The problem with archeologists is that the successful ones get a big head.

People used to think the Olmecs were not very old. In 1939 an archaeologist couple found the bottom half of this Olmec rock, which had part of a date carved on it. They guessed the date was 7.16.6.16.18, which is September 3, 32 BC. But the first digit was missing, so few believed them! (1/2)

Grrr, another recommendation website that's trying to test my intelligence.

Chaco Canyon should be called Choco Canyon, because researchers have found traces of *chocolate* in cups found at this site dating back to 1000 - 1125 AD. This is amazing: Chaco Canyon is in a dry part of New Mexico, 1900 kilometers north of where cacao grows. (1/n)

Here's why some algebraic geometers like to do math internal to the topos of sheaves on a scheme. Things look simpler when you're inside! Picture by Carina Willbold (CC BY-SA).

My wife and I are going to Gallup again this spring, where she studies Navajo-style weaving. We hope to finally visit Pueblo Bonito, one of the "great houses" of the Ancestral Puebloans, occupied circa AD 830-1130. When we tried before, it was very icy, and our car doesn't have 4-wheel drive!

Trying to see if I'm awake?

This is the coolest thread I've ever read

She discovered that the fundamental laws of physics distinguish between right and left!

Here's a tale of how nature triumphs in the end. And the Slag Queens! Steel mills dumped molten slag in parts of Chicago and nearby areas. The slag hardened in layers up to 15 feet deep. These places became barren wastelands. Other industries dumped hot ash and cinders there. (1/n)

People worldwide are starting to realize that billionaire-controlled social media are dangerous. But we still need lots of role models like Phil to shout it from the rooftops - and point people to better alternatives!

I'm not in the Epstein files - I checked. The name Baez shows up but it's mainly my cousin Joan. No bad reflection on her: some people went to a concert of hers. There's also a lawyer name Baez, etc. I've always just wanted to think and talk about math and physics.

This was my first talk where I played the piano and sang. math.ucr.edu/home/baez/tu...

It got so cold so fast in the US that freezing soil created frost quakes. Yes: FROST QUAKES - also called cryoseisms! They even cracked the wall in someone's house. www.npr.org/2026/01/30/n...

A single-line drawing that is simultaneously an open knight's tour of a 99x99 chessboard and a 3x3 Latin square.

Wow! Once the Rhine, the Thames and the Seine were all tributaries of the same big river that flowed out where the English Channel is now! It's called the Channel River: en.wikipedia.org/wiki/Channel... (hat-tip to @julesh.mathstodon.xyz.ap.brid.gy)

Fractions built from the numbers 2, 3 and 5 play a big role in music theory! Here I list those as close to 1 as possible for a given complexity. First we get a bunch of famous ones... then absurdly small ones like the quark of Baez: 2⁻⁵⁷³ 3²³⁷ 5⁸⁵ johncarlosbaez.wordpress.com/2026/01/28/t...

Progress in fundamental physics is often slow. The Oxford Calculators understood the motion of objects whose velocity changes at a constant rate back in 1335, yet it took centuries for Galileo to recover and further develop these insights. Patience! Enjoy the slow unfolding of mysteries.

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)

Wine descriptions have gone too far.

Wow, so Victor Hugo loved math! Or at least it made him less miserable.

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. 😢

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!

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...

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...

Umm, but in this theory space is 4-dimensional: one dimension too many. 😏 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

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...

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)

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.

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)

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...

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)

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

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!

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.

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...

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...

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...

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_...

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

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...

The time is back to nanosecond accuracy at the National Institute of Standards and Technology! 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...

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)