John C. Baez
@johncarlosbaez.bsky.social
Mathematical physicist
Pinned
Standard Model 5: Spin-1/2 Particles
YouTube video by John Baez
youtu.be
Another installment! This time I explain spin-1/2 particles, the Stern-Gerlach experiment that convinced people they exist, and why a quantum state of a spin-1/2 particle is the same as a vector of length 1/2 saying which way the spin points.
youtu.be/sZgjh_lqelY
youtu.be/sZgjh_lqelY
In the Divine Comedy, Dante says the planets move on larger and larger concentric spheres surrounding the Earth. Then comes a sphere for the stars, then one for the Prime Mover - and then come spheres of angels that get *smaller* as we go up, centered on God.
So his universe was a 3-sphere.
(1/2)
So his universe was a 3-sphere.
(1/2)
January 2, 2026 at 2:41 PM
In the Divine Comedy, Dante says the planets move on larger and larger concentric spheres surrounding the Earth. Then comes a sphere for the stars, then one for the Prime Mover - and then come spheres of angels that get *smaller* as we go up, centered on God.
So his universe was a 3-sphere.
(1/2)
So his universe was a 3-sphere.
(1/2)
Wow, someone discovered a more efficient way of multiplying two 3×3 matrices! It uses 23 multiplications and 58 addition/subtractions. The previous best used 23 and 60.
(Better methods exist if your matrix entries live in a *commutative* ring. This handles quaternions!)
arxiv.org/abs/2512.21980
(Better methods exist if your matrix entries live in a *commutative* ring. This handles quaternions!)
arxiv.org/abs/2512.21980
A 58-Addition, Rank-23 Scheme for General 3x3 Matrix Multiplication
This paper presents a new state-of-the-art algorithm for exact $3\times3$ matrix multiplication over general non-commutative rings, achieving a rank-23 scheme with only 58 scalar additions. This impro...
arxiv.org
January 1, 2026 at 11:05 PM
Wow, someone discovered a more efficient way of multiplying two 3×3 matrices! It uses 23 multiplications and 58 addition/subtractions. The previous best used 23 and 60.
(Better methods exist if your matrix entries live in a *commutative* ring. This handles quaternions!)
arxiv.org/abs/2512.21980
(Better methods exist if your matrix entries live in a *commutative* ring. This handles quaternions!)
arxiv.org/abs/2512.21980
Happy New Year!
Last year was so bad that I decided that what people need from me most is not more warnings, but a bit of happiness.
Like this design for a stained glass window by @gregegansf.bsky.social, based on a 4-dimensional lattice (see alt text).
You could build it with a few parts!
Last year was so bad that I decided that what people need from me most is not more warnings, but a bit of happiness.
Like this design for a stained glass window by @gregegansf.bsky.social, based on a 4-dimensional lattice (see alt text).
You could build it with a few parts!
January 1, 2026 at 11:57 AM
Happy New Year!
Last year was so bad that I decided that what people need from me most is not more warnings, but a bit of happiness.
Like this design for a stained glass window by @gregegansf.bsky.social, based on a 4-dimensional lattice (see alt text).
You could build it with a few parts!
Last year was so bad that I decided that what people need from me most is not more warnings, but a bit of happiness.
Like this design for a stained glass window by @gregegansf.bsky.social, based on a 4-dimensional lattice (see alt text).
You could build it with a few parts!
Harmony in music is the tango of rational and irrational numbers, coming close enough to kiss but never touching.
@gro-tsen.bsky.social illustrates it here.
@gro-tsen.bsky.social illustrates it here.
This almost-but-not-quite-periodic image is inspired by a recent blog post by @johncarlosbaez.bsky.social about musical scales: let me explain what I drew here, and what it has to do with music, but also with diophantine approximations of log(2), log(3) and log(5). 🧵⤵️ •1/19
December 31, 2025 at 12:13 AM
Harmony in music is the tango of rational and irrational numbers, coming close enough to kiss but never touching.
@gro-tsen.bsky.social illustrates it here.
@gro-tsen.bsky.social illustrates it here.
List equal-tempered scales with the best perfect fifths. You'll see those with 5, 7, 12, 17, 29, 41 and 53 tones are particularly good, and
12 = 7 + 5
17 = 12 + 5
29 = 12 + 12 + 5
41 = 12 + 12 + 12 + 5
53 = 12 + 12 + 12 + 12 + 5
Coincidence? No! 😈
johncarlosbaez.wordpress.com/2023/10/19/p...
12 = 7 + 5
17 = 12 + 5
29 = 12 + 12 + 5
41 = 12 + 12 + 12 + 5
53 = 12 + 12 + 12 + 12 + 5
Coincidence? No! 😈
johncarlosbaez.wordpress.com/2023/10/19/p...
Equal Temperament (Part 2)
When I listed some equal-tempered scales with good perfect fifths in Part 1, a reader named Sylvain noticed something interesting. The scales with 5, 7, 12, 17, 29, 41 and 53 tones are particularly…
johncarlosbaez.wordpress.com
December 30, 2025 at 4:51 PM
List equal-tempered scales with the best perfect fifths. You'll see those with 5, 7, 12, 17, 29, 41 and 53 tones are particularly good, and
12 = 7 + 5
17 = 12 + 5
29 = 12 + 12 + 5
41 = 12 + 12 + 12 + 5
53 = 12 + 12 + 12 + 12 + 5
Coincidence? No! 😈
johncarlosbaez.wordpress.com/2023/10/19/p...
12 = 7 + 5
17 = 12 + 5
29 = 12 + 12 + 5
41 = 12 + 12 + 12 + 5
53 = 12 + 12 + 12 + 12 + 5
Coincidence? No! 😈
johncarlosbaez.wordpress.com/2023/10/19/p...
At this time of year the UK is a dark and dreary land... except for the lighthouses. I need to go find a lighthouse.
mathstodon.xyz/@terence@fos...
mathstodon.xyz/@terence@fos...
December 30, 2025 at 11:25 AM
At this time of year the UK is a dark and dreary land... except for the lighthouses. I need to go find a lighthouse.
mathstodon.xyz/@terence@fos...
mathstodon.xyz/@terence@fos...
As a college student, Isaac Newton tried to figure out a scale where all the frequency ratios are simple fractions. How did it work, and how does it compare to the scales invented by Mercator, Mersenne and Euler?
I'll explain 174,240 such scales.
johncarlosbaez.wordpress.com/2025/12/29/j...
I'll explain 174,240 such scales.
johncarlosbaez.wordpress.com/2025/12/29/j...
Just Intonation (Part 6)
In this series I’ve been explaining 12-tone scales in just intonation—or more precisely, ‘5-limit’ just intonation, where all the frequency ratios are integer powers of the …
johncarlosbaez.wordpress.com
December 29, 2025 at 7:21 PM
As a college student, Isaac Newton tried to figure out a scale where all the frequency ratios are simple fractions. How did it work, and how does it compare to the scales invented by Mercator, Mersenne and Euler?
I'll explain 174,240 such scales.
johncarlosbaez.wordpress.com/2025/12/29/j...
I'll explain 174,240 such scales.
johncarlosbaez.wordpress.com/2025/12/29/j...
Brownian motion was studied by Norbert Wiener, so it's called the "Wiener process", and of course thickened Brownian motion - the set all points of distance < d from a Brownian path - is called the "Wiener sausage".
en.wikipedia.org/wiki/Wiener_...
en.wikipedia.org/wiki/Wiener_...
December 28, 2025 at 11:11 AM
Brownian motion was studied by Norbert Wiener, so it's called the "Wiener process", and of course thickened Brownian motion - the set all points of distance < d from a Brownian path - is called the "Wiener sausage".
en.wikipedia.org/wiki/Wiener_...
en.wikipedia.org/wiki/Wiener_...
Actual mathematician quotes:
"It was my lot to plant the harpoon of algebraic topology into the body of the whale of algebraic geometry" - Solomon Lefschetz
"It was my fate to implant the electrode of arithmetic geometry in the pleasure center of algebraic topology." - Jack Morava
"It was my lot to plant the harpoon of algebraic topology into the body of the whale of algebraic geometry" - Solomon Lefschetz
"It was my fate to implant the electrode of arithmetic geometry in the pleasure center of algebraic topology." - Jack Morava
December 27, 2025 at 3:26 PM
Actual mathematician quotes:
"It was my lot to plant the harpoon of algebraic topology into the body of the whale of algebraic geometry" - Solomon Lefschetz
"It was my fate to implant the electrode of arithmetic geometry in the pleasure center of algebraic topology." - Jack Morava
"It was my lot to plant the harpoon of algebraic topology into the body of the whale of algebraic geometry" - Solomon Lefschetz
"It was my fate to implant the electrode of arithmetic geometry in the pleasure center of algebraic topology." - Jack Morava
There's a giant star surrounded by so much dust we can't see its light - but the infrared radiation from this hot dust excites an enormous cloud of hydroxyl ions, creating an enormous MASER. That's Microwave Amplification by Stimulated Emission of Radiation!
mathstodon.xyz/@johncarlosb...
mathstodon.xyz/@johncarlosb...
John Carlos Baez (@johncarlosbaez@mathstodon.xyz)
There's a red giant star near the end of its life that's intensely pulsing every 4 years. It's spewing out a lot of gas with each pulse. The gases include hydroxyl ions (OH), carbon monoxide, sulfur...
mathstodon.xyz
December 27, 2025 at 3:09 PM
There's a giant star surrounded by so much dust we can't see its light - but the infrared radiation from this hot dust excites an enormous cloud of hydroxyl ions, creating an enormous MASER. That's Microwave Amplification by Stimulated Emission of Radiation!
mathstodon.xyz/@johncarlosb...
mathstodon.xyz/@johncarlosb...
The time is back to nanosecond accuracy at the National Institute of Standards and Technology!
groups.google.com/a/list.nist....
groups.google.com/a/list.nist....
December 26, 2025 at 10:12 PM
The time is back to nanosecond accuracy at the National Institute of Standards and Technology!
groups.google.com/a/list.nist....
groups.google.com/a/list.nist....
Ever wonder about the math hiding in music? There's a lot of it, and it's really fun! Here's a first take on some of it.
(When you're done, you'll know why musicians have special names for the numbers 81/80 and 128/125.)
johncarlosbaez.wordpress.com/2025/12/26/t...
(When you're done, you'll know why musicians have special names for the numbers 81/80 and 128/125.)
johncarlosbaez.wordpress.com/2025/12/26/t...
The Mathematics of Tuning Systems
I’m giving a talk on the math of tuning systems at Claremont McKenna College on January 30th at 11 am. If you’re around, please come! You can read my slides here: • The mathematics…
johncarlosbaez.wordpress.com
December 26, 2025 at 3:18 PM
Ever wonder about the math hiding in music? There's a lot of it, and it's really fun! Here's a first take on some of it.
(When you're done, you'll know why musicians have special names for the numbers 81/80 and 128/125.)
johncarlosbaez.wordpress.com/2025/12/26/t...
(When you're done, you'll know why musicians have special names for the numbers 81/80 and 128/125.)
johncarlosbaez.wordpress.com/2025/12/26/t...
New result: you can build a universal computer using a single billiard ball on a carefully crafted table!
More precisely: you can create a computer that can run any program, using just a single point moving frictionlessly in a region of the plane and bouncing off the walls elastically.
(1/n)
More precisely: you can create a computer that can run any program, using just a single point moving frictionlessly in a region of the plane and bouncing off the walls elastically.
(1/n)
December 23, 2025 at 10:21 AM
New result: you can build a universal computer using a single billiard ball on a carefully crafted table!
More precisely: you can create a computer that can run any program, using just a single point moving frictionlessly in a region of the plane and bouncing off the walls elastically.
(1/n)
More precisely: you can create a computer that can run any program, using just a single point moving frictionlessly in a region of the plane and bouncing off the walls elastically.
(1/n)
The atomic clocks in Boulder are back in service. But they could be as much as 5 microseconds off!
That's no big deal for transmitting time on the internet, where millisecond irregularities are common. But it'd be 5000 times the usual nanosecond errors!
groups.google.com/a/list.nist....
That's no big deal for transmitting time on the internet, where millisecond irregularities are common. But it'd be 5000 times the usual nanosecond errors!
groups.google.com/a/list.nist....
Update on Boulder Internet Time Services and atomic time scale
groups.google.com
December 22, 2025 at 12:15 PM
The atomic clocks in Boulder are back in service. But they could be as much as 5 microseconds off!
That's no big deal for transmitting time on the internet, where millisecond irregularities are common. But it'd be 5000 times the usual nanosecond errors!
groups.google.com/a/list.nist....
That's no big deal for transmitting time on the internet, where millisecond irregularities are common. But it'd be 5000 times the usual nanosecond errors!
groups.google.com/a/list.nist....
There's something satisfying about finding that two words you know are part of a doublet - like "regal" and "royal", the first coming directly from Latin and the second taking a detour through French.
Now in Spanish:
"Siesta" vs "sixta". Oh! "Siesta" happens at the 6th hour after dawn! 😀
Now in Spanish:
"Siesta" vs "sixta". Oh! "Siesta" happens at the 6th hour after dawn! 😀
December 22, 2025 at 10:54 AM
There's something satisfying about finding that two words you know are part of a doublet - like "regal" and "royal", the first coming directly from Latin and the second taking a detour through French.
Now in Spanish:
"Siesta" vs "sixta". Oh! "Siesta" happens at the 6th hour after dawn! 😀
Now in Spanish:
"Siesta" vs "sixta". Oh! "Siesta" happens at the 6th hour after dawn! 😀
Ouch! The most accurate standard of time in the USA may have gone down!
Yup, a bunch of atomic clocks at the National Institute of Standards and Technology (NIST) in Boulder, Colorado seem to have lost power.
(1/n)
Yup, a bunch of atomic clocks at the National Institute of Standards and Technology (NIST) in Boulder, Colorado seem to have lost power.
(1/n)
![Dear colleagues,
In short, the atomic ensemble time scale at our Boulder campus has failed due to a prolonged utility power outage. One impact is that the Boulder Internet Time Services no longer have an accurate time reference. At time of writing the Boulder servers are still available due a standby power generator, but I will attempt to disable them to avoid disseminating incorrect time.
The affected servers are:
time-a-b.nist.gov
time-b-b.nist.gov
time-c-b.nist.gov
time-d-b.nist.gov
time-e-b.nist.gov
ntp-b.nist.gov (authenticated NTP)
No time to repair estimate is available until we regain staff access and
power. Efforts are currently focused on obtaining an alternate source of
power so the hydrogen maser clocks survive beyond their battery backups.
More details follow.
Due to prolonged high wind gusts there have been a combination of utility power line damage and preemptive utility shutdowns (in the interest of wildfire prevention) in the Boulder, CO area. NIST's campus lost utility power Wednesday (Dec. 17 2025) around 22:23 UTC. At time of writing utility power is still off to the campus. Facility operators anticipated needing to shutdown the heat-exchange infrastructure providing air cooling to many parts of the building, including some internal networking closets. [....]
At initial power loss, there was no immediate impact to the NIST atomic
time scale or distribution services because the projects are afforded
standby power generators. However, we now have strong evidence one of the crucial generators has failed. In the downstream path is the primary signal distribution chain, including to the Boulder Internet Time Service. Another campus building houses additional clocks backed up by a different power generator; if these survive it will allow us to re-align the primary time scale when site stability returns without making use of external clocks or reference signals.
Best wishes,
-Jeff Sherman](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:fb5fjz7jxmbf3alhwh44ookd/bafkreicfacdabxxqyw77ujiuv2u6vow4vozmlmpgaeuemmxr422posq2ey@jpeg)
December 20, 2025 at 4:55 PM
Ouch! The most accurate standard of time in the USA may have gone down!
Yup, a bunch of atomic clocks at the National Institute of Standards and Technology (NIST) in Boulder, Colorado seem to have lost power.
(1/n)
Yup, a bunch of atomic clocks at the National Institute of Standards and Technology (NIST) in Boulder, Colorado seem to have lost power.
(1/n)
Another installment! This time I explain spin-1/2 particles, the Stern-Gerlach experiment that convinced people they exist, and why a quantum state of a spin-1/2 particle is the same as a vector of length 1/2 saying which way the spin points.
youtu.be/sZgjh_lqelY
youtu.be/sZgjh_lqelY
Standard Model 5: Spin-1/2 Particles
YouTube video by John Baez
youtu.be
December 18, 2025 at 2:33 PM
Another installment! This time I explain spin-1/2 particles, the Stern-Gerlach experiment that convinced people they exist, and why a quantum state of a spin-1/2 particle is the same as a vector of length 1/2 saying which way the spin points.
youtu.be/sZgjh_lqelY
youtu.be/sZgjh_lqelY
When Lee and Yang suggested there's a fundamental difference between left and right, Pauli wrote:
"Ich glaube aber nicht, daß der Herrgott ein schwacher Linkshänder ist."
(I do not believe that the Lord is a weak left-hander.)
2 days later, Wu's experiment showed that Lee and Yang were correct!
"Ich glaube aber nicht, daß der Herrgott ein schwacher Linkshänder ist."
(I do not believe that the Lord is a weak left-hander.)
2 days later, Wu's experiment showed that Lee and Yang were correct!
December 18, 2025 at 11:14 AM
When Lee and Yang suggested there's a fundamental difference between left and right, Pauli wrote:
"Ich glaube aber nicht, daß der Herrgott ein schwacher Linkshänder ist."
(I do not believe that the Lord is a weak left-hander.)
2 days later, Wu's experiment showed that Lee and Yang were correct!
"Ich glaube aber nicht, daß der Herrgott ein schwacher Linkshänder ist."
(I do not believe that the Lord is a weak left-hander.)
2 days later, Wu's experiment showed that Lee and Yang were correct!
Huge good news! They saw a "fisher" in a park in Cleveland Ohio! This is the is the first sighting of a fisher in Cuyahoga County since the 1800s. The sighting was recorded on a trail camera:
www.instagram.com/p/DSM9EpzjtR...
(The photo here is from Massachusetts.)
(1/n)
www.instagram.com/p/DSM9EpzjtR...
(The photo here is from Massachusetts.)
(1/n)
December 17, 2025 at 11:06 AM
Huge good news! They saw a "fisher" in a park in Cleveland Ohio! This is the is the first sighting of a fisher in Cuyahoga County since the 1800s. The sighting was recorded on a trail camera:
www.instagram.com/p/DSM9EpzjtR...
(The photo here is from Massachusetts.)
(1/n)
www.instagram.com/p/DSM9EpzjtR...
(The photo here is from Massachusetts.)
(1/n)
In 1973, Penrose conjectured a lower bound on the mass M of a black hole as a function of its area A:
M ≥ sqrt(A/16π)
Now someone claims to have a proof - 449 pages long. Do any experts have opinions on this?
arxiv.org/abs/2512.04137
M ≥ sqrt(A/16π)
Now someone claims to have a proof - 449 pages long. Do any experts have opinions on this?
arxiv.org/abs/2512.04137
The Unconditional Spacetime Penrose Inequality
We prove the unconditional spacetime Penrose inequality for three-dimensional asymptotically flat initial data sets satisfying the dominant energy condition. The result is unconditional in the sense t...
arxiv.org
December 15, 2025 at 6:42 PM
In 1973, Penrose conjectured a lower bound on the mass M of a black hole as a function of its area A:
M ≥ sqrt(A/16π)
Now someone claims to have a proof - 449 pages long. Do any experts have opinions on this?
arxiv.org/abs/2512.04137
M ≥ sqrt(A/16π)
Now someone claims to have a proof - 449 pages long. Do any experts have opinions on this?
arxiv.org/abs/2512.04137
Is anyone taking bets on when the Kessler syndrome starts? That's when a chain reaction of colliding satellites destroys many of them. All the experts say it's coming our way.
www.theverge.com/news/844502/...
www.theverge.com/news/844502/...
Starlink and Chinese satellites nearly collided last week
Kessler syndrome avoided for now.
www.theverge.com
December 15, 2025 at 3:23 PM
Is anyone taking bets on when the Kessler syndrome starts? That's when a chain reaction of colliding satellites destroys many of them. All the experts say it's coming our way.
www.theverge.com/news/844502/...
www.theverge.com/news/844502/...
This chart shows all objects in the Universe, arranged by mass (vertical) and radius (horizontal).
Let me explain it a bit...
• C. H. Lineweaver and V. M. Patel, “All objects and some questions”, American Journal of Physics 91 (2023), 819-825. www.researchgate.net/publication/...
(1/n)
Let me explain it a bit...
• C. H. Lineweaver and V. M. Patel, “All objects and some questions”, American Journal of Physics 91 (2023), 819-825. www.researchgate.net/publication/...
(1/n)
December 14, 2025 at 8:19 PM
This chart shows all objects in the Universe, arranged by mass (vertical) and radius (horizontal).
Let me explain it a bit...
• C. H. Lineweaver and V. M. Patel, “All objects and some questions”, American Journal of Physics 91 (2023), 819-825. www.researchgate.net/publication/...
(1/n)
Let me explain it a bit...
• C. H. Lineweaver and V. M. Patel, “All objects and some questions”, American Journal of Physics 91 (2023), 819-825. www.researchgate.net/publication/...
(1/n)
"I've never met any child prodigies, but I did meet many of their parents." - Daniel Baremboim
December 13, 2025 at 11:56 AM
"I've never met any child prodigies, but I did meet many of their parents." - Daniel Baremboim
At 17:00 UTC today I'm going to give a talk at the Topos Institute. You can watch it here!
youtube.com/watch?v=GzBV...
(1/n)
youtube.com/watch?v=GzBV...
(1/n)
John Baez: "Cospans of finite sets"
YouTube video by Topos Institute
youtube.com
December 11, 2025 at 1:40 PM
At 17:00 UTC today I'm going to give a talk at the Topos Institute. You can watch it here!
youtube.com/watch?v=GzBV...
(1/n)
youtube.com/watch?v=GzBV...
(1/n)