A theorem you'll never forget: Napoleon's theorem.

Take any triangle. Erect equilateral triangles on its three sides. The centers of these triangles are the vertices of another equilateral triangle!

While traditionally attributed to Napoleon, who *was* interested in geometry, there's no good evidence that he proved it. In reality, this result first appears in 𝘛𝘩𝘦 𝘓𝘢𝘥𝘪𝘦𝘴' 𝘑𝘰𝘶𝘳𝘯𝘢𝘭 in 1824, posed as a puzzle by one "Mr. W. Rutherford of Woodburn". This guy is famous for having computed pi to 208 digits... of which only the first 152 were correct.

The proof here claims to be "geometrical" but is strategy is algebraic:

https://www.slideserve.com/hogan/a-geometric-proof-of-napoleon-s-theorem

Find a formula for the length of one side of the would-be equilateral triangle that depends symmetrically on the lengths of all 3 sides of the original triangle. This implies all sides of the would-be equilateral triangle have the same length. So it's indeed equilateral!

Napoleon's theorem has an excellent generalization to n-gons: the Napoleon-Barlotti theorem. The centers of regular n-gons constructed over the sides of an n-gon P form a regular n-gon if and only if P is an affine image of a regular n-gon! Roughly speaking, this means P is a regular n-gon that's been stretched or squashed or sheared in a linear way.

Since every triangle is an affine image of an equilateral triangle, this reduces to Napoleon's theorem when n = 3. ✔️

I wonder how Barlotti felt about sharing a result with the Emperor of the French.

(1/2)

After the ecological collapse, many of the species we love will be extinct. But probably not liverworts.

Liverworts were among the first plants to bounce back after the world-wide rain of molten quartz set all the forests on fire when an asteroid hit the Earth 65 million years ago. They were among the first to bounce back after the much worse end-Permian extinction. They've been around for 420 million years, but with bursts of new species beginning in the mid-Jurassic and continuing through the Cenozoic.

Liverworts so deeply evolved that when you search the internet for info on them, half the articles you'll see are about how to *kill* them, because they're so fucking hard to kill! That's because a small piece of a liverwort can grow into a whole new plant. And some have little cup-shaped structures holding clusters of cells called gemmae. A gemma can be splashed out of one of these cups by falling raindrops, and if it lands in a suitable place it will grow into a new liverwort. It's as if flakes of your skin splashed off by rain could grow into new copies of you.

They like damp, poorly lit places. I was delighted to find a bunch along the River Leith in Edinburgh - shown in my photo here. They were mixed with ferns and moss: two other ancient types of plant that propagate using spores. After global crises, we see "spore spikes" in the fossil record: plants like these become very common, setting the stage for new forests.

Martin Schwartz wrote an article called "The importance of stupidity in scientific research":

https://fermatslibrary.com/s/the-importance-of-stupidity-in-scientific-research

I agree with most of it, but I think what he calls "stupidity" should not be called stupidity: he means "admitting our ignorance". For example, he writes:

"Second, we don’t do a good enough job of teaching our students
how to be productively stupid – that is, if we don’t feel stupid it
means we’re not really trying. I’m not talking about ‘relative
stupidity’, in which the other students in the class actually read
the material, think about it and ace the exam, whereas you don’t.
I’m also not talking about bright people who might be working
in areas that don’t match their talents. Science involves confronting
our ‘absolute stupidity’. That kind of stupidity is an existential
fact, inherent in our efforts to push our way into the unknown.
Preliminary and thesis exams have the right idea when the faculty
committee pushes until the student starts getting the answers wrong
or gives up and says, ‘I don’t know’. The point of the exam isn’t
to see if the student gets all the answers right. If they do, it’s the
faculty who failed the exam. The point is to identify the student’s
weaknesses, partly to see where they need to invest some effort
and partly to see whether the student’s knowledge fails at a
sufficiently high level that they are ready to take on a research
project.

Productive stupidity means being ignorant by choice. Focusing
on important questions puts us in the awkward position of being
ignorant. One of the beautiful things about science is that it allows
us to bumble along, getting it wrong time after time, and feel
perfectly fine as long as we learn something each time."

I hope folks make lots of copies of this map of 300,000 Flock cameras before Flock gets the "official" map taken down.

https://theintercept.com/2026/09/24/how-many-flock-devices-in-united-states-300000/

I went to Glasgow to give a talk on the 4d rotational symmetry of the hydrogen atom, but what sticks in my mind is the beauty of ferns, tree ferns, mosses, liverworts and other "primitive" plants on display at the Glasgow botanical garden!

(1/n)

If you drop a sloppy AI-generated proof of some theorem onto the internet hoping to be hailed as a hero, think twice. Mathematicians are sick of it.

https://arxiv.org/abs/2609.24779

Time is much more convenient than space. It takes up less room, and it lasts longer.

For a mathematical physicist such as myself, for whom "number" means a real or complex number, working with an algebraist is a bit hallucinatory. I say "the octonions" and they say "which octonions?" For them, an octonion algebra is a general concept. If I try to understand it and how you classify these things, I'm quickly sucked into etale cohomology and other eldritch horrors. Actually I'm starting to like this stuff - but I don't really understand it. So it's like being in a dream where you're trying to walk to the bathroom but for some reason you first need to clamber down a ladder while holding an octopus.

(1/n)

It's funny that the most massive black hole known is called TON 618. In fact it's mass is about 10³⁸ tonnes. Its event horizon is roughly 400 billion kilometers - over forty times the diameter of Neptune’s orbit. It sits proudly at the center of a distant galaxy.

Because it's so huge, you could fall into it without being ripped apart as you went through the event horizon.

And its surface gravity is *less* than most black holes! The surface gravity is the force per mass registered by a stationary faraway observer holding the mass right above the horizon via a long string. Surprisingly, this is inversely proportional to the black hole's mass.

For a 3-solar-mass black hole, the surface gravity is half a trillion times the surface gravity on Earth - half a trillion g's.

For the black hole at the center of the Galaxy, it's 3 million g's.

For TON 618, it's just 230 g's. Still damn strong, but much less!

The concept of surface gravity for black holes is a bit weird, and I didn't explain it carefully - but it does make sense, and it's important. The temperature of a black hole is proportional to its surface gravity!

It's explained more here:

https://en.wikipedia.org/wiki/Surface_gravity#Black_holes

The landscape of Mars is so much more diverse than you might think! Here you see tracks of dust devils in the Thyles Rupes region, near the south pole, formed when the surface warms up.

This picture was taken in the late summer. The dust devils move in a different direction as the the season progresses from spring to fall.

The University of Arizona has a massive gallery of Mars pictures taken by the HiRISE satellite:

https://www.uahirise.org/ESP_013751_1115

There's a slow drama going on beneath your feet. 660 kilometers down is the boundary between the upper and lower mantle. Enormous slabs of colder, denser rock are slowly sinking through the upper mantle until they hit this boundary. These slabs are 30-100 kilometers thick and up to a thousand kilometers across!

Some punch straight through into the lower mantle and keep sinking. But many flatten out when they hit the boundary, sometimes lying there and piling up for tens of millions of years. You can see this in seismic images beneath Japan and the Marianas. Numerical models suggest that they pile up until they overwhelm the barrier and flush down in a comparatively sudden avalanche - lasting mere millions of years.

For why this boundary exists and more on what's going on down there, read my blog article. Geology rocks!

https://johncarlosbaez.wordpress.com/2026/08/22/the-mantle/

When I was a kid I read Ionescu's play Rhinoceros, where the inhabitants of a small French town turn one by one into a herd of wild, rampaging rhinoceroses. I was told the play was "absurdist", and read it as such. But living in the United States now, I see it's completely realistic.