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John Baez
Works at Centre for Quantum Technologies
Attended Massachusetts Institute of Technology
Lives in Riverside, California
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John Baez

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Woohoo! (I hope)

The Laser Interferometric Gravitational-Wave Observatory or LIGO is designed to detect gravitational waves - ripples of curvature in spacetime moving at the speed of light. It's recently been upgraded, and it will either find gravitational waves soon or something really strange is going on.

Rumors are swirling that LIGO has seen gravitational waves produced by two black holes, of 29 and 36 solar masses, spiralling towards each other and then colliding to form a single 62-solar-mass black hole.

You'll notice that 29 + 36 is more than 62. So, it's possible that three solar masses were turned into energy, mostly in the form of gravitational waves!

According to these rumors, the statistical significance of the signal is very high: better than 5 sigma. That means there's at most a 0.000057% probability this event is a random fluke... assuming nobody made a mistake.

If these rumors are correct, we should soon see an official announcement. If the discovery holds up, someone will win a Nobel prize.

The discovery of gravitational waves is completely unsurprising, since they're predicted by general relativity, a theory that's passed many tests already. But it would open up a new window to the universe - and we're likely to see interesting new things, once gravitational wave astronomy becomes a thing.

To discuss, go here:
The Twitter image of the email seems to have disappeared from the Science news article and from Twitter, so here it is. edit: hmm, I can see it on my phone, but not on my laptop. Could be a cache thing?

(Thanks to for keeping a copy, where you can find the usual warnings that this is not confirmed or released etc.)

Personally, I would be totally :-D if the team put the paper on the arXiv, like all other sensible astrophysicists did. I don't think the glossy mags would pass up a chance to publish one of the major findings in physics merely because there was a preprint, no?

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John Baez

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What happens when matter emits light? Mainly, it boils down to electrons emitting photons. And there's a lot of ways this can happen.

The picture shows three of the simplest. An electron could emit a photon. It could emit two and then absorb one. Or, it could emit a photon which splits into an electron-positron pair which then recombines to give a photon!

Particles that don't make it to the edge of the picture are called virtual particles. We don't see them directly.

Feynman was the one who invented these pictures, so they're called Feynman diagrams. Each diagram stands for a process where some particles come in and some particles go out. But each diagram also stands for an integral. If you do the integral, you get the amplitude for that process to happen! From that, you can easily work out the probability that it happens.

But there's a catch: the integrals usually diverge. More simply put: they come out equal to infinity.

To deal with this, Feynman and others invented renormalization.

What is renormalization? How does it work? I explain that here, in simple terms:

At the end, I'll tell you about the most accurate prediction in all of science... and why nobody is completely sure why we should believe it.

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John Baez

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The Hoffman–Singleton graph

It's time for the twice-monthly Visual Insight post! This time it's a picture by +Félix de la Fuente, an architect and dedicated amateur mathematician in Barcelona who is in love with discrete geometry, polytopes and combinatorics.

He drew the Hoffman–Singleton graph by connecting 5 pentagons to 5 pentagrams. The picture at left shows the pentagons on the outside and the pentagrams on the inside. The picture at right shows how one of pentagons is connected to all 5 pentagrams. At Visual Insight you can see the whole construction and the final result:

The resulting graph has 252,000 symmetries! These symmetries form a group called PΣU(3,F₂₅), which I explain in the post.

For now let me just say that this group is built using the field with 25 elements, which is called F₂₅. To get this field, you can take the integers mod 5 and throw in a square root of some number that doesn't already have a square root. As a result, F₂₅ has an operation that acts a lot like complex conjugation, and this is used to define PΣU(3,F₂₅).

All this is nice... but it's not surprising that if we take 5 pentagons and 5 pentagrams and connect them up in a highly symmetrical way, we get a graph whose symmetries are connected to algebra involving the numbers 5 and 25.

I'm more tantalized by the mysterious connection between the Hoffman–Singleton graph and the Fano plane! The Fano plane has 7 points and 7 lines; it's not very 'five-ish'. And yet, you can build the Hoffman–Singleton graph starting from ideas involving the Fano plane. Why???
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John Baez

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Life among the bone eaters

A hyena can bite with a force of 220 pounds. But they are fiercely loyal to their friends. So Marcus Baynes-Rock became friends with some.... and ran with them through the streets of an ancient Ethiopian city at night.

It's quite a story! Read more here:

Mathematicians will be amused to hear that graph theory plays a role.
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The ultraviolet catastrophe

Classical physics seemed very good back in the early 1900s. But it made stunningly wrong predictions about what happens in a very hot oven.

You can see it here. In reality, the amount of light at different wavelengths peaks in the visible range - say, red in a red-hot oven. But classical physics says there's more and more radiation at shorter and shorter wavelengths - and an infinite amount of radiation in total! This is the ultraviolet catastrophe.

This is just one of the problems with infinity that physicists have had over the centuries. Planck, Einstein and others solved this one by inventing quantum mechanics and ultimately quantum field theory! For the full story, read my blog post:

and the next one, coming up soon.
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+John Baez​ Thanks for further clarifying the picture.
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John Baez

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Enjoying the snow

Not everyone likes the snow storm that hit DC today. But Tian Tian, the panda in the national zoo, seems to love it!

For more, see:
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Aggressively expanding civilizations

What will happen if some civilizations start aggressively expanding through the Universe at a reasonable fraction of the speed of light? Each such civilization will form a growing ‘bubble’: an expanding sphere of influence. And occasionally, these bubbles will collide.

Physicist S. Jay Olson has done some calculations, based on a range of assumptions, of what this will be like. Read more on my blog!

Here's the most surprising part.

If these civilizations are serious about expanding rapidly, they may convert a lot of matter into radiation to power their expansion. And while energy is conserved in this process, the pressure of radiation in space is a lot bigger than the pressure of matter, which is almost zero.

General relativity says that energy density slows the expansion of the Universe. But it also says that pressure has a similar effect. And as the Universe expands, the energy density and pressure of radiation drops at a different rate than the energy density of matter.

So, the expansion of the Universe itself, on a very large scale, could be affected by aggressively expanding civilizations! Olson does the math.

Ever since I became an environmentalist, the potential destruction wrought by aggressively expanding civilizations has been haunting my thoughts. Not just here and now, where it's easy to see, but ...
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John Baez

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City in the sky

This is so cool I'm not sure I believe it. It's a photo of the night sky over a city in Finland. A rare atmospheric phenomenon called light pillars created a map of the city itself, in the sky!

Street lights were reflected back down by ice crystals in the air. This only happens when flat hexagonal crystals are floating horizontally in still air. Light bounces back down from the crystals.

This was taken on Jan. 13, 2016, by Mia Heikkilä in Eura, Finland. For more, read Phil Plait's article here:

If he believes this is real, I guess I do too.

It's easier to compare this picture to a city map here:

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1+1 = 0

Math gets simpler in a world where 1+1=0, but it doesn't become self-contradictory and explode into nothing. We call this number system the field with 2 elements or F₂.

About a year ago, Greg Egan and I were studying a lattice in 8 dimensions called E8 lattice, and a lattice in 24 dimensions called the Leech lattice.

In the E8 lattice each point has 240 nearest neighbors. Let's call these the first shell. It also has 2160 second-nearest neighbors. Let's call these the second shell.

We noticed some cool things. For starters, you can take the first shell, rotate it, and expand it so that the resulting 240 points form a subset of the second shell!

In fact, there are 270 different subsets of this type. And if you pick two of them that happen to be disjoint, you can use them to create a copy of the Leech lattice inside E8⊕E8⊕E8 — that is, the direct sum of three copies of the E8 lattice! Egan showed that there are exactly 17,280 ways to do this.

Tim Silverman, a friend of mine in London, has been thinking about this ever since. And he found a nice way to understand it using the field with 2 elements.

As he explains:

“Everything is simpler mod p.” That is is the philosophy of the Mod People; and of all p, the simplest is 2. Washed in a bath of mod 2, that exotic object, the E8 lattice, dissolves into a modest orthogonal space, its Weyl group into an orthogonal group, its “large” E8 sublattices into some particularly nice subspaces, and the very Leech lattice itself shrinks into a few arrangements of points and lines that would not disgrace the pages of Euclid’s Elements. And when we have sufficiently examined these few bones that have fallen out of their matrix, we can lift them back up to Euclidean space in the most naive manner imaginable, and the full Leech springs out in all its glory like instant mashed potato.

Read the rest here:

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John Baez

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“Every good key must be a model of the lock it opens.”

That sentence states an obvious fact, but perhaps also a profound insight if we interpret it generally enough.

That sentence is also the title of a paper by Daniel L. Scholten. He gives a lot of examples, including these:

• A key is a model of a lock’s keyhole.

• A city street map is a model of the actual city streets

• A restaurant menu is a model of the food the restaurant prepares and sells.

• Honey bees use a kind of dance to model the location of a source of nectar.

• An understanding of some phenomenon (for example a physicist’s understanding of lightning) is a mental model of the actual phenomenon.

This line of thought has an interesting application to control theory. It suggests that to do the best job of regulating some system, a control apparatus should include a model of that system.

Indeed Conant and Ashby have tried to turn this idea into a theorem, the good regulator theorem.

But what does this theorem actually say??? It's not so easy to understand. For my attempt, and more discussion, visit my blog:

I think we're making some progress figuring it out.
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John Baez

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David Finkelstein, 1929-2016

If you're not a physicist you probably haven't heard of him, but in 1958 he made some big progress in understanding black holes.

Before that, most people thought that as a star collapsed, time would slow down for it, and it would become locked in place, a 'frozen star'. He realized this was a mistake due to taking a bad choice of coordinates too seriously. He realized that in fact an event horizon forms: a surface that you can fall into, but never get back out from.

He called it "a perfect unidirectional membrane: causal influences can cross it in only one direction".

He was always a radical, but he was always cheerful, with a twinkle in his eye. He gave me a copy of his book Quantum Relativity, which influenced some of my ideas on spin networks and spin foams. The last time I saw him was at an airport, by chance. Cheerful as ever!

According to my friend Predrag, the famous physicist Leonard Susskind wrote this on hearing of Finkelstein's death:

This is very sad news indeed; David was one of my greatest friends and a genuine hero to me. In recent years he and I did not have much opportunity to see each other, but just the fact that he was there was a inspiration. Whenever I felt frustrated by my colleagues it was great source of comfort to think of David's shining intellect, his curiosity, and his integrity. It is hard to accept that he is gone. David was truly unique.

I learned many things from David that played major roles in my thinking. I'll take the opportunity to remind myself of some of them. When I was a young physicist quantum field theory meant Feynman diagrams. It was the great work of Finkelstein and Rubinstein on topology in QFT that first made me understand the richness of the subject. It took quite a few years for the rest of the world to catch up. The great thing was that I got to hear about it straight from David himself.

For a long time I have been focused (some might say obsessed) on black holes. Again, it was David who first understood the nature of the horizon. That was a good deal before I knew him, but when I first became interested in gravity (1967 or so) it was David who said "Forget perturbation theory. Black holes are the key." Later he explained to me that the information in a region of space could not be as rich as the volume because most states would collapse to form a black hole. At the time--this was before Bekenstein--I didn't understand the importance of the remark. It was only after 't Hooft and I had thought about the holographic principle that I recalled the conversation. David understood the holographic principle long before anyone else.

There was lots more: his emphasis on the discrete nature of information predated John Wheeler's "It from Bit" philosophy, that today is playing such a central role in theoretical physics. In fact almost every time I start thinking about something new I recall something seminal that David said.

Here's an obituary:
Theoretical physicist David Ritz Finkelstein, Professor Emeritus in Georgia Tech School of Physics, died peacefully at home this morning, January 24, 2016. He was born in 1929, in New York City and has had a remarkable career. In 1958 he described what is now known as a black hole. He had done much since - his later years were dedicated to elucidating the quantum structure of space-time. His funeral will be on Tuesday the 26th, at Arlington cemetery in Atlanta.

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John Baez

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What is this place, and what makes it special?

Hint: the white stuff is not snow. Answer here:

There's another puzzle here, at the end, that I don't know the answer to. Help!
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I'm a mathematical physicist.
  • Centre for Quantum Technologies
    Visiting Researcher, 2011 - present
  • U.C. Riverside
    Professor, 1989 - present
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Riverside, California
Contributor to
I teach at U. C. Riverside and work on mathematical physics — which I interpret broadly as ‘math that could be of interest in physics, and physics that could be of interest in math’. I’ve spent a lot of time on quantum gravity and n-categories, but now I want to work on more practical things, too.

Why? I keep realizing more and more that our little planet is in deep trouble! The deep secrets of math and physics are endlessly engrossing — but they can wait, and other things can’t.

So, I’ve cooked up a plan to get scientists and engineers interested in saving the planet: it's called the Azimuth Project.  It includes a wiki, a blog, and a discussion forum.  I also have an Azimuth page here on Google+, where you can keep track of news related to energy, the environment and sustainability.

Check them out, and join the team!  Or drop me a line here.
  • Massachusetts Institute of Technology
    Mathematics, 1982 - 1986
  • Princeton University
    Mathematics, 1979 - 1982
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