John Carlos Baez

This circle represents an octave. Each new note is a perfect fifth above the last. There are always either 2 or 3 sizes of space between notes, listed at the bottom. When there are 2, it's called a "moment of symmetry". Then the number in the middle - the number of notes - turns red and we pause for a while.

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‪The list of happiest countries puts Finland at top, followed by Iceland, Denmark, Costa Rica, Sweden and Norway.

Clearly to be happy you need to freeze your ass off in the dark for half the year, and Costa Rica is a lot further north than I thought.

https://data.worldhappiness.report/table

(Apparently the huge number for Costa Rica's GDP per capita is bad, not good.)

The proton, 1836 times heavier than the electron, is made of two up quarks and a down, with two of their spins aligned and one pointing the other way.

The same quarks with all spins aligned give a new particle, the Δ⁺, that's 2411 times heavier than the electron!

But the Δ⁺ is just the first of many 'excited states' of the proton: particles made of two up quarks and a down, but arranged in different ways, with higher energy and thus more mass. They quickly decay, often turning back into a proton.

There are two main kinds:

• If two quarks have spin pointing the same way and one points the other way, you get a particle with total spin

1/2 + 1/2 - 1/2 = 1/2

It could be a proton, but there are lots of others. Any particle of this kind is called an N*⁺.

• If all three quarks have their spins aligned, you get a particle with spin

1/2 + 1/2 + 1/2 = 3/2

Any particle of this kind is called a Δ⁺.

When we want to be precise, the Δ⁺ I mentioned before is called Δ(1232)⁺, because its energy at rest is 1232 MeV. That corresponds to its mass being 2411 electron masses. But then come a family of increasingly overweight relatives: the Δ(1600)⁺, Δ(1620)⁺, Δ(1700)⁺, Δ(1750)⁺, and so on, all of spin 3/2.

Similarly the proton can be called N(939)⁺, though it'd be like calling water dihydrogen monoxide. Then come the N(1440)⁺, N(1520)⁺, N(1535)⁺, N(1650)⁺, N(1675)⁺, N(1680)⁺, and so on - a seemingly endless series of increasingly heavy relatives, this time all of spin 1/2.

Physicists started studying these excited states, or 'resonances', in 1952. By the late 1960s, people were cranking them out. How to understand them???

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The arXiv is separating from Cornell University, and is hiring a CEO, who will be paid roughly $300,000/year:

https://jobs.chronicle.com/job/37961678/chief-executive-officer

They say:

"After decades of productive partnership with Cornell University, and with support from the Simons Foundation, arXiv is establishing itself as an independent nonprofit organization, marking the next stage in its 35-year history as a pioneer of open-access science."

The arXiv’s current annual budget is approximately $6 million and they employ ~27 staff members, most of whom work remotely, primarily in the U.S. The new chief executive officer (CEO) will be responsible for all aspects of arXiv, including strategic planning, financial management, technical infrastructure, personnel oversight and stakeholder engagement. They will work closely with board member representatives of Cornell University and the Simons Foundation to establish the organization’s independence.

A firm called Spencer Stuart is recruiting the CEO. For confidential nominations and expressions of interest, you can contact them at arXivCEO@SpencerStuart.com.  The salary is expected to be around $300,000, though the actual salary offered may differ.

In the Swedish town of Södertälje, crows are being paid to pick up cigarette butts. When they take one and stick it in a machine, they get rewarded with some peanuts. It's called "Corvid Cleaning":

https://www.samodobrevijesti.com/en/news/wild-crows-in-sweden-help-clean-up-cigarette-butts/

If this works - if the crows don't get bored of it, etc. - it would be cheaper than paying people to pick up this litter.

It's cute. But I don't like us domesticating one of the most visible proudly independent species in the urban landscape. I also wonder if the nicotine could hurt them. Instead of training crows to pick up cigarette butts, how about training humans not to leave them lying around?

Manet's famous painting Un Bar aux Folies-Bergère never appealed to me. But now I realize its genius, and my spine tingles every time I see it.

The perspective looks all wrong. You're staring straight at this barmaid, but her reflection in the mirror is way off to right. Even worse, her reflection is facing a guy who doesn't appear in the main view!

But in 2000, a researcher showed this perspective is actually possible!!! To prove it, he did a photographic reconstruction of this scene. Check it out in my next post.

This blows my mind.

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Here’s a great, simple article on applied category theory! I’m glad that it explains a bit about ‘green mathematics’ and my struggle to do mathematics that will help the world.

Natalie Wolchover interviewed me twice for this, and it also features Matteo Capucci, Brendan Fong, Bob Coecke, David Spivak, Amar Hadzihasanovic, Nathaniel Osgood and Tom Leinster.

https://www.quantamagazine.org/can-the-most-abstract-math-make-the-world-a-better-place-20260304/

The flip-flop was so big that you can still buy this T-shirt from the US Director of National Intelligence, who is selling it on Etsy:

https://www.etsy.com/listing/1227682931/no-war-with-iran

Some of the news that's getting drowned out: last week, the House Oversight Committee's ranking member, Rep. Robert Garcia, said:

"Yesterday, I reviewed unredacted evidence logs at the Department of Justice. Oversight Democrats can confirm that the DOJ appears to have illegally withheld FBI interviews with this survivor who accused President Trump of heinous crimes," Garcia stated.

Details of one interview follow - watch out, it's unpleasant. But I think it's important to know what we're not supposed to pay attention to.

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RE: @TobyBartels@mathstodon.xyz

My trickiest student is eager to puzzle you.

EARTH-SIZED PLANETS EVERYWHERE!!!

The Radius Valley (also called the "Fulton gap" or "photoevaporation valley") is a name for this interesting phenomenon: when you plot the radii of planets orbiting fairly close to Sun-like stars, there are two peaks with a gap between them.

One peak is at roughly 1.2–1.4 Earth radii: super-Earths, essentially rocky planets.

The other is at roughly 2–3 Earth radii: sub-Neptunes, with large atmospheres of hydrogen/helium or water.

Relatively few planets sit in between.

BUT, that's for Sun-like stars. This new paper claims that when we get down to red dwarfs, the gap goes away. Almost all of them have Earth-sized planets!

This is a big deal in that red dwarfs are the most common stars. However, you shouldn't jump to the conclusion that Earth-sized planets orbiting red dwarfs are good places for life like we see here. Young red dwarfs tend to emit nasty flares! The heavier ones, 0.3 solar masses or more, calm down after 1-2 billion years. The lighter ones take longer.

https://arxiv.org/abs/2602.23364

I'm enjoying the novel "There is No Antimemetics Division" by qntm. I'll quote a bit:

.....
Item #: SCP-3125

Object Class: Keter

Special Containment Procedures: SCP-3125 is kept inside
Cognitohazard Containment Unit 3125 on the first floor of Site 41.

This containment unit is a 10m by 15m by 3m cuboidal room clad in layers of lead, soundproofing and telepathic shielding. Access is through an airlock system at one end of the containment unit. This airlock is programmed to allow only one person to enter the containment unit at a time, and to remain locked until this person exits before allowing another person to enter.

Under no circumstances may any coherent information be allowed to leave the containment unit. This includes written and electronic notes, photographs, audio and video recordings, sound, electromagnetic and particle-based signals and psi emanations. During the exit cycle, a purge system rigged to the airlock flushes the occupant's memory by flooding the airlock with amnestic gas for three minutes.

A senior Antimemetics Division staff member must visit SCP-3125 every six weeks (42 days).

END OF FILE
.....

"You're kidding me. That's the whole entry?"

"That's the whole entry," Wheeler says.

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https://www.youtube.com/watch?v=w-IiVeGAydE&t=280s

The 'mediant' of two fractions a/b and c/d is (a+c)/(b+d).

Beginners sometimes add fractions this way by mistake.

But it's important in number theory...

... and I just learned it has a simple geometrical meaning, too!

https://en.wikipedia.org/wiki/Mediant_(mathematics)

RE: @adapalmer@wandering.shop

You've seen them in cave paintings. But during the Middle Ages, bison became extinct in much of Europe, surviving into the 20th century only in northern-central Europe and the northern Caucasus Mountains. By 1927 there were just 60 left! But over the last decade, their number has increased from just over 2500 to around 9000. Species can bounce back.

In the northern hemisphere, the sun moves clockwise in the sky. This is why clocks, which were based on sundials, have hands that move clockwise.

In 2014 the Bolivians finally decided to break free of this colonial legacy. They're in the southern hemisphere, after all! So the clock on their parliament now looks like this.

I like it. But it must make a tempting target for counter-revolutionaries.

The problem with archeologists is that the successful ones get a big head.

But wait - don't leave just because of that joke! I have something more interesting to say about the Olmecs - the folks who made these enormous stone heads. Like: what do Olmecs have to do with emacs?

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RE: @cbaberle@mathstodon.xyz

My god! What an amazing blog article! For the last few weeks I've been studying the work of Gene Ward Smith, who discovered a connection between music theory and the Riemann zeta function. But it turns out @cbaberle@mathstodon.xyz has been thinking about his for years... and what she has discovered is much richer and more beautiful than I had imagined. This changes everything!

Briefly: while the Riemann hypothesis talks about where the Riemann zeta function is *zero* on a certain line, music focuses our attention on where this function is as *big as possible* on that line.

Trump now says the quiet part out loud: "The Republicans should say: 'We want to take over. We should take over the voting in at least 15 places.' The Republicans ought to nationalize the voting." https://www.bbc.com/news/articles/c0mke841zj0o For folks outside the US: Elections in the US are run, not nationally, but by the individual states - until the final stage. Thus, nationalizing the elections would be a way to end fair elections in the US. We can guess that the "15 places" include states where ICE thugs currently roam in large numbers: states with Democrat-dominated cities such as Minneapolis, Portland, Chicago, New York City, Los Angeles and Atlanta. Trump can use ICE, the National Guard and - if he invokes the Insurrection Act - the military to suppress demonstrations against this takeover. I've been telling people this for a long time. The question is what to do about it. Right now: tell your representatives that funding DHS, the department ICE belongs to, is a very bad idea. You can also protect the elections locally. The Election Protection site (https://protectthevote.net/) is a resource for nonpartisan election observation and voter assistance. Also: study how civic organizations in Minneapolis have linked up to resist ICE, and copy that pattern in your local area ahead of time: https://www.advocate.com/news/minnesota-mutual-aid-groups-ice There are lots of other things to do, especially if you're a politician or have other forms of power.

Here's a tale of how nature triumphs in the end. Steel mills dumped molten slag in parts of Chicago and nearby areas. The slag hardened in layers up to 15 feet deep. These places became barren wastelands. Other industries dumped hot ash and cinders there. But eventually the steel mills closed. The deep layers of hard, toxic material were not friendly to plants. Cottonwoods are usually 30 meters tall or more. In the slag fields, stunted cottonwoods grow to just 2 meters. But rare species that could handle these conditions began to thrive. The lakeside daisy, a federally threatened species lost to Illinois for decades, turns out to grow taller on slag than on topsoil! The capitate spike-rush, last recorded in Illinois in 1894 and considered locally extinct, was rediscovered growing on slag. And more! Native prairie grasses like little bluestem. Native milkweeds. Even tiny white orchids called sphinx ladies' tresses. A team of women ecologists began studying these unusual landscapes. They call themselves the Slag Queens. (1/n)

Some good news in a time of darkness: the European wildcat, Felis sylvestris, is making a comeback! This thoughtful-looking example was photographed in a mountainous region of the Czech republic. The European wildcat's extreme elusiveness may have helped it avoid hunters in places where a larger native cat, the lynx, has been killed off. There may be about 140,000 European wildcats spread across more than two dozen countries. But they are very hard to find! Wildlife photographer Andrea Giovanni, who made a video of one, writes: "I'd never even thought of taking photos of wildcats, for a simple reason: I thought it was impossible, or at least, extremely difficult. It's considered 'the ghost of the forests' because it's very, very elusive, and it's hard to predict where it can be spotted. Other animals tend to follow the same trails through the forest. The wildcat goes wherever she wants to." One reason the European wildcat is coming back is increased legal protections. But another is that villages in Italy and other regions are becoming depopulated! Some are very worried about declining human populations. But it does make room for other species. That gives me some hope for the future. I got this picture, taken by Vladimír Čech Jr in the Doupov mountains, from a very nice article on the European wildcat: https://www.bbc.com/future/article/20260112-rare-images-of-europes-ghost-cat For more on this species: https://en.wikipedia.org/wiki/European_wildcat Seven subspecies have been demarcated! #cats #catsOfFedi #catsOfMastodon

Brent Mckean captured this amazing moonlit scene with three combined exposures on an icy winter morning in Manitoba, Canada. The colorful rings are a corona caused by diffraction by ice crystals near the direction of the Moon. Outside those rings, a 22-degree halo was created by moonlight refracting through ice crystals shaped like hexagonal prisms. On the left and right are 'moon dogs', caused by light refracting through thin, flat, 6-sided ice crystals. At the top and bottom of the 22-degree halo are 'upper and lower tangent arcs', created by moonlight refracting through nearly horizontal hexagonal ice prisms. A few minutes later the halo and arcs disappeared and the sky returned to normal -- with the exception of a single faint moon dog. Image source: https://apod.nasa.gov/apod/ap200224.html For more on tangent arcs: https://en.wikipedia.org/wiki/Tangent_arc

RE: https://mathstodon.xyz/@johncarlosbaez/115677835220476940 My post on Axiom Math had major errors, which I've tried to correct in the new version below. It was not Axiom Math but another company, Harmonic, that first used AI to solve some Erdős problems! Harmonic's program is called Aristotle. As of today Harmonic's news page claims: "And just recently, a beta version of Aristotle was used to solve and formally verify multiple open Erdos problems, which had been unsolved for decades." https://harmonic.fun/news Two are the problems discussed below. For the third, #480, Aristotle found a trivial proof by exploiting a formalization error. So, I now feel Business Insider was misleading when they wrote "Axiom, which recently said it solved two Erdős math problems that eluded mathematicians for decades, announced a $64 million seed round in September." even if perhaps Axiom has redone these proofs. But I certainly should have checked the reality more carefully before coming out with my article, especially given the rather dramatic conclusions I jumped to. I was quite confused. The head of Axiom Math clarified these issues here: https://mathstodon.xyz/@carina1248@mastodon.social/115679850066865211 I would not be surprised if I haven't completely gotten to the bottom of what's going on in this story, but this is my current understanding, and I apologize for getting some big things wrong before.

The largest fundamental science agency in Europe will save €1.4 million annually by breaking free from Clavariate Analytics' commercial databases of references and citations.

Starting January 1st 2026, the CNRS will stop subscribing from the Web of Science, along with the Core Collection and Journal Citation Reports.

"We have worked for free to lock ourselves collectively into a paid system", one head of the CNRS explains - but no more! 🎉

https://www.cnrs.fr/en/update/cnrs-breaking-free-web-science

We need an internet archive outside the US! We can't put all our eggs in one basket.

Oh, wait: we *have* an internet archive outside the US. Let's support it.

And now they have an office - a physical presence!

https://flaminghydra.com/freedom-and-sharing-at-the-internet-archive-europe/

https://www.internetarchive.eu/

In this talk I show how to multiply quaternions AND ALSO OCTONIONS using the dot product and cross product of vectors in 3 dimensions:

https://www.youtube.com/watch?v=JI5xPGN_sWo

For quaternions this is well-known, but for octonions it's less so.

I sketched the proof that both these algebras have a norm obeying

|ab| = |a| |b|

To see the details, go to the proof of Theorem 2 here:

https://golem.ph.utexas.edu/category/2020/07/octonions_and_the_standard_mod_1.html

I do the proof for octonions, but the same argument also works for quaternions!

#quaternions #octonions