Beyond Euclid #226
Welcome to Beyond Euclid #226, the newsletter for the best mathematics and science stuff of the week. I am Ali, and I curate cool math and science stuff every week to help you have a better week.
Hello from Beyond Euclid 226. I noticed something odd this week. None of the things that stuck with me are answers, they’re all gaps. Cooper jumps out of a plane and for fifty years nobody knows where he landed. People leave Chornobyl and deer move into the space. The BBC switches off a frequency and a whole voice goes quiet. And then you read about the brain seeing a triangle that was never drawn in front of it. So we fill even the gaps ourselves. Maybe the only thing that really stays with us is the way we keep circling whatever was left unfinished.
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✶ Paul Klee taught at the Bauhaus between 1921 and 1931, and over those years he filled roughly 3,900 pages of teaching notes on pictorial form. And now nearly all of these pages online, full of colour studies, diagrams and geometric constructions. The page below shows, step by step, how a hexagon is built out of interlocking triangles, the small numbers along the edges marking the proportions. Klee can no longer teach anyone, but the notes he left behind can still be an inspiration to whoever cares to look.
△ Turn three notched circles, like little pac-men, so their bites point toward the centre, and a pure white triangle appears where nothing was ever drawn. It has no edges, and it is no brighter than the paper around it, yet the eye sees it plainly. This is the Kanizsa triangle, an illusion described in 1955 by the Italian psychologist Gaetano Kanizsa. The brain fills a gap with the most likely explanation, decides there must be a triangle sitting on top of three notched circles, and draws the missing edge itself. It is a reminder that what we see is less the light falling on our eye than the guess the brain builds from it.
⬡ There is a giant hexagon at Saturn’s north pole. This six-sided cloud pattern, wider than the diameter of Earth, forms at the edge of a jet stream in the planet’s atmosphere and has held its shape for decades. It was first spotted by Voyager in 1981, then photographed up close by Cassini in 2006.
🪂 This week I learned about a strange episode. In 1971 a man calling himself Dan Cooper hijacks a plane flying from Portland to Seattle, claiming there is a bomb in his bag. In Seattle he has the thirty-six passengers released in exchange for two hundred thousand dollars and four parachutes, then has the plane take off again and jumps out the rear stairs somewhere over Washington. After that, no trace of him is ever found. He most likely fell into the forest he jumped over and did not survive, because years later a decayed bundle of banknotes with serial numbers matching the ransom was found buried by a riverbank. The FBI examined hundreds of suspects and suspended the investigation in 2016. It is still the world’s only unsolved skyjacking.
🦌 The Chornobyl Exclusion Zone, one of the most radioactive places on Earth, is now home to the richest wildlife in northern Ukraine. Deer, wolves, lynx, brown bears and wild horses roam the area freely. According to data from a hundred and seventy-four camera traps, diversity peaks exactly where people enter least, on the large and unbroken land. What is interesting is not that the animals tolerate radiation. What brought them back was not a resistance to radiation but the fact that no one was there to disturb them.
✨ The European Space Agency’s Euclid telescope captured the highest-resolution photograph ever taken of the bright centre of the Milky Way. More than sixty million stars are visible in a single frame, along with nebulae, star clusters and molecular clouds that sit like dark blotches in front of the brighter regions. The area Euclid scans in a few hours is two hundred and seventy times wider than Hubble’s field of view. To image the same mosaic, the Keck Observatory would need around two thousand hours. There is also a video version that travels through the depth of the image.
🕳️ At a science high school in Seoul, three students wrote a paper on black holes without any university support, and the paper was published in an international physics journal. The work looks in a new way at the rules governing how heat and energy behave in black holes, and shows that these rules also hold for more complex cases such as rotating or charged black holes. It is a rare thing for high school students to appear in a journal like this.
🎓 Of course not everyone has to be this successful. At Brown University, the economist Roberto Serrano found conclusive evidence that at least fifty students used AI to cheat on the midterm of his advanced mathematical economics course. The exam was the take-home kind. The average score of the eighty-nine students who sat it came out at ninety-six out of a hundred, and forty students got a perfect score. When Serrano announced the final would be held in person, the average dropped to forty-eight, and most of the students who had scored perfectly on the midterm never showed up for the final. The course, which had never drawn more than thirty students, had eighty-six enrolled this term.
📱 Google added a feature called “Short Video Overviews” to NotebookLM. It turns the documents you upload into sixty-second, vertical videos, with paper-cutout style visuals, diagrams and narration. Google describes it as making “endless scrolling educational.” What sets NotebookLM apart is that it draws only on the sources you upload rather than the open internet, which reduces the risk of made-up information.
✍️ In 1959 C.S. Lewis replied to an American schoolgirl’s letter asking for writing advice with eight points. Turn off the radio, avoid magazines, read good books. Write every sentence with the ear rather than the eye, and if it sounds bad read aloud, try again. Write only about what really interests you. Take great pains to be clear, because you know what you mean from the start but the reader does not. Do not throw away a piece you have abandoned, put it in a drawer, it may come in useful later. Know the meaning of every word you use.
⚽ Who would win the 2026 World Cup if something other than football decided it? This interactive bracket starts with the sixty-four teams of the real tournament, but settles matches by data you pick rather than by goals. Choose the democracy index, GDP per capita, life expectancy, happiness, gender equality or carbon emissions per capita, and let the matchups play out accordingly.
📞 Steve Wozniak, one of Apple’s co-founders, once got hold of a memorable number like 888-888-8888. But then he had to deal with a very big problem. More than a hundred calls came in every day, most of them from children pressing the keys at random and running the same digit down the row. Wozniak eventually had to give the number up.
📻 By shutting down its 198 kHz longwave broadcast on 27 June, the BBC brought an entire era to a close. Since the equipment used for the longwave service had reached the end of its life, they decided not to invest in a platform that now brings almost no return. I doubt any of you still listens, but you can hear the broadcast over here.
🐋 The Kuroshio Sea tank at Okinawa’s Churaumi Aquarium is one of the largest aquarium tanks in the world. Behind seven and a half million litres of water, whale sharks and manta rays glide past a sixty-centimetre-thick acrylic panel. This one-hour video shows the tank quietly, without any visitors.
🎵 Most people don’t realise it, but music is one of the main things that shape a young person’s character. A study that analysed the lyrics of more than three hundred and eighty thousand songs released between 1960 and 2023 found that popular music has grown steadily darker over sixty years. As references to virtues like care and decency declined, expressions tied to harm, cheating, subversion and degradation rose, along with a tone of anger and disgust. The researchers read this like a cultural barometer, lyrics being a record of how a society tells its own story.
🚶 I mentioned before that you can minimise the harm of prolonged sitting by taking short movement breaks. But how should the break be timed for best effect? A two-week study with more than nineteen thousand adults had participants take a five-minute walking break every 30, 60 or 120 minutes. In every group fatigue and negative feeling fell and positive feeling rose, but the best balance came with a break every hour. A break every half hour showed the strongest effect but was the hardest to keep up.
🔒 There really are dramas in the world you could never imagine. Someone who built a vast system for defrauding people is not only holding more than five thousand three hundred people by force in an area near Myanmar’s border with Thailand, but forcing them every day to defraud people around the world. According to data from a human rights group, about one thousand six hundred of those held are Chinese nationals, the rest from the Philippines, Taiwan, Malaysia, Brazil, Russia, Kenya and many other countries. Most were tricked or trafficked into these compounds.
📷 GoPro all but invented the action-camera category in 2002, and in 2022 it still held roughly three quarters of the global market. In three years that share fell to about eighteen percent, and this month the company disclosed there is substantial doubt about its ability to keep operating. Its place was taken by two Chinese brands, DJI and Insta360, which together hold more than eighty percent of the market. The same story played out in robot vacuums, where iRobot, the company that invented the Roomba, filed for bankruptcy last December and, in the end, was bought by a Chinese manufacturer too.
🍔 Eating a truly delicious burger is hard, because there are estimated to be around 10⁴³ possible burger recipes in the world right now. I have no wish to learn how they arrived at that estimate. Anyway, back to our subject. A team at Stanford had an AI tool they call BurgerAI design new burger recipes based on age, taste, nutritional need and even environmental impact. The point is to pick the best one out of that vast space of possibilities I just mentioned. In a blind taste test with more than a hundred diners in San Francisco, the burgers the AI designed scored higher than a popular fast-food burger. The mushroom one cut environmental impact by more than a factor of ten.
🌀 The Lithuanian Žilvinas Kempinas makes his works out of magnetic tape. When he lays hundreds of metres of VHS tape on the ground and sets it moving with fans, the tape undulates and shimmers, and a form appears that sits somewhere between solid and fluid, giving the impression of gushing water. This piece, called “Fountain,” is a kinetic construction that plays with the limits of movement and material.
⛳ This week I discovered a game called Alphaputt. It combines typography with mini golf, sinking the ball on a separate course designed for each letter of the alphabet. A is an airport, Z is a zen garden, each with its own theme and sound. You can type any word and build your own course out of those letters. It comes as a separate app on Apple devices, but you can also play it in the browser.
🇺🇸 Since 2001 Gallup has asked Americans how proud they are to be American. The share who see themselves as extremely or very proud has fallen to fifty-three percent this year, the lowest level in twenty-five years. Those saying only “extremely proud” have slipped to thirty-three percent.
📺 A new version of LG’s viral wheeled screen, the StanbyME, is out. The StanbyME 2 Max is thirty-two inches and 4K, and thanks to a built-in battery that lasts four and a half hours it can roll from room to room without needing an outlet. The screen detaches from its wheeled stand with a single click and moves to a tabletop base, and works in both landscape and portrait. It costs thirteen hundred dollars. A very sweet piece for small, minimally furnished homes.
🎬 The first Star Wars film was made for eleven million dollars, money that today would barely cover the set of a single scene. A team called Secondhand Movie Co. asked the question in reverse. What if all you had was twelve dollars? With cardboard spaceships and duct-tape costumes they shot a full parody of the film from start to finish. Then they trimmed the budget by another dollar and tried eleven-, then ten-dollar versions too.