It s long been a mystery why there are 3 generations of quarks and leptons: three sets of particles, apparently identical except for how they interact with the Higgs boson. It would be nice if there were some good physical explanation. Nobody knows one. Barring that, it would be nice if some beautiful mathematical structure made [ ]
I don t usually talk about particle physics here. I have a whole series of articles about octonions and the Standard Model on my other blog. But I m kind of excited about this new paper, so I ll talk about it here too: John Baez, Endre Bokor and Latham Boyle, Jordan pair quantum theory and the [ ]
Maxwell s equations are invariant under Lorentz transformations. The usual equations of fluid flow are not! Like the rest of Newtonian mechanics, they re invariant under Galilean transformations like So, if we simply slap these two theories together, we get a mess! How can we study electrically conductive fluids like plasma without bringing special relativity into the game? We [ ]
Paul Schwahn and I have come out with a new paper about octonions and the Standard Model: The Standard Model gauge group from the exceptional Jordan algebra It builds on things I ve discussed here, but it goes further. Let me explain a bit. A bit is just a binary alternative: 1 or 0, true [ ]
I m not completely happy with this interview with Micah Zarin. It was nothing he did, it was me. I forgot to say that current-day AI wastes a lot of energy, and companies hope to use it to lay off people, and oligarchs are using it to extract lots of money from everyone. While obvious, these [ ]
The sum of the reciprocals of the primes diverges, but very slowly. The sum of the reciprocals of the first 100 primes is 2.106 The sum of the reciprocals of the first 1,000 primes is 2.457 For the first 10,000 it s 2.709 And it keeps creeping up, ever more slowly. To get the sum to [ ]
I recently showed you that if you take the regular icosahedron: considered in a coordinate system based on the golden ratio, and then replace √5 by -√5 in all your formulas, you get the great icosahedron: But this fact isn t an isolated one-off! If we do the same for the regular dodecahedron: we get the [ ]
I never knew what was so great about the great icosahedron . Now I do. Take a regular icosahedron whose vertices have coordinates in the field ℚ[√5], which consists of numbers a + b√5 with a and b rational. Apply the nontrivial element of the Galois group of this field: that is, simply replace √5 by [ ]
This post starts with the the slides of an elementary talk I gave at Sloans Bar and Grill, in Glasgow, as part of a wonderful series called A Pint of Science. At the end I include some fascinating details which I only had time to briefly touch on in my talk. If you already know [ ]
In 2025, researchers studied a quasicrystal forged in a hypervelocity asteroid collision 600 million years ago and found that it contains phasons ! It s not a perfect icosahedral quasicrystal: it s slightly distorted. 6 gentle phason waves run through it, oriented along the 6 fivefold symmetry axes of an icosahedron. These waves were locked in when the alloy [ ]
I love quasicrystals like crystals, but with patterns that never repeat, like Penrose tiles. But they ve very rare in nature. They re created only by the most exotic and violent events: a high-speed collision of asteroids, lightning hitting a downed power cable in a sand dune or an atomic bomb! Amazingly, the first 3 kinds of naturally occurring [ ]
I recently asked people on Mastodon What s the most surprising fact you ve learned in the last couple of weeks? It was a nice way to learn a lot of interesting things. My own biggest recent surprise was this: the number plays a fundamental role in number theory! For any irrational we define its Lagrange [ ]
In the last episode of my column in Notices of the American Mathematical Society, we looked at a particle moving in an attractive central force whose strength is proportional to the inverse cube of the distance from the origin. Among other things, we saw that a particle moving in such a force can spiral in [ ]
We have left the Holocene and entered a new epoch, the Anthropocene, in which the biosphere is rapidly changing due to human activities. We do not need to decide to address these issues. They are already addressing us: grabbing us by the collar, so to speak. Our only choice is how to respond. In [ ]
Returning from a trip to New Mexico to explore some Puebloan ruins, I picked up this beautiful chunk of labradorite in the town of Quartzsite. This mineral creates an eerie blue shimmer in the sunlight: a phenomenon called labradorescence . Reading up on it, I discovered it s a form of feldspar. I didn t know at [ ]
I m only now learning about vector meson dominance a big idea put forth by Sakurai and others around 1960. Here s a family of 9 mesons called the vector nonet’. Each one is made of an up, down or strange quark and an antiup, antidown or antistrange antiquark. That s 3 × 3 = 9 choices. In this [ ]
I m giving a talk online tomorrow at the 2026 Spring Southeastern Sectional Meeting of the American Mathematical Society, in the Special Session on Non-Associative Rings and Algebras. The organizers are Layla Sorkatti and Kenneth Price. I doubt the talk will be recorded, but here are my slides: Projective geometry and the exceptional Jordan algebra. [ ]
This time I m talking about pions: Pions were a revolutionary discovery in the 1930s part of the first wave of the particle zoo but I m explaining them as a way to work toward the math and physics concepts needed for the Standard Model. As soon as the neutron was discovered in 1932, Heisenberg invented the idea of [ ]
guest post by William Waites The previous post introduced the plumbing calculus: typed channels, structural morphisms, two forms of composition, and agents as stateful morphisms with a protocol for managing their state. The examples were simple. This post is about what happens when the algebra handles something genuinely complex. To get there, we need to [ ]
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. We tend to take them for granted. But where do they come from? Here I derive them from scratch! There are lots of ways to derive them, and [ ]