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John Baez’s Stuff

I'm a mathematical physicist. I work at the math department at U. C. Riverside in California, and also at the Centre for Quantum Technologies in Singapore. I'm working on network theory, information theory, and the Azimuth Project, which is a way for scientists, engineers and mathematicians to do something about the global ecological crisis. If you want to help save the planet, please send me an email or say hi on my blog.

What’s New?

Roger Penrose won the Nobel prize! See my my explanation of what he did.

I've been blogging about octonions and the Standard Model:

• Part 1. How to define octonion multiplication using complex scalars and vectors, much as quaternion multiplication can be defined using real scalars and vectors. This description requires singling out a specific unit imaginary octonion, and it shows that octonion multiplication is invariant under $$\mathrm{SU}(3)$$.
• Part 2. A more polished way to think about octonion multiplication in terms of complex scalars and vectors, and a similar-looking way to describe it using the cross product in 7 dimensions.
• Part 3. How a lepton and a quark fit together into an octonion - at least if we only consider them as representations of $$\mathrm{SU}(3)$$, the gauge group of the strong force. Proof that the symmetries of the octonions fixing an imaginary octonion form precisely the group $$\mathrm{SU}(3)$$.
• Part 4. Introducing the exceptional Jordan algebra: the 3×3 self-adjoint octonionic matrices. A result of Dubois-Violette and Todorov: the symmetries of the exceptional Jordan algebra preserving their splitting into complex scalar and vector parts and preserving a copy of the 2×2 adjoint octonionic matrices form precisely the Standard Model gauge group.
I didn't give the proof of that result. Instead I moved in a different direction, which should eventually loop back:
• Part 5. How to think of the 2×2 self-adjoint octonionic matrices as 10-dimensional Minkowski space, and pairs of octonions as left- or right-handed Majorana-Weyl spinors in 10 dimensional spacetime.
• Part 6.  The linear transformations of the exceptional Jordan algebra that preserve the determinant form the exceptional Lie group $$\mathrm{E}_6$$. How to compute this determinant in terms of 10-dimensional spacetime geometry: that is, scalars, vectors and left-handed spinors in 10d Minkowski spacetime.
• Part 7. How to describe the Lie group $$\mathrm{E}_6$$ using 10-dimensional spacetime geometry. This group is built from the double cover of the Lorentz group, left-handed and right-handed spinors, and scalars in 10d Minkowski spacetime.
• Part 8.  A geometrical way to see how $$\mathrm{E}_6$$ is connected to 10d spacetime, based on the octonionic projective plane.

Read my study of Fisher's fundamental theorem of natural selection: part 1, part 2, part 3.

Kenny Courser's thesis is out: Open Systems: a Double Categorical Approach. Read about it here and here.

And while you're at it: take a copy of my diary from 2003 to July 2020!

Can we actually remove carbon dioxide from the air? Yes! Can we remove enough to make a difference? Yes! But what are the best ways, and how much can they accomplish? I explain that in my article in Nautilus, an online science magazine.

Six random permutations of a 500-element set, where
circle areas are drawn in proportion to cycle lengths.
From Analytic Combinatorics by Flajolet and Sedgewick.

Learn about random permutations, and learn how to understand their properties using a blend of old-fashioned combinatorics, category theory, and complex analysis!

Check out the slides of the talks at the special session on applied category theory at UCR, and also my research team's talks at the Fourth Symposium on Compositional Structures:

and at Quantum Physics and Logic 2019:

Also try my more heavy-duty talk on structured cospans at CT2019.

Read my article in Nautilus magazine: The math that takes Newton into the quantum world. And for more of the technical details, read this.

Check out this interview: A quest for beauty and clear thinking. Also check out my talk on Unsolved mysteries of fundamental physics, which is now available on video.

Together with three students at Applied Category Theory 2018, I wrote a paper on biochemical coupling through emergent conservation laws. Check out our blog posts about this!

Learn about quantum mechanics and the dodecahedron:

Then read about the glories of the 600-cell (Part 1, Part 2, Part 3), which is a 4-dimensional analogue of the icosahedron:

These posts lead up to a grand conclusion: the Kepler problem and the 600-cell!

And while you're at it, check out my new paper on the icosahedron and E8, and learn about excitonium, Wigner crystals, the universal snake-like continuum, and the connection between braids, entropy and the golden ratio!

We recently had a special session on Applied Category Theory here at UCR, and you can see slides and videos of lots of talks. And this summer at a conference on applied algebraic topology I gave an overview of algebraic topology and how it's changed our thinking about math: The Rise and Spread of Algebraic Topology. Check it out!

Learn about diamondoids and phosphorus sulfides:

A while back I gave a talk at the Stanford Complexity Group on Biology as Information Dynamics, and here's a video:

The Azimuth Backup Project is saving about 40 terabytes of US government climate data from the threat of deletion. Our Kickstarter campaign exceeded its goal by a factor of 4, so we will be well funded to store this data and copy it to many safe locations.

Check out this talk on networks and category theory:

Try my articles on 'struggles with the continuum' — that is, problems with infinities in physics arising from our assumption that spacetime is a continuum:

• Part 1 - Problems with infinity. Point particles interacting gravitationally.
• Part 2 - The quantum mechanics of nonrelativistic charged point particles.
• Part 3 - The relativistic electrodynamics of point particles.
• Part 4 - The ultraviolet catastrophe, and quantum field theory.
• Part 5 - Quantum field theory: renormalization.
• Part 6 - Quantum field theory: summing over Feynman diagrams.
• Part 7 - General relativity: the singularity theorems.
• Part 8 - General relativity: the cosmic censorship hypothesis. Conclusion.

• Part 1 - the basic idea.
• Part 2 - the maximal abelian cover of a graph.
• Part 3 - embedding topological crystals.
• Part 4 - examples of topological crystals.

Take a road trip to infinity:

• Part 1: up to εo.
• Part 2: up to the Feferman–Schütte ordinal.
• Part 3: up to the small Veblen ordinal.

There's a mysterious relation between the discriminant of the icosahedral group:

and the involutes of the cubic parabola:

Learn a bit about quantum gravity, n-categories, crackpots and climate change in my interview on Physics Forums.

Anita Chowdry and I gave a joint lecture on The Harmonograph at the University of Waterloo. You can watch a video of it!

Read the tale of a doomed galaxy:

or if you prefer, read about the mysteries of the inverse cube force law:

Then learn about the butterfly, the gyroid and the neutrino:

Nature and the world of human technology are full of networks. People like to draw diagrams of networks: flow charts, electrical circuit diagrams, signal-flow graphs, Bayesian networks, Feynman diagrams and the like. Mathematically minded people know that in principle these diagrams fit into a common framework: category theory. But we are still far from a unified theory of networks! Learn more here:

Network Theory: overview. Video on YouTube.

Last year I gave talks on What Is Climate Change and What To Do About It? at the Balsillie School of International Affairs. You can see the slides.

But if you prefer biology and algebraic topology, try my talk on Operads and the Tree of Life.

For fun, I've been continuing to study the octonions, E8, the exceptional Jordan algebra, the Leech lattice and related structures:

• Part 1: integral octonions and the Coxeter group E10.
• Part 2: the integral octonions, 11d supergravity, and cosmological billiards.
• Part 3: the integer octonions in their guise as the E8 lattice.
• Part 4: the 240 smallest integer octonions, also known as the root vectors of E8.
• Part 5: the geometry of the root polytope of E8.
• Part 6: how to multiply octonions, and the Cayley integral octonions.
• Part 7: Greg Egan's proof that 2 × 2 self-adjoint matrices with integral octonion entries form a copy of the E10 lattice.
• Part 8 - my proof that 3 × 3 self-adjoint matrices with integral octonion entries form a copy of the K27 lattice.
• Part 9 - Egan's construction of the Leech lattice from the E8 lattice.
• Part 10 - fitting the Leech lattice into the exceptional Jordan algebra.

I also love Coxeter theory. Here's the Coxeter complex for the symmetry group of a dodecahedron:

You can learn more about this in my series "Platonic solids and the fourth dimension": part 1, part 2, part 3, part 4, part 5, part 6, part 7, part 8, part 9, part 10, part 11, part 12, and part 13.

Douglas Adams said the answer to life, the universe and everything is 42. But you may not know why. Now I have found out. The answer is related to Egyptian fractions and Archimedean tilings.

You may enjoy kaleidocycles and collidocycles:

Click the boxes to hear and read about some pieces I made with Greg Egan's QuasiMusic program, which translates quasicrystals into sound:

Read my series on the mathematical delights of rolling circles and balls!

There's a math puzzle whose answer is a really huge number. How huge? According to Harvey Friedman, it's incomprehensibly huge. Now Friedman is an expert on enormous infinite numbers and how their existence affects ordinary math. So when he says a finite number is incomprehensibly huge, that's scary. It's like seeing a seasoned tiger hunter running through the jungle with his shotgun, yelling "Help! It's a giant ant!" For more, read this.

In week319 of This Week's Finds, learn about catastrophe theory in climate physics! This issue features a program you can play with on your browser. It's a simple climate model that illustrates how a small increase in the amount of sunlight hitting the Earth could have a big effects on the climate, by melting snow and revealing darker soil. It was made by Allan Erskine.

Also on my blog, learn about ice, its many forms and crystal structures, how it resembles diamonds, and what scientists do with a machine that uses 80 times the world's electrical power for the few nanoseconds it's running.

What's on This Site

Also try my blogging here:

For common questions about physics, you can't beat this:

I don't maintain this Physics FAQ - Don Koks does, so please send any comments about it to him, not me!

If reading my stuff makes you want to ask questions, take a look at this.

The universe is full of magical things, patiently waiting for our wits to grow sharper. - Eden Philpotts