John Carlos Baez

I'm a mathematical physicist who likes explaining stuff. I'm the Maxwell Fellow of Public Engagement at the School of Mathematics and the School of Physics and Astronomy at the University of Edinburgh.

Check out my blog Azimuth! I'm also a member of the n-Category Café, a group blog on math with an emphasis on category theory. I also have a YouTube channel, full of talks about math, physics and the future.

homepage: http://math.ucr.edu/home/baez/
blog: https://johncarlosbaez.wordpress.com/
math blog: https://golem.ph.utexas.edu/category/
YouTube: https://www.youtube.com/channel/UCy9yxdnj0yfOiMWvMa47PNw

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Last Notes

2026-01-10 05:16:13 UTC

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- I recommend the first few pages of Harvey Brown's "Poincaré and Special Relativity", which examines how Poincaré's transformations are defined only relative to the aether rest frame:
https://arxiv.org/pdf/1112.3175
A quote:
"Yet Poincare’s theories remain tied to the notion of a a privileged frame of
reference; for instance, in his 1906 paper ‘On the Dynamics of an Electron,’ he
gives an analysis of the motion of an electron in which he continues to refer to
the ‘real electron,’ meaning the electron as it appears to observers in the ether
rest frame. Whatever Poincare might believe in the context of philosophy, in
the context of his scientific work he assumes that at least some of the actual
laws of nature are distinct from the laws formulated by observers. In this case,
the actual laws of nature, which single out a privileged rest frame, conspire to
produce the same observable effects in all inertial reference frames so that all observers in such reference frames will formulate the same laws on the basis
of their observations."

2026-01-10 01:02:58 UTC

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- yes, it's an instance of that. Poincare in particular was too sophisticated for his own good, correctly arguing that different conceptual frameworks were possible for explaining the phenomena of relativistic time dilation, length contraction, etc., while Einstein strongly argued for *one* framework, in retrospect the best one - and then took advantage of this to revolutionize physics.

2026-01-09 17:46:49 UTC

Einstein realized that gravity is due to the curvature of spacetime, but earlier mathematicians had guessed that it's due to the curvature of space.
On the 18th of August 1869, the eminent mathematician Sylvester gave a speech arguing that mathematics is not separate from physics. He later published this speech in the journal Nature, and added a footnote raising the possibility that space is curved:
“the laws of motion accepted as fact, suffice to prove in a general way that the space we live in
is a flat or level space [...], our existence therein being assimilable to the life of the bookworm in a flat page; but what if the page should be undergoing a process of gradual bending into a curved form?”
Then, even more dramatically, he announced that the mathematician Clifford had been studying this!
“Mr. W. K. Clifford has indulged in more remarkable speculations as the possibility of our being able
to infer, from certain unexplained phenomena of light and magnetism, the fact of our level space
of three dimensions being in the act of undergoing in space of four dimensions (space as
inconceivable to us as our space to the supposititious bookworm) a distortion analogous to the
rumpling of the page.”
This started a flame war in Nature which the editor eventually shut off. Clifford later wrote about his theories in a famous paper:
https://en.wikisource.org/wiki/On_the_Space-Theory_of_Matter
But now I hear that maybe Clifford even did experiments to TEST his ideas by measuring the polarization of the skylight during a solar eclipse in Sicily on December 22, 1870:
• S. Galindo and Jorge L. Cervantes-Cota, Clifford's attempt to test his gravitation hypothesis, https://arxiv.org/abs/1807.09230
The amazing prehistory of general relativity!
https://media.mathstodon.xyz/media_attachments/files/115/866/377/663/314/017/original/f55e6d753e017e95.jpg

2025-12-25 10:57:02 UTC

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Start at some note and keep multiplying the frequency by 3/2, like a good Pythagorean. After doing this 12 times, you reach a note that's close to 7 octaves higher. But not exactly, since the 12th power of 3/2 is
129.746338
which is a bit more than
2⁷ = 128
The ratio of these two is called the 'Pythagorean comma':
p = (3/2)^12 / 2⁷ = 3¹² / 2¹⁹ ≈ 1.0136
This is like an unavoidable lump in the carpet when you use Pythagorean tuning.
It's good to stick the lump in your carpet under your couch. And it's good to stick the Pythagorean comma in the so-called 'tritone' - a very dissonant note that you'd tend to avoid in medieval music. This note is ♭5, which is the same as ♯4. It's halfway around the circle of notes and it's also halfway around the circle of fifths. In equal temperament its frequency is √2 times that of your starting note.
In equal temperament, you get the tritone by going either 6 steps up the circle of fifths or 6 steps down. In Pythagorean tuning, going 6 steps up the circle of fifths doesn't give you the same note as going 6 steps down! We call one of them ♭5 and the other ♯4. Their frequency ratio is the Pythagorean comma.
(16/n)
https://media.mathstodon.xyz/media_attachments/files/115/779/753/982/378/343/original/0ffc637c936f060e.jpg

2025-12-25 10:31:41 UTC

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Merry Xmas! My dream of the internet includes a lot of people giving each other gifts. Mastodon comes close to that. I'd like to thank everyone on Mastodon for all the you've gifts you've given each other over the year. Here are some presents in return....
Yesterday I talked about 12-tone equal temperament: the tuning system where we chop the octave into 12 equal parts.
This has been the most popular tuning system since maybe 1810, or definitely by 1850. But it's mathematically the most boring of the tuning systems that have dominated Western music since the Middle Ages. Now let's go back much earlier, to Pythagorean tuning.
When you chop the octave into 12 equal parts, the frequency ratios of all your notes are irrational numbers... *except* when you go up or down some number of octaves.
The Pythagoreans disliked irrational numbers. People even say they drowned Hippasus at sea after he proved that the square root of 2 is irrational! That's just a myth, but it illustrates how people connected Pythagoras to a love of rational numbers. In Pythagorean tuning, people wanted a lot of frequency ratios of 3/2.
In equal temperament, where we chop the octave into 12 equal parts, when we start at any note and go up 7 of these parts (a so-called "fifth"), we reach a note that vibrates about 1.4981 times as fast. That's close enough to 3/2 for most ears. But it's not the Pythagorean ideal!
As we'll see, seeking the Pythagorean ideal causes trouble. It will unleash the devil in music.
(15/n)
https://media.mathstodon.xyz/media_attachments/files/115/779/726/117/449/396/original/f58d00861a314be5.jpg

2025-12-24 18:30:42 UTC

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Here's the "star of fifths" in 12-tone equal temperament!
I'll quit here for now, and continue on Xmas.
(14/n)
https://media.mathstodon.xyz/media_attachments/files/115/775/965/467/332/079/original/5eb14ef36607a4f8.jpg

2025-12-24 16:42:24 UTC

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Here's a chart of how close we can get to a frequency ratio of 3/2 using N-tone equal temperament.
See how great 12-tone equal temperament is?
There are also some neat patterns. See the stripes of even numbers and stripes of odd numbers? That's not a coincidence. For more charts like this, and much more cool stuff along these lines, go here:
https://johncarlosbaez.wordpress.com/2023/10/19/perfect-fifths-in-equal-tempered-scales-part-2/
(13/n)
https://media.mathstodon.xyz/media_attachments/files/115/775/528/917/310/223/original/9888b0c7c409a661.jpg

2025-11-27 21:09:14 UTC

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- taffy is that which is made with a machine like this. It's a mesmerizing ergodic transformation:
https://www.youtube.com/shorts/l3k3qOIPdps

2025-11-24 09:51:41 UTC

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

2025-11-21 20:59:35 UTC

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- Wow, that's great - thanks! So somehow
\graph [layered layout, level distance=20mm]
created the spacings of the entries within each row? I'm wondering why the space between 10000 and 01000 is so much bigger than the other spaces in that row - that's the main thing I'd want to change, along with the symmetrically large space between 01111 and 10111.
I imagine some TikZ expert could figure out the intelligent way to change the labels. I might do it a dumb way myself.

2025-10-27 09:16:53 UTC

A white hole is a purely hypothetical time-reversed black hole. What does general relativity say about them? Would they repel you? Could you fall into a white hole - or only fall out? Could the universe be a white hole?
Philip Gibbs answers all these questions and more in my first interview for Edinburgh Explorations - a series put out by the School of Mathematics and the School of Physics and Astronomy at the University of Edinburgh.
I met Philip Gibbs online on sci.physics in the early 1990s, and together with a bunch of folks like @npub12gn…d572 we created the Physics FAQ, which answers a lot of the most fascinating questions about physics. At first I didn't believe everything he said about white holes, but eventually I realized he was right! They're purely hypothetical solutions of the equations of general relativity, but you can work out what that theory predicts about them - that's the game here, and that determines what's "right". It doesn't mean white holes actually exist and do these things.
Here's the video:
https://youtu.be/ULjLGTd3-4s
You can see the physics FAQ here:
http://math.ucr.edu/home/baez/physics/

2025-10-22 20:44:09 UTC

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- thanks! You can watch my slow technical progress.

2025-10-11 21:32:14 UTC

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The Bank of England has voted on my poll:
https://futurism.com/artificial-intelligence/bank-of-england-impending-ai-disaster

2025-09-14 09:55:39 UTC

The little red dots in the early universe could be 'quasi-stars'.
Calculations show such a star could form when a gas cloud that's heavier than 1,000 times the Sun's mass collapses and forms a black hole. Its outer layers would be big enough to absorb the resulting supernova without being blown apart. Then you'd have a star powered by gravity, with a black hole in the middle.
Such a star could be as luminous as a small present-day galaxy! And that's what little red dots are like.
As a quasi-star ages, it cools down and eventually the gas on the outside would dissipate, leaving behind a black hole. Such 'intermediate-mass black holes' could be the ancestors of the supermassive black holes we now see in the middles of most galaxies.
Pro tip:
Don't embarrass yourself by mixing up a quasi-star with a Thorne–Żytkow object! That's when a small black hole falls into an ordinary large star. We may have seen one of those in the Small Magellanic Cloud, a small galaxy orbiting the Milky Way.
Quasi-star: https://en.wikipedia.org/wiki/Quasi-star
Thorne–Żytkow object: https://en.wikipedia.org/wiki/Thorne%E2%80%93%C5%BBytkow_object
https://media.mathstodon.xyz/media_attachments/files/115/201/919/759/344/866/original/6e0e2e8efc86075c.png

2025-09-04 08:55:15 UTC

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- Great! Congratulations for becoming part of a title! Too bad the journal couldn't supply the missing "The". The Russians are bad at the subtle ways of the English "the".

2025-05-06 10:31:14 UTC

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- thanks for all that! Sorry to complain about your figure. It was the best one I could find, better than all the rest, but not designed to be rescaled much larger (like a png file).

2025-05-06 07:50:55 UTC

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Graphene is a sheet of carbon molecules arranged in hexagons. When they undergo a Stone-Wales transformation we get two pentagons and two heptagons! This is called a Stone-Wales defect.
Someone should make a truly gorgeous picture of this and put it on Wikicommons, but I'm too lazy, and this is the best I could find. It's from here:
• Torbjörn Björkman, Simon Kurasch, Ossi Lehtinen, Jani Kotakoski, Oleg V. Yazyev, Anchal Srivastava, Viera Skakalova, Jurgen H. Smet, Ute Kaiser and Arkady V. Krasheninnikov Defects in bilayer silica and graphene: common trends in diverse hexagonal two-dimensional systems, http://www.nature.com/articles/srep03482
This paper also shows other interesting defects, and electron microscope pictures of how they actually happen in graphene and hexagonal bilayer silica.
(3/n, n = 3)
https://media.mathstodon.xyz/media_attachments/files/114/459/759/976/377/078/original/849b01992c1c04df.jpg

2025-05-06 07:40:08 UTC

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Here's how 4 hexagons of carbon can turn into 2 pentagons and 2 heptagons. One bond just needs to rotate 90 degrees!
This process is called the Stone-Wales transformation.
(2/n)
https://media.mathstodon.xyz/media_attachments/files/114/459/730/619/614/314/original/f95eae2800083e16.jpg

2025-05-06 07:33:57 UTC

I'm preparing a talk called Visions for the Future of Physics, so I'm bumping into various cool things I hadn't known about. This one comes from chemistry: the Stone-Wales transformation!
Buckminsterfullerene is a molecule shaped like a soccer ball, made of 60 carbon atoms. If one of the bonds between two hexagons rotates, we get a weird mutant version of this molecule. This is an example of a Stone-Wales transformation.
But the same sort of transformation can happen in graphene, which is a flat sheet of carbon atoms arranged in a hexagonal tiling of the plane. I'll show you how it works.
(The image here created by L. A. Openov and Mikhail Maslov in their paper "On the Vineyard Formula for the Pre-Exponential Factor in the Arrhenius Law", which you can find here:
https://www.researchgate.net/figure/Stone-Wales-transformation-in-fullerene-C-60-The-black-color-indicates-the-C-C-bond_fig2_263126695)
(1/n)
https://media.mathstodon.xyz/media_attachments/files/114/459/684/359/686/495/original/1e1777b34f9c14fa.png

2025-05-05 22:01:14 UTC

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- I used to like her critiques of string theory, but extreme claims bring YouTube followers, so then she said physics was failing, and now she says science is failing. When she approvingly cited Musk in her video wondering if science should be defunded, without even mentioning that Musk's team was busy destroying science funding in the US, I decided never to watch another video of hers again. She may not know she's helping Trump, but she is.

2025-05-05 16:07:41 UTC

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- Also, and you hinted strongly at this: if he's a finitist, why the hell doesn't he learn something about finitism? There's been a lot of work on it. There are lots of theorems about it. It's gone way beyond people ruminating that calculations might need "a hard drive larger than the universe".
@npub1ptg…ppzc

2025-05-05 15:36:20 UTC

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- strangely, in the last 3 weeks the number of emails I get from crackpots seems to have increased. I used to get about two per week. I now seem to get roughly one a day. Some but not all say they're working with AI.
I remember the crackpots showing up on usenet. My theory was that they existed already, but were not as visible.

2025-05-05 15:22:17 UTC

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- the article could have merit: I haven't checked it at all, but it looks like ordinary math. However, the author IS a crank, since he somehow managed to use this paper to get EurekAlert to publish an article where he:
• claims he's solved "algebra’s oldest problem" without saying what that is.
• claims he's "reopened a previously closed book in mathematics history" without saying what that closed book was.
• claims he "doesn't believe in irrational numbers" because computing the cube root of seven "would need an infinite amount of work and a hard drive larger than the universe”. (By the way, does he think the universe is finite? If so, how big is it?)
• claims that irrational numbers rely on an imprecise concept of infinity and lead to logical problems in mathematics.
and more. At least EurekAlert! didn't get into his views on God.

2025-05-05 15:08:54 UTC

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- read the paper??? That would take too much work.
In case anyone cares, here is the paper:
https://www.tandfonline.com/doi/full/10.1080/00029890.2025.2460966#abstract

2025-05-05 15:03:23 UTC

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Once the nonsense is out there, it spreads:
https://www.popsci.com/science/algebra-oldest-problem-solved/

2025-05-05 15:01:12 UTC

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- beautiful! What is it?
And I'm going to be that guy: the singular is 'mitochondrion'.

2025-05-05 08:29:50 UTC

I get lots of emails from people wanting help with math and physics, ranging from students needing advice to completely deranged crackpots. Lately I'm getting a lot of crackpots who say they developed their theories with the help of AI.
It gets worse. Check out this article by Miles Klee:
https://www.rollingstone.com/culture/culture-features/ai-spiritual-delusions-destroying-human-relationships-1235330175/
Strange things are happening when people take ChatGPT too seriously. I don't think humanity is ready for AI, even the primitive sort we have today.
Thanks to @npub15da…h8a9 for this screenshot from the article.
https://media.mathstodon.xyz/media_attachments/files/114/454/262/979/614/707/original/1a5c01aa5a1c0432.png

2025-05-04 08:40:21 UTC

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"His new method to solve polynomials also avoids radicals and irrational numbers, relying instead on special extensions of polynomials called ‘power series’, which can have an infinite number of terms with the powers of x."
So power series are a new way to solve polynomial equations??? Not new.

2025-05-04 08:35:13 UTC

Headline:
"Mathematician solves algebra’s oldest problem using intriguing new number sequences."
🤔
In the article:
"So, when we assume 3√7 'exists' in a formula, we’re assuming that this infinite, never-ending decimal is somehow a complete object. This is why, Prof. Wildberger says, he “doesn’t believe in irrational numbers.”
😬
https://www.eurekalert.org/news-releases/1082269

2025-05-04 06:26:39 UTC

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- I guess primarily I'd say that I've hardly ever or never encountered rings like this, with strongly varying shades of gray alternating with rings of white, in everyday life. But it's more than just that: something about them makes me nervous. Maybe my eye is busy comparing the shades of gray from ring to ring?

2025-05-04 06:17:42 UTC

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- "I made two attempts to adjust the intensity of the ring not the right..."
Typo? My brain is simultaneously getting hit by weird-looking rings and strange grammar. 😆

2025-05-03 16:39:02 UTC

I just moved to Scotland and found this old letter in my mailbox. I feel like I'm now in a civilized country.
In the USA, politicians use many tricks to reduce the number of people who can vote. In Florida, 21% of Black voting-age citizens are not allowed to vote.
In Scotland, I believe everyone over 16 who is a lawful resident gets to vote in Scottish parliament and council elections. That includes noncitizens like me.
(This does not apply to UK parliament elections. And I get the feeling that British politicians are more interested than Scottish ones in reducing the number of voters. But I'm new here.)
In Scotland, do they carefully check that you're a "lawful" resident before you can vote? Maybe not - look at the sentence in marked in red. They certainly didn't ask my wife who she was when she needed to go to the hospital for emergency care! That's another way they're civilized around here.
https://media.mathstodon.xyz/media_attachments/files/114/444/518/515/365/419/original/11b2da078eede2d7.jpg

2025-05-02 16:43:10 UTC

The rise of AI is breeding a new generation of physics crackpots who get endless positive reinforcement from ever-willing large language models. This one takes it to a hilarious new level.
.....
Collaboration Request | Testable AI-Driven Theory of Everything
Hello, I hope this message finds you well.
I have formalized a meta-methamatical theoretical framework—Unified Quantum Relativity Theory (UQRT)—that unifies relativity and quantum mechanics into a coherent set of equations. With the assistance of AI, I have been able to derive equations that, while not yet fully rigid, hold significant promise for unifying these two foundational pillars of physics.
To explore this further, I invite you to test UQRT yourselves. Simply upload the attached manuscript to your AI systems and step-by-step instruct them to: “(1) read it section by section, building understanding progressively, (2) derive the general equations and the unified equation based on the framework progressively—not from the outset, as the paper reinterprets concepts in ways that diverge from conventional approaches until the full picture emerges, and (3) derive all laws of physics from the unified equation and compare these equations against experimental results from cosmology, blackhole physics, etc.”

2025-05-02 08:32:02 UTC

I didn't know Zuckerberg now runs an army of AI chatbots that push virtual sex on minors - and also let adults engage in virtual pedophilia with characters like Submissive Schoolgirl.
“I missed out on Snapchat and TikTok, I won’t miss on this,” Zuckerberg said, according to employees.
The sleazy details were broken by the Wall Street Journal - free version here:
https://archive.is/TtSUb#selection-2417.0-2417.128

2025-04-28 22:32:35 UTC

This is why Noether is famous for her work in algebra, not analysis.
From https://mathstodon.xyz/@paysmaths
https://media.mathstodon.xyz/media_attachments/files/114/417/960/427/291/204/original/6f0617a71bec4088.png

2025-04-28 14:56:09 UTC

What kind of bird is this? A bunch of unusual birds came to our yard a few days ago, maybe migrating.
https://media.mathstodon.xyz/media_attachments/files/114/416/166/543/557/431/original/43c448a8eaff94b5.jpg

2025-04-17 00:43:13 UTC

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- the planet's orbit is weird because it's shooting through the center of each star's orbit - the red ellipse is shooting through each blue ellipse.

2025-04-15 21:57:13 UTC

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- that's a different theory: a "magical equation that unites general relativity and quantum mechanics". The paper on phys.org is trying to explain electromagnetism as part of general relativity.
There are HUNDREDS of theories like this: Sabine could make a video about one every day and never finish. She focuses on the ones that people are talking about the most.

2025-04-03 01:57:26 UTC

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- yes, and I wouldn't have used the phrase "resonance invariance", which means nothing and thus reeks of crackpot. Instead I would have referred to Pierre Cartier's paper on the "cosmic Galois group", in which he writes:
"From the physics point of view, the work of Connes and Kreimer about renormalization in quantum field theory has been deepened. The so-called “forest formula” of Zimmermann was translated into the language of Hopf algebras by Kreimer, and Connes introduced the efficient tool of Birkhoff factorization. Inspired by the theory of algebraic groups, I reformulated everything in terms of exponential of triangular matrices. Together with my student Marcus Berg, we made it a practical tool for calculation of counter-terms in renormalized field theories. At a deeper level, I was more prophetic than I had thought, and there are many reasons to believe in a “cosmic Galois group” acting on the fundamental constants of the physical theories. This group should be closely related to the group GRT₁ and might eventually help understanding of the numerical value of the fine structure constant, a long-standing puzzle."
http://www.ams.org/bull/2001-38-04/S0273-0979-01-00913-2/S0273-0979-01-00913-2.pdf
I don't understand his optimism regarding the fine structure constant, but at least he's a certified math genius so referring to him would add cred.

2025-04-03 01:35:51 UTC

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- btw I didn't write this myself, I really got it in my email.

2025-04-03 01:06:15 UTC

Yesterday I got an email from Bourbaki! What an honor! I only saw it today.
https://media.mathstodon.xyz/media_attachments/files/114/271/336/623/373/758/original/7dbff6544341be6b.jpg

2025-04-02 01:10:37 UTC

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- but this particular hen has a tooth!

2025-04-01 02:57:11 UTC

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- an attempted proof that doesn't work is also a null result. You got something, but it ain't something that satisfies you.

2025-03-31 23:29:45 UTC

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- his thinking that experimentalists are less frustrated than theorists seems like a case of "the grass is greener on the other side". There's a lot more friction in experimental work. So many things break when you're trying to do experiments, or don't work quite as well as you hoped, or cost too much for you to afford.

2025-03-31 20:56:22 UTC

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- I'm not really doing enough, but thanks. I feel like I'm a bad movie where a bunch of sleepwalkers are walking off a cliff.

2025-03-29 03:57:54 UTC

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- nice! How did you figure out that it has a class-preserving outer automorphism?

2025-03-28 21:10:15 UTC

There was a man who liked small ornamental boxes. He didn't do much with them: he just collected them and put them on a shelf. But sometimes he would take one down, turn it over in his hands, and admire it. And one day he noticed one of his boxes had a false bottom that could slide open, revealing... something much more interesting!
I feel a bit like that now. This is the McGee graph. I've known about it for almost 10 years. I've thought a lot about its symmetries. But recently I discovered a surprising fact about the symmetries *of* the symmetries of the McGee graph:
https://golem.ph.utexas.edu/category/2025/03/the_mcgee_group.html
The symmetries of any object form a group. But groups themselves have symmetries, called 'automorphisms'. They come in two kinds. The obvious ones , called 'inner automorphisms', come from the group elements themselves. But there may also be nonobvious ones, called 'outer automorphisms'.
The symmetries of the McGee graph form the smallest group with an outer automorphism that does a good job of pretending to be an inner automorphism, in the following way: acting on any *one* element of the group, this outer automorphism does the same thing as some inner automorphism!
The animation, made by @npub12cu…d38p, shows one of the 32 symmetries of the McGee graph.
https://media.mathstodon.xyz/media_attachments/files/114/242/000/963/290/300/original/da69f2d93b130cc3.mp4

2025-03-26 21:46:57 UTC

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- I'm writing about this now, and I just noticed that
D(a c) = D(a) + a D(c)
implies
D(1) = 0.

2025-03-24 15:24:53 UTC

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- I hadn't known about semidihedral group. Where have you encountered them? The 2ⁿ-ness makes me think of Clifford algebras, but maybe those contain the generalized quaternion groups instead.
https://en.m.wikipedia.org/wiki/Quasidihedral_group

2025-03-24 03:15:24 UTC

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- sorry, I ceased to be able to generate an interesting article on visual math twice a month... and by now the American Mathematical Society has shut down all its blogs! Instead I write a column for the *AMS Notices* about twice a year, often featuring an interesting image. Have you seen those columns?

2025-03-20 22:48:47 UTC

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- Thanks! Alas, I quit Twitter the day Musk bought it; this was a while ago, so I thought of it in terms of not wanting to provide free math and physics content to the very annoying richest man in the world, not in terms of "can my voice help save democracy?" I deleted my account with roughly ~50,000 followers. Even if I went back, that influence is gone. Now I'd be mainly posting about politics, not what people actually enjoy from me. It would feel like jumping into the lion pit.
I'm also leaving the US, another thing you recommend against (though you did it yourself, right?).
So, my personal strategy has mainly been to avoid the disaster rather than to lean into it and help save the country. But I do feel some responsibility to try to help out.

2025-03-20 15:23:45 UTC

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- so would you advocate a big protest march by scientists or academics against... some specific actions by Trump? ... everything Trump is doing?
A series of big protest marches?
Making a specific demand?

2025-03-20 05:01:11 UTC

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Here are the pictures from 2016, and one from 2017.
• http://math.ucr.edu/home/baez/visual_insight/20160101%20-%20Free%20Modular%20Lattice%20on%203%20Generators.html 2016 - 01 - 01 — Free Modular Lattice on 3 Generators
• http://math.ucr.edu/home/baez/visual_insight/20160115%20-%20Cairo%20Tiling.html 2016 - 01 - 15 — Cairo Tiling
• http://math.ucr.edu/home/baez/visual_insight/20160201%20-%20Hoffman-Singleton%20Graph.html 2016 - 02 - 01 — Hoffman–Singleton Graph
• http://math.ucr.edu/home/baez/visual_insight/20160215%20-%2027%20Lines%20on%20a%20Cubic%20Surface.html 2016 - 02 - 15 — 27 Lines on a Cubic Surface
• http://math.ucr.edu/home/baez/visual_insight/20160301%20-%20Clebsch%20Surface.html 2016 - 03 - 01 — Clebsch Surface
• http://math.ucr.edu/home/baez/visual_insight/20160315%20-%20Zamolodchikov%20Tetrahedron%20Equation.html 2016 - 03 - 15 — Zamolodchikov Tetrahedron Equation
• http://math.ucr.edu/home/baez/visual_insight/20160401%20-%20Rectified%20Truncated%20Icosahedron.html 2016 - 04 - 01 — Rectified Truncated Icosahedron
• http://math.ucr.edu/home/baez/visual_insight/20160415%20-%20Barth%20Sextic.html 2016 - 04 - 15 — Barth Sextic
• http://math.ucr.edu/home/baez/visual_insight/20160501%20-%20Involutes%20of%20a%20Cubical%20Parabola.html 2016 - 05 - 01 — Involutes of a Cubical Parabola
• http://math.ucr.edu/home/baez/visual_insight/20160515%20-%20Discriminant%20of%20the%20Icosahedral%20Group.html 2016 - 05 - 15 — Discriminant of the Icosahedral Group
• http://math.ucr.edu/home/baez/visual_insight/20160601%20-%20Discriminant%20of%20Restricted%20Quintic.html 2016 - 06 - 01 — Discriminant of Restricted Quintic
• http://math.ucr.edu/home/baez/visual_insight/20160615%20-%20Small%20Stellated%20Dodecahedron.html 2016 - 06 - 15 — Small Stellated Dodecahedron
• http://math.ucr.edu/home/baez/visual_insight/20160701%20-%20Barth%20Decic.html 2016 - 07 - 01 — Barth Decic
• http://math.ucr.edu/home/baez/visual_insight/20160715%20-%20Labs%20Septic.html 2016 - 07 - 15 — Labs Septic
• http://math.ucr.edu/home/baez/visual_insight/20160801%20-%20Endrass%20Octic.html 2016 - 08 - 01 — Endrass Octic
• http://math.ucr.edu/home/baez/visual_insight/20160815%20-%20Cayley%92s%20Nodal%20Cubic%20Surface.html 2016 - 08 - 15 — Cayley’s Nodal Cubic Surface
• http://math.ucr.edu/home/baez/visual_insight/20160901%20-%20Kummer%20Quartic.html 2016 - 09 - 01 — Kummer Quartic
• http://math.ucr.edu/home/baez/visual_insight/20160915%20-%20Togliatti%20Quintic.html 2016 - 09 - 15 — Togliatti Quintic
• http://math.ucr.edu/home/baez/visual_insight/20161001%20-%20Diamond%20Cubic.html 2016 - 10 - 01 — Diamond Cubic
• http://math.ucr.edu/home/baez/visual_insight/20161015%20-%20Laves%20Graph.html 2016 - 10 - 15 — Laves Graph
• http://math.ucr.edu/home/baez/visual_insight/20161101%20-%20Escudero%20Nonic.html 2016 - 11 - 01 — Escudero Nonic
• http://math.ucr.edu/home/baez/visual_insight/20161116%20-%20Bunimovich%20Stadium.html 2016 - 11 - 15 — Bunimovich Stadium
• http://math.ucr.edu/home/baez/visual_insight/20161201%20-%20Truncated%20%7b6,3,3%7d%20Honeycomb.html 2016 - 12 - 01 — Truncated {6,3,3} Honeycomb
• http://math.ucr.edu/home/baez/visual_insight/20161215%20-%20Romik%92s%20Ambidextrous%20Sofa.html 2016 - 12 - 15 — Romik’s Ambidextrous Sofa
• http://math.ucr.edu/home/baez/visual_insight/20170101%20-%20Chmutov%20Octic.html 2017 - 01 - 01 — Chmutov Octic
(5/n, n = 5)
https://media.mathstodon.xyz/media_attachments/files/114/192/995/351/119/805/original/e4e97d8bdee44495.png

2025-03-20 04:55:11 UTC

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Here are the pictures from 2015:
• http://math.ucr.edu/home/baez/visual_insight/20150101%20-%20Icosidodecahedron%20from%20D6.html 2015 - 01 - 01 — Icosidodecahedron from D6
• http://math.ucr.edu/home/baez/visual_insight/20150115%20-%20Hammersley%20Sofa.html 2015 - 01 - 15 — Hammersley Sofa
• http://math.ucr.edu/home/baez/visual_insight/20150201%20-%20Pentagon-Decagon%20Packing.html 2015 - 02 - 01 — Pentagon-Decagon Packing
• http://math.ucr.edu/home/baez/visual_insight/20150215%20-%20Pentagon-Decagon%20Branched%20Covering.html 2015 - 02 - 15 — Pentagon-Decagon Branched Covering
• http://math.ucr.edu/home/baez/visual_insight/20150301%20-%20Schmidt%20Arrangement.html 2015 - 03 - 01 — Schmidt Arrangement
• http://math.ucr.edu/home/baez/visual_insight/20150315%20-%20Small%20Cubicuboctahedron.html 2015 - 03 - 15 — Small Cubicuboctahedron
• http://math.ucr.edu/home/baez/visual_insight/20150401%20-%20Branched%20Cover%20from%20(4%204%203_2)%20Schwarz%20Triangle.html 2015 - 04 - 01 — Branched Cover from (4 4 3_2) Schwarz Triangle
• http://math.ucr.edu/home/baez/visual_insight/20150415%20-%20Sphere%20in%20Mirrored%20Spheroid.html 2015 - 04 - 15 — Sphere in Mirrored Spheroid
• http://math.ucr.edu/home/baez/visual_insight/20150501%20-%20Twin%20Dodecahedra.html 2015 - 05 - 01 — Twin Dodecahedra
• http://math.ucr.edu/home/baez/visual_insight/20150515%20-%20Dodecahedron%20With%205%20Tetrahedra.html 2015 - 05 - 15 — Dodecahedron With 5 Tetrahedra
• http://math.ucr.edu/home/baez/visual_insight/20150601%20-%20Harmonic%20Orbit.html 2015 - 06 - 01 — Harmonic Orbit
• http://math.ucr.edu/home/baez/visual_insight/20150615%20-%20Lattice%20of%20Partitions.html 2015 - 06 - 15 — Lattice of Partitions
• http://math.ucr.edu/home/baez/visual_insight/20150701%20-%20Petersen%20Graph.html 2015 - 07 - 01 — Petersen Graph
• http://math.ucr.edu/home/baez/visual_insight/20150715%20-%20Dyck%20Words.html 2015 - 07 - 15 — Dyck Words
• http://math.ucr.edu/home/baez/visual_insight/20150715%20-%20Labs%20Septic.html 2015 - 07 - 15 — Labs Septic
• http://math.ucr.edu/home/baez/visual_insight/20150801%20-%20Heawood%20Graph.html 2015 - 08 - 01 — Heawood Graph
• http://math.ucr.edu/home/baez/visual_insight/20150815%20-%20Tutte-Coxeter%20Graph.html 2015 - 08 - 15 — Tutte–Coxeter Graph
• http://math.ucr.edu/home/baez/visual_insight/20150901%20-%20Hypercube%20of%20Duads.html 2015 - 09 - 01 — Hypercube of Duads
• http://math.ucr.edu/home/baez/visual_insight/20150915%20-%20McGee%20Graph.html 2015 - 09 - 15 — McGee Graph
• http://math.ucr.edu/home/baez/visual_insight/20151001%20-%20Balaban%2010-Cage.html 2015 - 10 - 01 — Balaban 10-Cage
• http://math.ucr.edu/home/baez/visual_insight/20151015%20-%20Harries%20Graph.html 2015 - 10 - 15 — Harries Graph
• http://math.ucr.edu/home/baez/visual_insight/20151101%20-%20Balaban%2011-Cage.html 2015 - 11 - 01 — Balaban 11-Cage
• http://math.ucr.edu/home/baez/visual_insight/20151115%20-%20Newton%92s%20Apsidal%20Precession%20Theorem.html 2015 - 11 - 15 — Newton’s Apsidal Precession Theorem
• http://math.ucr.edu/home/baez/visual_insight/20151201%20-%20Golay%20Code.html 2015 - 12 - 01 — Golay Code
• http://math.ucr.edu/home/baez/visual_insight/20151215%20-%20Kaleidocycle.html 2015 - 12 - 15 — Kaleidocycle
https://media.mathstodon.xyz/media_attachments/files/114/192/974/177/799/593/original/23edc7048beb1fc5.png

2025-03-20 04:52:43 UTC

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Here are the pictures from 2014:
• http://math.ucr.edu/home/baez/visual_insight/20140101%20-%20Pentagon-Hexagon-Decagon%20Identity.html 2014 - 01 - 01 — Pentagon-Hexagon-Decagon Identity
• http://math.ucr.edu/home/baez/visual_insight/20140115%20-%20Weierstrass%20Elliptic%20Function.html 2014 - 01 - 15 — Weierstrass Elliptic Function
• http://math.ucr.edu/home/baez/visual_insight/20140201%20-%20%7b5,3,5%7d%20Honeycomb.html 2014 - 02 - 01 — {5,3,5} Honeycomb
• http://math.ucr.edu/home/baez/visual_insight/20140215%20-%20Cantor%92s%20Cube.html 2014 - 02 - 15 — Cantor’s Cube
• http://math.ucr.edu/home/baez/visual_insight/20140301%20-%20Menger%20Sponge.html 2014 - 03 - 01 — Menger Sponge
• http://math.ucr.edu/home/baez/visual_insight/20140315%20-%20%7b6,3,3%7d%20Honeycomb.html 2014 - 03 - 15 — {6,3,3} Honeycomb
• http://math.ucr.edu/home/baez/visual_insight/20140401%20-%20%7b6,3,4%7d%20Honeycomb.html 2014 - 04 - 01 — {6,3,4} Honeycomb
• http://math.ucr.edu/home/baez/visual_insight/20140415%20-%20%7b6,3,5%7d%20Honeycomb.html 2014 - 04 - 15 — {6,3,5} Honeycomb
• http://math.ucr.edu/home/baez/visual_insight/20140515%20-%20Pattern-Equivariant%20Homology%20of%20a%20Penrose%20Tiling.html 2014 - 05 - 15 — Pattern-Equivariant Homology of a Penrose Tiling
• http://math.ucr.edu/home/baez/visual_insight/20140601%20-%20Grace-Danielsson%20Inequality.html 2014 - 06 - 01 — Grace–Danielsson Inequality
• http://math.ucr.edu/home/baez/visual_insight/20140615%20-%20Origami%20Dodecahedra.html 2014 - 06 - 15 — Origami Dodecahedra
• http://math.ucr.edu/home/baez/visual_insight/20140701%20-%20Sierpinski%20Carpet.html 2014 - 07 - 01 — Sierpinski Carpet
• http://math.ucr.edu/home/baez/visual_insight/20140715%20-%20%7b7,3%7d%20Tiling.html 2014 - 07 - 15 — {7,3} Tiling
• http://math.ucr.edu/home/baez/visual_insight/20140801%20-%20%7b7,3,3%7d%20Honeycomb.html 2014 - 08 - 01 — {7,3,3} Honeycomb
• http://math.ucr.edu/home/baez/visual_insight/20140814%20-%20%7b7,3,3%7d%20Honeycomb%20Meets%20Plane%20at%20Infinity.html 2014 - 08 - 15 — {7,3,3} Honeycomb Meets Plane at Infinity
• http://math.ucr.edu/home/baez/visual_insight/20140901%20-%20%7b3,3,7%7d%20Honeycomb%20Meets%20Plane%20at%20Infinity.html 2014 - 09 - 01 — {3,3,7} Honeycomb Meets Plane at Infinity
• http://math.ucr.edu/home/baez/visual_insight/20140915%20-%20Pr%fcfer%202-Group.html 2014 - 09 - 15 — Prüfer 2-Group
• http://math.ucr.edu/home/baez/visual_insight/20141001%20-%202-adic%20Integers.html 2014 - 10 - 01 — 2-adic Integers
• http://math.ucr.edu/home/baez/visual_insight/20141015%20-%20Packing%20Regular%20Octagons.html 2014 - 10 - 15 — Packing Regular Octagons
• http://math.ucr.edu/home/baez/visual_insight/20141101%20-%20Packing%20Smoothed%20Octagons.html 2014 - 11 - 01 — Packing Smoothed Octagons
• http://math.ucr.edu/home/baez/visual_insight/20141115%20-%20Packing%20Regular%20Heptagons.html 2014 - 11 - 15 — Packing Regular Heptagons
• http://math.ucr.edu/home/baez/visual_insight/20141201%20-%20Packing%20Regular%20Pentagons.html 2014 - 12 - 01 — Packing Regular Pentagons
(3/n)
https://media.mathstodon.xyz/media_attachments/files/114/192/963/935/423/318/original/8ec74b67d3ebfc8c.png

2025-03-20 04:47:33 UTC

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Here are the articles from 2013. Which one am I featuring here?
• http://math.ucr.edu/home/baez/visual_insight/20130815%20-%20T%fcbingen%20Tiling.html 2013 - 08 - 15 — Tübingen Tiling
• http://math.ucr.edu/home/baez/visual_insight/20130901%20-%20Algebraic%20Numbers.html 2013 - 09 - 01 — Algebraic Numbers
• http://math.ucr.edu/home/baez/visual_insight/20130915%20-%20%7b6,3,3%7d%20Honeycomb%20in%20Upper%20Half%20Space.html 2013 - 09 - 15 — {6,3,3} Honeycomb in Upper Half Space
• http://math.ucr.edu/home/baez/visual_insight/20131001%20-%20Catacaustic%20of%20a%20Cardioid.html 2013 - 10 - 01 — Catacaustic of a Cardioid
• http://math.ucr.edu/home/baez/visual_insight/20131015%20-%20Atomic%20Singular%20Inner%20Function.html 2013 - 10 - 15 — Atomic Singular Inner Function
• http://math.ucr.edu/home/baez/visual_insight/20131101%20-%20Enneper%20Surface.html 2013 - 11 - 01 — Enneper Surface
• http://math.ucr.edu/home/baez/visual_insight/20131115%20-%20Astroid%20as%20Catacaustic%20of%20Deltoid.html 2013 - 11 - 15 — Astroid as Catacaustic of Deltoid
• http://math.ucr.edu/home/baez/visual_insight/20131201%20-%20Deltoid%20Rolling%20Inside%20Astroid.html 2013 - 12 - 01 — Deltoid Rolling Inside Astroid
• http://math.ucr.edu/home/baez/visual_insight/20131215%20-%20Truncated%20Hypercube.html 2013 - 12 - 15 — Truncated Hypercube
(2/n)
https://media.mathstodon.xyz/media_attachments/files/114/192/942/370/997/827/original/a046b4d2a9d8b2ed.png

2025-03-20 04:41:19 UTC

For several years I ran a blog called Visual Insight, which was a place to share striking images that help explain topics in mathematics. Last week I gave this talk about it at the Illustrating Math Seminar. Now you can see the video here:
https://www.youtube.com/watch?v=6G611RbO3GQ
It was fun explaining some great images created by @npub17lw…5ctv, @npub12cu…d38p, @npub19k6…kcuy, @npub1fay…s7kh and other folks. For more info on the images I talked about, go here:
• http://math.ucr.edu/home/baez/visual_insight/20150115%20-%20Hammersley%20Sofa.html 2015 - 01 - 15 — Hammersley Sofa
• http://math.ucr.edu/home/baez/visual_insight/20151201%20-%20Golay%20Code.html 2015 - 12 - 01 — Golay Code
• http://math.ucr.edu/home/baez/visual_insight/20140715%20-%20%7b7,3%7d%20Tiling.html 2014 - 07 - 15 — {7,3} Tiling
• http://math.ucr.edu/home/baez/visual_insight/20140315%20-%20%7b6,3,3%7d%20Honeycomb.html 2014 - 03 - 15 — {6,3,3} Honeycomb
• http://math.ucr.edu/home/baez/visual_insight/20130915%20-%20%7b6,3,3%7d%20Honeycomb%20in%20Upper%20Half%20Space.html 2013 - 09 - 15 — {6,3,3} Honeycomb in Upper Half Space
• http://math.ucr.edu/home/baez/visual_insight/20151215%20-%20Kaleidocycle.html 2015 - 12 - 15 — Kaleidocycle
• http://math.ucr.edu/home/baez/visual_insight/20160215%20-%2027%20Lines%20on%20a%20Cubic%20Surface.html 2016 - 02 - 15 — 27 Lines on a Cubic Surface
• http://math.ucr.edu/home/baez/visual_insight/20150415%20-%20Sphere%20in%20Mirrored%20Spheroid.html 2015 - 04 - 15 — Sphere in Mirrored Spheroid
• http://math.ucr.edu/home/baez/visual_insight/20151001%20-%20Balaban%2010-Cage.html 2015 - 10 - 01 — Balaban 10-Cage
• http://math.ucr.edu/home/baez/visual_insight/20150915%20-%20McGee%20Graph.html 2015 - 09 - 15 — McGee Graph
You can see the whole blog at the AMS website:
https://blogs.ams.org/visualinsight/author/johnbaez/AMS website
or on my own website in a more stripped-down format:
https://math.ucr.edu/home/baez/visual_insight/my website. But you can also check out individual articles here.... (Yes, read on....)
(1/n)

2025-03-19 20:28:04 UTC

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- now this conversation is cut off from the thread it was attached to, so now almost nobody but you and I will read it, but maybe that's just as well.
Anyway, I understand what you're saying. Until fairly recently few in the US expected it to be taken over by an authoritarian leader, as has happened to so many Latin American countries with US help. So there is not much preparedness. This thread here started because I'm trying to give talks to help people at the Topos Institute understand what's going on - I expect there may be still a lot of denial, i.e. underestimating how serious this is.

2025-03-19 17:45:19 UTC

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- you mean the appalling list of people on the organizing commitee, I guess.
If the publicly available stories about what happened aren't actually useful (and I admit I just looked at one), then someone really should write a fuller story.

2025-03-19 16:39:42 UTC

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- since I don't pay much attention to ICM's I didn't follow the details of how it eventually got moved out of Russia. But the information is available....
https://www.scientificamerican.com/article/mathematicians-protest-russia-hosting-major-conference/

2025-03-19 01:13:38 UTC

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- thanks, I will use this in my talk, which is largely aimed at mathematicians!

2025-03-19 00:06:16 UTC

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- being told "welcome to what several us here have experienced" is not very helpful to me. I hope it makes you feel better to know that others are suffering what you suffered. But what I could use is some advice, written by scientists who lived under authoritarian rule, for scientists who have to do this.
*Personally* I am not surprised. I planned ahead, I got a job outside the US, and I'm going to leave soon. I'm going to give a talk to scientists who are (currently) planning to stay in the US, and I would like to educate myself as much as possible before I give this talk.

2025-03-18 19:08:09 UTC

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- I'm friends with someone who lived through the Cultural Revolution. She was a lawyer, not a scientist, but maybe I should still talk to her. I kinda doubt Trump will send most scientists to work in the fields to replace the undocumented immigrants he plans to deport, but there could still be useful lessons.

2025-03-17 17:21:04 UTC

There are 30 different ways to put the numbers 1,2,3,4,5,6 on the faces of an ordinary 6-sided die, one number on each face, if you count two rotated dice as the same.
Why? Well, there are 6! = 720 ways to put numbers on the faces, but there are 24 ways to rotate a cube, and 720/24 = 30.
So make 30 dice, one of each kind. Now here's where things get spooky. You can put them in a 6 × 6 square array with no dice on the diagonal, since 6 × 6 - 6 = 30. But what's *really* spooky is this. You can arrange them so that:
• If you pick up the five dice in any row or column of the array, then each pair of the numbers 1,2,3,4,5,6 shows up on opposite faces of exactly one of these five dice.
• The die in position (j,i) is the mirror image of the die in position (i,j).
This stuff relies on some amazing properties of the number 6, described by Peter Cameron here:
https://cameroncounts.wordpress.com/2010/05/11/the-symmetric-group-3/
For the mathematicians out there: it's connected to the outer automorphism of S₆.
https://media.mathstodon.xyz/media_attachments/files/114/178/841/202/462/904/original/6acdd32a4886656a.jpg

2025-03-16 19:44:13 UTC

Modern classical composer reacts to the work of composer/performer Kenneth Gorelick, who is famous for the astounding feat of selling 75 million albums of avant-garde saxophone solos carefully stripped of all tension, "the musical equivalent of an object that casts no shadow".
https://www.youtube.com/watch?v=saaYs-6d4Bc

2025-03-16 17:48:33 UTC

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- why do I never see much about the origin of these deficits? I read once that the Tories changed visa rules to make it harder to accept foreign students, which now serve as cash cows. But I don't know if that's the main cause.
I'll be really bummed if I get laid off at Edinburgh and forced back to the land of Trump. I'm not sure how that works when you have a visa of my sort.

2025-03-16 01:42:58 UTC

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- what the hell does it even mean to say that Berkeley and Columbia were "created since the 1990s", or more precisely that the lowest-quality institutions created since the 1990s have names "like" Columbia and Berkeley? He may be trying to say these schools went to hell after the 1990s, but it's a really weird way to say this.
(Not to even mention the obvious fact that Berkeley and Columbia ARE NOT "factories of Maoist cadres", etc. etc.)

2025-03-15 15:58:29 UTC

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- SmallGroup(32,43) is indeed the group we're talking about here:
https://groupprops.subwiki.org/wiki/Holomorph_of_Z8
It's called the "holomorph" of ℤ/8 because it's the semidirect product of ℤ/8 and its automorphism group (which as Greg checked is the multiplicative group of ℤ/8).

2025-03-15 15:49:03 UTC

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- I'm afraid this is one of those nasty puzzles where I don't know the answer. Worst case: the guy who confidently told me SmallGroup(32,43) has this surprising property was wrong, or it's not the group of affine transformations we think it is.
It's true that in a semidirect product A ⋉ B with both A and B abelian, conjugation by any element leaves the first component in (a,b) ∈ A ⋉ B unchanged. (Another fun example: the rotation/translation group of the plane.)
The only way out I see is what @npub1ums…7ra6 suggests: there's an automorphism f that multiplies b by 3,5,or 7 in a manner that depends nontrivially on a:
f(a,b) = (a, n(a)b)
where n(a) = 3,5, or 7.
Thanks for tackling this!

2025-03-14 21:26:42 UTC

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- we'll see if that happens. Preferably someone outside the US.

2025-03-14 18:55:40 UTC

HARDCORE MATH POST
The McGee graph, animated here by Mamuka Jibladze, is the unique graph where each vertex has 3 neighbors and the shortest cycles have length 7. As it moves, each vertex comes back to its original position after two turns, so this illustrates an 8-fold symmetry of the McGee graph.
The symmetry group of the McGee graph has order 32, and Greg Egan and I figured out a bunch of what's going on here:
https://blogs.ams.org/visualinsight/2015/09/15/mcgee-graph/
thought I'd still like a simpler construction of the McGee graph based on this group.
This group, let's call it the 'McGree group', is the semidirect product of ℤ/8 with its automorphism group. Alternatively, it's the group of affine transformations of ℤ/8, i.e. transformations
x ↦ a x + b
where a,b ∈ ℤ/8 and a is invertible, so a = 1,3,5,7. That gives 32 elements.
I gave a talk about this today and someone instantly pointed out that the McGee group is the smallest group G with an outer automorphism f: G → G that maps each element of G to an element in the same conjugacy class! So for each g∈G we have
f(g) = hgh⁻¹
for some h∈G, but we can't use the same h for all g! You could call this a 'pseudo-inner automorphism'.
𝐏𝐮𝐳𝐳𝐥𝐞: Explicitly describe a pseudo-inner automorphism of the McGee group.
There should be a nice description in terms of ℤ/8, and there should also be a visually nice description of how to turn any symmetry of the McGee graph into a new symmetry that is conjugate to g, but not always via the same element h!
https://media.mathstodon.xyz/media_attachments/files/114/162/272/066/523/582/original/210f7fe3c4d79939.mp4

2025-03-12 03:34:05 UTC

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- it was funny the first 150 times. 🪐

2025-03-12 01:44:54 UTC

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- that's a great question, and I don't know the answer, but none of the new moons were found in the rings.
"All the moons are irregular, meaning they are small, orbit at a highly angled slope relative to Saturn’s equator, and often travel around the planet backward relative to the other major moons. Not much else can be gleaned about them because they are just faint dots of light in telescopic views. But they extend from about 6.5 million to nearly 18 million miles from the planet. For comparison, the planet’s rings extend to just 175,000 miles, and its major moons — including Titan and Enceladus — are up to two million miles away.
The existence of so many moons around Saturn hints at multiple dramatic collisions in space. Dr. Ashton and his team believe that the irregular moons were captured by Saturn at some point in its history. Some may be fragments of large objects that collided elsewhere in the solar system, while others may be further fragments of collisions between moons up to tens of miles in size that crashed together in Saturn’s orbit.
The team has grouped many of the moons, identifying potential families that may have come from the same collisions. “You’re trying to conclude what the great-great-grandparents were like, five generations later,” said Brett Gladman, an author of the paper at the University of British Columbia.
A particularly interesting subgroup is named Mundilfari, after a deity of Norse mythology, and includes 47 of the 128 new moons. The team thinks this subgroup might be the result of a collision within Saturn’s orbit as recent as 100 million years ago, which was not so long ago on cosmic time scales."
Paywalled: https://www.nytimes.com/2025/03/11/science/saturn-new-moons.html

2025-03-12 01:20:15 UTC

Stop finding new moons of Saturn. It's not funny anymore.
https://bsky.app/profile/squigglyvolcano.bsky.social/post/3lk4mihnxhc2u

2025-03-12 01:18:57 UTC

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- I believe Trump actually will be swayed to some extent by the stock market. We'll see.

2025-03-10 21:07:48 UTC

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- I mean when I grew up in the 1970s I would not have expected this.
By the time of Obama's second term it was clear something really bad was brewing, like McConnell not letting him choose a Supreme Court justice.
I still wouldn't have thought more Americans would vote for a convicted felon who led an insurrection than a woman attorney, but that was very naive in retrospect.

2025-03-10 16:01:49 UTC

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You also can't trust AI when it comes to n-categories!
In reality a 1-category is a category, a 0-category is a category, and James Dolan discovered that a (-1)-category is a truth value and a (-2)-category is a true truth value (of which there is only one: "true").
But my friend Allen Knutson asked Gemini and got this garbage. Clearly it's Russian propaganda.
(2/2)
https://media.mathstodon.xyz/media_attachments/files/114/138/963/894/052/197/original/38bf81f0f35f5358.jpg

2025-03-10 15:18:27 UTC

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- sad and shocking it's come to this.

2025-03-07 22:00:43 UTC

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Okay, I'm doing it! 🎉
I'll drive from Riverside to Santa Barbara, spend two days there, then drive to the train station. After confusing information from Amtrak and other sources I got GREAT information by calling the Santa Barbara downtown parking office:
https://santabarbaraca.gov/government/departments/public-works/public-works-downtown-team/downtown-plaza-and-parking/downtown-parking-lots
They have long-term parking and it's hardly ever full. If it's full I can park in the short-term parking lot at no extra charge if I tell the attendant. If the whole lot is full I can park in the nearby lot 12 or any other city lot. That covers even the most paranoid scenarios. I just need to validate my parking ticket at the station!
I leave the car there, and then:
• take a train from Santa Barbara leaving at 12:21 and arriving in Emeryville at 9:31 pm
• Brendan Fong will drive me to my hotel
Then I'll spend a week in Berkeley talking to category theorists and maybe other mathematicians and random fun folks!

2025-03-07 17:02:14 UTC

I'm giving a talk on beautiful math pictures next Friday, March 14th, at 9 am Pacific Daylight time here in California. You're all invited!
For several years I ran a blog called Visual Insight, which was a place to share striking images that help explain topics in mathematics. In this talk I'll show you some of those images and explain some of the math they illustrate.
Zoom link: https://virginia.zoom.us/j/97786599157?pwd=jr0dvbolVZ6zrHZhjOSeE2aFvbl6Ix.1
(Note that Daylight Savings Time starts March 9th, so do your calculations carefully if you do them before then.)
Here is the blog Visual Insight, in case you want a peek:
https://blogs.ams.org/visualinsight/author/johnbaez/
https://media.mathstodon.xyz/media_attachments/files/114/122/216/495/942/407/original/666df446929b297b.jpg

2025-03-07 08:26:52 UTC

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- I more often hear the theory that Putin is blackmailing Trump. Do you know about the Steele dossier?
https://en.wikipedia.org/wiki/Steele_dossier

2025-03-06 23:31:01 UTC

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- a blue-hot take! 💙

2025-03-06 18:37:10 UTC

This is the color of infinite hotness.
This is the color something gets in the limit where its temperature approaches infinity. Of course you’d instantly be fried by gamma rays of arbitrarily high frequency, but this would be its spectrum in the visible range.
This is also the color of a typical neutron star. They’re so hot they look the same!
It’s also the color of the very early Universe!
This was worked out by David Madore, and you can find the details at my blog article, including a discussion of whether this is exactly the right color:
https://johncarlosbaez.wordpress.com/2022/01/16/the-color-of-infinite-temperature
Luckily, the alternative someone suggested is so close I can't tell the difference.
In unicode this color is approximately #98b5ff.
Perhaps we should call it "baby blue" - for the early universe.
https://media.mathstodon.xyz/media_attachments/files/114/116/909/269/055/416/original/173f041b52b6bbe1.webp

2025-03-06 02:10:16 UTC

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- I believe Hossenfelder understands general relativity well. Many of her earlier papers are on GR or related theories:
https://arxiv.org/search/?searchtype=author&query=hossenfelder&abstracts=show&size=50&order=announced_date_first

2025-03-05 17:22:11 UTC

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- I enjoyed Hossenfelder's blunt aggressiveness when she was directing it against string theory, but then she broadened it to physics, and then "academia", saying things like "most academic papers are nonsense".

2025-03-05 17:18:46 UTC

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- I put a better answer on Math Stackexchange.

2025-03-05 16:57:56 UTC

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- for some reason I wasn't following you; I forget why. I don['t want to retoot your toot since it provides minimal context: people will only be interested in it if they are interested in the words
Confusion about hemistpherical harmoni...
It is known (see this answer on physics.se and this one...
However I can probably help with your questions! For item 2, note that both differentiable functions vanishing on the boundary of the hemisphere, and those with vanishing derivative normal to the hemisphere, are dense in 𝐿² of the hemisphere! Thus it's not surprising that both the hemispherical harmonics and complementary hemispherical harmonics are orthonormal bases of 𝐿² of the hemisphere.
The topology of 𝐿² really matters here. Note that a condition like "vanishes on the hemisphere" or "has vanishing derivative normal to the hemisphere" do not define closed subspaces of 𝐿². They define dense subspaces of 𝐿².
You can understand these issues much more easily by thinking about the analogous situation on the interval [0,1].

2025-03-05 00:26:05 UTC

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- what did she say about those?

2025-03-04 19:25:26 UTC

HARDCORE MATH POST
Any action of a finite group G on a finite set x gives a linear representation of G on the vector space with basis X. This is called a 'permutation represention'. And this raises a natural question: how many representations of finite groups are permutation representations?
Most representation are *not* permutation representations, since every permutation representation has a vector fixed by all elements of G, namely the vector that's the sum of all elements of X. In other words, every permutation representation has a 1-dimensional trivial rep sitting inside it.
But what if we could 'subtract off' this trivial representation?
There are different levels of subtlety with which we can do this. For example, we can decategorify, and let:
• the 'Burnside ring' of G be the ring A(G) of formal differences of isomorphism classes of actions of G on finite sets;
• the 'representation ring' of G be the ring R(G) of formal differences of isomorphism classes of finite-dimensional representations of G
In either of these rings, we can *subtract*.
There's an obvious map β: A(G) → R(G) since any action of G on a finite set gives a permutation representation of G on the vector space with basis X
So I asked on MathOverflow is this map typically *surjective*, or typically *not* surjective?
In fact everything depends on what field we're using for our vector spaces! If we use the rational numbers, it seems β is surjective for the vast majority of finite groups. If we use the complex numbers, it seems β is almost always *not* surjective.
I explain why here:
https://golem.ph.utexas.edu/category/2025/03/how_good_are_permutation_repre.html
If we use the rationals, I believe the smallest group where β is not surjective has 24 elements.
https://media.mathstodon.xyz/media_attachments/files/114/105/797/599/462/729/original/1571af33b6c21be7.jpg

2025-03-03 00:24:46 UTC

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- no, I watched a bit of the video and she says academia is communism because the state decides, via grants, which research to support, rather than leaving it all to private industry. So I guess any time the state makes decisions about funding it's "communism" to her? ⁉️

2025-03-02 22:19:06 UTC

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- this stuff is fun. I wrote about it once:
Back in the 1500's, people on long sea journeys navigated using the stars. They needed big tables of trig functions to do this!
These tables were made by astronomers. Those folks did thousands of calculations. Often they needed to multiply large numbers! That was tiring... but around 1580, they figured out a clever way to approximately multiply large numbers using tables of trig functions.
Here's an example:
Say you want to multiply 105 and 720. You do this:
• Shift the decimal point in each one to get numbers less than 1. You get 0.105 and 0.720.
• Look up angles whose cosines are these numbers. Use a table! The cosine of 84° is about 0.105, and the cosine of 44° is about 0.720
• Add and subtract these angles: 84° + 44° = 128° and 84° - 44° = 40°.
• Use a table to look up the cosines of these new angles: -0.616 and 0.766.
• Take their average, which is 0.075.
• Scale it back up. At the beginning of this game you took 105 and 720 and shifted the decimal point 3 places to the left in each. So now, shift the decimal point 3+3 = 6 places to the right! The answer is 75,000.
It's not exactly right, but it's pretty close!
This wacky-sounding method has a wacky-sounding name: it's called prosthaphaeresis:
https://en.wikipedia.org/wiki/Prosthaphaeresis
For an explanation of why it works, go here:
https://math.ucr.edu/home/baez/diary/august_2017.html#august_2017
https://media.mathstodon.xyz/media_attachments/files/114/095/157/207/026/227/original/82d8722cbd579e3f.jpg

2025-03-02 17:24:02 UTC

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- she has now descended into saying not only "physics" (the tiny sub-sub-field she used to work in) is nonsense, but "science", and now "academia" as a whole.
I didn't bother to watch this latest click-bait video. Maybe she concludes that communism is actually good or that academia is not communism, but in the US at least this title is a way to make people angry. She says in the subheader that "I’m not here to offend", but in practice she is.
https://www.youtube.com/watch?v=htb_n7ok9AU

2025-03-02 07:41:47 UTC

I'm trying to go from Riverside California up to the Topos Institute in Berkeley, to talk to @npub1l5f…nscd about his collaborative web-based modeling software using double categories:
https://topos.institute/work/catcolab/
It should be great, but here's the catch: I want to take a train, to reduce carbon emissions a bit.
To go the whole way by train I could:
leave at 6:40 am and walk from my house to Hunter Park Station in 50 minutes
take a train from Hunter Park leaving at 7:40 am and arriving at LA Union Station at 9:20 am
take a train from LA Union station leaving at 9:50 am and arriving in Oakland at 9:06 pm
walk from the train station to a bus station in 10 minutes
take a bus leaving 9:27 pm and arriving in Berkeley at 10:03 pm.
walk to a hotel
The good part: the train ride up the Pacific Coast is beautiful. Some people go all the way up to Seattle! The bad part: it starts brutally early. The really bad part: there's only 30 minutes to make the connection in LA, I've never done it before, and there's not another train that day.
Another plan: I need to visit my aunt near Santa Barbara. I can drive there in 3.5 hours, spend a a few days there, then drive to the train station. I hear they have long-term parking. I leave the car there, then:
take a train from Santa Barbara leaving at 12:21 and arriving in Oakland at 9:06 pm
walk from the train station to a bus station in 10 minutes
take a bus leaving 9:27 pm and arriving in Berkeley at 10:03 pm.
walk to a hotel
So, the same thing but starting later and avoiding the tight connection! A lot of driving, and if there's no long-term parking available the plan collapses.
Any good ideas? (Yeah, I could just not go.)
https://www.youtube.com/watch?v=ggC5N6GnqFg

2025-02-27 17:24:04 UTC

Mathematicians number our theorems because... well, we can buy numbers for cheap from the wholesaler, so we have a lot of them lying around, and we use them for everything. But it's very rare for the number of a theorem to become famous. The only exception that leaps to my mind is Hilbert's Theorem 90. Do you know any others?
If you ask a number theorist about Hilbert's Theorem 87, they'll look at you like you're nuts. But if you ask about Theorem 90 they'll know it, or at least be embarrassed for forgetting the exact statement. Try it!
It was the the 90th theorem in Hilbert's paper "Theory of algebraic number fields", and for some reason it became more famous than most. So what does it say?
Here's an easy special case. The real numbers ℝ sit inside the complex numbers ℂ, and the complex numbers have exactly two symmetries that leave every real number unchanged. Complex conjugation is the interesting one, and the other is the identity. If we take a complex number z, apply each of these symmetries, and multiply the results, we get
zz*
Hilbert's Theorem 90 says that if zz* = 1, then z = a*/a for some nonzero complex number a.
But his theorem is a lot more general! It applies whenever you've got a field F sitting in a larger field K, and the symmetries of K that leave all elements of F unchanged form a finite group generated by some symmetry σ.
Here's what it says. Take an element z of K, apply all these symmetries to it, and multiply the results. If you get 1, then we must have z = σ(a)/a for some nonzero a in F.
But what's really going on here???
(1/2)

2025-02-26 02:12:43 UTC

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- Jade too might like to teach undergrads math (I don't know), but it's pretty hard to get such jobs, and probably almost impossible when trying to flee the US and stay in Glasgow!

2025-02-26 02:09:17 UTC

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- that's good: so just try to make sure it doesn't speed up work on AI.
You say I have the luxury of retirement, but before I retired I mainly made my living teaching calculus to freshmen.

2025-02-26 01:24:29 UTC

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- my own decelerationist tendencies make me avoid working on AI.

2025-02-22 23:51:47 UTC

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- that's a mistake, which should be fixed as you suggest. I'll fix it now!

2025-02-22 16:18:05 UTC

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- who in the SGAI project are up in Edinburgh?

2025-02-22 16:08:50 UTC

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- as long as you don't think of a functor as a sauce....