Notices by John Carlos Baez (johncarlosbaez@mathstodon.xyz)

I don't really listen to music analytically, so I'd never noticed the infectious groove of Steely Dan's "Peg" was based on the moving the Hendryx chord around the circle of fifths, and then applying tritone substitutions!

Those are three music theory nerd delights wrapped up in a neat package. Charles Cornell explains them pretty clearly here. Then he gets sidetracked by *other* cool things going on in this song....

https://www.youtube.com/watch?v=PHHAzbKR3xk

RE: https://mamot.fr/@pluralistic/116407495473802207

In 2022 the Spanish parliament recognized the legal personhood of a lagoon called the Mar Menor, thus making it the first ecosystem in Europe with its own rights. This follows similar decisions in some other countries:

https://en.wikipedia.org/wiki/Rights_of_nature_law

I've been thinking about a lot of stuff lately. Literally. Humans are currently using over 1,100 gigatonnes of stuff, which exceeds the mass of all living creatures on Earth. Also, since the dawn of agriculture we have roughly halved the mass of life on Earth, mainly by killing trees.

Animals count for only about 0.5% of all biomass, and mammals only about 0.4% of all animal biomass. Wild mammal biomass has probably declined by something like 85% since the late Pleistocene. Now over 90% of mammal biomass is livestock!

https://www.visualcapitalist.com/all-the-biomass-of-earth-in-one-graphic/
https://refubium.fu-berlin.de/handle/fub188/32375

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!

🌱 By mass, plants rule. 🌱

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

https://doi.org/10.1073/pnas.1711842115

If you're in Glasgow on May 19th you can see me explain subatomic particles at Sloans Bar and Grill, as part of an event taking place across Scotland called "A Pint of Science". And please ask questions and say hi! It costs just £5.

https://pintofscience.co.uk/event/seeing-the-unseeable/

Wow! While I was driving to New Mexico to visit archaeological sites, Ukraine was hammering oil refineries in Russia with drones! How come none of you told me? 😜

Urkaine sent drones over 1000 kilometers to destroy oil storage tanks and shipping infrastructure at the ports of Ust-Luga and Primorsk on the Gulf of Finland. These ports were rendered unable to ship any cargo. Reuters estimated that Russia lost about 40% percent of its oil export capacity while Bloomberg put the figure at 43%.

As you know, Putin had been enjoying the oil price spike due to the Iran war. Not so much now. He'll try to rebuild the ports and/or reroute the oil. But until he gets an effective defense against these drones, the Ukrainians can just repeat what they've done.

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https://www.aljazeera.com/features/2026/4/6/smell-of-war-comes-to-st-petersburg-as-ukraine-hammers-russian-refineries

In case like me you hadn't been keeping up on the Ukraine war, you might like to watch Anders Puck Nielsen analyze the current situation and prognosticate here.

He and the host agree that Ukraine is now killing off Russian soldiers faster than Russia can recruit them. I never know if I can trust these numbers, but Ukraine claims to have killed ~35,000 last month and has a plan to kill 50,000/month - mainly by increasing its production and use of drones, I guess. Anders and the host say that Putin will be driven to conscript troops, which so far he's avoided doing - and that this will force him to increase various repressive measures. But they think this will only *slow* the decline of Russian military power: the only way out for Putin will be to stop the EU from supporting Ukraine financially. The defeat of Orbán makes that harder for him. So, they expect Putin will eventually threaten the Baltic states with war unless the EU stops supporting Ukraine. He's already threatened Estonia, Finland and Latvia for letting Ukrainian drones fly over their territory.

In short, they think Putin is losing but he'll become more dangerous as he becomes more desperate.

(2/2)

https://www.youtube.com/watch?v=iqkWf12Wcxo

@pr_ret_lutz - who is rejoicing about that?

The eldritch horrors of math.

This is the graph of a simply defined sequence - yet it looks like a demonic force is shredding a diagonal line.

Details are here, and in the alt text: https://oeis.org/A394413

Hurrah! I hope the Hungarians can reconstitute their democracy now, and hold on to it.

It's also great that both Vance and Putin failed to get Orbán re-elected. It's insane that Vance went to Budapest and campaigned for Orbán.

https://apnews.com/article/hungary-election-orban-magyar-trump-1a4eb0ba6b94e0c80c3cd18bd36254ab

#hungary2026 #ViktorOrban

15% of Oklahoma is under jurisdiction of the Choctaw Nation. Now the Choctaw have used their power to prevent ICE from getting their hands on a big detention center.

https://www.projectsaltbox.com/p/choctaw-nation-buys-former-big-lots

The Higgs boson gives elementary particles their mass, but 98% of the visible mass in the Universe (not dark matter) comes from a less famous mechanism: chiral symmetry breaking. This is why protons and neutrons are so much heavier than their quarks!

Briefly, protons and neutrons act like bags full of a soup of virtual quark-antiquark pairs, which give them most of their mass. This soup, called a 'quark condensate', breaks a certain symmetry that exists outside the bag: 'chiral symmetry', where you change the phase of the clockwise and counterclockwise rotating quarks separately. In the quark condensate, the clockwise spinning virtual quarks are entangled with counterclockwise spinning virtual antiquarks.

https://en.wikipedia.org/wiki/Chiral_symmetry_breaking

RE: https://mastodon.social/@sundogplanets/116246751176637018

What? 40 old nuclear reactors in orbit?!?

RE: https://mathstodon.xyz/@mjb/116234235269708669

I knew a petition like this would appear. Now it's here.

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.

RE: https://mstdn.social/@ComicContext/116202288465898190

That feeling when you get the email saying your paper has been reviewed.

Two referee reports are attached.

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.

(1/3)

Wow - two new mammal species have just been discovered! We had thought we'd found them all. Even better, one is of an entirely new genus.

One is a striped possum with an extraordinarily long fourth finger - twice as long as the rest - that it uses to get wood-boring insect larvae to eat. It was known to have lived in West Papua until about 6,000 years ago... but it's actually still there!

The other is a ring-tailed glider: a marsupial that can soar downwards through the air. Fossils from this genus have been found in eastern Australia and New Guinea, but those are hundreds of thousands of years old! This one was also found in West Papua.

Both these new mammal species were found by the same team, who must be feeling ecstatic right now. Of course they're not really "new": they were just flying under the radar.

https://www.theguardian.com/environment/2026/mar/05/marsupials-discovered-new-guinea

Let's look at this wacky procedure in a more interesting case: when the number in boldface is even bigger, like 5. See below! This fraction is

1/(1 + 1/(1 + 1/(2 + 1/(2 + 1/(3 + 1/(1 + 1/5)))))) = 179/306

The previous fraction on the list is

1/(1 + 1/(1 + 1/(2 + 1/(2 + 1/(3 + 1/1))))) = 31/53

and before that is

1/(1 + 1/(1 + 1/(2 + 1/(2 + 1/3)))) = 24/41

To look for good approximations to log₂(3/2) that aren't on our list, we write down this funny thing:

(31n + 24)/(53n + 41)

Taking n = 0,1,2,3,4,5 this gives 24/41 (on our list), 55/94, 86/147, 117/200, 148/253, 179/306 (on our list).

The four not on our list are new candidates for fractions closer to log₂(3/2) than any fraction with a smaller denominator! But only the last two actually have this good property: 117/200 and 148/253.

In general, when we get an even number of new candidates this way, the last half have this good property. The first half do not.

When we get an odd number of new candidates, it becomes more tricky. The middle one can go either way - but all those after it are closer to log₂(3/2) than any fraction with a smaller denominator, and none before are.

There is a rule to decide this tricky middle case but you've probably had enough by now!

Again: what makes all this stuff worth knowing is that it gives the best rational approximations of *any* positive irrational number, not just log₂(3/2). And this is relevant to resonance problems like the rings of Saturn, which have gaps at orbital periods that are close to simple rational multiples of the periods of the big moons.

(2/n)

There are various places to read more about this stuff. I haven't read them yet, I'm ashamed to say!

But first, three useful buzzwords.

• The best approximations to an irrational number coming from truncating its continued fraction are called 'convergents'.

• The other candidates for being best approximations, obtained by the weird procedure I described, are called 'semiconvergents'. These include convergents as a special case.

• Given two fractions a/b and c/d their 'mediant' is (a + c)/(b + d). The weird procedure I described is based on mediants. Starting from the numbers 0/1 and 1/0 you can build a tree of numbers by taking mediants, called the 'Stern-Brocot tree'. See below!

Here are some books:

• Khinchin's "Continued Fractions" covers best approximations and semiconvergents carefully, including the delicate middle case.

• Rockett and Szüsz's "Continued Fractions" goes into the best-approximation theory in lots of detail.

• If you like the Stern–Brocot tree, you may like to think about how semiconvergents are connected to that. For this, see Conway and Guy's "The Book of Numbers", and Graham, Knuth, and Patashnik's "Concrete Mathematics".

Or read this:

https://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree

All this from trying to understand equal-tempered scales!

(3/n, n = 3)