but the Cubs have been going well of late. Hovering over all this is the pending free agency of the Cubs’ splashy offseason pickup, outfielder Kyle Tucker.


Pressure Index: 99.4

Last pennant: 2023

Last World Series title: 2001

You could argue the Diamondbacks don’t really have any pressure on them at all. For one thing, they have been a fringe hopeful, one that needed the continuing cooperation of the flagging Mets to stay alive. Arizona also waved a faint white flag in advance of the deadline by dealing Naylor and Suarez. But the Diamondbacks’ offense has been strong over the past few weeks, keeping their postseason window ajar. After all this, if Arizona ends up in the bracket, it’ll have to feel like it’s got nothing to lose.


Pressure Index: 99.3

Last pennant: 2018

Last World Series title: 2018

The Red Sox, once synonymous with the concept of “championship drought,” are good in that crucial column at present. Sure, in Boston, seven years without a banner might feel like a long time after the Red Sox’s run of four titles in 15 years, but it’s really not. Boston does rank high in the knock-knock column (fourth) and in the flickering star category (second). That’s because of possible free agencies for Alex Bregman and Trevor Story (both have player options). It would be worse if Aroldis Chapman had not signed an extension.


Pressure Index: 97.3

Last pennant: 2022

Last World Series title: 2022

The Astros aren’t dealing with a drought. But the roster is on the old side (ninth in average age) and is looking at the possible free agent departure of Framber Valdez. The window felt like it was starting to close when we entered the season, and the Astros have teetered more than once through the summer. After a weekend sweep by the rival Mariners, those wishing for a Houston-less postseason are feeling more hopeful than ever. The declining Astros need to take advantage of this contention opportunity while they can.


Pressure Index: 90.0

Last pennant: 2024

Last World Series title: 2024

The Dodgers’ drought points reset with last season’s title. At the same time, Los Angeles puts pressure on itself by funneling so many resources into building and managing its roster. There is no pressure on the Dodgers in the context of the Pressure Index approach, but L.A. has created a dynamic where you have to consider any season in which it doesn’t win as at least a mild disappointment. Then of course there is one major soft factor for this year’s Dodgers: This is the last go-round for Clayton Kershaw, and the desire to send him into retirement with a second straight championship has to be looming large in the organization.


Pressure Index for all other teams

16. Baltimore Orioles (103.3)
17. Pittsburgh Pirates (102.2)
18. Athletics (102.1)
19. Minnesota Twins (101.6)
20. St. Louis Cardinals (100.5)
21. Los Angeles Angels (99.7)
22. Tampa Bay Rays (98.2)
23. Atlanta Braves (97.6)
24. San Francisco Giants (96.9)
25. Chicago White Sox (96.5)
26. Texas Rangers (95.6)
27. Colorado Rockies (96.1)
28. Kansas City Royals (95.1)
29. Washington Nationals (92.9)
30. Miami Marlins (84.1)

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