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.
Alibaba announced plans to release a pair of smart glasses powered by its AI models. The Quark AI Glasses are Alibaba’s first foray into the smart glasses product category.
Alibaba
Alibaba on Monday unveiled a pair of smart glasses powered by its artificial intelligence models, marking the Chinese firm’s first foray into the product category.
The e-commerce giant said the Quark AI Glasses will be launched in China by the end of 2025 with hardware powered by the firm’s Qwen large language model and its advanced AI assistant called Quark.
The Hangzhou, headquartered company is one of the leaders in China’s AI space, aggressively launching new models with capabilities that compete with Western counterparts like OpenAI.
Many tech companies see wearables, specifically glasses, as the next frontier in computing alongside the smartphone. Quark, which was updated this year, is currently available as an app in China. Alibaba is stepping into the hardware game as a way to distribute the app more widely.
The Quark AI Glasses are Alibaba’s answer to Meta’s smart glasses that were designed in collaboration with Ray-Ban. The Chinese tech giant will also now compete with Chinese consumer electronics player Xiaomi who this year released its own AI glasses.
Alibaba said its glasses will support hands-free calling, music streaming, real-time language translation, and meeting transcription. The glasses also feature a built-in camera.
Alibaba owns a range of different services in China from mapping to an online travel agent. Its affiliate company Ant Group also runs the widely-used Alipay mobile service. Alibaba said users will be able to use a navigation service via the glasses, pay with Alipay and compare prices on Taobao, its China e-commerce platform.
The firm has yet to release other details such as the price and technical specifications.
Tesla will use Samsung for as a supplier for its self-driving computer’s next-gen hardware in a $16.5 billion deal, according to Tesla CEO Elon Musk.
But despite planning two generations ahead, the company still doesn’t have a solution to bring the promised full autonomy to hardware that it’s been promising that capability to since 2016.
Earlier today, Samsung announced a 22.8 trillion won ($16.5 billion) deal that would run through 2033. In that filing, Samsung did not name the customer, only that it is a “large global company”. Later, Bloomberg reported that the customer is Tesla, and Musk confirmed this on twitter. Then in his usual bravado, he stated that the deal is “likely much more than that.”
Samsung makes the chips for the self-driving computers in Tesla’s current vehicles, but the next generation will be made by TSMC, first in Taiwan and then later in Arizona. Then the next-next generation will be covered by this new Samsung deal.
The new deal is significant due to TSMC’s global dominance of chipmaking. Samsung has had significant unused capacity, so the Tesla deal is a big boost for the company’s chip foundry business.
Tesla has gone through several generations of chips, previous referred to as “HW,” standing for “hardware,” with a number indicating their generation. More recently, Tesla started referring to its chips with “AI” instead of “HW,” in order to incorporate the tech buzzword du jour.
Currently Tesla is on HW4/AI4, and TSMC will make HW5, then Samsung will make HW6 again.
These generations of hardware each get successively more capable, and can handle more data and thus theoretically become better at self-driving tasks.
Current Tesla HW4 vehicles cannot drive themselves, and are only capable of SAE level 2 operation, which requires an attentive driver behind the steering wheel (though Tesla’s solution does work better than most others). Tesla’s ‘Robotaxi’ system is currently operating in Austin without anyone in the driver’s seat, but has a “safety rider” who can take control of the vehicle, blurring the line somewhat on which SAE level it is operating at.
But what about HW3?
There’s a problem with the differentiation between these generations of hardware: ever since 2016, when Tesla was on version 2 of its hardware, it has promised full self-driving capability on all of its vehicles.
Tesla stated, at the time, that every single Tesla vehicle produced after that date had the hardware that would allow for full self-driving.
It eventually became apparent that HW2 would not be capable of full self-driving tasks, and Tesla upgraded to HW3, promising all HW2 customers that they would get a free upgrade to HW3 if they bought Tesla’s Full Self-Driving system, which has varied in price over time but once cost $15,000.
Now, with the change from HW3 to HW4, we’re seeing indications of a similar run-around.
We’ve already seen differing FSD software versions based on which hardware level vehicles have, with HW3 vehicles getting updates later than HW4 vehicles do. On last week’s Q2 earnings call, Tesla CFO Vaibhav Taneja said:
What we want to do is get unsupervised done on hardware four first. Once it’s done, then we’ll go back and look at what we need to do with the hardware three cars. Like I said, the focus is first to get unsupervised out and then we’ll go back and see what more work we need to do.
“Unsupervised” is Tesla’s new name for actual full self-driving, which would allow a vehicle to drive without the supervision of someone in the driver’s seat. This as opposed to “supervised FSD,” a phrase Tesla started using after about a decade of promising full self-driving without delivering it.
Here, Taneja said that HW3 cars will eventually get FSD, but Tesla hasn’t really figured out the path to that, and it’s focusing on new cars first, then will go back around to see what needs to happen.
Previously, Musk had stated that Tesla “will have to upgrade people’s hardware 3 computer,” but more recently it has become apparent that Tesla really doesn’t have a plan for that upgrade. And Taneja’s comments suggest that Tesla will still try to wedge FSD onto HW3, despite previously admitting that the system is not capable of it.
The existence of future HW5 and even HW6 chips also suggest that current systems are not capable of full self-driving. If HW4 is FSD-capable, then why would Tesla need two more generations of chip in the next two years in order to do the tasks that it promised all of its cars could do a full decade prior?
So, much more than having no solution for HW3 cars (or even HW2 cars, some of which have gotten free upgrades, but others who have been charged $1,000 to upgrade to a computer they already paid for), does this mean that Tesla is going to kick the can further down the road, and eventually have no solution for HW4 and HW5 either?
And, when will we know about these solutions? Tesla has sold millions of vehicles with the promise of self-driving which will seemingly need an upgrade at some point. And many of those vehicles are old enough, at this point, to be retired, despite spending up to $15,000 on a piece of software that has never been delivered to them.
An HW6/AI6 computer will surely have all sorts of new whizbang capabilities, but we were promised those capabilities years ago, and they’re still not delivered yet.
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A group of Senate Democrats has probed Federal Housing Finance Agency director William Pulte over his order to propose how to consider crypto in mortgage applications.