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Look carefully! Mathematicians have invented a new 13-sided shape that can be tiled infinitely without ever repeating a pattern. They call it “the einstein.”

For decades, mathematicians wondered if it was possible to find a single special shape that could perfectly tile a surface, without leaving any gaps or causing any overlaps, with the pattern never repeating. Of course, this is trivial to do with a pattern that repeats — just look at a bathroom or kitchen floor, which is probably made up of simple rectangular tiles. If you were to pick up your floor and move it (called a “translation” in mathematics), you could find a position where the floor looks exactly the same as before, proving that it’s a repeating pattern.

In 1961, mathematician Hao Wang conjectured that aperiodic tilings, or tilings that never become a repeating pattern, were impossible. But his own student, Robert Berger, outwitted him, finding a set of 20,426 shapes that, when carefully arranged, never repeated. He then slimmed that down to a set of 104 tiles. That means that if you were to buy a set of those tiles, you could arrange them on your kitchen floor and never find a repeating pattern.

In the 1970s, Nobel prize-winning physicist Roger Penrose found a set of only two tiles that could be arranged together in a nonrepeating pattern, now known as a Penrose tiling.

Here we see the first four iterations of the H metatile and its supertiles. (Image credit: Smith el at. (2023))

Since then, mathematicians around the world have searched for the aperiodic tiling holy grail, called “the einstein.” The word doesn’t come from the famous Albert but from the German translation of his last name: one stone. Could a single tile — one “stone” — fill a two-dimensional space without ever repeating the pattern it creates?

The answer was just discovered by David Smith, a retired printing technician from East Yorkshire, England. How did he come across this remarkable solution? “I’m always messing about and experimenting with shapes,” Smith told The New York Times (opens in new tab) . “It’s always nice to get hands-on. It can be quite meditative.”

Smith and his co-authors dubbed the new shape “the hat,” mostly because it vaguely resembles a fedora. Although mathematicians have known about the shape, which has 13 sides, they had never considered it a candidate for aperiodic tiling.

“In a certain sense, it has been sitting there all this time, waiting for somebody to find it,” Marjorie Senechal (opens in new tab) , a mathematician at Smith College who was not part of the study, told The Times.Related stories—Mathematicians make rare breakthrough on notoriously tricky ‘Ramsey problem’

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Smith worked closely with two computer scientists and another mathematician to develop two proofs showing that “the hat” is an aperiodic monotile — an einstein. One proof relied on building larger and larger hierarchical sets of the tiles, showing how the pattern never repeats as the surface area grows. The other proof relied on the team’s discovery that there wasn’t just one of these tiles, but an infinite set of related shapes that could all do the trick. The team’s paper is available on the preprint server arXiv (opens in new tab) but has not yet been peer-reviewed, and the proofs have not yet been scrutinized.

These kinds of aperiodic tilings are more than mathematical curiosities. For one, they serve as a springboard for works of art, like the Penrose tiling found at the Salesforce Transit Center (opens in new tab) in San Francisco, and reveal that some medieval Islamic mosaics employed similar nonrepeating patterns.

Aperiodic tilings also help physicists and chemists understand the structure and behavior of quasicrystals, structures in which the atoms are ordered but do not have a repeating pattern.

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Tesla shares drop 7% in premarket trading after Elon Musk says he is launching a political party

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Tesla shares drop 7% in premarket trading after Elon Musk says he is launching a political party

White House Senior Advisor Elon Musk walks to the White House after landing in Marine One on the South Lawn with U.S. President Donald Trump (not pictured) on March 9, 2025 in Washington, DC.

Samuel Corum | Getty Images News | Getty Images

Tesla shares fell in premarket trade on Monday after CEO Elon Musk announced plans to form a new political party.

The stock was down 7.13% by 4:27 a.m. E.T.

Musk said over the weekend that the party would be called the “America Party” and could focus “on just 2 or 3 Senate seats and 8 to 10 House districts.” He suggested this would be “enough to serve as the deciding vote on contentious laws, ensuring that they serve the true will of the people.”

The billionaire’s involvement in politics has been a point of contention for investors. Musk earlier this year was part of the so-called Department of Government Efficiency and worked closely with President Donald Trump — a move seen as potentially hurting Tesla’s brand.

Musk left DOGE in May, which helped Tesla’s stock.

Now tech billionaire’s reinvolvement in the political arena is making investors nervous.

“Very simply Musk diving deeper into politics and now trying to take on the Beltway establishment is exactly the opposite direction that Tesla investors/shareholders want him to take during this crucial period for the Tesla story,” Dan Ives, global head of technology research at Wedbush Securities, said in a note on Sunday.

“While the core Musk supporters will back Musk at every turn no matter what, there is broader sense of exhaustion from many Tesla investors that Musk keeps heading down the political track.”

Musk’s previous political foray earned him Trump’s praise in the early days, but he has since drawn the ire of the U.S. president.

The two have clashed over various areas of policy, including Trump’s spending bill which Musk has said would increase America’s debt burden. Musk has taken issue to particular cuts to tax credits and support for solar and wind energy and electric vehicles.

Trump on Sunday called Musk’s move to form a political party “ridiculous,” adding that the Tesla boss had gone “completely off the rails.”

Musk is contending with more than just political turmoil. Tesla reported a 14% year-on-year decline in car deliveries in the second quarter, missing expectations. The company is facing rising competition, especially in its key market, China.

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Environment

Paris’ popular bike share program has a big sticky finger problem

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Paris' popular bike share program has a big sticky finger problem

Paris’ bike-share system, Vélib has long been considered one of the shining success stories of urban micromobility. With a massive fleet of over 20,000 pedal and electric-assist bicycles around Paris, the service has helped millions of residents and tourists get around the City of Light without needing a car or scooter. But lately, a growing problem is threatening to knock the wheels off this urban mobility marvel: theft and joyriding.

According to city officials and the service operator, more than 600 Vélib bikes are now going missing every single week. That’s over 30 bikes a day simply vanishing from the system – some stolen outright, others taken on “joy rides” and never returned.

“At the moment we’re missing 3,000 bikes,” explained Sylvain Raifaud, head of the Agemob company that currently operates the Velib system. That’s nearly 15% of over 20,000 Vélib bikes across Paris.

The sticky-fingered culprits aren’t necessarily professional thieves or organized crime rings. Instead, they’re often regular users who treat the shared bikes like disposable toys.

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The city estimates that many people have figured out how to pry the bikes out of the system’s parking docks, unlocking one for a casual cruise and then ditching it somewhere far from a docking station.

Once pried free, the bikes are technically usable for the next 24 hours until their automatic locking feature kicks in. At that point, the bikes are often simply abandoned. Some end up in alleyways. Others get tossed in rivers. A few just disappear completely.

And since the bikes are intended to be parked at their many docking stations around the city, they don’t have GPS chips, further complicating recovery of “liberated” bikes.

The issue started small but has grown into more than an inconvenience – it’s beginning to undermine the entire purpose of the service. With bikes going missing at such a high rate, many Vélib docking stations are left empty, especially during rush hours.

Riders looking for a quick commute or a convenient hop across town are increasingly finding themselves without available bikes, or having to walk long distances to find a functioning one.

That kind of unreliability chips away at user confidence and threatens to drive potential riders back into cars, cabs, or other less sustainable forms of transport at a time when Paris has already made great strides to dramatically reduce car usage in the city.

The losses are financially painful, too. Replacing stolen or vandalized bikes isn’t cheap, and the resources spent on tracking down missing equipment or reinforcing anti-theft measures are stretching thin. Vélib has faced theft and vandalism issues before, especially during its early years, but this latest surge has officials sounding the alarm with renewed urgency.

Officials acknowledge that there’s no easy fix. Paris, like many cities with bike-share systems, walks a fine line between accessibility and accountability. Part of what makes Vélib so successful is its ease of use and widespread availability. But those same features make it vulnerable to misuse – especially when enforcement is limited and the consequences for abuse are minimal.

The timing of the problem is especially unfortunate. In recent years, Paris has seen impressive results in reducing car traffic, expanding bike lanes, and promoting cycling as a key part of its sustainable transport strategy. Vélib is a cornerstone of that plan. But if the system becomes too unreliable, it risks losing the very people it was designed to serve.

Meanwhile, as Parisians increasingly find themselves staring at empty docks, the challenge for the city and Vélib will be to restore confidence in the system without making it harder to use. That means striking the right balance between freedom and responsibility, between open access and protection against abuse.

In a city where cycling is supposed to be the future of mobility, losing thousands of bikes to joyriders and sticky fingers isn’t just frustrating; it’s unsustainable.

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Environment

CNBC Daily Open: Elon Musk, founder of companies and political parties

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CNBC Daily Open: Elon Musk, founder of companies and political parties

U.S. President Donald Trump and Elon Musk attend a press event in the Oval Office of the White House in Washington, D.C., U.S., May 30, 2025.

Nathan Howard | Reuters

When they lose a significant other, most men do indeed become a “TRAIN WRECK.” Then they pick up the pieces of their lives and start living again — paying attention to their personal grooming, hitting the gym and discovering new hobbies.

What does the world’s richest man do? He starts a political party.

Last weekend, as the United States celebrated its independence from the British in 1776, Elon Musk enshrined his sovereignty from U.S. President Donald Trump by establishing the creatively named “American Party.”

Few details have been revealed, but Musk said the party will focus on “just 2 or 3 Senate seats and 8 to 10 House districts,” and will have legislative discussions “with both parties” — referring to the U.S. Democratic and Republican Parties.

It might be easier to realize Musk’s dream of colonizing Mars than to bridge the political aisle in the U.S. government today.

To be fair, some thought appeared to be behind the move. Musk decided to form the party after holding a poll on X in which 65.4% of respondents voted in favor.

Folks, here’s direct democracy — and the powerful post-separation motivation — in action.

 — CNBC’s Erin Doherty contributed to this report.

What you need to know today

And finally…

An investor sits in front of a board showing stock information at a brokerage office in Beijing, China.

Thomas Peter | Reuters

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