John Carlos Baez

I'm a mathematical physicist who likes explaining stuff. Sometimes I work at the Topos Institute. Check out my blog! 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.

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Last Notes

2025-04-06 01:48:47

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I'm glad people are standing up for the CDC.
https://media.mathstodon.xyz/media_attachments/files/114/288/497/652/009/625/original/51d0bf04fa890bdc.jpg

2025-03-15 16:55:33

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@nprofile…l9jz @nprofile…yu7x - okay, I read that this paper shows the semidirect product of ℤ/n and its multiplicative group has an outer automorphism that preserves conjugacy classes whenever n is divisible by 8:
G. E. Wall, Finite groups with class-preserving outer automorphisms, J. London Math. Soc. 22 (1947), 315–320.
I read it here:
Peter A. Brooksbank and Matthew S. Mizuhara, On groups with a class-preserving outer automorphism, https://msp.org/involve/2014/7-2/involve-v7-n2-p04-p.pdf

2025-01-11 17:43:46

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@nprofile…guz4 - I left a comment on your blog. I agree with you!

2025-01-03 07:12:53

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@nprofile…hn0f - Be careful: if you slip up, you'll be a millennium off.
https://media.mathstodon.xyz/media_attachments/files/113/763/179/705/233/205/original/44636a0e74a8bb87.jpg

2024-12-28 16:40:34

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@nprofile…m2ru - thanks! Weirdly, people have become a lot more eager to talk about stuff with me here in the last week. It could be a holiday effect - I'll see if it goes away - or it could be that some attempts to tweak my style to be more inviting have paid off. I'm (belatedly) guessing that just explaining things clearly is not a good way to start a conversation.
For example, admitting that I'm confused about the magnetic and electric limits of Maxwell's equations seems to have brought out people who know about them, or at least want to talk about them.

2024-12-28 16:29:07

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@nprofile…cnse @nprofile…mqrg - yes, this claim is on Wikipedia, and it puzzled me:
"Cassini believed that a planet orbiting around another body traveled on one of these ovals, with the body it orbited around at one focus of the oval.[2]"
where [2] is
Cohen, I. Bernard (1962). "Leibniz on elliptical orbits: as seen in his correspondence with the Académie Royale des Sciences in 1700". Journal of the History of Medicine and Allied Sciences. 17 (1): 72–82. doi:10.1093/jhmas/xvii.1.72.
and now I'm going to read it here:
https://www.jstor.org/stable/24620858
It's about a guy named Fontenelle telling Leibniz that Cassini has abandoned Kepler's ellipses. I don't see anything about *why* Cassini did this. It mentions that Leibniz might have liked this because it could invalidate Newton's mechanics, but it doesn't provide any evidence that Leibniz actually *did*. So, fairly mysterious.

2024-12-27 02:22:33

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@nprofile…m2ru - there's no point in whining, so I usually don't, but after most of my Twitter math pals came here and then left, I've mainly been hanging on due to sheer stubbornness and my love of talking... but I quit talking about serious math because there's little point.
Then they went to Bluesky and tried to get me to go there.

2024-12-24 17:37:22

Today the Parker Solar Probe will get 7 times closer to the Sun than anything spacecraft ever has, going faster than any spacecraft ever has - 690,000 kilometers per hour. WHEEEEEE!!!!!!!
But the really cool part is what the newspapers are barely talking about: what it's like down there. The Sun doesn't have a surface like the Earth does, since it's all just hot gas. But it has an "Alfvén surface" - and the probe has penetrated that.
What's the Alfvén surface? In simple terms, it's where the solar wind - the hot gas emitted by the Sun - breaks free of the Sun and shoots out into space. But to understand how cool it is, we need to dig a bit deeper.
After all, how can we say where the solar wind "breaks free of the Sun"?
Hot gas shoots up from the Sun, faster and faster due to its pressure, even though it's pulled down by gravity. At some point it goes faster than the speed of sound! This is the Alfvén surface. Above this surface, the solar wind becomes supersonic, so no disturbances in its flow can affect the Sun below.
But it's even cooler than that, because "sound" in the solar wind is very different from sound on Earth. Here we have air. The Sun has ions - atoms of gas so hot that electrons have been ripped off - interacting with powerful magnetic fields. You can visualize these fields as tight rubber bands, with the ions stuck to them. They vibrate back and forth together!
You could call these vibrations "sound", but the technical term is "Alfvén waves". Alfvén was the one who figured out how fast these waves move. Parker studied the surface where the solar wind's speed exceeds the speed of the Alfvén waves.
And now we've gone deep below that surface!
(1/n)
https://media.mathstodon.xyz/media_attachments/files/113/708/838/045/816/269/original/56bcd57bc6216c76.jpg

2024-12-08 19:36:29

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@nprofile…c4xk - Sure! The final, clearest story came much later, thanks to calculations by @nprofile…yu7x. I summarized them in words here:
https://johncarlosbaez.wordpress.com/2024/11/30/black-hole-puzzle/#comment-186085

2024-12-01 17:25:04

If fundamental physics were making big progress, I'd be all over it - that's what I wanted to do ever since I was a kid. But it's stagnant: the action is elsewhere, like using category theory to design radical new kinds of software. So these days I get some of my physics fix by studying the *history* of physics.
After studying the hell out of particle physics and general relativity, I went back and dug into the history of electromagnetism, which is really just as fascinating. Now I'm going back to medieval physics - because the idea that everyone was an idiot until Galileo is just plain wrong.
"Natural philosophers" in the 1200s and 1300s developed key concepts, utterly necessary for modern physics, but almost invisible now because we're so used to them - except for students, who find physics really hard because we don't bother to CLEARLY EXPLAIN those concepts: we act like children are born knowing them.
I'm talking about concepts like "the speed of an object at a moment of time". What the hell does that even mean? How can you figure out how fast something is going in one instant of time, when doesn't have time to go anywhere?
Well, that was clarified by calculus, and we credit it to Newton and Leibniz. But they had to have the idea already, in order to clarify it! And the idea of "instantaneous velocity" was developed around 1340 by the Oxford Calculators, a school of thinkers like Heytesbury and Swineshead - geniuses we never hear about.
Now I'm going back further. Did you know that back in 420 AD Martianus Capella had a theory where Mercury and Venus revolved around the Sun? And this was known to thinkers in Charlemagne's day... and also Copernicus! Wow!
https://media.mathstodon.xyz/media_attachments/files/113/578/696/013/956/853/original/ac0cf3f472c321a3.jpeg

2024-11-30 18:08:33

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@nprofile…e5g0 -
"Ok but I still don't get why he should "see her at a close distance"?"
"Because he sees her in front of the horizon, and he's nearing the horizon himself. So her image is in between."
You're still not reckoning with the fact that the horizon is not a location in space: it's a lightlike surface and it contains very long light rays, like the light ray going from you now back to the Andromeda Galaxy. You can see lots of things between you and Andromeda, and they're all on the same light ray, but you don't say that therefore you see them at a close distance. Nor do you say this for things that are *almost* on this light ray - that's what's analogous to "nearing the horizon".
@nprofile…su6u @nprofile…yu7x

2024-11-22 05:16:59

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@nprofile…j7pr @nprofile…e5g0 - right, as you approach the horizon it looks dark except perhaps for Einstein rings.
From a distance:
https://media.mathstodon.xyz/media_attachments/files/113/524/904/054/037/448/original/65841f4666e67bdd.mp4

2024-11-17 17:29:53

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@nprofile…wlc0 - actually I think people would accept other systems that made it easy for them to find interesting people. It's quite hard here, and I dislike that, but I stick around because I'm tired of social media run by corporations.
@nprofile…ccu8

2024-11-17 17:24:59

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@nprofile…wlc0 - You asked "Why do people need followpacks?" and I answered. Perhaps your intent was to persuade people to be different (by insulting them, calling them "sheep").
@nprofile…ccu8

2024-11-17 17:19:32

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@nprofile…wlc0 - people
are hungering for these followpacks because they come to Mastodon and find it to be dead as a graveyard until they slowly, laboriously stumble across interesting people. They are primarily interested in people, not hashtags.
@nprofile…ccu8

2024-11-09 00:57:40

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@npub13g2…m4y3 - I agree with most of those tactical remarks. I wouldn't quite say the Cheney family stuff "totally failed" - it may have helped - but I'd say it's way too little for just a couple of Republicans to speak out. Nikki Haley made a great case against Trump and won plenty of votes in the primary, even in states where she wasn't running! But then she knuckled under and kissed the ring. Mitt Romney spoke out for a while but then became quiet.
There's a reason: death threats.
https://www.vox.com/23899688/2024-election-republican-primary-death-threats-trump

2024-11-08 20:21:50

21% of the total US population voted for Trump - not really more than when he lost in 2020. But fewer people voted against him this time, so our crappy system put him in charge.
We can't expect he surrounds himself with people who will restrain him. The Senate and probably the House will be controlled by people under his thumb. The Supreme Court, three of whose members he appointed, has already announced he'll have wide powers to act without being legally liable. He's lining up a cabinet full of billionaires.
My worries are not for my family. We planned ahead and are ready to move to Scotland. My worries are for the US, Ukraine, democracy worldwide, and the biosphere.
We need to organize to resist Trump's bad ideas and criminality. Unfortunately I am not good at politics. So, just a couple of thoughts:
The Democratic Party needs to figure out what it's doing wrong. Many Americans are desperate. 50% live paycheck to paycheck all the time. 25% have negative net worth: they owe more than they own. If Democrats don't take this seriously, people will keep voting for demagogues.
I wish there were a serious constituency for constitutional change: abolishing the electoral college, introducing ranked choice voting, limiting or banning campaign contributions and other forms of pay-to-play politics, increasing taxes on the super-rich, etc. I don't see how these changes have a chance in the current environment. How can we improve the overall framework when politics devolves to a culture war, a battle between two dominant parties, a plutocracy - or actually all of the above? How could a new party devoted to true reform gain power? Time for me to study more history.

2024-10-18 04:34:48

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@npub1305…k70l - super-cool!
This statue of a Buddha was found in a Viking's grave in Sweden. It dates from about the 6th century AD and was probably made in Kashmir.
http://irisharchaeology.ie/2013/12/the-helgo-treasure-a-viking-age-buddha/
https://media.mathstodon.xyz/media_attachments/files/113/326/560/541/148/109/original/62d67fa9e7f6d296.webp

2024-10-17 00:21:42

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@npub1eu3…d529 - whoops! I'll kill that post and try again.

2024-10-17 00:21:00

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@npub1ky2…ghvc @npub1c9m…9xjd - the Bayesian interpretation of probability is a way to think about probabilities of one-time events, like "what's the probability that Donald Trump will win the 2024 election?"
"QFT, the best theory of the micro-world we currently have, can only answer a very narrow selection of questions. It lets us calculate the S-matrix: a transition function between asymptotically free states."
Now you're changing the subject and making some unrelated claim, which happens to be false. QFT lets us compute many things other than S-matrices. I invite people here to list four.

2024-10-16 17:48:40

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@npub1c9m…9xjd @npub1ky2…ghvc -
Second,, I talked to an early-universe cosmologist, John Peacock, about this because I wanted to be sure that the standard theory *does* say that the inhomogeneities in the early Universe arise as quantum fluctuations, rather than being put in "by hand". He said that it does. I pressed him on it, and in the end I felt a bit dumb because it turns out to be textbook stuff. He wrote:
"Many textbooks treat inflationary generation of fluctuations at various levels of detail. You can have an entire (excellent) book dedicated to the subject:
https://www.amazon.co.uk/Primordial-Density-Perturbation-Cosmology-Inflation/dp/052182849X
Or for a briefer tour of the key elements you could look at Section 6.5 of my cosmology notes:
https://www.roe.ac.uk/japwww/teaching/cos5_1213/cos5_full.pdf
The basic idea is to say that the inflaton field must experience quantum fluctuations (since it’s just a harmonic oscillator in the ground state). These fluctuations can be computed on small scales using standard flat-space QFT. Now follow a given field mode in comoving terms, so that the mode becomes stretches beyond the inflationary de Sitter horizon, transforming the inflaton fluctuation into a classical perturbation to the energy density."
Of course this being the standard theory doesn't mean it's true. But I'm greatly relieved that the fluctuations aren't stuck in by hand: the initial conditions are translation and rotation invariant; all inhomogeneities result from spontaneous symmetry breaking, i.e. the total wavefunction retains its symmetry but it's a superposition of inhomogeneous states.
(2/2)

2024-10-15 16:10:45

The Europa Clipper finally took off! It'll reach this icy, water-filled moon of Jupiter in 2030. But it won't orbit that moon. Instead, it will fly by Europa 44 times, tracing out these wonderful tracks.
Why not orbit Europa? There's too much radiation that close to Jupiter, and the Clipper can collect data much faster than it can transmit it to Earth. If it orbited Europa it would fry before it could tell us very much!
Thus, the Clipper will spend most of its time farther from Jupiter, transmitting data to Earth. Using gravity assists from 2 other moons, it will occasionally swoop down, zip past Europa, and make more measurements.
In its 3.5 year life, it will get 2.8 megarads of radiation from Jupiter. This is about 280 times the amount that would kill you! Jupiter is pretty, but don't try to live nearby.
(1/n)
https://media.mathstodon.xyz/media_attachments/files/113/312/253/329/856/740/original/3031a2f4b7e17066.jpg

2024-10-15 15:52:01

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@npub1c9m…9xjd - I think people like you are just quieter than people who love or hate LLM. It's hard to YELL a balanced opinion.

2024-10-15 15:36:10

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@npub1ulf…sjfm - there is one, but they've never gotten around to publishing it.

2024-10-13 16:13:27

Hey, they did it! People have known for a long time that in 4 dimensions you can pack equal-sized balls so that each touches 24 others. In 2008 someone showed that's the best you can do. But in April this year, three mathematicians showed there's essentially just 𝑜𝑛𝑒 𝑤𝑎𝑦 you can accomplish this!
This way is easy to describe. Take the points whose coordinates are all integers, together with those whose coordinates are all integers plus 1/2. Use each of these points as the center of a ball of radius 1/2. Then each ball touches 24 others. This pattern is called the 𝗗𝟰 𝗹𝗮𝘁𝘁𝗶𝗰𝗲.
The new result shows that if you pack equal-sized balls in 4 dimensions so that each touches 24 others, you're 𝑓𝑜𝑟𝑐𝑒𝑑 to create a version of the D4 lattice - possibly rotated, translated and/or rescaled.
This sort of thing is hard to show: it took a lot of fancy math and two weeks of calculation on a standard desktop computer. I believe only other dimensions where we know a result like this are 1, 2, 8 and 24. In dimension 3 it's just false: you can pack equal-sized balls so that each touches 12 others, and this is the best you can do, but there are uncountably many different ways to do it!
https://arxiv.org/abs/2404.18794

2024-10-11 04:54:17

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@npub1g7v…scf3 - a bunch of people were wondering about such things on the Zulip.

2024-10-09 15:34:09

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@npub1jzg…hc3a - a lot of people don't like AI and they think machine learning researchers have essentially managed to steal the physics Nobel.

2024-10-09 02:35:47

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@npub1hv3…rjmk - I wrote this:
https://mathstodon.xyz/@johncarlosbaez/113272834785880929

2024-10-09 01:06:31

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@npub10l8…ath2 - I don't really care much who wins Nobel prizes, but yes a lot of my favorite physicists never won Nobels. Heck, Einstein didn't win one for special relativity, and he didn't win one for general relativity - just one for understanding photons, which is fine as far as it goes.
Btw, actually the point of LIGO was not to test our theory of gravitational waves: that would be a lot of money to check something we all believed in. The real point was to revolutionize astronomy by creating a new kind of telescope: we've seen a lot of unexpected things, like lots of black holes of masses 30-60 times the Sun, and we don't know where they came from.

2024-10-08 23:36:43

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@npub10l8…ath2 - You mean like LIGO's detection of gravitational waves? The physics was worked out long before, and we all knew gravitational waves existed - Hulse and Taylor had already won the Nobel for that - so one could argue that all LIGO did was an (magnificent, awesome) application of physics. But apparently the Nobel committee thought that was fine.
Or how about the 1912 physics prize, to Gustaf Dalén, for "for his invention of automatic valves designed to be used in combination with gas accumulators in lighthouses and buoys"? 😆

2024-10-08 22:52:57

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@npub10l8…ath2 - but I didn't say "neural networks research is physics". I say "it all started with physics", and gave evidence for how Hopfield and Hinton were using ideas from statistical mechanics. Yes, neural network research has grown beyond this now - it's now its own subject. I don't think anyone is claiming otherwise.
Wikipedia on Hopfield networks:
"The Sherrington–Kirkpatrick model of spin glass, published in 1975, is the Hopfield network with random initialization. Sherrington and Kirkpatrick found that it is highly likely for the energy function of the SK model to have many local minima. In the 1982 paper, Hopfield applied this recently developed theory to study the Hopfield network with binary activation functions. In a 1984 paper he extended this to continuous activation functions. It became a standard model for the study of neural networks through statistical mechanics."
Wikipedia on Boltzmann machines:
"They are named after the Boltzmann distribution in statistical mechanics, which is used in their sampling function. They were heavily popularized and promoted by Geoffrey Hinton, Terry Sejnowski and Yann LeCun in cognitive sciences communities, particularly in machine learning, as part of "energy-based models" (EBM), because Hamiltonians of spin glasses as energy are used as a starting point to define the learning task."

2024-10-08 16:51:32

Yay! John Hopfield and Geoffrey Hinton won the physics Nobel for their work on neural networks... work that ultimately led to modern-day machine learning.
Some of you are wondering why this counts as physics.
In the 1980s, Hopfield invented the 'Hopfield network' based on how atomic spins interact in a chunk of solid matter. Each atom's spin makes it into a tiny magnet. Depending on the material, these spins may tend to line up, or point opposite to their neighbors, or interact in even more complicated ways. No matter what they do, at very low temperatures they tend to minimize energy.
A Hopfield network is a simulation of such a system that's been cleverly set up so that the spins store data, like images. When the Hopfield network is fed a distorted or incomplete image, it keeps updating the spins so that the energy decreases... and works its way toward the saved image that's most like the imperfect one it was fed with.
Later, Geoffrey Hinton and others generalized Hopfield's ideas and developed 'Boltzmann machines'. These exploit a discovery by the famous physicist Boltzmann!
Boltzmann realized that the probability that a chunk of matter in equilibrium has energy E is proportional to
exp(-E/kT)
where E is its energy, T is its temperature and k is a number now called Boltzmann's constant. When the temperature is low, this makes it overwhelmingly probable that the stuff will have low energy. But at higher temperatures this is less true. By exploiting this formula and cleverly adjusting the temperature, we can make neural networks do very useful things.
There's been a vast amount of work since then, refining these ideas. But it started with physics.
https://arstechnica.com/ai/2024/10/in-stunning-nobel-win-ai-researchers-hopfield-and-hinton-take-2024-physics-prize/

2024-10-05 16:18:12

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@npub1t76…4l63 - what's the idea of 'tidal squeeze' and 'tidal stretch'? I guess these are forces that appear when we have a central gravitational field and we're working in a rotating coordinate system, maybe with its origin at a point that's rotating in a circular orbit around the center?

2024-10-04 04:46:41

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@npub12n4…5dmn - interesting puzzle! To me the obvious approach is to start with some low 𝑛.
Let's compare SO(3) to S² × SO(2). Since π₁(SO(3)) ≅ ℤ/2 while π₁(S² × SO(2)) ≅ ℤ, they're not homeomorphic. By the way, SO(3) ≅ ℝP² since it's SU(2)/±1 and SU(2) is a 3-sphere.
Next let's compare SO(4) to S³ × SO(3). Since π₁(SO(n)) ≅ ℤ/2 for all n ≥ 3 now both these spaces have π₁ ≅ ℤ/2 so the above fundamental group argument no longer works, and we see it will never work to distinguish SO(n+1) from Sⁿ × SO(n) for n ≥ 3: they both have π₁ ≅ ℤ/2.
So we need a deeper idea.
(1/n)

2024-10-03 17:17:44

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"“The Neptunian Ridge stands tall above the Desert and Savanna. It provides us with a key to understanding the physical mechanisms shaping the Desert,” points out Vincent Bourrier. Most Neptunes may be distributed over the Savanna and Desert early in their life by migrating within the disk in which they formed. The existence of the Ridge suggests that some Neptune-size planets are brought to this region by a type of migration called high-eccentricity migration, which occurs later in their life and allow them to survive erosion from the stars.
These migration processes, coupled with photoevaporation, likely shape the distinct features observed in the Neptunian landscape. The similarities between the Neptunian Ridge and another feature in the exoplanet distribution, the hot Jupiter pileup, suggests that similar evolutionary processes may influence both groups of planets."
(2/2)
https://astrobiology.com/2024/09/hidden-exoplanets-between-the-neptunian-desert-and-savanna.html

2024-10-03 17:14:10

𝗧𝗛𝗘 𝗡𝗘𝗣𝗧𝗨𝗡𝗜𝗔𝗡 𝗗𝗘𝗦𝗘𝗥𝗧
Once we knew just one solar system. Now we can see 𝘩𝘶𝘯𝘥𝘳𝘦𝘥𝘴 of solar systems and learn how they work on average! We are learning amazing things, like this:
Here's a chart showing planets whose year is 25 Earth days long or less. These planets are very close to their suns. There's a big blob of planets with mass 10-20 times the Earth's mass, whose year is just 1-10 days long! These are called 𝗵𝗼𝘁 𝗡𝗲𝗽𝘁𝘂𝗻𝗲𝘀.
But there's an interesting shortage of slightly less massive planets that are quite close to their sun. This is called the 𝗡𝗲𝗽𝘁𝘂𝗻𝗶𝗮𝗻 𝗱𝗲𝘀𝗲𝗿𝘁. And there's another less dramatic shortage of planets in the 𝗡𝗲𝗽𝘁𝘂𝗻𝗶𝗮𝗻 𝘀𝗮𝘃𝗮𝗻𝗻𝗮. Both these need to be explained.
Now we have a new clue. This September astronomers discovered that the desert and savanna are separated by a region with more planets, which they call the 𝗡𝗲𝗽𝘁𝘂𝗻𝗶𝗮𝗻 𝗿𝗶𝗱𝗴𝗲.
What does it all mean? There are lots of theories, but it will take a while to figure it out.
Here's the new paper:
• A. Castro-González, V. Bourrier, J. Lillo-Box, J.-B. Delisle, D. J. Armstrong, D. Barrado and A. C. M. Correia, Mapping the exo-Neptunian landscape: a ridge between the desert and savanna, https://arxiv.org/abs/2409.10517.
With all this work we need to think about 'selection bias': some planets are easier to see than others, so what we see is not the same as the real situation! So, a lot of this paper is about compensating for selection bias.
For a link to a gentler, less technical article, read on.
(1/2)
https://media.mathstodon.xyz/media_attachments/files/113/244/496/342/306/359/original/8a986be8fc9391fa.jpg

2024-10-02 05:36:34

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@npub1g7f…25j2 - meaning?

2024-10-01 05:32:37

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@npub1rf7…zwfx - this is cool, but the subtitle "This is very unexpected" is an exaggeration: the issue has been around for a long time:
https://en.wikipedia.org/wiki/Cuspy_halo_problem

2024-09-30 16:34:21

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@npub1z4e…aasw - yes, I hope you read the image description.

2024-09-29 16:02:08

𝗤𝗨𝗜𝗖𝗞! 𝗙𝗘𝗘𝗗 𝗧𝗛𝗘 𝗕𝗘𝗔𝗦𝗧!
Academic publisher Taylor & Francis recently sold many of its authors’ works to Microsoft for $10 million, without asking or paying the authors — to train Microsoft’s large language models!
Taylor & Francis asked their journal "Learning, Media and Technology" to cut peer review time to 15 days — absurdly little time — to crank out more content.
And Taylor & Francis's subsidiary Routledge told staff that it was “extra important” to meet publishing targets for 2024. It moved some book deadlines from 2025 to 2024. Why? To meet its deadline with Microsoft.
Another academic publisher, Wiley, made a $44 million deal to feed academic books to LLMs — with no way for authors to opt out. They say “it is in the public interest for these emerging technologies to be trained on high-quality, reliable information.”
When you publish with one of the big academic publishers, they try to make you sign a contract saying they can do whatever they want with your work. That means anything.
Hat-tip to @npub1yt3…wmv6 for pointing this out.
These articles have links to original sources:
https://pivot-to-ai.com/2024/08/04/more-academic-publishers-are-doing-ai-deals/
https://pivot-to-ai.com/2024/09/28/routledge-nags-academics-to-finish-books-asap-to-feed-microsofts-ai/
https://media.mathstodon.xyz/media_attachments/files/113/221/626/482/836/881/original/b380b5ccede93fcd.jpg

2024-09-28 21:22:47

In 1596, Kepler claimed that the planetary orbits would only follow "God's design" if there were two more planets: one between Mars and Jupiter and one between Mercury and Venus. Later folks came up with the Titius-Bode rule. This says there should be a planet whose orbital radius is
0.4 + 0.3 × 2ⁿ
times the distance between the Earth and Sun.
• For n = 0 we get Venus.
• For n = 1 we get Earth.
• For n = 2 we get Mars.
• For n = 3 we get... NOTHING???
• For n = 4 we get Jupiter.
• For n = 5 we get Saturn.
• For n = 6, this rule correctly predicted the location of another planet: Uranus.
So, 24 astronomers called the "celestial police" looked between Mars and Jupiter. And in 1801 one of them found a small object in the right place! Later people found other asteroids in a belt between Mars and Jupiter, but the first was the biggest: Ceres.
In 2015 we sent a probe called Dawn to investigate Ceres. It found something wonderful.
The surface of Ceres is a mix of ice and hydrated minerals like carbonates and CLAY! It probably doesn't have an internal ocean of liquid water like Jupiter's moon Europa. But it seems that brine still flows through its outer mantle and reaches the surface!
A new paper argues that Ceres contains a 𝑙𝑜𝑡 of ice, and was once an ocean-covered world:
"We think that there's lots of water-ice near Ceres surface.... People used to think that if Ceres was very icy, the craters would deform quickly over time, like glaciers flowing on Earth, or like gooey flowing honey. However, we've shown through our simulations that ice can be much stronger in conditions on Ceres than previously predicted if you mix in just a little bit of solid rock.""
https://phys.org/news/2024-09-asteroid-ceres-ocean-world-slowly.html

2024-09-27 22:28:44

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@npub19eg…ma58 @npub1kqz…8s5l - I don't primarily post interesting things for other people: fundamentally I post for myself. I explain things so I understand them. But people seem to like it.
I don't follow anyone who posts stupid memes or political rage bait, so I don't see that, though I'm sure it's there.
And so, I'm happy.

2024-09-26 21:06:31

Professor emeritus blues:
You know you've been retired a while when the department staff accidentally puts some new lecturer's photo up with your name on it, because they don't remember who you are.
https://media.mathstodon.xyz/media_attachments/files/113/205/883/520/123/142/original/0547f399b930ed86.jpg

2024-09-24 05:35:29

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@npub1l40…nm43 - our ancestors probably killed them off, and they may have passed down tales about how in the old days there were a lot more dangerous predators but the game was more plentiful.

2024-09-20 06:12:47

What would happen if the Moon broke into seven pieces? It would not be good. Check out this simulation, including a video.
https://jasmcole.com/2017/09/20/the-moon-blew-up-without-warning-and-for-no-apparent-reason/

2024-09-18 00:09:11

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@npub1d8z…zgls - "no" to the first question, in the math department at U. C. Riverside.

2024-09-17 20:23:56

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@npub12nw…uegw - I used the "explosion" feature in Irfanview, setting it to different levels in different patches of the picture.

2024-09-17 16:46:23

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@npub18vx…zvkl - if it's easy, why do you have to steel youself for it? 😈

2024-09-16 15:19:49

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@npub1sqt…hmh2 - beautiful!

2024-09-15 20:36:51

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@npub1ld9…04d0 - No problem! Btw, if the size of a rainbow depended on the droplet size, rainbows would probably look 'fuzzed-out', since different drops often have different sizes.

2024-09-14 17:03:46

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@npub1vt8…yzq0 - I guess you know about the new "Beat" tour:
https://www.youtube.com/watch?v=QIgxRK914WI

2024-09-13 23:38:25

This feature in Hudson Bay is probably 𝑛𝑜𝑡 a large meteor crater. It's called the Nastapoka Arc. While the dark red part of the circle really is the coastline, the rest of the circle is not anything real.
Details:
https://en.wikipedia.org/wiki/Nastapoka_arc
https://media.mathstodon.xyz/media_attachments/files/113/132/859/294/301/181/original/abfcc15cc37dda23.png

2024-09-13 21:53:21

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@npub1wnd…8kcm - they should repeat it with people who aren't Dutch. 🙃

2024-09-13 16:08:36

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@npub17q0…je6u - if you block the email address and/or journal/conference name for each one, does it help at all? Or are they mostly new instances?

2024-09-13 01:43:08

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@npub1cd3…paq7 - you meant to say rig.

2024-09-10 21:17:08

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@npub1ez3…xnxl - I was in a debate society for a little while, and I've never seen an official debate where the debaters get to ask each other questions. Maybe you're thinking of a 'discussion'?

2024-09-10 14:20:09

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@npub1crf…qu6y - presumably you are thinking, talking and pacing around a lot?

2024-09-09 15:12:21

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@npub1a0a…kv03 - yikes!

2024-09-08 19:29:49

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@npub1syx…d6lz - I haven't seen one in my 2007 Honda. Let me just get in the trunk and see....

2024-09-08 19:11:09

I rented a car, but they didn't have the sensible compact I wanted so they gave me Ford Mustang. What the hell does this pull chain in the trunk do? "Pull to let kidnapped victim scamper to freedom?" Is this something Mustang owners regularly need? And why the improbably high arc on the parabola? "Pull to let gremlin jump out and run off gibbering manically?"
https://media.mathstodon.xyz/media_attachments/files/113/103/490/851/026/152/original/d9b55de735b5579b.jpg

2024-09-08 14:52:35

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@npub16sc…hgj4 - yes, indeed it's a bummer.

2024-09-05 03:04:53

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@npub10m3…j2yt - thanks, it looks interesting - and not just because my middle name is Carlos. 😉

2024-08-31 20:43:18

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@npub14fs…qegk - one Nobel coming up!

2024-08-30 14:03:36

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@npub1a0a…kv03 - I'm keeping it real.

2024-08-29 23:21:09

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@npub16at…f89f - Richard Feynman's pop book "QED: Strange Theory of Matter".

2024-08-28 06:17:31

No, there are no "new, exceptionally bright photons". The energy of a photon is still related to its frequency by
E = h ν
Please go back to what you were doing.
https://www.livescience.com/technology/communications/unbreakable-quantum-communication-closer-to-reality-thanks-to-new-exceptionally-bright-photons

2024-08-25 09:34:45

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@npub1kc7…tf4u - give in! Whenever I see a Bayesian I give them a swift kick in the posterior.

2024-08-25 09:33:36

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@npub15v3…luuc - I'd never heard it's name until today, when the New York Times acted like we all knew it.

2024-08-25 08:04:44

What's a "Bayesian luxury yacht" and why does this New York Times article assume we all know what it is?
Is it a yacht whose probability of sinking is computed by Bayesian methods?
Full article free here:
https://archive.is/tjAvY
https://media.mathstodon.xyz/media_attachments/files/113/021/612/342/309/376/original/5032eefa6aee11fa.jpg

2024-08-24 23:50:29

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@npub10gg…a0hf
https://en.wikipedia.org/wiki/Mouse_(set_theory)

2024-08-24 13:04:37

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@npub1lap…elxz - there are lots of theories to test, but so far most of them look pretty implausible. So yes, we should take a break on big particle colliders. There's also this thing called climate change....

2024-08-24 10:54:26

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@npub10l8…ath2 - yeah, I almost stopped reading the thesis in disgust when I saw this:
2.2 Hypercharge Quantisation and Fermat’s Last Theorem

2024-08-24 09:17:09

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@npub1683…qclh - excellent caption!

2024-08-20 09:55:13

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@npub16hx…u6hc - did he seem to be following it?

2024-08-18 11:12:29

Have you ever tried learning physics from a teacher who does calculations without explaining them well? Setting constants equal to 1, then putting them back later using dimensional analysis. Setting terms equal to zero when they're negligible - which you're supposed to know by instinct. And so on. You're just supposed to watch it and pick it up.
It's like trying to learn gymnastics by watching Simona Biles.
https://media.mathstodon.xyz/media_attachments/files/112/982/718/965/079/950/original/527ba85e06bdcf32.mp4

2024-08-17 13:31:21

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@npub1jw3…h3v8 - that's great! It's so fun when reality changes and suddenly makes *more* sense, not less. But for me that usually requires coffee.

2024-08-17 13:22:46

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@npub1swy…snpj - thanks, your praise is worth more than most.

2024-08-17 07:31:02

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@npub1uth…wprp - read the comments: it did not modify its own code.

2024-08-12 08:26:03

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@npub12hx…78rc - it's called Gargannock House, near Stirling in Scotland.

2024-08-11 14:19:24

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@npub1c0s…pcx4 - I haven't found a cellar! But lots of staircases.
https://media.mathstodon.xyz/media_attachments/files/112/943/822/339/915/378/original/c34756ba2d1fc5e9.jpg

2024-08-10 14:07:06

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I explored it today. It led to another stairway, and then....
https://media.mathstodon.xyz/media_attachments/files/112/938/089/098/925/171/original/8538bf42714d8280.jpg

2024-08-10 13:50:18

I often have dreams of getting lost in complicated old houses with many stairways and secret rooms. I am now spending a weekend in such a house. To get to my room I need to climb this spiral staircase. Last night to get to the bathroom I needed to climb to the next floor. There I found two bathrooms side by side, and a short dark hallway with an eerie green glow at the other end. I was tempted to explore but... no, in movies that usually ends badly.
https://media.mathstodon.xyz/media_attachments/files/112/938/004/282/421/342/original/af23aa4c1744b7d5.jpg

2024-08-09 13:25:15

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@npub10p4…jn3m the "expected" or average momentum of a standing wave is zero, because it's a sum of two or more waves going in different directions, whose momenta cancel out.

2024-08-09 10:50:40

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@npub1e6h…ckhy - this is what drove me to think physics was pretty cool.

2024-08-07 13:32:19

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@npub19g8…9445 - I'm surprised it would take that long. I'm willing to believe Wikipedia, but I should look up estimates of the wave speed and distance.

2024-08-07 12:32:54

Don't call these "protests", the word is "riots".
https://www.bbc.co.uk/news/live/cy8497l7dx8t

2024-08-06 11:12:26

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@npub1r7e…0vz0 - welcome! Follow me if you like fun pop explanations of science and music theory (and, sorry, occasional hardcore math).

2024-08-06 07:30:03

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@npub178l…zhp6 - which lies is Google spreading that benefit my side - the side that's against installing right-wing extremists across the globe? I'm against lies even for a good cause.

2024-08-05 15:50:27

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@npub1ykh…cmt4 - agreed!

2024-08-05 11:33:54

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@npub1nad…2adz - it's not published yet! That's why I said "is coming out with" instead of "has come out with".

2024-08-05 11:08:05

The AMS Notices is coming out with a collection of papers on how to publish and present math. I'm glad I persuaded them to not put my paper first, like they did here.
It would be amusing, but not really a good idea. It looks like a subtitle.
https://media.mathstodon.xyz/media_attachments/files/112/909/083/481/946/456/original/aea3c5911d487a85.jpg

2024-08-05 08:42:54

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@npub1lau…uqs9 - Nice! By the way, US college students would understand "sine wave" better than "sinusoidal" - pretty much nobody in the US says "sinusoidal", even though it's more accurate.

2024-08-04 21:53:57

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@npub19g8…9445 - wow, cool. I hadn't known they'd gotten evidence that the asteroid hit in the spring.

2024-08-04 16:16:18

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@npub1c9m…9xjd - I thought you pretending to be a physicist. Physicists love quoting theorems, but they forget the hypotheses.

2024-08-04 15:55:22

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@npub17hm…7lag - punctured disk.

2024-08-04 15:51:20

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@npub1c9m…9xjd - if two unproved conjecture are true!

2024-08-04 11:34:32

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@npub1tkz…cd27 - where did you find that? I couldn't instantly find a Dutch etymological dictionary, but I guess you did.

2024-08-03 14:59:02

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@npub1kvv…y5yh - are you talking about the origin of life? LUCA came quite a while after that, so it's almost a separate question.

2024-08-03 11:50:34

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@npub1e6h…ckhy - Americans have evolved to require ketchup.

2024-08-03 11:48:47

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@npub1ky2…ghvc - thanks, I hadn't spotted that!
But was there spam?