John C. Baez
@johncarlosbaez.bsky.social
Mathematical physicist
Pinned
Standard Model 6: Pauli Matrices
YouTube video by John Baez
youtu.be
Wolfgang Pauli invented his famous matrices to describe the angular momentum of a spin-1/2 particle back in 1927. You'll see them in most courses on quantum mechanics. But where do they come from? Here I derive them from scratch.
youtu.be/M4QwKz1wvn4
youtu.be/M4QwKz1wvn4
Not very urgent compared to some things, but okay:
On MathOverflow I asked what has been done to make this question-answer website more welcoming to women:
meta.mathoverflow.net/questions/63...
Answer: something once, but nothing now. But some preferred to argue against the very question.
(1/2)
On MathOverflow I asked what has been done to make this question-answer website more welcoming to women:
meta.mathoverflow.net/questions/63...
Answer: something once, but nothing now. But some preferred to argue against the very question.
(1/2)
What is being done to make MathOverflow more welcoming to women?
What if anything has been done, or is being done, to make MathOverflow more welcoming to women - or at least less unwelcoming to women?
meta.mathoverflow.net
January 26, 2026 at 11:27 PM
Not very urgent compared to some things, but okay:
On MathOverflow I asked what has been done to make this question-answer website more welcoming to women:
meta.mathoverflow.net/questions/63...
Answer: something once, but nothing now. But some preferred to argue against the very question.
(1/2)
On MathOverflow I asked what has been done to make this question-answer website more welcoming to women:
meta.mathoverflow.net/questions/63...
Answer: something once, but nothing now. But some preferred to argue against the very question.
(1/2)
Wine descriptions have gone too far.
January 22, 2026 at 2:51 AM
Wine descriptions have gone too far.
Wow, so Victor Hugo loved math!
Or at least it made him less miserable.
Or at least it made him less miserable.
!["Mathematics is no less immense than the sea. [...] What algebraist could cast anchor in the infinitesimal calculus? Conic sections plunge into the azure as deeply as the Milky Way. The asymptote and the hyperbola are manifestations of the incomprehensible in geometric form." – Victor Hugo (1802-1885)](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:fb5fjz7jxmbf3alhwh44ookd/bafkreicvxaxsixiouvikfgvmt46uxvwpcxogqzmh2fyo34ta7eywpve5eq@jpeg)
January 21, 2026 at 11:28 PM
Wow, so Victor Hugo loved math!
Or at least it made him less miserable.
Or at least it made him less miserable.
There's a danger that our technology could fail if it fails!!!
AI boom could falter without wider adoption, Microsoft chief Satya Nadella warns
AI boom could falter without wider adoption, Microsoft chief Satya Nadella warns
Big tech boss tells delegates at Davos that broader global use is essential if technology is to deliver lasting growth
www.irishtimes.com
January 20, 2026 at 5:03 PM
There's a danger that our technology could fail if it fails!!!
The article is okay, but the headline bugs me.
🎉🎉🎉 String theory can now describe a universe that has dark energy!!! 🎉🎉🎉
Just one little problem: this universe has the wrong dimension. 😢
🎉🎉🎉 String theory can now describe a universe that has dark energy!!! 🎉🎉🎉
Just one little problem: this universe has the wrong dimension. 😢
Eva Silverstein, a physicist at Stanford University, co-authored a 2021 paper that has inspired several recent reformulations of string theory.
String Theory Can Now Describe a Universe That Has Dark Energy | Quanta Magazine
In an unprecedented step, researchers crafted a detailed model compatible with the universe’s accelerated expansion.
www.quantamagazine.org
January 19, 2026 at 4:37 AM
The article is okay, but the headline bugs me.
🎉🎉🎉 String theory can now describe a universe that has dark energy!!! 🎉🎉🎉
Just one little problem: this universe has the wrong dimension. 😢
🎉🎉🎉 String theory can now describe a universe that has dark energy!!! 🎉🎉🎉
Just one little problem: this universe has the wrong dimension. 😢
Dante was doing some impressive topology back in 1320 AD, compactifying space with God as the point at infinity!
Cool geometry in the Inferno from @johncarlosbaez.bsky.social johncarlosbaez.wordpress.com/2026/01/18/d... also #mathart!
Dante and the 3-Sphere
Apparently Dante conceived of the universe as a 3-sphere! That’s a 3-dimensional space formed by taking two solid 3-dimensional balls and completely gluing their surfaces together. In his Div…
johncarlosbaez.wordpress.com
January 19, 2026 at 12:06 AM
Dante was doing some impressive topology back in 1320 AD, compactifying space with God as the point at infinity!
Fun beginner-level category puzzle:
Suppose f: x → y and g: y → x are morphisms such that fg and gf are both isomorphisms.
Show that f and g must both be isomorphisms!
Suppose f: x → y and g: y → x are morphisms such that fg and gf are both isomorphisms.
Show that f and g must both be isomorphisms!
January 17, 2026 at 6:59 PM
Fun beginner-level category puzzle:
Suppose f: x → y and g: y → x are morphisms such that fg and gf are both isomorphisms.
Show that f and g must both be isomorphisms!
Suppose f: x → y and g: y → x are morphisms such that fg and gf are both isomorphisms.
Show that f and g must both be isomorphisms!
Reposted by John C. Baez
Just 1 ship packed with solar panels delivers the means to produce as much electricity as 57 LNG tankers or 120 coal ships
A world with little fossil fuels has an enormously lighter footprint - starting with zero need for 40% of global shipping
reneweconomy.com.au/one-ship-loa...
A world with little fossil fuels has an enormously lighter footprint - starting with zero need for 40% of global shipping
reneweconomy.com.au/one-ship-loa...
January 15, 2026 at 11:49 PM
Just 1 ship packed with solar panels delivers the means to produce as much electricity as 57 LNG tankers or 120 coal ships
A world with little fossil fuels has an enormously lighter footprint - starting with zero need for 40% of global shipping
reneweconomy.com.au/one-ship-loa...
A world with little fossil fuels has an enormously lighter footprint - starting with zero need for 40% of global shipping
reneweconomy.com.au/one-ship-loa...
This famous ranking of universities is now put out by Elsevier.
Is that why their brain-dead website seems to list MIT as the top university in the world for arts and humanities?
www.timeshighereducation.com/world-univer...
Is that why their brain-dead website seems to list MIT as the top university in the world for arts and humanities?
www.timeshighereducation.com/world-univer...
World University Rankings by Subject 2025: Arts and Humanities | Times Higher Education
Explore the 2025 Times Higher Education Subject Rankings for Arts and Humanities below. This year, we have evaluated 750 universities from 72 countries and territories on their contributions to the b...
www.timeshighereducation.com
January 15, 2026 at 5:42 PM
This famous ranking of universities is now put out by Elsevier.
Is that why their brain-dead website seems to list MIT as the top university in the world for arts and humanities?
www.timeshighereducation.com/world-univer...
Is that why their brain-dead website seems to list MIT as the top university in the world for arts and humanities?
www.timeshighereducation.com/world-univer...
Umm, but in this theory space is 4-dimensional: one dimension too many. 😏
www.quantamagazine.org/string-theor...
www.quantamagazine.org/string-theor...
String Theory Can Now Describe a Universe That Has Dark Energy | Quanta Magazine
In an unprecedented step, researchers crafted a detailed model compatible with the universe’s accelerated expansion.
www.quantamagazine.org
January 14, 2026 at 10:11 PM
Umm, but in this theory space is 4-dimensional: one dimension too many. 😏
www.quantamagazine.org/string-theor...
www.quantamagazine.org/string-theor...
Wolfgang Pauli invented his famous matrices to describe the angular momentum of a spin-1/2 particle back in 1927. You'll see them in most courses on quantum mechanics. But where do they come from? Here I derive them from scratch.
youtu.be/M4QwKz1wvn4
youtu.be/M4QwKz1wvn4
Standard Model 6: Pauli Matrices
YouTube video by John Baez
youtu.be
January 13, 2026 at 4:47 PM
Wolfgang Pauli invented his famous matrices to describe the angular momentum of a spin-1/2 particle back in 1927. You'll see them in most courses on quantum mechanics. But where do they come from? Here I derive them from scratch.
youtu.be/M4QwKz1wvn4
youtu.be/M4QwKz1wvn4
Our old system of energy production will shock future generations with its inefficiency - if there are future generations left to be shocked.
In a newly published paper, we find that in 2025 human emissions of greenhouse gases added around 23 billion trillion joules of heat to the world's oceans – 39 times as much as the annual energy produced by all human activity on Earth. link.springer.com/ar...
January 13, 2026 at 3:53 AM
Our old system of energy production will shock future generations with its inefficiency - if there are future generations left to be shocked.
In 1869, long before Einstein, the famous mathematician Sylvester announced that Clifford - the Clifford algebra guy - was studying whether space is curved!
Clifford's mission to study this was almost thwarted by a shipwreck. But that didn't stop him!
johncarlosbaez.wordpress.com/2026/01/10/s...
Clifford's mission to study this was almost thwarted by a shipwreck. But that didn't stop him!
johncarlosbaez.wordpress.com/2026/01/10/s...
Sylvester and Clifford on Curved Space
Einstein realized that gravity is due to the curvature of spacetime, but let’s go back earlier: On the 18th of August 1869, the eminent mathematician Sylvester gave a speech arguing that geom…
johncarlosbaez.wordpress.com
January 11, 2026 at 9:56 PM
In 1869, long before Einstein, the famous mathematician Sylvester announced that Clifford - the Clifford algebra guy - was studying whether space is curved!
Clifford's mission to study this was almost thwarted by a shipwreck. But that didn't stop him!
johncarlosbaez.wordpress.com/2026/01/10/s...
Clifford's mission to study this was almost thwarted by a shipwreck. But that didn't stop him!
johncarlosbaez.wordpress.com/2026/01/10/s...
Cloud-9 is a galaxy with practically no stars in it - just gas and a large amount of dark matter! It seems to be a remnant of early galaxy formation: a 'failed galaxy' that never produced any stars.
But there's more to it than this.
(1/n)
But there's more to it than this.
(1/n)
January 10, 2026 at 11:07 PM
Cloud-9 is a galaxy with practically no stars in it - just gas and a large amount of dark matter! It seems to be a remnant of early galaxy formation: a 'failed galaxy' that never produced any stars.
But there's more to it than this.
(1/n)
But there's more to it than this.
(1/n)
If you stare at this movie it looks like nothing is happening, but then you notice groups of lines appearing and disappearing.
No, you're not on drugs - it's math.
No, you're not on drugs - it's math.
And now I'm trying my hand at animating them by making the phase shift:
(Mathematical art, I guess.)
Phase-shifting the Fourier transform of a regular 14-gon. Or equivalently: seven plane waves meeting at equal angles.
Phase-shifting the Fourier transform of a regular 14-gon. Or equivalently: seven plane waves meeting at equal angles.
January 10, 2026 at 10:20 PM
If you stare at this movie it looks like nothing is happening, but then you notice groups of lines appearing and disappearing.
No, you're not on drugs - it's math.
No, you're not on drugs - it's math.
I try to avoid talking about religion and politics here, but this time they've gone too far - I gotta say something.
The Bishop of Paris banned a list of 219 propositions, and number 68 goes too far: I must object.
(1/n)
The Bishop of Paris banned a list of 219 propositions, and number 68 goes too far: I must object.
(1/n)
January 7, 2026 at 10:56 AM
I try to avoid talking about religion and politics here, but this time they've gone too far - I gotta say something.
The Bishop of Paris banned a list of 219 propositions, and number 68 goes too far: I must object.
(1/n)
The Bishop of Paris banned a list of 219 propositions, and number 68 goes too far: I must object.
(1/n)
Biology is so wild. It doesn't respect the rules we make up.
We all know children inherit genes from their parents. But it turns out mothers also inherit genes from their children!
And it can help them. (But it can also hurt them.)
For more, read this:
www.nature.com/articles/d41...
We all know children inherit genes from their parents. But it turns out mothers also inherit genes from their children!
And it can help them. (But it can also hurt them.)
For more, read this:
www.nature.com/articles/d41...
January 6, 2026 at 9:18 AM
Biology is so wild. It doesn't respect the rules we make up.
We all know children inherit genes from their parents. But it turns out mothers also inherit genes from their children!
And it can help them. (But it can also hurt them.)
For more, read this:
www.nature.com/articles/d41...
We all know children inherit genes from their parents. But it turns out mothers also inherit genes from their children!
And it can help them. (But it can also hurt them.)
For more, read this:
www.nature.com/articles/d41...
One of those days....
January 6, 2026 at 12:06 AM
One of those days....
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...