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

Horrifying but not surprising:

Under Trump, the so-called Environmental Protection Agency has decided to completely give up on all carbon regulations. They're delaying approvals of wind and solar energy projects on federal lands. And they're halting the measurement of greenhouse gases in the atmosphere.

A bit more surprising to me:

Meanwhile, in the UK, the government has launched a £21.7 billion scheme to catch carbon dioxide from major industries and pump it into rocks under the North Sea, so gas companies can claim to be clean. This price is only for the building, not the operation. Paid for by us. If it's anything like 3 previous UK-run carbon capture and storage schemes, it will go over budget and then fail. It's an attempt at greenwashing launched by the previous Tory government. Now Labour is foolishly going along with it.

"A more reliable and cost-effective means of sequestering carbon would be to bundle up the money (roughly 1,100 tonnes in £20 notes) and shove it down a pipe."

The power of the fossil fuel industries to capture governments is unparalleled. They don't care about the future: they just want money now. And they know it pays to spend money, to buy politicians.

https://www.theguardian.com/commentisfree/2024/oct/11/labour-carbon-capture-climate-breakdown

@GreenSkyOverMe - none yet, so I resort to social media. If that doesn't work I may have to publish a paper.

Here's a shocking consequence of Newton's law of gravity. I've been running around Edinburgh saying this to anyone who will listen to me:

A planet orbiting the Sun has the same symmetries as a massless particle moving at the speed of light around a sphere in 4-dimensional space!

Their energies are different, but related. If the massless particle has energy A, the energy of the planet is

E = -1/8A²

(in some convenient units).

The same thing works in quantum mechanics, where now we have a hydrogen atom instead of a planet. But there's a slight twist!

Now, if the massless particle has energy A, the energy of the hydrogen atom is

E = -1/8(A² + ¼)

(in some convenient units where Planck's constant is 1).

Where does the extra ¼ come from?

There are a lot of little twists like this in quantum mechanics. For example, the energy of a harmonic oscillator has an extra ½. Mathematicians have thought a lot about these things, and a lot is known about them... but the big picture still seems mysterious to me.

In this blog article, I sketch how the ¼ in the hydrogen atom is related to the 1/24 that's so important in string theory... the twist that makes bosonic strings only work in 26 dimensions.

https://johncarlosbaez.wordpress.com/2025/07/26/the-kepler-problem-part-6/

List all the numbers

1, 2, 3, 4, 5, 6, 7, ...

skip every second one:

1, 3, 5, 7, ...

form the partial sums like this:

1, 1+3, 1+3+5, 1+3+5+7, ...

Hey, you get the square numbers!

1, 4, 9, 16, ...

Lots of people know that. But now list all the numbers

1, 2, 3, 4, 5, 6, 7, ...

skip every *third* one:

1, 2, 4, 5, 7, 8, 10, 11, ...

then form the partial sums:

1, 1+2, 1+2+4, 1+2+4+5, 1+2+4+5+7 ...

=

1, 3, 7, 12, 19, ...

then skip every *second* one:

1, 7, 19, ....

then form the partial sums again:

1, 8, 27, ...

Hey, now you get the cubes! You shouldn't trust me based on so little evidence, so do some more, or prove it works.

But the cool part is that this pattern goes on forever. If you list all the natural numbers starting from 1, skip every nth one, form the list of partial sums, skip every (n-1)st one, form the list of partial sums, skip every (n-2)nd one, ... blah di blah di blah... skip every 2nd one, then form the list of partial sums, you get the nth powers!

This is called Moessner's theorem, and I learned about it from Michael Fourman. It's in Chapter 7.5 here:

• Jan Rutten, The Method of Coalgebra: Exercises in Coinduction, https://ir.cwi.nl/pub/28550/rutten.pdf

Moral: anytime you see a pattern in mathematics - one that goes on infinitely, not a coincidence! - it's probably just the tip of a bigger iceberg.

@plyspomitox - wow, that's a nice pattern. I've known

1/7 = 0.142857142857...

and the 14 and 28 struck me, but I never thought about it too hard.

China is carrying out the solar revolution at scale. Here panels float on a reservoir in northwest China. This is just one of many massive solar power plants. The photos look like something out of science fiction:

https://www.theatlantic.com/photography/archive/2025/07/photos-china-solar-power-energy/683488/

Meanwhile Trump is fighting solar and pushing coal. The Chinese call him 特朗普, "Comrade Jianguo". Literally this means "Comrade Building the Nation". It's a joke about "Make America Great Again". It hints that Trump is actually helping China.

@pneumaculturist - I'm reading a bit more about it:

“My friend, the way it is with us Bushmen,” he [Tomazo, a Kalahari San] began, “is that we love meat. And even more than that, we love fat. When we hunt we always search for the fat ones, the ones dripping with layers of white fat: fat that turns into a clear, thick oil in the cooking pot, fat that slides down your gullet, fills your stomach and gives you a roaring diarrhea."

“If the Ihalmiut [Inuit, NW Canada] hunter shoots a deer for food when he is on a trip far from the camps, he seldom bothers to go to the trouble of building a fire. Usually his first act is to cut off the lower legs of the deer, strip away the meat, and crack the bones for marrow. Marrow is fat, and an eternal craving for fat is part of the price of living on an all-meat diet."

https://www.sciencedirect.com/science/article/abs/pii/S0278416520300544

@pneumaculturist - it's believed that Neanderthals had language. So once someone, somewhere, somehow, figured out that it was not good to only eat lean meat, this knowledge would get passed down. And natural selection is a powerful force too.

But it would be really cool to be there and watch how they figured this stuff out.

Now I better understand how humans domesticated wolves. In winter we weren't just giving them scraps. We were giving them steaks and roasts.

Today I learned about 'rabbit starvation' and how Neanderthals avoided it.

When you're a hunter-gatherer and it's winter, you may try to survive by eating only meat - like rabbits, but also deer and other game. But this gives you too much protein and not enough carbohydrates and fat: most of this meat is very lean. If you eat enough lean meat to get all the calories you need, you can die from an overdose of protein! It's called 'protein toxicity'.

Hunter-gatherers in this situation sometimes throw away the 'steaks' and 'roasts' - the thighs and shoulders of the animals they kill - or feed them to their dogs. They need FAT to survive! So they focus on eating the fatty parts, including bone marrow.

So, in some cultures, while the men are out hunting, the women spend time making bone grease. This takes a lot of work. They take bones and break them into small pieces with a stone hammer. They boil them for several hours. The fat floats to the top. Then they let the water cool and skim off the fat.

There's been evidence for people doing this as far back as 28,000 BC. But now some scientists have found a Neanderthal 'bone grease factory' that's 125,000 years old!

This was during the last interglacial, in Germany. In a site near a lake, called Neumark-Nord, Neanderthals killed a lot of bison, horses and deer and crushed their bones, leaving behind tens of thousands of small bone fragments.

• Lutz Kindler et al, Large-scale processing of within-bone nutrients by Neanderthals, 125,000 years ago, https://www.science.org/doi/10.1126/sciadv.adv1257

Thanks to @sarahtaber for spotting this!

Edward Teller is mainly famous as the "father of the hydrogen bomb". But he wasn't dumb. He was invited to speak in 1959 at a big party in New York for the 100th birthday of the oil industry - a party put on by the American Petroleum Institute. Over 300 government officials, economists, historians, scientists, and industry executives were there.

And this is what he said:

‘”Whenever you burn conventional fuel, you create carbon dioxide.... The carbon dioxide is invisible, it is transparent, you can’t smell it, it is not dangerous to health, so why should one worry about it?"

“Carbon dioxide has a strange property. It transmits visible light but it absorbs the infrared radiation which is emitted from the earth. Its presence in the atmosphere causes a greenhouse effect …. a temperature rise …. sufficient to melt the icecaps and submerge New York. All the coastal cities would be covered, and since a considerable percentage of the human race lives in coastal regions, I think that this chemical contamination is more serious than most people tend to believe.”

https://www.theguardian.com/environment/climate-consensus-97-per-cent/2018/jan/01/on-its-hundredth-birthday-in-1959-edward-teller-warned-the-oil-industry-about-global-warming

@ArchaeoIain - yes, the web page from the Institut de Ciències del Mar extrapolates a bit:

"The upwelling of deep, warm, CO₂-rich waters is believed to be driving the accelerated melting of sea ice in the Southern Ocean. In the long term, this process could double current atmospheric CO₂ concentrations by releasing carbon that has been stored in the deep ocean for centuries—potentially with catastrophic consequences for the global climate."

@matejpp - the usual story is that surface water gets very cold due to contact with the air near the Arctic and Antarctic. Cold water is more dense, so it sinks. This pulls in warmer surface water, which comes from nearer to the equator. That in turn pulls cold deep water up near the equator. It goes round and round.

If you want clear diagrams, science journalism is not very good, since many journalists don't understand the science, so they don't know what diagrams would help. Instead, you should read something like Wikipedia. Or you ask someone like me. 🙂

I agree that this is an unsatisfactory state of affairs! But saying "science still lacks communication skills" neglects that there are many people involved: research scientists, professional journalists, freelance science explainers, and so on. Improving the situation requires finding a way to change the motivations for these different groups of people. For example, research scientists are mainly paid to discover new things, and explain those things to other research scientists - not to ordinary people or even journalists.

The Trump gang just took down a bunch of legally mandated climate assessments. Meanwhile, here in actual reality, the climate doesn't give a fuck:

“We are witnessing a true reversal of ocean circulation in the Southern Hemisphere—something we’ve never seen before. While the world is debating the potential collapse of the AMOC [Atlantic Meridional Overturning Circulation] in the North Atlantic, we’re seeing that the SMOC is not just weakening, but has reversed. This could have unprecedented global climate impacts.”

That's a quote from Antonio Turiel, one author of a new paper in the Proceedings of the National Academy of Sciences. This study found that since 2016, the water in parts of the Antarctic Ocean is getting more salty. This suggests that the deep ocean circulation in the southern hemisphere—known as the SMOC—has reversed. Instead of sinking into the depths, surface water is now being replaced by deep water coming up to the surface.

This reversal seems to be driving the accelerated melting of sea ice in the Antarctic Ocean:

https://www.icm.csic.es/en/news/major-reversal-ocean-circulation-detected-southern-ocean-key-climate-implications

This web page from the Institut de Ciències del Mar says

"In the long term, this process could double current atmospheric CO₂ concentrations by releasing carbon that has been stored in the deep ocean for centuries—potentially with catastrophic consequences for the global climate."

But never mind, just close your eyes. Don't read this:

• Alessandro Silvano et al, Rising surface salinity and declining sea ice: A new Southern Ocean state revealed by satellites, https://www.pnas.org/doi/10.1073/pnas.2500440122

This shape is the 3d associahedron.

To get it, take a hexagon and triangulate it by drawing red lines between its corners - lines that don't cross each other. There are 14 ways to do this, and these are the vertices of the associahedron. You get an edge of the associahedron from two triangulations that differ by changing just one red line. You get a face from all the triangulations that have one red line in common.

If you replace the hexagon by a polygon with more sides, you get a higher-dimensional associahedron. The associahedra have many magical properties, and here's one of the most astounding.

Take a formal power series like this:

𝐶(𝑥) = 𝑥+𝑐₁𝑥²+𝑐₂𝑥³+⋯

If you take its inverse under composition, meaning the power series 𝐷 with

𝐶(𝐷(𝑥))=𝑥

you get another formal power series of the same type:

𝐷(𝑥) = 𝑥+𝑑₁𝑥²+𝑑₂𝑥³+⋯

How are the numbers 𝑑ₙ related to the numbers 𝑐ₙ? Do some calculations:

𝑑₁ = −𝑐₁
𝑑₂ = −𝑐₂ + 2𝑐₁²
𝑑₃ = −𝑐₃ + 5𝑐₂𝑐₁ − 5𝑐₁³
𝑑₄ =−𝑐₄ + 6𝑐₃𝑐₁ + 3𝑐₂² − 21𝑐₂𝑐₁² + 14𝑐₁⁴

What are these coefficients? They're controlled by the associahedra! I'll show how it works for 𝑑₄.

Call the n-dimensional associahedron 𝑐ₙ₋₁, so that 𝑐₁ is a point, 𝑐₂ is an interval, 𝑐₃ is a pentagon, and so on. From the picture notice that the 3d associahedron 𝑐₄ has

• 1 face shaped like 𝑐₄ (the whole thing)

• 6 faces shaped like 𝑐₃ × 𝑐₁ (pentagons) and 3 faces shaped like 𝑐₂ × 𝑐₂ (squares)

• 21 faces shaped like 𝑐₂ × 𝑐₁× 𝑐₁ (the edges)

• 14 faces shaped like 𝑐₁ × 𝑐₁ × 𝑐₁ × 𝑐₁ (the vertices)

All this information is packed into here:

𝑑₄ = −𝑐₄ + 6𝑐₃𝑐₁ + 3𝑐₂² − 21𝑐₂𝑐₁² + 14𝑐₁⁴

Look at it!

We get the other 𝑑ₙ from the associahedra of other dimensions, in the same way!

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If you're the kind of person who follows me, you may know about the Kessler Syndrome.

That's when collisions between satellites and space junk create enough debris to cause *more* collisions, leading to a runaway chain reaction. This could render certain regions of near-Earth space unusable!

It's one of nature's ways of containing stupid civilizations, sort of like how inflammation contains infections. So don't be surprised:

A new study by Lewis and Kessler argues that we've hit the "runaway threshold" - the point where a chain reaction is expected - at nearly all altitudes between 520 and 1000 kilometers.

Below that, or above that, space could remain usable. So we could still get out and ruin other layers of space - or go to other planets and mess up those. Luckily, planned deployments of large satellite constellations like Starlink, Amazon's Project Kuiper, etc. will reduce the risk of such a breakout.

Yes, I'm joking - we can differ on whether the expansion of stupidity into the cosmos would be a good or bad thing compared to a mostly dead cosmos, and I don't really have an opinion on that. But the study is for real, and worth checking out:

• Hugh G. Lewis and Donald J. Kessler, Critical number of spacecraft in low Earth orbit: a new assessment of the stability of the orbital debris environment, https://conference.sdo.esoc.esa.int/proceedings/sdc9/paper/305/SDC9-paper305.pdf

Thanks to @michael_w_busch for pointing this out.

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Crown Prince Mohammed bin Salman, wanted to have at least 5 Saudi universities among the top 200 in the world by 2030.

So he paid lots of scientists to falsely claim they worked in Saudi Arabia! Some got up to €70,000 a year just for pretending to work there.

Unfortunately they got caught. The magazine El Pais exposed the scam.

In 2022, Saudi Arabia had 109 professors on the List of Highly Cited Researchers, compiled by Clarivate. After they got caught, the number dropped to 26. 🤣

In fact this list is a fraud magnet. As of 2024, one in three apparently highly cited researchers has been caught engaging in bad practices.

This is a great example of Goodhart's Law: ""When a measure becomes a target, it ceases to be a good measure".

Thanks to @mansr for pointing this out.

https://english.elpais.com/science-tech/2024-12-05/dozens-of-the-worlds-most-cited-scientists-stop-falsely-claiming-to-work-in-saudi-arabia.html

Great! A bunch of us here wanted it. Now it exists. 👍

It's a "dark archive" of the arXiv - a non-public backup to save the data in case of attack by hackers or the US government. The arXiv, I hope you know, is the biggest source of modern math and physics papers.

Who got the job done? The TIB: the Technische Informationsbibliothek, run by the Leibniz Information Centre for Science and Technology, in Hannover, Germany.

They write:

"The TIB has now set up a so-called dark archive for the arXiv content in order to be able to make the backed-up data accessible if the data stored in the USA is lost. The archive functions as a silent reserve: the complete copy of the content is stored decentrally at the TIB, but is not publicly accessible. This means that the data stock – almost 10 terabytes – is protected against potential outages and can be activated in an emergency.

The TIB is currently working on processes to keep the archive up to date: new submissions and updated versions must be backed up regularly in order to preserve the state of research as completely as possible.

“Building a Dark Archive is an expression of our longstanding commitment for a reliable, international academic provision, and as a partner of arXiv. Even though the Dark Archive today only works in the background, it is a key element in safeguarding digital research contents in the long term, because in case of a crisis, we could open the archive,” explains Dr Irina Sens, Deputy Director of the TIB."

We should call it the darXiv.

More details here:

https://blog.tib.eu/2025/05/14/protecting-science-tib-builds-dark-archive-for-arxiv/

Here's another system archetype. There are two ways to solve a problem: a fundamental solution that takes a while to have any effect, and a quicker symptomatic solution... which unfortunately has a side-effect that makes the problem worse!

Like scratching a rash.

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The more cash you need, the more you want to borrow. In the short term this reduces your need for cash, but after a while it increases it. A vicious cycle.

The same pattern shows up all over the place, where a short-term fix makes the problem worse in the long run. This pattern is an example of a 'system archetype', and we can draw it as a diagram. Different instances of this archetype will give diagrams with different words - but the archetype is the pattern of edges with plus signs, minus signs and delays.

I've been working on the math of this stuff. I'm fascinated by how a general problem that haunts my life - I like to put off solving problems, and wind up making them worse - can be summarized as a simple diagram.

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