John Carlos Baez Profile picture
I'm a mathematical physicist interested in saving the planet.
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Jan 19 10 tweets 3 min read
Hardcore math tweet:

I've been explaining how spaces with dimension is a multiple of 4 are special. Using these ideas you can build a 4d topological manifold that can't be made smooth - starting from the Dynkin diagram of E8! But we won't get that far today.

(1/n) For today, 'manifold' will mean 'compact smooth oriented manifold'. If such a manifold M has dimension n = 2k, its deRham cohomology comes with a bilinear map

b: Hᵏ(M) × Hᵏ(M) → R

which is symmetric when k itself is even:

b(x,y) = b(y,x)

Jan 18 6 tweets 3 min read
Why is mercury liquid at room temperature? Its innermost electrons are moving at 58% the speed of light, so thanks to special relativity they become 23% heavier and their orbit about 23% smaller!

Without special relativity, mercury would melt at about 80 °C.

(1/n) The transition metals on the far right are weird compared to the rest because the d subshell is completely full at this point. Zinc is hard, cadmium is soft and mercury is actually liquid.

Jan 16 6 tweets 3 min read
A 𝘀𝘂𝗯𝘄𝗮𝗿𝗳 𝗕 𝘀𝘁𝗮𝗿 is a blue-hot star smaller than the Sun.

Some of these crazy stars pulse in brightness as fast as every 90 seconds! Waves of ionizing iron pulse through their thin surface atmosphere.

What's up with these weird stars?

(1/n) Image Sometimes a red giant loses most of its outer hydrogen... nobody knows why... leaving just a thin layer of hydrogen over its helium core.

We get a star with at most 1/4 the diameter of the Sun, but really hot.

It's the blue-hot heart of a red giant, stripped bare.

(2/n) Image
Jan 13 5 tweets 4 min read
This is the color of something infinitely hot.

Of course you'd instantly be fried by gamma rays of arbitrarily high frequency, but still.

This is also the color of a typical neutron star. They're so hot they look the same.

This was worked out by @gro_tsen.

(1/n) Image @gro_tsen As a blackbody gets hotter and hotter, its spectrum approaches the classical "Rayleigh-Jeans law".

That is, its true spectrum as predicted by quantum mechanics - the "Planck law" - approaches the classical prediction over a larger and larger range of frequencies.

(2/n) Image
Jan 11 6 tweets 2 min read
Przybylski's star. Almost no iron in its atmosphere. Instead: rare earth metals... and even radioactive elements like plutonium, curium, einsteinium!

It's so weird some astronomers have wondered if it's a nuclear waste dump for an alien civilization.

(1/n) Image The problem is, how do you get a lot of short-lived radioactive elements into the upper atmosphere of a big hot star? They're usually formed by merging neutron stars. So I'll make my own wacky guess: Przybylski's star zipped through the remains of such an event.

(2/n) Image
Jan 8 12 tweets 5 min read
One of the big surprises when you're first learning about topology is that a lot depends on the dimension of space mod 4.

Who would have guessed that 9d spaces are more like 17d spaces than 10d spaces? Just saying this makes me feel like a mad scientist.

(1/n) Here's the basic idea. If you have two smooth 1d curves drawn on a 2d plane, they can intersect - and if they do, usually moving them slightly won't stop them from intersecting. How does this generalize to higher dimensions?

Jan 6 10 tweets 4 min read
In April 2021, this study estimated about 10 million Americans believe Biden "stole" the presidential election and are willing to engage in violent protests over this:…

It tries to figure out who these people are.

(1/n) First, note that 10 million people is a lot. Even if you limit yourself to the small fraction of these people who are also white male gun owners who have military training since they're veterans, that's 360,000 people - almost the size of the entire US National Guard.

Jan 2 9 tweets 4 min read
Hardcore math tweet:

I have a crush on the 𝗕𝗼𝗹𝘇𝗮 𝗰𝘂𝗿𝘃𝗲:

y² = x⁵ - x

The space of complex solutions of this equation, plus a point at infinity, is the 2-holed Riemann surface with the biggest possible symmetry group! Its universal cover looks like this:

(1/n) More precisely you get the complex version of the Bolza curve, with its correct conformal structure, by gluing together opposite sides of this patch of the hyperbolic plane.

Dec 13, 2021 7 tweets 3 min read
Light can bounce off light by exchanging virtual charged particles! This makes Maxwell's equations nonlinear, even in the vacuum - but only noticeably so when the electric field is over 10¹⁸ volts/meter.

People want to test this at the Extreme Light Infrastructure.

(1/n) This is an enormous electric field, able to accelerate a proton from rest to Large Hadron Collider energies in just 5 micrometers!

Light-on-light scattering has been seen at the LHC by colliding lead ions. But people want to do it with lasers.

Dec 12, 2021 6 tweets 3 min read
The heaviest element: oganesson, the noble gas under radon in the periodic table.

Second heaviest: tennessine.

Combine them: get oganesson tetratenneside, OgTn₄ - so heavy that special relativity makes it weird!

Alas, it decays radioactively in less than a millisecond.

(1/n) It's purely theoretical, since both elements are artificially made in tiny amounts, and short-lived. But we can study the compounds they make using computer calculations. And it seems OgTn₄ should be possible, shaped like the tetrahedron at left.

Dec 11, 2021 5 tweets 2 min read
James Dolan and I did a lot of good math together, like discover the periodic table of n-categories and the cobordism hypothesis. Lately he's been talking to Todd Trimble about principally polarized abelian varieties, Eisenstein series, etc.

(1/n) James and I haven't talked much in about 10 years - I went off a different direction, and thought about climate change and ultimately applied category theory. But I'll give it a try today. I'll join him and Todd at 4 this afternoon and see what it's like.

Dec 6, 2021 12 tweets 4 min read
The surface of a doughnut is called the "boundary" of the doughnut.

Indeed any compact oriented 2-manifold, like these here, can be filled in - it's the boundary of some compact oriented 3-manifold.

The same thing is true one dimension up - but not two dimensions up!

(1/n) I say "oriented" because the Klein bottle is a compact 2-manifold that's not orientable, and it's not the boundary of any compact 3-manifold.

Basically the problem is that you can't tell what its "inside" would be.

(2/n )
Dec 4, 2021 4 tweets 2 min read
If you've met Cauchy formula's in complex analysis, guess what: there's a version for the quaternions too!

It works for any function that obeys the quaternionic version of Cauchy's equation ∂f/∂x + i∂f/∂y = 0.

It was discovered by Rudolf Fueter in 1935.

(1/n) Quaternionic analysis isn't quite as beautiful as complex analysis, mainly because if you multiply two solutions of

∂f/∂x + i∂f/∂y + j∂f/∂z + k∂f/∂z = 0

you don't get another solution! 😢

But it ain't chopped liver, neither. It deserves a little love.

Nov 30, 2021 4 tweets 2 min read
The 𝘁𝗿𝗮𝗻𝘀𝗶𝘁𝗶𝗼𝗻 𝗺𝗲𝘁𝗮𝗹𝘀 like iron, copper and zinc are more complicated than lighter elements.


Because they're the first whose electron wavefunctions are described by 𝘲𝘶𝘢𝘥𝘳𝘢𝘵𝘪𝘤 functions of x,y, and z - not just linear or constant.

(1/n) Image More precisely: these waves involve a 'spherical harmonic', a function on the sphere that can be described using a polynomial in x,y,z. The jargon goes like this:

constant: s orbital
linear: p orbital
quadratic: d orbital


(2/n) Image
Nov 27, 2021 6 tweets 3 min read
Benzene has 6 electrons whose wavefunctions are smeared out in a ring.

When you turn on a magnetic field, it automatically lines up at right angles to the field, and the electrons start moving around!

This current loop creates its own magnetic field (in purple).

(1/n) Image What does this current loop look like, exactly?

To understand this, you have to know that the 6 smeared out or "delocalized" electrons lie above and below the plane of the benzene molecule.

(They come from "p orbitals" of the carbon atoms, which point up & down.)

(2/n) Image
Nov 24, 2021 5 tweets 2 min read
If we give Mars an atmosphere, it'll slowly get knocked away by the solar wind unless we also give Mars a magnetic field.

Ruth Bamford's plan: create a torus of plasma around the orbit of Mars' moon Phobos, carrying electric current!

(1/n) The electric current, going around a loop, could create a magnetic field that protects Mars from the solar wind... just as Earth's magnetic field protects us!

The solar wind hitting Mars now creates radiation 12,000 times that on Earth - not good for your health.

Nov 23, 2021 5 tweets 2 min read
There are infinitely many ways an electron (straight line) can emit virtual photons (wiggly lines), which in turn split into virtual electron-positron pairs, and so on. All these affect the electron's mass.

There 104 ways where the picture has 4 loops.

(1/n) By attaching another photon line to those pictures, Stefano Laporta found all 891 diagrams with 4 loops where an electron absorbs a photon. Here are examples.

An electron is a little magnet. He computed the effect of all 891 diagrams on the strength of that magnet.

Nov 22, 2021 4 tweets 3 min read
Owen Lynch, @Joe_DoesMath and I have combined classical thermodynamics, classical statistical mechanics and quantum stat mech in a unified framework based on entropy maximization! The key trick is the operad of convex spaces.

(1/n)… @Joe_DoesMath Owen blogged about this project early on, and this post explains the physics intuitions behind our work:…

It's all about equilibrium thermodynamics, or "thermostatics". Thermostatic systems maximize entropy. Entropy is a concave function.

(2/n) Image
Nov 19, 2021 6 tweets 3 min read
The structure of benzene is fascinating. In 1865 Kekulé guessed it has a ring of 6 carbons with alternating single and double bonds. But this led to big problems, which were only solved with quantum mechanics.

(1/n) If benzene looks like Kekulé first thought, there would be 4 ways to replace two hydrogens with chlorine! You could have two chlorines next to each other with a single bond between them as shown here... or a double bond.

But there aren't 4, just 3.

Nov 18, 2021 6 tweets 3 min read
What's a "topos", and why do people care? I wrote a short answer here:…

But since this is twitter let me say it even shorter... and more vaguely. A topos is a mathematical universe kinda like the one you grew up in, but maybe different. (1/n) Topos theory arose from the collision of Lawvere and Grothendieck, two great mathematicians with very different goals. Lawvere wanted to find foundations of mathematics more closely connected to actual practice, with the help of category theory. (2/n)
Nov 11, 2021 6 tweets 3 min read
Wow! Rare earth elements, or 'lanthanides', aren't really very rare. But only in 2011 was a bacterium found that requires rare earths to live. It even has a special protein for dealing with them, called 'lanmodulin'. And now scientists have used it to make a sensor.

(1/n) The bacterium lives in bubbling hot mud in a volcano. It survives by metabolizing methane.

It can use any of the 4 lightest lanthanides to do this: lanthanum (Ln), cerium (Ce), praseodymium (Pr) and neodymium (Nd). These are chemically very similar.