I never knew what was so great about the "great icosahedron". Now I do! Take a regular icosahedron whose vertices have coordinates in ℚ[√5]. Apply the nontrivial element of the Galois group: that is, replace √5 by -√5. You get the great icosahedron! (1/n)

Asteroid collisions. Downed power lines in thunderstorms. Atomic bomb tests in deserts. These are the only things we know that make quasicrystals outside the lab! johncarlosbaez.wordpress.com/2026/05/14/n...

In 2025, researchers studied a quasicrystal forged in a hypervelocity asteroid impact roughly 4.5 billion years ago in the early solar system - and found that it contains 'phasons'! (1/n)

Is music secretly a kind of geometry? An 'augmented triad' is a kind of chord with 3 notes. If you draw the 12 notes of the scale as equally spaced dots on a circle, an augmented triad is simply any 3 notes that form an equilateral triangle! (1/2)

There are a few more tickets to see me talk about "What Are Atoms Made Of?" next Tuesday evening at Sloans Bar and Grill on 108 Argyle Street in Glasgow. It costs just £5.00 - and you can also hear two more talks. Tickets here: pintofscience.co.uk/event/seeing... Doors open at 18:30. Say hi!

The most creative thinkers manage to do just 4-5 hours of work a day, and take a lot of walks. Very hard for most of us. academic.oup.com/edited-volum...

Wow! Three kinds of quasicrystals come from a single meteorite found in Khatyrka, Russia, and nowhere else. It's the the only meteorite containing metallic aluminum. It was formed in a ultra-high-velocity collision between asteroids. en.wikipedia.org/wiki/Khatyrk... (1/2)

Most of the mathematicians I hang out with are opposed to AI, many rather virulently. The mathematicians who make *news* are the ones who are using AI to do surprising stuff: Tao and Gowers, for example. The ones who refuse to have anything to do with AI aren't making news.

'Blue bronze' is a crystal that can vibrate in two ways: ordinary sound waves, and charge density waves that move faster than sound. Thanks to quantum mechanics, waves act like particles. So in blue bronze we get 'phonons' - particles of sound - but also 'phasons' moving faster than sound! (1/n)

Archaea are single-celled life forms that prospered when the Earth was hot. You can still find them in extreme environments like hot springs, hydrothermal vents and salt lakes. Turns out they also make methane in your gut! Now when I fart, I'll blame archaea. en.wikipedia.org/wiki/Methano...

Fields medalist @wtgowers.bsky.social on how large language models are improving, and the dangers this poses. "I would judge the level of the result that ChatGPT found in under two hours to be that of a perfectly reasonable chapter in a combinatorics PhD." gowers.wordpress.com/2026/05/08/a...

When God ran out of simple, beautiful constants, it started making up really crazy ones. Here's my explanation of one that blew my mind - not why it is what it is, just what it means. johncarlosbaez.wordpress.com/2026/05/07/f...

Voted for the Scottish Parliament today! We get two votes: one for our local representative, winner-take-all, and one for regional representatives, who get seats in proportion to the fraction of people voting for that party. So, lots of game theory. These are the projections, not the reality:

Rudolph Laban used the icosahedron to map movement in expressionist dance and developed a notation for choreography based on it.

What's the most surprising fact you've learned in the last couple of weeks? I don't mind if it's quite technical. I just want to hear what you folks are being surprised by!

Jason Hise thinks he's found a way to partition the tetrahedral faces of the 600-cell into 6 isomorphic sheets of 100 tetrahedra! Each sheet is organized like a 10 × 10 lattice on a Clifford torus. Animation below. He asks: has anyone seen this before? entropygames.net/files/shared...

A brand new diamond open-access math journal for students to publish in. Could be good! journals.calstate.edu/im

Now, for Chrome users: a 4 gigabyte LLM sneaks onto your computer whether you ask for it or not! It's possible to remove it, but hard to keep it out. www.thatprivacyguy.com/blog/chrome-...

I have never seen it open.

The nerdy delights of the Edinburgh Hacklab.

The war first cut oil supply and then oil *demand*. Total world oil demand is now down 5 million barrels per day, or 5%. “Demand destruction will come and is coming in waves. Asia was first in line…” www.worldoil.com/news/2026/4/...

The silver lining: especially if it keeps on going, the Iran war will hasten the collapse of fossil fuels. Solar power adoption is already shooting up.

Stanislaw Ulam on the dangers of too much concentration when working in cafes with Banach, Mazur and other mathematicians. This was the mathematical scene in pre-WW2 Poland - more specifically, Lwów. Then the Nazis invaded and exterminated many people. Now it's the Ukrainian city of Lviv.

With the old system of COP climate summits fatally compromised by the fossil fuel industry, some are starting an alternative.

At first I was grumpy about this book by @richardelwes.bsky.social, because I wanted to write a book about big numbers myself someday. But this book is so entertaining and intelligent I couldn't stay upset for long. Enjoy it!

If you are in Oxford and would like to learn about applied category theory, I'm giving a course at the Maths department. It's on Fridays, at 11 am, in L5. We start this week with monoidal categories and string diagrams.

Some people get a bit carried away in their veneration of Grothendieck.

They forgot to add "which is idiotic".

The wild donkeys near my house can be a nuisance, but the young ones are awfully cute. And hey, we humans are a bigger nuisance. What's this little guy doing crossing its legs?

As we descend from the crust into the mantle, it seems matter becomes simpler - more easy to describe mathematically. For example, 520–660 km down we get a lot of ringwoodite, made of octahedra and tetrahedra. I guess under pressure there's just less room to fuck around.

The academic equivalent of that hand sign you make to quietly alert others that you're being threatened.

Algebraic geometry mind-blower of the day, from my friend Todd Trimble: a product of zero-dimensional schemes can have as large a dimension as you want. (Of course I don't know much commutative algebra, so my mind is easily blown on matters such as this.) (1/2)

The Standard Model keeps winning in experiments here on Earth. The seeming conflict between different measurements of the proton radius puzzle got solved, and so has the problem with the magnetic moment of the muon! The big mysteries seem to be in outer space... like dark matter.

You see a Dürer-drawn celestial chart: an analog computer from the early sixteenth century. Have a seat, and enjoy this short paper instrument's 🧵 for #skystorians and more:

1/1² - 1/2² + 1/4² - 1/5² + 1/7² - 1/8² + ⋯ Together with Calegari, Dimitrov and Tang proved this number is irrational... and they just won a $100,000 prize! It's the biggest advance in irrationality since Apéry showed ζ(3) = 1/1³ + 1/2³ + 1/3³ + ⋯ is irrational back in 1978. (1/n)

Alex Bores is running for Congress in New York. Ads are coming out attacking him because he worked for Palantir. These ads are funded by... Palantir. Why? Because Bores wants to regulate AI. youtu.be/znKb71kLG5c

People are talking about a lot of things here - but sadly, not about how most of the Earth is made of peridotite.

Käthe Seidel was a pioneer in ecological engineering, using plants like reeds and rushes to clean waste water. This is standard now, but mocked when she started doing it in the 1950s. Elisabet Sahtouris interviewed her in the 1990s - before her work caught on:

In a famous paper, some economists estimated the annual value of the whole biosphere's ecosystem services at $33 trillion per year. Michael Toman quipped that this is "a serious underestimate of infinity". www.nature.com/articles/387...

The good old days, when the midwestern US had a large network of electric railways... wait, what? Yes: this map shows what it looked like! By 1920 there were 25,000 kilometers of electric passenger trains, called 'interurbans'. en.wikipedia.org/wiki/Interur...

While AI attracts a lot of loud hype and haters, some mathematicians are happily using it to knock off unsolved problems. (I'm not saying anyone is "right" here.) Recently Liam Price prompted GPT-5.4 Pro to prove this conjecture of Erdős: chatgpt.com/share/69dd1c... (1/n)

A newly discovered newt: Tylototriton vietnamirabilis. Its last name is a Latin version of "Vietnamazing". And this fellow, so far only seen in the evergreen forests of Phu Xai Lai Leng Mountain in Vietnam, is indeed an amazingly cute new newt. zookeys.pensoft.net/article/1738...

This paper claims you can define all elementary functions using just two primitive operations: the constant 1, and the binary function eml(x, y) := e^x - log(y). It's probably true but it may need work because it effectively defines i as √-1 ignoring the two choices here. arxiv.org/abs/2603.21852

Since the dawn of agriculture we have roughly halved the mass of life on Earth. Wild mammal biomass has probably declined by something like 85% since the late Pleistocene! But mammals are only about 0.4% of all animal biomass, which is only about 0.5% of all biomass. Plants still rule! 🌿

President Bush and Secretary Rumsfeld thought that the Iraq-Afghanistan war would cost $40 billion and it ended up costing at least $3 trillion, not counting at least this much in veteran's benefits. So they were off by a factor of ~100. How much will this one cost? www.wbur.org/onpoint/2026...

The biggest things we do as a species, measured in sheer mass: Last year we dug up about 50 gigatonnes of rocks, gravel and sand and made about 30 gigatonnes of concrete. We also dug up 10 gigatonnes of carbon and burned it. www.nature.com/articles/d41...

Don't mix up "gold open access", where you have to pay a lot to let people read your articles for free, with "diamond open access", where you don't.

We like animals. But the entire animal kingdom - every insect, fish, bird, mammal, worm, crustacean, etc. - counts for less than 0.4% of global biomass! Plants: ~450 gigatonnes of carbon Bacteria: ~70 Gt C Fungi: ~12 Gt C Archaea: ~7 Gt C Protists: ~4 Gt C Animals: ~2 Gt C

If you're in Glasgow on May 19th you can see me explain subatomic particles at Sloan's Bar and Grill, as part of an event taking place across Scotland called "A Pint of Science". And please ask questions! pintofscience.co.uk/event/seeing...