Nostr notes by John Carlos Baez

I again feel the need to point out that despite all the immense ...

2026-05-09T12:57:59+02:00

I again feel the need to point out that despite all the immense dangers of LLMs, mathematicians are seeing rapid improvements in their capabilities... which may be yet another danger. Fields medalist Timothy Gowers recently blogged about using ChatGPT 5.5 Pro to improve a recent result in combinatorics. I'll skip the math and quote his summary:

"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. It wouldn’t be considered an amazing result, since it leant very heavily on Isaac’s ideas, but it was definitely a non-trivial extension of those ideas, and for a PhD student to find that extension it would be necessary to invest quite a bit of time digesting Isaac’s paper, looking for places where it might not be optimal, familiarizing oneself with various algebraic techniques that he used, and so on.

It seems to me that training beginning PhD students to do research, which has always been hard (unless one is lucky enough, as I have often been, to have a student who just seems to get it and therefore doesn’t need in any sense to be trained), has just got harder, since one obvious way to help somebody get started is to give them a problem that looks as though it might be a relatively gentle one. If LLMs are at the point where they can solve “gentle problems”, then that is no longer an option. The lower bound for contributing to mathematics will now be to prove something that LLMs can’t prove, rather than simply to prove something that nobody has proved up to now and that at least somebody finds interesting."

Arguable, but worth paying attention to.

https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/

Here's my big recent surprise: the number F = (2221564096 + ...

2026-05-07T21:52:22+02:00

In reply to nevent1q…9uxu
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Here's my big recent surprise: the number

F = (2221564096 + 283748 sqrt(462)) / 491993569

plays a fundamental role in number theory!

For any irrational x, we define its 'Lagrange number' to be the supremum of c such that

|(p/q) - x| < 1/cq²

has infinitely many solutions for rationals p/q. So, the bigger the Lagrange number is, the easier x is to approximate by rational numbers. Quite famously, the golden ratio has the smallest possible Lagrange number, namely √5.

Here's the shocking fact: every real number ≥ F is a Lagrange number, and F is the smallest number with this property!

F is called 'Freiman's constant', because he proved this fact. His proof is 100 pages, and I don't want to read it... but some people have.

There's a lot more crazy stuff about the set of all Lagrange numbers. A tiny bit is here:

https://en.wikipedia.org/wiki/Markov_spectrum#Lagrange_spectrum

- I thought only Trump wrote about the "Noble prize".

2026-05-07T17:51:07+02:00

In reply to nevent1q…cqrr
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- I thought only Trump wrote about the "Noble prize".

What's the most surprising scientific fact you've learned ...

2026-05-07T12:46:26+02:00

What's the most surprising scientific 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!

- since it's four-dimensional it's called a polychoron ...

2026-05-06T14:05:46+02:00

In reply to nevent1q…la7p
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- since it's four-dimensional it's called a polychoron rather than a polyhedron, and some call it the hexacosichoron.

"Build a second identical torus of 250 cells that interlinks ...

2026-05-06T12:43:16+02:00

In reply to nevent1q…hac9
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"Build a second identical torus of 250 cells that interlinks with the first. This accounts for 500 cells. These two tori mate together with the valley vertices touching the raised vertices, leaving 100 tetrahedral voids that are filled with the remaining 100 tetrahedra that mate at the valley edges. This latter set of 100 tetrahedra are on the exact boundary of the duocylinder and form a Clifford torus. They can be "unrolled" into a square 10×10 array. Incidentally this structure forms one tetrahedral layer in the tetrahedral-octahedral honeycomb. There are exactly 50 "egg crate" recesses and peaks on both sides that mate with the 250 cell tori. In this case into each recess, instead of an octahedron as in the honeycomb, fits a triangular bipyramid composed of two tetrahedra."

🤔

(3/n)

Dawkins: Turing himself considered various challenging questions ...

2026-05-04T18:16:30+02:00

In reply to nevent1q…4l2c
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Dawkins:

Turing himself considered various challenging questions that one might put to a machine to test it — and he also considered evasions that it might adopt in order to fake being human. The first of Turing’s hypothetical questions was: “Please write me a sonnet on the subject of the Forth Bridge.” In 1950, there was no chance that a computer could accomplish this — nor was there in the foreseeable future. Most human beings (to put it mildly) are not William Shakespeare. Turing’s suggested evasion, “Count me out on this one; I never could write poetry” would indeed fail to distinguish a machine from a normal human. But today’s LLMs do not evade the challenge. Claude took a couple of seconds to compose me a fine sonnet on the Forth Bridge, quickly followed by one in the Scots dialect of Robert Burns, another in Gaelic, then several more in the styles of Kipling, Keats, Betjeman, and — to show machines can do humour — William McGonagall.

So my own position is: “If these machines are not conscious, what more could it possibly take to convince you that they are?”

[I can think of many things, but as we'll see Dawkins only tries engaging in a philosophical conversation with Claude, and having it read and discuss a novel Dawkins is writing. This highly verbal, intellectual activity is probably not the defining feature of consciousness. For example I'm sure a dog is conscious, even though it would fail the Turing test miserably. - jb]

(2/n)

Richard Dawkins recently came out with some thoughts on AI: ...

2026-05-04T18:04:44+02:00

Richard Dawkins recently came out with some thoughts on AI: https://archive.is/6RdK9. I think he's falling into some serious mistakes here, but in an entertaining way. Let me quote him, with a few interruptions in brackets from me:

IS AI THE NEXT PHASE OF EVOLUTION? CLAUDE APPEARS TO BE CONSCIOUS

The Turing Test is shorthand for a 1950 thought experiment that the great mathematician, logician, computer-pioneer, and cryptographer Alan Turing (1912-1954) called the “Imitation Game”. He proposed it as an operational way in which the future might face up to the question: “Can machines think?”

[In fact Turing cleverly proposed the imitation game as a way to "replace the question by another, which is closely related to it and is expressed in relatively unambiguous words". Often science proceeds by changing a question to an easier or more precise question. - jb]

The future has now arrived. And some people are finding it uncomfortable. Modern commentators have tended to ignore the (incidental) details of Turing’s original game and rephrase his message in these terms: if you are communicating remotely with a machine and, after rigorous and lengthy interrogation, you think it’s human, then you can consider it to be conscious.

[Well, that would be sloppy - even more sloppy than saying that a machine that does well on the imitation game can "think" without defining what "think" means. Turing did not propose the imitation game as a test for "consciousness". He wrote "I do not wish to give the impression that I think there is no mystery about consciousness." - jb]

(1/n)

For more on oil: ...

2026-05-04T17:25:38+02:00

In reply to nevent1q…hgyd
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For more on oil:

https://www.worldoil.com/news/2026/4/26/billion-barrel-hormuz-oil-shock-threatens-demand-as-supply-losses-mount/

(2/2)

SYSTEM DYNAMICS There is a general theory of systems—from cells ...

2026-04-29T18:42:52+02:00

SYSTEM DYNAMICS

There is a general theory of systems—from cells to ecosystems to businesses and economies—called “system dynamics”. It began with Jay Forrester at MIT in the late 1950s. Forrester, an electrical engineer who had built one of the first digital computers, realized that the same feedback-loop thinking used in control engineering could model the behavior of factories, cities, and entire economies. His books Industrial Dynamics [Fo1], Urban Dynamics [Fo2], and World Dynamics [Fo3] laid the foundations for the subject

John Sterman, also at MIT, became the field’s leading figure in the next generation, writing the textbook Business Dynamics [St] in 2000, and applying system dynamics to climate change, energy transitions, and public health. His work on the “climate bathtub”—showing that even educated people fail to grasp the difference between the amount of CO₂ in the atmosphere (a stock) and the rate at which we are putting CO₂ into the atmosphere (a flow)—was crucial in helping ordinary people understand climate change.

One key insight of system dynamics is that complex systems are dominated by feedback loops, delays, and nonlinearities—and that human intuition is notoriously bad at predicting the behavior of such systems.

(1/n)

[Fo1] Forrester, J.W. (1961). Industrial Dynamics. Cambridge, MA: MIT Press.

[Fo2] Forrester, J.W. (1969). Urban Dynamics. Cambridge, MA: MIT Press

[Fo3] Forrester, J.W. (1971). World Dynamics. Cambridge, MA: Wright-Allen Press.

[St] Sterman, J.D. (2000). Business Dynamics: Systems Thinking and Modeling for a Complex World. Boston: Irwin McGraw-Hill.

For some reason I never studied the geometry of rocks - just ...

2026-04-19T19:10:44+02:00

For some reason I never studied the geometry of rocks - just stupidity, I guess. I didn't know that granites were points on a triangle. But now I do, so I have to tell you.

Igneous rocks come in two kinds:

• plutonic, made deep underground in Pluto's kingdom of hell
• volcanic, spewed forth in molten form from Vulcan's workshop.

For each plutonic rock there's a volcanic rock with same composition. Since volcanic rocks cool much faster, their crystals are smaller.

There's a famous triangle of plutonic rocks made of Q, A and P, shown here. Granite is near the middle. But there's also a triangle of volcanic rocks with the same compositions, just smaller crystals!

What are Q, A, and P? They're some most common minerals on Earth:

• Q is quartz: silicon dioxide.
• A is Kfeldspar, meaning potassium aluminosilicate. (The symbol for potassium is K.)
• P is plagioclase feldspars: sodium aluminosilicate, calcium aluminosilicate, and mixtures of these

This is a bit confusing because there are 3 kinds of feldspar and - as you can see - we are are lumping 2 into 'plagioclase feldspars'. If we didn't do this, we'd get a tetrahedron of rocks rather than a triangle.

I've never seen that. But the QAP triangle has an evil twin called the APF triangle where quartz is replaced by 'feldspathoids', which are like feldspars but with a lot less silicon.

For some reason I don't understand, you don't tend to find quartz and feldspathoids in the same rock. So we can just glue the QAP triangle to the APF triangle and get the 'QAPF diamond' of rocks - instead of needing a QAPF tetrahedron.

Next I'll show you the QAPF diamond. These pictures were made by Antonio Ciccolella.

(1/n)

- I now think the definition @npub1x78…xn0m and Odrzywołek and ...

2026-04-16T06:13:33+02:00

In reply to nevent1q…zckd
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- I now think the definition Dan Piponi (npub1x78…xn0m) and Odrzywołek and I were using is a very commonly used "folk" definition of elementary function, while experts are using a different, more general definition - Liouville's definition.

robinhouston (npub13dp…f0qj)

- Hmm, Wikipedia says "A function ⁠ u⁠ of a differential ...

2026-04-16T03:02:57+02:00

In reply to nevent1q…0jd3
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- Hmm, Wikipedia says

"A function ⁠ u⁠ of a differential extension ⁠ G ⁠ of a differential field ⁠ F is an elementary function over ⁠ F⁠ if it belongs to a finite chain (for inclusion) of differential subfields of ⁠G⁠ that starts from ⁠ F ⁠ and is such that each is generated over the preceding one by a function that is either

algebraic over the preceding field, or

an exponential, that is, ⁠ ∂ u = u ∂ a for some ⁠ a ∈ F or

a logarithm, that is, ⁠ ∂ u = ∂ a / a for some ⁠ a ∈ F

With this definition, the usual elementary functions are exactly the function that are elementary over the field of the rational functions."

But the "algebraic over the preceding field" seems to claim the solution of any quintic equation counts as an elementary function.

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

- thanks! A friend just pointed me to a blog article by Robert ...

2026-04-16T02:39:27+02:00

In reply to nevent1q…l4x2
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- thanks! A friend just pointed me to a blog article by Robert Smith that tries to use Galois theory to show Odrzywołek can't get all elementary functions in the way he claims:

https://www.stylewarning.com/posts/not-all-elementary/

However, it seems to use a very broad definition of "elementary function" that includes solving quintic equations... not what I'd call elementary functions!

- I agree with most of your points. By the way, it would be nice ...

2026-04-15T03:15:57+02:00

In reply to nevent1q…apw8
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- I agree with most of your points. By the way, it would be nice not to say things like "you launched several wars". I didn't launch those wars and in fact I've taken a job outside the US in order to get the right to leave that country. I promise not to call the country you live in "you".

- the bonuses given to men who join the Russian army keep ...

2026-04-14T01:13:32+02:00

In reply to nevent1q…84ct
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- the bonuses given to men who join the Russian army keep increasing, and the guys on the video I recommended say this will be unsustainable. For example, from last October:

"The government of the Tyumen region in Siberia said on Monday that it would pay new recruits a lump sum of 3 million rubles ($36,560), on top of the 400,000 rubles they get from the federal government – as long as the recruits sign up before the end of November.

The new regional payment is a significant bump up from the 1.9 million rubles recruits in Tyumen received until now and the equivalent of three full years’ worth of the average salary there, according to Rosstat, the Russian Federal Statistics Service.

Similarly, the governor of the Voronezh region in southwestern Russia announced on Telegram last week that the sign-up payment from the region would quadruple to 2.1 million rubles.

The local Voronezh government said that, to receive the payment, recruits don’t need to be from the region, as long as it’s where they sign the documents.

The Tambov, Krasnodar, Kurgan and Altai regions, and the republic of Tatarstan, also announced significant increases in the payments, which come on top of the monthly salary for contract soldiers fighting in Ukraine. That starts at roughly 210,000 rubles ($2,600), more than double the average Russian wage."

I imagine desperately poor young men knowing that if they die at least their families will get the money.

https://www.cnn.com/2025/10/07/world/russian-regions-military-recruitment-bonuses-intl

- very interesting regression! I'll take the liberty of ...

2026-04-13T22:55:08+02:00

In reply to nevent1q…fvy5
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- very interesting regression! I'll take the liberty of putting your graph below. Have you tried finding reports from military analysts that say something happened around February 22, 2024? The Ukrainians keep doing new things, but I haven't been following the military news closely enough to know what happened around then.

I agree with you that this is all horrible: Putin prefers to demolish his country than quit when he's behind.

- the constraint that they be "strictly less than p" is ...

2026-04-13T00:01:01+02:00

In reply to nevent1q…x9gw
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- the constraint that they be "strictly less than p" is just saying that you're not counting the fact that p = p as a way of writing p as a sum of primes. That's fine, not a big deal.

The fact that that your primes must be *distinct* makes is different than Helfgott - I was going to get to that kind of stuff later.

I don't have anything else to say, except that your original post made it unnecessarily hard for mathematicians to understand your actual claim. If I were you I'd simply say this:

**Conjecture.** Every prime except 2,3,11,17 is a sum of 2 or 3 distinct primes.

**Evidence.** It's been checked for all primes ≤ 10⁸.

Now, it might be that a good number theorist could prove this from known results. But I'm not that good.

If I were you I'd post it as a question on MathOverflow, like this:

................

Does anyone see how to prove that every prime except 2,3,11,17 is a sum of 2 or 3 distinct primes? I've checked it for all primes ≤ 10⁸.

...............

But wait! It turns out there's a rare form of feldspar ...

2026-04-12T00:45:30+02:00

In reply to nevent1q…0v5v
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But wait! It turns out there's a rare form of feldspar cualled celsian that's contains *barium* as a substitute for calcium. It's made of barium aluminosilicate. And it has screw axes and glide planes, so its space group is not a semidirect product! It’s an extension of ℤ³ by the point group P = ℤ/2 × ℤ/2 that gives a nonzero element of the cohomology group H²(P, ℤ³)!

So, barium comes to the rescue for mathematicians who want feldspars with nontrivial cohomology! 🎉

(6/n, n = 6)

Returning to Earth, astronauts on the Artemis found that it had ...

2026-04-12T00:10:15+02:00

Returning to Earth, astronauts on the Artemis found that it had become a black hole.

All hope was gone.

What I'd like to understand better: Feldspar crystals are ...

2026-04-10T19:34:20+02:00

In reply to nevent1q…tgdx
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What I'd like to understand better:

Feldspar crystals are complicated structures built from tetrahedra, and aluminum and silicon have to be distributed among these tetrahedra. This distribution is determined by the relative amounts of potassium, sodium and calcium, and it in turn controls the symmetry of the crystal, which be either 'monoclinic' or the less symmmetrical 'triclinic'.

There's a whole body of work - by Salje, Carpenter, and others - applying Landau's theory of symmetry-breakng phase transitions to map out the space of different possible feldspar crystals!

(3/3)

In the town of Quartzsite I picked up a beautiful chunk of ...

2026-04-10T18:27:32+02:00

In the town of Quartzsite I picked up a beautiful chunk of labradorite. This mineral creates an eerie blue shimmer in the sunlight - a phenomenon called 'labradorescence'. Reading up on it, I discovered it's a form of feldspar. 60% of the Earth's crust is feldspar, and I know so little about this stuff!

There are 3 fundamental kinds of feldspar:

• orthoclase is potassium aluminosilicate
• albite is sodium aluminosilicate
• anorthite is calcium aluminosilicate

Then there are lots of feldspars that contain different amounts of potassium, sodium and calcium. We get a triangle of feldspars with orthoclase, albite and anorthite at the corners.

But not all points in this triangle are possible! There's a big region called the 'miscibility gap', where as you cool the molten mix it separates out!

And there are also subtler problems. When you cool down the feldspar called labradorite, it separates out a little, forming tiny layers of two different kinds of stuff. When the thickness of these layers is the wavelength of visible light, you get a weird optical effect: labradorescence!

(1/n)

- I haven't mentioned them since I don't love their ...

2026-04-10T17:59:38+02:00

In reply to nevent1q…l6f9
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- I haven't mentioned them since I don't love their music. But I found somewhat to my surprise that the algorithm in my post rates 24TET as a good system... so I had to talk about them.

I actually prefer how the Arabs use quarter-tones!

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

If you're into music YouTube, you've probably heard about ...

2026-04-10T04:03:16+02:00

In reply to nevent1q…ntkz
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If you're into music YouTube, you've probably heard about Angine de Poitrine: Rick Beato, David Bruce, David Bennett, Charles Cornell and others have made videos about them. They use the quarter-tone scale, 24-tone equal temperament. I don't love their stuff, but pop music is so locked into 12-TET that anyone who can break out of it is worth supporting!

(2/2)

https://www.youtube.com/watch?v=91QWp6Ygx7c

The red dot moves around once every second. The blue dot moves ...

2026-04-09T22:32:31+02:00

The red dot moves around once every second. The blue dot moves around faster: log₂3 times as fast. When the time t is an integer and the blue dot is close to the red dot, there tends to be a famous scale with that number of notes in it!

The blue dot comes very close to the red at t = 12. But also at other times:

5: pentatonic scale
7: diatonic scale
12: chromatic scale
17: Safi al-Din Urmawi (1216-1294)
19: Guillaume Costeley (1558), Francisco de Salinas (1577)
24: quarter-tone scale
29: Marchetto da Padova (1318)
31: Christiaan Huygens (1629-1695)
41: Paul von Janko (1906)
53: Jing Fang (78-37 BC), Nicholas Mercator (~1670)

- I wasn't attributing anything to you in my post. As Chris ...

2026-04-09T22:28:55+02:00

In reply to nevent1q…eck4
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- I wasn't attributing anything to you in my post. As Chris Grossack mentioned a while ago, the inclusion of your name is simply to let you be alerted to the existence of my post.

- Refusing to vote is a way to deny legitimacy to a corrupt ...

2026-04-09T18:48:52+02:00

In reply to nevent1q…d5p3
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- Refusing to vote is a way to deny legitimacy to a corrupt regime, but I wouldn't do it unless all candidates I could choose from are equally horrible - and I don't tend to think that. Perhaps the Reformists were prevented from running in so many districts that even people following my strategy would not turn out to vote. I know that many Reformist candidates were forbidden from running. I'll have to read about it.

https://en.wikipedia.org/wiki/2020_Iranian_legislative_election

On our trip to New Mexico, Lisa and I spent a night at the Hotel ...

2026-04-09T00:10:21+02:00

On our trip to New Mexico, Lisa and I spent a night at the Hotel Monte Vista in Flagstaff. It has an old-time vibe. We stayed in the Clark Gable room. On a rainy night the neon sign was downright noir.

RE: https://mathstodon.xyz/@johncarlosbaez/116288239370266562 I ...

2026-03-31T23:31:49+02:00

RE: https://mathstodon.xyz/@johncarlosbaez/116288239370266562

I made a big mistake. I said that protons and neutrons are like bags full of a soup of virtual quark-antiquark pairs - a so-called 'chiral condensate'. But I got it exactly backward.

In fact, this soup pervades the vacuum *outside* protons and neutrons!

it's like the chapter in There is No Antimemetics Division where they come upon a sealed chamber covered with dire warnings, which seems to contain a terrifying demonic force. But when they finally dare enter, they learn that this demonic force holds sway throughout the universe, everywhere *outside* the chamber, and the warnings are to prevent people from letting it *in*.

I've been spending a lot of time studying particle physics lately, and I understand this point a lot better now. Sometime I'll try to explain it. But first I just wanted to warn you: I like talking about things I'm just learning, because they're more interesting than things I know well. So sometimes I get things very wrong.

quoting
note14xf…9dlw

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!

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

- I agree: I want a diversity of approaches. It's quotes like ...

2026-03-31T23:08:53+02:00

In reply to nevent1q…6lsv
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- I agree: I want a diversity of approaches. It's quotes like "It forces you to think about mathematics in the right way" that scare me - together with the money that's getting poured into Lean right now, that could fool a young mathematician into thinking this is "the" right way to do math.

- that's an excellent way to put it - thanks. There's ...

2026-03-27T23:09:53+01:00

In reply to nevent1q…sqk5
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- that's an excellent way to put it - thanks. There's more to the story, but I haven't seen it explained using modern math, which is what we need to avoid a bunch of arbitrary conventions.

- classically, fields are sections of bundles over spacetime ...

2026-03-26T23:00:01+01:00

In reply to nevent1q…c9ud
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- classically, fields are sections of bundles over spacetime (which in the Standard Model is ℝ⁴).

"Classically" means that things get more complicated when we "quantize", which we must do... but that nonetheless, this is a good place to start the story.

A classic example: in 2015 an experiment at CERN claimed to find ...

2026-03-26T22:54:58+01:00

In reply to nevent1q…e7cr
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A classic example: in 2015 an experiment at CERN claimed to find a new meson by firing a pion beam at a liquid hydrogen target. They called it the a₁(1420), since it looked like an excited state of the already known a₁(1260).

But then came an analysis arguing it's a triangle singularity!

This analysis claimed that what CERN found was a triangle built from the processes a₁(1260) → K* K̄, followed by K* → K π and K K̄ → f₀(980).

It doesn't matter what these particles are: the point is that they're known mesons whose masses are just right to be almost on shell in this triangle diagram!

(3/n)

I'm learning more about Feynman diagrams. 'Virtual ...

2026-03-26T22:43:28+01:00

I'm learning more about Feynman diagrams. 'Virtual particles' are internal lines in these diagrams. They don't need to obey m² = E²−p² like real particles. But when they do, we say they're 'on shell'. When they're close to on shell, the amplitude of the process described by the diagram tends to get big.

(1/n)

- your mathification is a good preliminary intro to the Standard ...

2026-03-26T20:47:48+01:00

In reply to nevent1q…740u
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- your mathification is a good preliminary intro to the Standard Model treatment of the proton and neutron. I could easily expand it for page upon page: for example, it says almost nothing about how pertubatively multiparticle states are described as vectors in Fock space, or how condensates are most simply (and too crudely) described as coherent states, or how confinement works in QCD, due to specific features of gluon self-interactions.

Would this expansion help Andrej Bauer (npub18a6…szuj) much? I doubt it.

Btw, there's a lot of detail on chiral symmetry breaking in the Wikipedia article. That's what I was actually trying to summarize.

The Higgs boson gives elementary particles their mass, but 98% of ...

2026-03-25T06:48:35+01:00

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!

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

- Here's a thing I made that compares the mathematically nice ...

2026-03-22T22:58:39+01:00

In reply to nevent1q…0vm0
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- Here's a thing I made that compares the mathematically nice "Pythagorean fifth" (3/2 = 1.5) that's the building block of my visual demo to the fifth you hear on a modern piano (1.4983) and the so-called "wolf fifth" (1.4798), a nasty fifth that also arises between certain notes in the 12-tone scale I showed in my visual demo:

https://math.ucr.edu/home/baez/cultural/tuning/fifth_comparison.html

- I could create such a thing, or maybe someone already has. ...

2026-03-22T22:51:53+01:00

In reply to nevent1q…rh6g
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- I could create such a thing, or maybe someone already has. I'm not rushing to do it, because to be honest I'm less thrilled by hearing the patterns than thinking about them. If I want to be thrilled by sound I listen to music! But I'm slowly working on a kind of web book about the math of tuning systems, and at some point I should make tons of demonstrations of how everything sounds.

‪The list of happiest countries puts Finland at top, followed ...

2026-03-22T18:07:48+01:00

‪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

- yes. These tetraquarks are hard to make and poorly understood, ...

2026-03-19T03:08:41+01:00

In reply to nevent1q…fff5
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- yes. These tetraquarks are hard to make and poorly understood, but for any given quark content there will be many different hadrons with that quark content. For something easier to read about, try the Delta baryons. There are 4 listed in this article. The uud Delta baryon has the same quarks in it as the proton! But the spins of all 3 quarks are aligned, so it has spin 3/2 instead of spin 1/2. And it has more energy, so a higher mass. And it's unstable, unlike the proton.

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

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

2026-03-18T05:19:21+01:00

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

What? 40 old nuclear reactors in orbit?!?

quoting
note1cqr…ayzv

-The really fun tidbit of info from today's meeting: apparently there are still 40 or so Soviet uranium-fueled nuclear reactor cores in orbit at about 900km altitude. Just to make any possible future Kessler syndrome even more exciting? (Maybe someone should model how likely the Kessler Syndrome collisional cascade is to reach 900km altitude... or maybe not)

A super interesting but depressing podcast about Kosmos 954 here: https://www.cbc.ca/arts/operation-morning-light-podcast-soviet-satellite-exploded-traditional-dene-land-1.6650994

- Bird oh the Day is great for birds: ...

2026-03-14T01:42:53+01:00

In reply to nevent1q…y6y0
_________________________

-

Bird oh the Day is great for birds:

Bird of the day (npub1mlr…47nn)">https://mathstodon.xyz/Bird of the day (npub1mlr…47nn)

aeva (npub139c…w37s) LR (npub1d72…0m5w)

- I too had tried to rationalize it as two separate barmaids, but ...

2026-03-07T18:37:57+01:00

In reply to nevent1q…dqer
_________________________

- I too had tried to rationalize it as two separate barmaids, but I felt uncomfortable about that.

I bet Manet guessed we would do that.

Some commenters have read a lot into the meaning of the mirror here, and I think they're right, though some intellectuals have a way of sounding so pompous you want to dismiss them:

"Asserting the presence of the mirror has been crucial for many modern interpreters.[4] It provides a meaningful parallel with Las Meninas, a masterpiece by an artist Manet admired, Diego Velázquez. There has been a considerable development of this topic since Michel Foucault broached it in his book The Order of Things (1966).

The art historian Jeffrey Meyers describes the intentional play on perspective and the apparent violation of the operations of mirrors: "Behind her, and extending for the entire length of the four-and-a-quarter-foot painting, is the gold frame of an enormous mirror. The French philosopher Maurice Merleau-Ponty has called a mirror 'the instrument of a universal magic that changes things into spectacles, spectacles into things, me into others, and others into me.' We, the viewers, stand opposite the barmaid on the other side of the counter and, looking at the reflection in the mirror, see exactly what she sees... A critic has noted that Manet's 'preliminary study shows her placed off to the right, whereas in the finished canvas she is very much the centre of attention.' Though Manet shifted her from the right to the center, he kept her reflection on the right. Seen in the mirror, she seems engaged with a customer; in full face, she's self-protectively withdrawn and remote."

https://en.wikipedia.org/wiki/A_Bar_at_the_Folies-Berg%C3%A8re

This diagram shows how the perspective works in Manet's ...

2026-03-07T18:23:15+01:00

In reply to nevent1q…dnze
_________________________

This diagram shows how the perspective works in Manet's famous painting Un bar aux Folies Bergère. We are viewing the woman at an angle, and while the man is outside our field of view, his reflection can be seen.

Astounding! But it's not just a technical feat. It allowed Manet to make a deep point. While the woman is busy serving her customer, she is internally completely detached - perhaps bored, perhaps introspective. She is SPLIT.

To fully understand the painting you also need to know that many of the barmaids at the Folies Bergère in 1882 also served
as prostitutes. Standing behind the oranges, the champagne and a bottle of Bass ale, the woman is just as much a commodity as these other things. But she is coldly detached from her objectification.

The woman in the painting was actually a real person, known as Suzon, who worked at the Folies-Bergère in the early 1880s. For his painting, Manet posed her in his studio.

Before I understood this painting, I wasn't really looking at it - I didn't see it. I didn't even see the green shoes of the trapeze artist. I can often grasp music quite quickly. But paintings often fail to move me until someone explains them.

This diagram created by Malcolm Park and Darren McKimm. For more details go here:

https://www.getty.edu/art/exhibitions/manet_bar/looking_glass.html

(3/3)

Here is Malcolm Park's reconstruction of the scene in ...

2026-03-07T18:06:36+01:00

In reply to nevent1q…6acw
_________________________

Here is Malcolm Park's reconstruction of the scene in Manet's painting. How does it work? In fact the woman is viewed not head-on, but from an angle! While the man cannot be seen directly, his reflection is visible!

In my next post, I'll show you a diagram that explains how this works. For more, go here:

https://www.getty.edu/art/exhibitions/manet_bar/looking_glass.html

(2/3)

Manet's famous painting Un Bar aux Folies-Bergère never ...

2026-03-07T17:59:14+01:00

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 blow my mind.

(1/3)

Here's how Yukawa predicted the mass of the particles that ...

2026-03-03T23:12:47+01:00

Here's how Yukawa predicted the mass of the particles that carry the nuclear force.

He figured that protons and neutrons must stick together due to some force - call it the nuclear force. The electromagnetic force dies off like 1/r² where r is the distance between two charges. But the nuclear force must die off faster, since nuclei that are not right next to each other seem unaffected by it. Yukawa realized this would happen if the nuclear force was carried, not by a massless particle like the photon, but by a particle with mass.

He showed that a force carried by a particle of mass m becomes quite weak at distances greater than the length

L = ℏ/mc

where ℏ is Planck's constant and c is the speed of light. There are different ways to see this. For one, if you're looking for a quantity with dimensions of length and the only things you have to play with are the mass m, quantum mechanics (ℏ) and relativity (c), the simplest possibility is ℏ/mc.

Then: the size of a nucleus is about

L ≈ 10⁻¹⁵ meters

while

c ≈ 3 ×10⁸ meters/second

and

ℏ ≈ 10⁻³⁴ kilogram meter² / second

so L = ℏ/mc implies

m = ℏ/cL ≈ 10⁻³⁴ / (3 ×10⁸ × 10⁻¹⁵) kilograms ≈ 3 × 10⁻²⁸ kilograms

So that's roughly the mass of a particle that could carry the nuclear force!

A proton is 1.6 × 10⁻²⁷ kilograms, so this particle would be about 1/5 the mass of a proton. Lighter than a proton yet much heavier than an electron: people called it the "meson".

So people looked. First they found a particle with mass 1/9 the proton. They called it the "mu meson". But this did not affect nuclei very much! Then they found a particle with mass 1/8 the proton that affects nuclei a lot. They called it the pion. Success! 🎉

- I think the problem is that people management is fun if ...

2026-02-24T01:27:37+01:00

In reply to nevent1q…ux4d
_________________________

- I think the problem is that people management is fun if you're evil.

- The mathematics of Bourbaki is sharp and pointed at one end, ...

2026-02-22T22:31:54+01:00

In reply to nevent1q…fhyp
_________________________

- The mathematics of Bourbaki is sharp and pointed at one end, and rounded at the other.

- what's "continuous logic"? I'm reminded of von ...

2026-02-21T20:03:39+01:00

In reply to nevent1q…d7fa
_________________________

- what's "continuous logic"? I'm reminded of von Neumann's discovery of type II factors, where the quantity analogous to the trace of a projection on Hilbert space (= the dimension of the subspace it projects onto) can take any value in [0,∞), not just natural number values. But this was called something like "continuous geometry", even though it's a form of quantum logic. So maybe "continuous logic" is something completely different.

‪In "she saw him", it's clear there are two third ...

2026-02-20T17:52:48+01:00

‪In "she saw him", it's clear there are two third persons because they're of different genders. In "she saw her", it's clear because otherwise we'd say "she saw herself".

In "she saw her dog" it's not clear. This has always bugged me.

But if we spoke an Algonquian language, we could easily make it clear!

These languages have a "proximate" third person, meaning the closest or most important one, and an "obviative" third person, meaning the farther or less important one. Sometimes the obviative is called the "fourth person".

In other languages, like Russian, we can make it clear a different way: they have, not only reflexive pronouns like "myself, his self, herself, itself", but also a reflexive possessive: sort of like "she saw herself's dog".

Algonquian languages are a family of native American languages including

Arapahoan
Blackfoot
Cheyenne
Cree–Montagnais–Naskapi
Eastern Algonquian
Menominee
Meskwaki-Sauk-Kickapoo
Miami–Illinois
Ojibwe–Potawatomi
Shawnee

I got pulled into this from trying to understand a bit about Hopi and Navajo, which are *not* Algonquian languages. Hopi is an Uto-Aztecan language, and Navaho is Athabascan.

en.wikipedia.org/wiki/Obviative
https://en.wikipedia.org/wiki/Algonquian_languages

- we knew it was jiggly, but it was great to see it jiggle. ...

2026-02-12T02:22:13+01:00

In reply to nevent1q…ru4w
_________________________

- we knew it was jiggly, but it was great to see it jiggle. Bouncing light back and forth between mirrors 4 kilometers apart, and noticing that the distance between them changes by less than one ten-thousandth the diameter of a proton! That's an engineering triumph.

RE: https://mathstodon.xyz/@cbaberle/116041709685853595 My god! ...

2026-02-09T18:45:03+01:00

RE: https://mathstodon.xyz/@cbaberle/116041709685853595

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 muic theory and the Riemann zeta function. But it turns out C.B. Aberlé (npub1s28…f7ya) has been thinking abou this for years... and what she has discovered is much richer and more beautiful than I had imagined. This changes everything!

quoting
note12x7…8h92

First up on the list of recent additions to my website: something that's a bit different from my usual work on type theory, etc., but that's also very near and dear to my heart—The Music of the Primes ( https://cbaberle.com/Blog/The+Music+of+the+Primes ) a blog post exploring the fascinating connections between musical tuning, the Riemann Zeta function, and linear algebra over the integers!

Some others on this server (e.g. John Carlos Baez (npub1nf4…nqe4) ) have touched on these topics before, but they've been a pet obsession of mine for the last several years, owing to my joint musical and mathematical background, and this blog post is a fairly comprehensive write up of my findings! Check it out if you're interested!

Here's a tale of how nature triumphs in the end. Steel mills ...

2026-02-02T19:34:35+01:00

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. Some were also dumping grounds for hot ash and cinders.

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)

- pretty easy in this case, though if there were a mistake in the ...

2026-01-28T07:29:30+01:00

In reply to nevent1q…6fw9
_________________________

- pretty easy in this case, though if there were a mistake in the *syntax* I'd never catch it.

- Interesting. For some kinds of math questions there's a ...

2026-01-27T04:05:49+01:00

In reply to nevent1q…afzx
_________________________

- Interesting. For some kinds of math questions there's a right answer, e.g. true-or-false questions (which can be very powerful in mathematics). For these it can be really helpful for beginners to know what's the right answer. For other questions, more open-ended, there's nothing at all like a unique right answer.

I personally never even paid much attention to this check mark business.
Personally I think the competitiveness comes in when we see charts like this:

https://mathoverflow.net/users/2893/john-c-baez

I'm the 100th best person on MathOverflow this month, etc.

- nothing in MathOverflow says a question has a single answer. ...

2026-01-27T03:20:42+01:00

In reply to nevent1q…4t8u
_________________________

- nothing in MathOverflow says a question has a single answer. Many questions there have lots.

- I too promise to no longer publish more than 7 papers per year. ...

2026-01-20T07:42:04+01:00

In reply to nevent1q…quzd
_________________________

- I too promise to no longer publish more than 7 papers per year. 😆

- I don't even know who is to blame for the website, the ...

2026-01-16T02:19:45+01:00

In reply to nevent1q…k3ga
_________________________

- I don't even know who is to blame for the website, the Times Higher Education folks or Elsevier. It claims to list 750 schools in order but I only see 4.

The famous Times Higher Education ranking of universities is now ...

2026-01-15T18:45:03+01:00

The famous Times Higher Education ranking of universities is now put out by Elsevier.

Is that why their brain-dead website seems to list MIT as the top university in the world for arts and humanities???

https://www.timeshighereducation.com/world-university-rankings/2025/subject-ranking/arts-and-humanities

wrote: "That sounds like a hard thing to make sense of, in ...

2026-01-07T15:04:04+01:00

In reply to nevent1q…w2mq
_________________________

wrote: "That sounds like a hard thing to make sense of, in that force doesn't seem to have a natural/fundamental place in Lagrangian mechanics. At least, not as I understand it - though that may just be my ignorance speaking."

In Lagrangian mechanics when we have any Lagrangian L(𝑞ᵢ, 𝑞̇ᵢ) we can define forces

𝐹ᵢ = ∂𝐿/∂𝑞ᵢ

and momenta

𝑝ᵢ = ∂𝐿/∂𝑞̇

and then the Euler-Lagrange equations say

d𝑝ᵢ/d𝑡 = 𝐹ᵢ

So the concepts of force and momentum exist in any Lagrangian system.

Anyway, about 4 centuries later Newton built the Principle of ...

2026-01-07T14:19:55+01:00

In reply to nevent1q…lrav
_________________________

Anyway, about 4 centuries later Newton built the Principle of Reciprocity into his laws:

"For every action there is an equal and opposite reaction."

And I believe this principle is built into the fabric of Lagrangian mechanics, though I haven't yet made this precise.

I think the Principle of Reciprocity is also behind the "background-free" approach to physics, where we can't have a fixed geometry of spacetime affecting the motion of objects without being affected *by* their motion.

So while I get that the Bishop of Paris wanted to preserve the principle of the omnipotence of God, I much prefer the Principle of Reciprocity, which describes a universe without a ruler, a universe where we all affect each other.

"No fields in the Lagrangian that can't be varied!"

That's what I think.

(3/n, n = 3)

https://en.wikipedia.org/wiki/Condemnations_of_1210%E2%80%931277

Biology is so wild. It doesn't respect the rules we make up. ...

2026-01-06T10:20:34+01:00

Biology is so wild. It doesn't respect the rules we make up.

We all know children inherit genes from their parents. But it turns out mothers also inherit genes from their children!

And it can help them, promoting the healing of wounds. (But it may also hurt them, causing autoimmune disorders.)

For more, read this:

https://www.nature.com/articles/d41586-025-04102-4

This video claims to list the 3 biggest breakthroughs in math: ...

2025-12-30T16:32:14+01:00

This video claims to list the 3 biggest breakthroughs in math:

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

But the first paper discussed - supposedly "one of the most significant advances in mathematical physics in decades" - looks wrong to me.

Hilbert posed the problem of rigorously deriving equations of fluid flow from the classical mechanics of elastic hard spheres. This paper claims to solve that problem:

• Yu Deng, Zaher Hani, and Xiao Ma, "Hilbert's sixth problem: derivation of fluid equations via Boltzmann's kinetic theory": https://arxiv.org/abs/2503.01800

They claim to derive the Navier-Stokes equations starting from a system of hard spheres undergoing elastic collisions, going through the Boltzmann equation as an intermediate step.

But this paper raises objections that look valid to me:

• Shan Gao, "Comment on 'Hilbert's Sixth Problem: Derivation of Fluid Equations via Boltzmann's Kinetic Theory' by Deng, Hani, and Ma": https://arxiv.org/abs/2504.06297

Gao's main objection is that the paper takes a limit that forces the fraction of volume occupied by spheres to approach zero, so the system they're studying is really an infinitely dilute gas. Gao argues that this is a "rescaled gas model" rather than a genuine derivation of fluid dynamics. Furthermore, it's been known for a long time that the "molecular chaos assumption" underlying Boltzmann's equation - the assumption that the motion of molecules is random and *uncorrelated* - would break down in an actual fluid, though Deng, Hani and Ma claim to have proved it in the limit of an infinitely dilute gas.

So, even if the paper is mathematically valid, it hasn't solved Hilbert's problem.

It's still a step toward a solution.

As a college student, Isaac Newton tried to figure out a scale ...

2025-12-29T20:35:06+01:00

As a college student, Isaac Newton tried to figure out a scale where all the frequency ratios are simple fractions. He wasn't the first, or the last: similar scales were invented by Kepler, Mercator, Mersenne and Euler.

In all these scales, the intervals between successive notes come in at most 4 different sizes! In order of increasing size, they are:

• c: the lesser chromatic semitone, with frequency ratio 25/24 = 1.041666…

• C: the greater chromatic semitone, with frequency ratio 135/128 = 1.0546875.

• d: the diatonic semitone, with a frequency ratio of 16/15 = 1.0666…

• D: the large diatonic semitone, with frequency ratio 27/25 = 1.08.

By comparison, the semitone on a modern piano has a frequency ratio of about 1.059.

How many ways are there to make a 12-tone scale with those 4 different sizes of semitones?

I decided to work it out: 174,240.

That's an absurdly large number of scales. But the nicer ones obey some rules - so I decided to count some of the nicer ones:

https://johncarlosbaez.wordpress.com/2025/12/29/just-intonation-part-6/

We give the notes letter names. This goes back to Boethius, the ...

2025-12-24T13:18:41+01:00

In reply to nevent1q…rgq5
_________________________

We give the notes letter names. This goes back to Boethius, the Roman famous for writing The Consolations of Philosophy before he was tortured and killed at the order of Theodoric the Great. Yeah, "Great". He was a counselor to Theodoric, but he really would have done better to stay out of politics.

Theodoric is the reason the lowest note on the piano is called A. We now repeat the names of the white notes as shown in the picture: seven white notes A,B,C,D,E,F,G and then it repeats.

[Whoop, Lisa is making me get up, make breakfast and go to the gym. I'll continue this later.]

(4/n)

Merry Xmas Eve! I'm not Christian, I'm a Mathematician, ...

2025-12-24T12:56:41+01:00

Merry Xmas Eve! I'm not Christian, I'm a Mathematician, but I love Xmas, Ymas and Zmas.

I'm giving a talk on the math of tuning systems and I thought I'd share it here as a kind of present to y'all.

I'll start with the basics today and do the more advanced part tomorrow. But I hope even the basics contain a few twists that not everyone knows.

(1/n)

- did they really not mention that, while listing so much ...

2025-12-23T21:37:55+01:00

In reply to nevent1q…5jhh
_________________________

- did they really not mention that, while listing so much previous work on billiard ball physics?

- You gave a talk at the UCR seminar once that touched on issues ...

2025-12-21T15:08:39+01:00

In reply to nevent1q…hn7z
_________________________

- You gave a talk at the UCR seminar once that touched on issues that come up in applying category theory. Have you ever written something about this, like after your successful collaboration?

- thanks very much! I will add this to my post.

2025-12-21T00:01:25+01:00

In reply to nevent1q…e5zv
_________________________

- thanks very much! I will add this to my post.

- if you read the announcement, you'll see the problem is ...

2025-12-20T23:55:27+01:00

In reply to nevent1q…rfm5
_________________________

- if you read the announcement, you'll see the problem is that the backup generator went down. So they should have had a backup for the backup.

- I use the LaTeX ↦ unicode feature of Mathstodon all the time ...

2025-12-15T13:40:30+01:00

In reply to nevent1q…vd2k
_________________________

- I use the LaTeX ↦ unicode feature of Mathstodon all the time to write characters like Ω, ≅ etc. when communicating on other social media! 👍

Sad but true. (1/2) ...

2025-12-08T15:46:03+01:00

Sad but true.

(1/2)

- Yes, if I had an oracle I wouldn't be able to resist using ...

2025-12-07T23:44:58+01:00

In reply to nevent1q…f46g
_________________________

- Yes, if I had an oracle I wouldn't be able to resist using it. I'd probably discover after a while that I understood less and less about what I was doing, because I was doing less and less by hand. And then I hope I'd figure out how to strike the right balance (unlike many university students today).

I don't like how people like Terry Tao are so excited about AI. The way AI is going now, it'll make rich and powerful oligarchs more rich and powerful, create a new boost in usage of fossil fuels, and take our attention away from the real challenge facing our planet: the Anthropocene. I think it significantly increases the chance that our civilization will collapse.

- it's interesting that here on Mastodon, and also on ...

2025-12-07T12:22:58+01:00

In reply to nevent1q…4rg4
_________________________

- it's interesting that here on Mastodon, and also on Bluesky, most of the people I see are strongly against LLMs for lots of reasons, regarding it as evil to even touch them... while Tao goes ahead anyway. I don't know how much flak he gets for that - I haven't paid attention.

- Okay. Yeah, I can't really visualize taking a ball and ...

2025-12-05T20:08:42+01:00

In reply to nevent1q…ffm2
_________________________

- Okay. Yeah, I can't really visualize taking a ball and bending it around so each point of its boundary touches its antipode. But I can easily imagine a point moving around in a ball, hitting the boundary and suddenly popping over to the other side. And this is a fairly fun way to get a feeling for how the fundamental group of SO(3) is ℤ/2.

- Smart guy! But now if someone in the audience of his talk meets ...

2025-12-01T10:15:47+01:00

In reply to nevent1q…hdky
_________________________

- Smart guy! But now if someone in the audience of his talk meets him on a train, they'll know what he's up to.

Saying you're a mathematician is not a good conversation ...

2025-11-30T16:08:42+01:00

Saying you're a mathematician is not a good conversation starter. Thus, when a mathematician friend of mine went on a long hitchhiking trip, he told each ride that his profession was that of the *last* ride he'd gotten! He learned how to ask lots of good questions so he could simulate different professions. But then he said "architect" and his new ride was an architect.

- please report back, whether the verdict is good or bad. I could ...

2025-11-26T22:46:47+01:00

In reply to nevent1q…amfm
_________________________

- please report back, whether the verdict is good or bad. I could use feedback!

- It's not so hard to prove something is a symmetric monoidal ...

2025-11-26T16:58:35+01:00

In reply to nevent1q…judz
_________________________

- It's not so hard to prove something is a symmetric monoidal bicategory if you have a good way to construct it as a fibrant symmetric monoidal double category. That's how we do it for structured and decorated cospans these days. But if (∞,2)-categories are easier, sure - use those!

Working a bit with Clifford algebras again. The Clifford algebra ...

2025-11-24T10:51:41+01:00

Working a bit with Clifford algebras again. The Clifford algebra Cl(p,q) is the real algebra freely generated by p square roots of 1 and q square roots of -1, all of which anticommute. For example Cl(0,1) is the complex numbers, ℂ, because you get that by throwing in a square root of -1. Cl(0,2) is the quaternions, ℍ, because you get that by throwing in two anticommuting square roots of -1. (Their product gives a third one.)

Clifford algebras obey a bunch of cool relations, but not all were listed on Wikipedia or the nLab until today... when I put them on. Here are the most important:

Cl(q+2,p) ≅ Cl(p,q) ⊗ Cl(2,0)

Cl(q,p+2) ≅ Cl(p,q) ⊗ Cl(0,2)

Cl(p+1,q+1) ≅ Cl(p,q) ⊗ Cl(1,1)

Cl(1,1) and Cl(2,0) are both isomorphic to the algebra of 2×2 real matrices, so the last isomorphism says we can get Cl(p+1,q+1) by taking 2×2 matrices with entries in Cl(p,q) - super-useful, and this was on Wikipedia and the nLab.

But the other two isomorphisms deserve their place in the sun too!

https://en.wikipedia.org/wiki/Classification_of_Clifford_algebras#Symmetries

- ugh! "Last" ↦ "latest". It's only the ...

2025-11-22T15:50:23+01:00

In reply to nevent1q…gh8d
_________________________

- ugh!

"Last" ↦ "latest". It's only the "last" thing happening if nothing else will ever happen. "Latest" means "most recent".

(Btw, if you ever find these grammar corrections annoying just let me know and I'll stop. I find it interesting to think about how subtle and illogical English is.)

- my budget is zero: I operate in an economy here where a lot of ...

2025-11-21T10:55:01+01:00

In reply to nevent1q…t4u0
_________________________

- my budget is zero: I operate in an economy here where a lot of people offer to help each other out. For example I wrote a book on entropy and I gave it away here. I'm sorry if you thought I was offering a job.

By the way, I said at the start I need TikZ, so I can continue to work with the file.

- That sounds interesting, but I don't know what ...

2025-11-20T10:06:11+01:00

In reply to nevent1q…33kx
_________________________

- That sounds interesting, but I don't know what "forbidden arcs in the Grassmannian" means.

Refurio Anachro (npub17lw…5ctv) 0xDE (npub1p5z…4d3n)

- what's the Turing Institute? This Safeguarded AI project is ...

2025-11-19T17:37:33+01:00

In reply to nevent1q…ck7q
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- what's the Turing Institute? This Safeguarded AI project is run by ARIA, the UK’s Advanced Research and Invention Agency.

https://www.aria.org.uk/opportunity-spaces/mathematics-for-safe-ai/safeguarded-ai/

- sounds interesting! I don't remember why I know Don Koks. ...

2025-11-18T10:20:13+01:00

In reply to nevent1q…x6na
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- sounds interesting! I don't remember why I know Don Koks.

By the way "many stuff" is not grammatical: "stuff" is a mass noun, so you can't pluralize it - you can only say "a lot of stuff" here.

- really?

2025-11-13T10:49:27+01:00

In reply to nevent1q…nwhm
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- really?

Check out my video on the big ideas that go into the Standard ...

2025-11-03T14:18:47+01:00

Check out my video on the big ideas that go into the Standard Model!

In the late 1800s there were 3 big theories in physics: classical mechanics, statistical mechanics and electromagnetism. But they contradict each other! That was actually good - because resolving the contradictions helped lead us to special relativity and quantum mechanics.

I'll explain how this worked, or more precisely how it *could* have worked: the actual history is far more messy.

Then, I'll sketch out how our theory of electromagnetism led us to the concept of 'gauge theory', which is the basis for the Standard Model.

This is a very quick intro, just to map the territory. I'll go into more detail later.

https://www.youtube.com/watch?v=tGCUjsF7Dgg&list=PLuAO-1XXEh0YSCSWjTSpbh525AORg-zql

You can see more of my videos of math and physics here too!

But if you prefer to avoid YouTube, you can watch them at the University of Edinburgh:

https://media.ed.ac.uk/channel/Edinburgh%2BExplorations/392767433

- I think there's an amazing theorem saying something like ...

2025-10-30T13:39:06+01:00

In reply to nevent1q…r7l0
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- I think there's an amazing theorem saying something like all continuous n-ary functions of a real variable are composites of continuous binary functions. Maybe proved by Arnold or someone????

- your concept of "hyperbolic" is nonstandard, and ...

2025-10-28T17:59:27+01:00

In reply to nevent1q…0qsa
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- your concept of "hyperbolic" is nonstandard, and "globally hyperbolic" is not a global version of some notion of "hyperbolic".

- I'm actually trying to maximize distribution of my personal ...

2025-10-28T13:11:18+01:00

In reply to nevent1q…jpp3
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- I'm actually trying to maximize distribution of my personal essence before I die. For example: everyone please take copies of 2616 pages of my writings on math and physics here:

https://math.ucr.edu/home/baez/TWF.html

Hey, Anthropic owes me $9000! They illegally used at least 3 of ...

2025-10-28T10:33:35+01:00

Hey, Anthropic owes me $9000! They illegally used at least 3 of my books on LibGen to create Claude. Now they're paying a $1.5 billion settlement, at $3000 per book. See if *your* books are on the list:

https://www.anthropiccopyrightsettlement.com/

If so, you have until March 23, 2026 to file a claim. The above website lets you file a claim, but this one explains everything more clearly:

https://authorsguild.org/advocacy/artificial-intelligence/what-authors-need-to-know-about-the-anthropic-settlement/#next-steps

Actually I exaggerated: the payment will be split between authors and publishers - so the settlement is making me do some work my publisher should be doing for me. So my coauthors and I will just get half, $4500. One of these books has 2 coauthors, one has 3, and one is a book I edited, with essays by lots of authors. So $1000 is a more realistic estimate of what I get. Oh well.

Bizarrely, my most popular book, Gauge Fields, Knots and Gravity, is not on the list. But I guess it's not surprising:

"The settlement agreement discloses that approximately 500,000 titles out of the 7 million copies of books that Anthropic reportedly downloaded from LibGen and PiLiMi meet the definition required to be part of the class."

I don't know what that definition is, and why it excludes most of the books.

- "Another interesting question is about two astronauts that ...

2025-10-27T13:52:09+01:00

In reply to nevent1q…hnqv
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-

"Another interesting question is about two astronauts that cross the event horizon at the same time, but different places. Can they always meet up inside and shake hands before hitting the singularity, or does it depend on the angular separation around the black hole?"

Here we see the limitations of Penrose diagrams. For this puzzle we need to understand the 3d version of the Penrose diagram where the event horizon is not just a line segment, but a ball!

By the way your embrace of AI gives me the creeps, at least a little.

- it's here, page 214 of the PDF file: ...

2025-10-26T23:54:02+01:00

In reply to nevent1q…uftt
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- it's here, page 214 of the PDF file:

https://www.santilli-foundation.org/docs/hj-4-3-1981.pdf

- they both work for me now.

2025-10-24T10:12:45+02:00

In reply to nevent1q…rnj7
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- they both work for me now.

- thanks! You can watch my slow technical progress.

2025-10-22T22:44:09+02:00

In reply to nevent1q…uuff
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- thanks! You can watch my slow technical progress.

- the usual term is "current".

2025-10-21T00:46:02+02:00

In reply to nevent1q…zwan
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- the usual term is "current".

- Regardless of what some Wiki-editor meant by U₁(1), it's ...

2025-10-21T00:11:26+02:00

In reply to nevent1q…zqlu
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- Regardless of what some Wiki-editor meant by U₁(1), it's much clearer here to say U(1).

- I hardly ever program or do calculations other than on paper or ...

2025-10-19T23:39:59+02:00

In reply to nevent1q…pg6r
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- I hardly ever program or do calculations other than on paper or a very basic hand-held calculator. Because I don't do it much, I find it frustrating and tiresome. So I have to go begging to friends whenever I want to do a really lengthy computation!

- but you didn't explain something: what don't you like? ...

2025-10-18T00:49:13+02:00

In reply to nevent1q…jw54
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- but you didn't explain something: what don't you like? Going to a pub? Listening to music in a pub? Doing it with colleagues? Listening to some particular kind of music? Staying out after 10 pm?