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## John Baez

Occupation: I'm a mathematical physicist.

Followers: 57,665

Views: 53,859,287

Cream of the Crop: 11/05/2011

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### Most comments: 92

2016-09-17 02:15:36 (92 comments; 22 reshares; 110 +1s; )

**Exploring black holes - with cats!**

There should be a series of videos exploring black holes with cats.

So far all we have is this gif made by +Dragana Biocanin. A cat can orbit just above the photon sphere of a non-rotating black hole, moving at almost the speed of light. It's impossible for a cat to orbit below the photon sphere. As long as it's outside the event horizon it can accelerate upwards and escape the black hole's gravitational pull. But if it crosses the event horizon, it's doomed!

The **event horizon** is an imaginary surface in spacetime that's defined by this property: once a cat crosses this surface, it can't come back without going faster than light! This property involves events in the future, so there's no guaranteed way for the cat to tell when it's crossing an event horizon.

Forexample, if... more »

### Most reshares: 91

2016-11-25 00:56:54 (52 comments; 91 reshares; 206 +1s; )

**Thanksgiving**

Today I give thanks for my childhood. I grew up on a planet where global warming had just begun — a place your children will never know.

It was a beautiful planet. It seems like a long time ago. This was before the drought killed 100 million trees in California, a third of all trees in the state. New Orleans had not yet drowned under flood waters. The Great Barrier Reef off the coast of Australia was still healthy, not yet bleached by the raging heat.

But the biggest difference was near the North Pole. Back when I started college in 1979, the volume of Arctic sea ice in summer was 4 times what is now!

Last winter was especially shocking. In February, the climate scientist Peter Gleick wrote:

What is happening in the Arctic now is unprecedented and possibly catastrophic.

The extent of Arctic sea icehad shrunk ... more »

### Most plusones: 412

2016-11-04 16:53:42 (28 comments; 43 reshares; 412 +1s; )

**Thorny devil**

So cute! This small lizard, called the **thorny devil** or **Moloch horridus**, lives in the deserts and scrub lands of Australia.

It may look fierce, but it's not dangerous. It only eats ants. It's spiny so it doesn't get eaten. It can change color, for camouflage! And it has a "false head" on the back of its neck, which it shows to potential predators by dipping its real head. I'm not sure why.

It's also called a **thorny dragon**:

https://en.wikipedia.org/wiki/Thorny_dragon

I thank +rasha kamel for introducing me to this beast. She pointed out this article:

http://m.phys.org/news/2016-11-thorny-devil-skin-gravity.html

Scientists have recently figured out more about how this lizard gets water:

Researchers discovered long ago that because itsmouth ha... more »

Latest 50 posts

2016-12-05 16:21:42 (16 comments; 16 reshares; 71 +1s; )

**The math of networks**

In 2007 Jim Simons, the mathematician who helped invent Chern–Simons theory and then went on to make billions using math to run a hedge fund, founded a research center for geometry and physics on Long Island. More recently he also set up an institute for theoretical computer science in Berkeley. I’ve never been there - until today!

This week a bunch of mathematicians and computer scientists are meeting here to talk about **compositionality**. That means: how big complicated things are built from small simple things.

The show starts at 9:00 am today. We'll begi with talks from Gordon Plotkin (from Edinburgh, an expert on using category theory to study computer science and biology), David Spivak (who is proselytizing for applied category theory at MIT, and whose work on operads helped launch the project I'm working on withMetron)... more »

2016-12-01 17:15:59 (0 comments; 22 reshares; 104 +1s; )

**The writing is on the wall**

This is the main building of the Environmental Protection Agency or **EPA**, right across from Trump Hotel in Washington DC. A lot of us are protesting the guy Trump hired to demolish the EPA. His name is **Myron Ebell**. He's a climate change denier whose work has long been funded by fossil fuel industries.

Join the battle:

http://climatetruth.org/rebelagainstebell

http://petitions.moveon.org/sign/keep-myron-ebell-from

George Monbiot provides more detail:

Yes, Donald Trump’s politics are incoherent. But those who surround him know just what they want, and his lack of clarity enhances their power. To understand what is coming, we need to understand who they are. I know all too well, because I have spent the past 15 years fighting them.

Over this time, I have watched as tobacco, coal,oil, ... more »

2016-11-29 18:54:29 (20 comments; 6 reshares; 80 +1s; )

**I hate writing grant proposals. Luckily I don't need to anymore!**

2016-11-27 17:08:04 (26 comments; 18 reshares; 119 +1s; )

**Jarzynski on thermodynamics**

In the old days, despite its name, thermodynamics was mainly about thermodynamic equilibrium. Thermodynamic equilibrium is a situation where nothing interesting happens. For example, if you were in thermodynamic equilibrium right now, you'd be dead. Not very dynamic!

Sure, there were a few absolutely fundamental results like the second law, which says that entropy cannot decrease as we carry a system from one equilibrium state to another. But the complications you see when you boil a pot of water... those were largely out of bounds.

This has changed in the last 50 years. One example is the **Jarzynski equality**, discovered by Christopher Jarzynski in 1997.

The second law implies that the change in 'free energy' of a system is less than or equal to the amount of work done on it. But the Jarzynskiequali... more »

2016-11-26 17:13:28 (14 comments; 20 reshares; 153 +1s; )

**Fireflies in bamboo**

This photo by **Kei Nomiyama** shows fireflies just above the ground in a bamboo forest.

Photographing fireflies is popular in Japan, and this article by Courtney Constable shows some other nice examples:

http://www.thecoolist.com/japan-summer-firefly-phenomenon/

She writes:

Japan is a beautiful country full of breathtaking buildings, landscapes, and scenery any time of year. In the height of summer, however, something particularly magical happens. Throughout the countryside, twinkling fireflies take to the evening skies in search of a mate. This natural phenomenon creates a beautifully ethereal glow through trees and leaves that is nothing short of breathtaking.

Of course, in this phenomenon, Japanese and visiting photographers have found a gorgeous source of inspiration. Capturing the lights of the fireflies,... more »

2016-11-25 00:56:54 (52 comments; 91 reshares; 206 +1s; )

**Thanksgiving**

Today I give thanks for my childhood. I grew up on a planet where global warming had just begun — a place your children will never know.

It was a beautiful planet. It seems like a long time ago. This was before the drought killed 100 million trees in California, a third of all trees in the state. New Orleans had not yet drowned under flood waters. The Great Barrier Reef off the coast of Australia was still healthy, not yet bleached by the raging heat.

But the biggest difference was near the North Pole. Back when I started college in 1979, the volume of Arctic sea ice in summer was 4 times what is now!

Last winter was especially shocking. In February, the climate scientist Peter Gleick wrote:

What is happening in the Arctic now is unprecedented and possibly catastrophic.

The extent of Arctic sea icehad shrunk ... more »

2016-11-22 08:09:05 (0 comments; 11 reshares; 71 +1s; )

**Hail Trump!**

At a meeting of the **National Policy Institute** in Washington DC this Saturday, neo-Nazis in the so-called 'alt-right' movement raised their hands and cried "Hail Trump!". Watch the video.

From the New York Times:

WASHINGTON — By the time Richard B. Spencer, the leading ideologue of the alt-right movement and the final speaker of the night, rose to address a gathering of his followers on Saturday, the crowd was restless.

In 11 hours of speeches and panel discussions in a federal building named after Ronald Reagan a few blocks from the White House, a succession of speakers had laid out a harsh vision for the future, but had denounced violence and said that Hispanic citizens and black Americans had nothing to fear. Earlier in the day, Mr. Spencer himself had urged the group to start acting less like an undergroundorg... more »

2016-11-20 17:44:52 (30 comments; 9 reshares; 65 +1s; )

**Completely integrable billiards**

Check out +Carlos Scheidegger's great website that lets you play around with billiards on two tables:

https://cscheid.net/projects/bunimovich_stadium/

The table here is elliptical, and you'll see that the billiards trace out nice patterns - not at all random. Often there's a region of the table that they never enter! Not in this particular example, but try others and you'll see what I mean.**Puzzle 1.** What shape is this 'forbidden region', and why?

It will be easier to answer if you experiment a bit.

The other table is a rectangle with rounded ends, called the **Bunimovitch stadium**. For that one the billiards move chaotically. After a while they seem randomized.

This illustrates two very different kinds of dynamical systems. The'com... more »

2016-11-19 12:29:14 (19 comments; 10 reshares; 101 +1s; )

**Mammal-like reptiles**

These are **Pristerognathus**, very ancient mammal-like reptiles. They lived in the middle Permian, around 260 million years ago. That's long before the dinosaurs!

These animals were roughly dog-sized, and had long, narrow skulls and large canine teeth. They probably lived in woodlands, and preyed on smaller animals.

There were many kinds of mammal-liked reptiles back then. In general they're called **therapsids**. Some of them evolved to become mammals - like you and me! Fur has been found in the fossilized poop of some of these animals. So, at least some of them had hair.

https://en.wikipedia.org/wiki/Therapsid

https://en.wikipedia.org/wiki/Pristerognathus

These particular guys are called Pristerognathus vanderbyli. This picture is from Wikicommons and was apparently made byУчастник:Д... more »

2016-11-18 14:34:30 (48 comments; 13 reshares; 91 +1s; )

**Fighting climate change in the Trump era**

There's some good news and some bad news.

Read this article and you'll see the bad news - we have our work cut out for us! Not only Trump but also Republicans in the House and Senate have repeatedly stated they want to:

1) Kill Obama’s Clean Power Plan

2) Withdraw from the Paris Climate Agreement

3) Dismantle US environmental rules around coal power

4) Weaken fuel economy standards for cars and trucks

5) Open up new public lands to oil and gas drilling

6) Scale back federal support for wind and solar power

7) Limit the Environmental Protection Agency

8) Reverse the White House’s climate guidance to federal agencies

9) Make the Supreme Court hostile to environmental regulation

10) Pack the executive branch with carbon dioxide lovers

These things aren'tinevit... more »

2016-11-17 15:17:08 (18 comments; 29 reshares; 72 +1s; )

**What is a random string of bits?**

Shannon invented a way to measure **information**, but it doesn't let you measure the information in a specific message. Shannon's information is really just the **entropy** of a random source of messages. Later came **Kolmogorov complexity**, which gives a concept of how much information is contained in a specific message. It's basically the length of the shortest program that prints out this message.

But there's a catch: you can't usually compute the Kolmogorov complexity of a message! In fact there's a **complexity barrier**: there's a constant C such that you can't prove messages have complexity > C.

The precise statement of this result is a bit more complicated, because what you can prove depends on what system of math you use, but still: there are limits on how muchinfo... more »

2016-11-15 14:00:34 (40 comments; 15 reshares; 92 +1s; )

**Chaotic billiards**

Nice animation by Phillipe Roux! Take some balls moving in the same direction and let them bounce around in this shape: a rectangle with ends rounded into semicircles. They will soon start moving in dramatically different ways. (To keep things simple we don't let the balls collide - they pass right through each other.) In a while they will be almost evenly spread over the whole billiard table.

This is an example of **chaos**: slightly different initial conditions lead to dramatically different trajectories.

It's also an example of **ergodicity**: for almost every choice of initial conditions, the trajectory of a ball will have an equal chance of visiting each tiny little region. **Puzzle:** why did I say "almost" every choice? Can you find some exceptions?

Check out more ofPhillipe ... more »

2016-11-14 15:07:49 (17 comments; 13 reshares; 76 +1s; )

**The math of networks**

I'm at the +Santa Fe Institute! I got here just now - it's a beautiful place. Tomorrow I'll give a talk on the math of networks.

The idea: nature and the world of human technology are full of networks. People like to draw diagrams of networks: flow charts, electrical circuit diagrams, chemical reaction networks, signal-flow graphs, Bayesian networks, food webs, Feynman diagrams and the like. People often treat these diagram languages as informal tools - not true mathematics. But in fact, many of these languages fit into a rigorous framework: monoidal categories.

Don't be scared if you don't know what a monoidal category is - my talk explains that. Here are the slides:

http://tinyurl.com/santa-fe-network-talk

Eventually a video will be available.

The main new thing here is work withBlake... more »

2016-11-12 23:44:55 (40 comments; 5 reshares; 120 +1s; )

**California — heading toward a greener future**

Just 8 days ago, the Paris Climate Agreement came into force. Trump will try to get out of it. But California has other plans. And today our Governor, Jerry Brown, told the world we're not backing down. He made this announcement:

Today we saw the beginning of the transfer of power to the President-elect.

While the prerogatives of victory are clear, so also are the responsibilities to ensure a strong and unified America. As President Lincoln said, 'A house divided against itself cannot stand.' With the deep divisions in our country, it is incumbent on all of us - especially the new leadership in Washington — to take steps that heal those divisions, not deepen them. In California, we will do our part to find common ground whenever possible.

But as Californians, we will also stay true toour basi... more »

2016-11-12 03:40:57 (0 comments; 19 reshares; 143 +1s; )

**The loser won**

Having lost the popular vote, Trump is busy deleting tweets from 2012 in which he falsely claimed that Obama did the same - and argued that therefore "We should have a revolution in this country!"

http://www.theverge.com/2016/11/11/13596932/trump-protestors-electoral-college-tweets

2016-11-10 15:01:39 (69 comments; 48 reshares; 156 +1s; )

**This man must be stopped**

Trump has said on Twitter that:

the concept of global warming was created by and for the Chinese in order to make US manufacturing non-competitive.

While he later denied saying this, he is now threatening to put **Myron Ebell** in charge of his Environmental Protection Agency transition team. "Transition team"? Yes, apparently Trump wants to weaken or destroy this agency. And if you don't know Myron Ebell, you'd better learn about him now!

Myron Ebell has said:

I don’t want to say it’s a disaster, but I think it is potentially a disaster for humankind and not necessarily any good for the planet.

What's he talking about? Global warming? No, he's talking about the Paris agreement to fight global warming. He claims global warming is, on the whole, a good thing. Why?

... more »

2016-11-09 06:58:57 (0 comments; 11 reshares; 149 +1s; )

2016-11-07 17:43:29 (18 comments; 10 reshares; 83 +1s; )

**Learn advanced math at a ski resort - for free!****Homotopy type theory** takes modern ideas on logic, computation, topology and category theory and weaves them into an elegant new set of axioms for math in which spaces replace sets. In this brave new world, things like 'the space of all loops in a space' become just as simple and fundamental as the number 2.

On June 4-10, 2017, the American Mathematical Society will run a workshop on Homotopy Type Theory at the Snowbird Resort in Utah.

This workshop will bring together advanced graduate students and postdocs with some background in algebraic topology, category theory, mathematical logic, or computer science, with the goal of learning how these areas come together in homotopy type theory, and working together to prove new results. You'll need basic knowledge of one of these areas to be a successfulpa... more »

2016-11-06 20:53:28 (73 comments; 10 reshares; 87 +1s; )

**Predicting the election: a bit of information**

Right now Nate Silver's site **FiveThirtyEight** says Hillary Clinton has a 64.2% chance of winning the election, while Sam Wang's **Princeton Election Consortium** says she has more than a 99% chance.

The obvious question is: who is right?

But that's not a very good question. For starters, maybe neither is right!

A more reasonable question is: who is closer to being right?

But even this is very tricky. For starters, it's hard to define what it means to be right about such an estimate. Probability and statistics are slippery subjects. And a probabilistic prediction about a single event that will never be repeated is about as slippery as it gets.

If you have technical ideas about why Nate Silver and Sam Wang get such different results, I'd behappy to ... more »

2016-11-05 16:10:49 (5 comments; 9 reshares; 65 +1s; )

**Escudero nonic**

An amazing surface! This was recently discovered by Juan García Escudero, a mathematician in Spain, in his quest for surfaces with large numbers of 'ordinary double points'. These are points where two cones meet at their tips.

This particular surface is a **nonic**: it's described by a polynomial equation of degree 9. And it has 220 ordinary double points. You can see more pictures of it here:

http://blogs.ams.org/visualinsight/2016/11/01/escudero-nonic/

Some, like this, were created by +Abdelaziz Nait Merzouk. Others were created by Escudero himself.

In his 2005 thesis, Oliver Labs described a nonic surface with 226 nodes; however, they live in 3-dimensional complex space, so you can't draw them like this. Escudero's nonic is the record-holder for real ordinary double points.

Escud... more »

2016-11-04 16:53:42 (28 comments; 43 reshares; 412 +1s; )

**Thorny devil**

So cute! This small lizard, called the **thorny devil** or **Moloch horridus**, lives in the deserts and scrub lands of Australia.

It may look fierce, but it's not dangerous. It only eats ants. It's spiny so it doesn't get eaten. It can change color, for camouflage! And it has a "false head" on the back of its neck, which it shows to potential predators by dipping its real head. I'm not sure why.

It's also called a **thorny dragon**:

https://en.wikipedia.org/wiki/Thorny_dragon

I thank +rasha kamel for introducing me to this beast. She pointed out this article:

http://m.phys.org/news/2016-11-thorny-devil-skin-gravity.html

Scientists have recently figured out more about how this lizard gets water:

Researchers discovered long ago that because itsmouth ha... more »

2016-10-28 20:07:45 (8 comments; 11 reshares; 99 +1s; )

**Higher Structures**

There's a brand new journal called **Higher Structures**, run by top experts on n-categories and such stuff. It's free to read and free to publish in. That's called **diamond open access**. Pay-to-publish is strictly for suckers!

Check it out:

https://journals.mq.edu.au/index.php/higher_structures**Focus and Scope**

This journal publishes articles that make significant new contributions to mathematical science using higher structures, or that significantly advance our understanding of the foundational aspects of the theory of such structures. The scope of the journal includes: higher categories, operads and their generalisations, and applications of these to Algebra, Geometry, Topology, Combinatorics, Logic and Mathematical Physics.**Peer Review Process**

Articles appearingin... more »

2016-10-25 20:32:38 (76 comments; 62 reshares; 190 +1s; )

**Dark energy not real? Don't jump to conclusions!**

A new study from researchers at Oxford may indicate that the universe is expanding at a constant rate. Before, scientists thought it was expanding faster and faster. The most popular explanation was 'dark energy'. Does this mean we can forget about dark energy, and everything makes perfect sense without it?

Not so fast! Without dark energy pushing the galaxies apart, their gravity would tend to pull them together and make the expansion slow down. If they're expanding at a constant rate, there is still a mystery to solve.

If this is confusing, imagine throwing a ball up in the air. Suppose you see the ball shoot up faster and faster. That would be weird. It requires explanation!

But then suppose you discover that no, the ball moves upward at a constant rate. Doesthat mean the... more »

2016-10-24 17:59:12 (20 comments; 24 reshares; 83 +1s; )

**Category theory - an advanced online course!**

Hurrah! My student +Brendan Fong has teamed up with Alexander Campbell and +Emily Riehl to teach an advanced reading course on category theory to 8 students chosen from around the world. We'll all benefit, because these students will write essays on The n-Category Café, a popular math blog. And next summer, they'll give talks at the 2017 International Category Theory Conference at the University of British Columbia.

Emily Riehl has done this before, and it's worked well. So if you're a grad student interested in category theory, you should apply to take this course!

The course is about "functorial semantics" - a great idea going back to Lawvere. The students will read 8 papers on this topic. Here's the ad for the course, written by Emily Riehl.

------------

In ea... more »

2016-10-21 15:07:05 (38 comments; 12 reshares; 59 +1s; )

**Activity diagrams****Unified modeling language** or **UML** is a commonly used way to create models of complex systems. Among other things it lets you draw **activity diagrams** with boxes connected by wires, like this one here. To me this is clearly part of "applied category theory". It deserves to be recognized as such!

On Monday, my grad student +Blake Pollard and I went down to San Diego to visit the west coast headquarters of Metron. We're working with this company to develop new mathematics and apply it to Coast Guard search and rescue missions. They have a software environment that resembles UML. We got a tutorial about that, and now I'm going to think about it mathematically. It's an interesting new twist in my attempts to understand complex networks.

For the whole story, go here:

https://... more »

2016-10-19 16:25:09 (21 comments; 17 reshares; 184 +1s; )

**Mini Saturn****Chariklo** orbits the Sun between Saturn and Uranus. Just 250 kilometers across, it has two tiny rings!

Is it an asteroid? Not quite: it's a 'centaur'. In Greek mythology, a centaur was half-human, half-horse. In astronomy, a **centaur** is halfway between an asteroid and a comet. Centaurs live in the outer solar system between Jupiter and Neptune. They don't stay there long - at most a million years. They come from further out, pulled in by the gravity of Neptune, but their orbits are chaotic and they eventually move in toward Jupiter.

Over 300 centaurs have been seen, and scientists believe there are over 40,000 that are bigger than a kilometer across. But Chariklo is the biggest. And it has two rings!

A while ago I told you about a 'super Saturn' - an object in another solar system with ringsalmost a... more »

2016-10-18 15:31:45 (0 comments; 5 reshares; 39 +1s; )

**Trump's Russian connections**

This page created by the Financial Times is a good overview of Donald Trump's connections to Russia over the last 30 years:

https://ig.ft.com/sites/trumps-russian-connections/

2016-10-16 17:24:21 (30 comments; 8 reshares; 64 +1s; )

**Triamond**

The structure of a diamond crystal is fascinating. But there’s an equally fascinating form of carbon, called the **triamond**, that’s theoretically possible but never yet seen in nature.

In the triamond, each carbon atom is bonded to three others at 120° angles, with one double bond and two single bonds. Its bonds lie in a plane, so we get a plane for each atom.

But here’s the tricky part: for any two neighboring atoms, these planes are different. In fact, if we draw the bond planes for all the atoms in the triamond, they come in four kinds, parallel to the faces of a regular tetrahedron!

If we discount the difference between single and double bonds, the triamond is highly symmetrical. There’s a symmetry carrying any atom and any of its bonds to any other atom and any of its bonds. However, the triamond has an inherent handedness, orchirality... more »

2016-10-15 16:35:08 (28 comments; 45 reshares; 138 +1s; )

**Alien machinery**

That's what it looks like to me. But it's an image created by Greg Egan, the science fiction author. And there's a story behind it.

Egan and I figured out a bunch of stuff about the **McGee graph**, a highly symmetrical graph with 24 vertices and 36 edges. I wrote an article about it on Visual Insight, my blog for beautiful math pictures.

Later I got an email from Ed Pegg, Jr saying he'd worked out a **unit-distance embedding** of the McGee graph: a way of drawing it in the plane so that any two vertices connected by an edge are distance 1 apart. He wanted to know if this was **rigid** or **flexible**. In other words, he wanted to know whether you can change its shape slightly while it remains a unit-distance embedding.

Egan thought about it a lot and did a lot of computations and discovered thatthis un... more »

2016-10-14 01:28:44 (18 comments; 18 reshares; 180 +1s; )

**Super Saturn**

About 400 light years away, there's something with rings like Saturn — but much, much bigger!

It's called **J1407b**. It could be a huge planet. Or it could be a star so small that it never lit up: a **brown dwarf**.

One of Saturn's largest visible rings, the **F ring**, is about 140 thousand kilometers in radius. But J1407b's rings are almost a thousand times bigger. It has rings 90 million kilometers in radius!

That's 2/3 as big as the Earth's orbit around the Sun. That's insane! It's so huge that scientists don't know why the ring doesn’t get ripped apart by the gravity of the star it orbits.

One theory is that the rings are spinning in a **retrograde** way — in other words, backwards. If you have a planet moving clockwise around a star, andits rings are turning ... more »

2016-10-12 14:39:40 (13 comments; 16 reshares; 138 +1s; )

**Gas, Solid, Liquid, Darkness**

Here Canadian photographer David Burdeny captured an iceberg rising straight out of the ocean. It seems to divide the world into four parts.

He took this photo in 2007 in the **Weddell Sea**, one of the two big dents in Antarctica separated by the huge peninsula called **West Antarctica**. Scientists have found that the Weddell Sea has the clearest water of any sea. But it's a dangerous place, according to historian Thomas R. Henry's book White Continent:

The Weddell Sea is, according to the testimony of all who have sailed through its berg-filled waters, the most treacherous and dismal region on earth. The Ross Sea is relatively peaceful, predictable, and safe.

The **Ross Sea** is the other big dent in Antarctica - look at the map here:

https://en.wikipedia.org/wiki/Weddell_Sea

Da... more »

2016-10-11 18:03:22 (10 comments; 26 reshares; 127 +1s; )

**The golden ratio**

When I was in junior high, my uncle the physicist Albert Baez gave me a calculator. This was back in 1971, when electronic calculators were pretty rare. I immediately got turned on to experimental math, at a very basic level.

One thing I did was hit "1 + 1/(1 + 1/(1 + ..." and watch the numbers get closer and closer to the **golden ratio**:

Φ = (1 + sqrt(5))/2 = 1.6180339...

I tried other options, like this:

2 + 1/(2 + 1/(2 + ...

and this:

3 + 5/(3 + 5/(3 + ...

Eventually I figured out a nice formula for all expressions like these. I was very proud of it.**Puzzle 1:** what's the formula?

Only much later did I learn that people know how to find a formula for such infinite fractions, called **continued fractions**, whenever they repeat. Forexamp... more »

2016-10-10 17:09:45 (13 comments; 37 reshares; 145 +1s; )

**McGee graph**

On MathOverflow, someone named მამუკა ჯიბლაძე made this cool animation of the McGee graph, which has 24 dots and 36 edges.

This movie illustrates a **symmetry** of the McGee graph. In other words, if you let the picture make a quarter turn, it looks just the same, even though the dots have moved.

In fact, even if you let the graph make a full turn, the dots have moved from their original position! Why? Because the red edges have flipped upside down. So, you need to let the graph make 2 full turns before everything returns to its original position.

So, this movie illustrates 2 × 4 = 8 symmetries of the McGee graph. But the McGee graph actually has a total of 32 symmetries. These symmetries are precisely the transformations of the "affine line over Z/8". For details, try this:

http://blogs.ams.org/visualinsight... more »

2016-10-08 20:46:35 (25 comments; 39 reshares; 159 +1s; )

**Vortex versus antivortex**

No, I'm not trying to hypnotize you! These animations by Greg Egan show a **vortex** at left and an **antivortex** at right - two patterns that frequently occur in a 2-dimensional magnet if the spins are forced to lie in a plane. Kosterlitz and Thouless just won the Nobel prize for their work on such magnets.

The pictures are changing with time, with each little vector rotating at a constant rate - but that's just to show that there are many different possible vortex configurations, and also many different antivortex configurations.

For a better explanation, read my article:

https://johncarlosbaez.wordpress.com/2016/10/07/kosterlitz-thouless-transition/

I just wanted to show you these cool animations, which Egan added to the comments. Also check out +Simon Willerton's animations and SimonBurt... more »

2016-10-07 20:36:03 (18 comments; 37 reshares; 122 +1s; )

**Nobel Prize in Physics Goes to Another Weird Thing Nobody Understands**

That was the headline this week in Wired. Kosterlitz, Thouless and Haldane won the Nobel for their work on topological phase transitions. It's beautiful, truly fundamental work - but journalists were unable to explain it.

Indeed, most of them could barely pronounce 'topological'. In case you're wondering, the stress is on the third syllable.

So what's a 'topological phase transition'? It's not so complicated. Check out my blog article, which is graced by wonderful illustrations created by Brian Skinner, a physics postdoc at MIT.

And for more details, read his blog:

http://www.ribbonfarm.com/2015/09/24/samuel-becketts-guide-to-particles-and-antiparticles/

He doesn't explain topological phase transitions, but hedescribes... more »

2016-10-07 00:39:12 (31 comments; 14 reshares; 81 +1s; )

**Diamonds are forever?**

This shows the pattern of carbon atoms in a diamond. Each atom is connected to 4 neighbors. Its neighbors are the corners of a regular tetrahedron!

The mathematics of this pattern is beautiful, and I explain it here:

http://blogs.ams.org/visualinsight/2016/10/01/diamond-cubic/

I also explain **hyperdiamonds** in 4 or more dimensions. The hyperdiamond in 8 dimensions is especially awesome: it's called the **E8 lattice**, and it's connected to string theory, the octonions and more.

In 1888, Cecil Rhodes started a company called De Beers to sell the diamonds dug up by slaves in Botswana, Namibia, and South Africa. De Beers got a total monopoly on diamonds. To keep the price up, they wanted a slogan to make diamonds into the jewel of choice for weddings:

After unsuccessfuly trying tocreate a... more »

2016-10-05 16:23:05 (31 comments; 8 reshares; 94 +1s; )

**Systems of systems**

In January of this year, I was contacted by a company called Metron Scientific Solutions. They asked if I’d like to join them in a project to use category theory to design and evaluate complex, adaptive systems of systems.

What’s a **system of systems**?

It’s a system made of many disparate parts, each of which is a complex system in its own right. The biosphere is a system of systems. But so far, people usually use this buzzword for large human-engineered systems where the different components are made by different organizations, perhaps over a long period of time, with changing and/or incompatible standards. This makes it impossible to fine-tune everything in a top-down way and have everything fit together seamlessly.

So, systems of systems are inherently messy. And yet we need them.

Metron was applying for a grantfrom ... more »

2016-10-04 17:53:10 (15 comments; 13 reshares; 61 +1s; )

**Mathematical Enchantments**

If you like math, and you haven't tried my friend's +James Propp's blog, give a try! He's an expert on combinatorics who enjoys explaining things. The article here is about Ramanujan and his mysterious formulas.

Propp also looking for someone who is good at making animated gifs of mathematics! If that person could be you, leave a comment here, or email me.

Here's an example of what he wants:

------------------------

In the Spring I'll be posting a Mathematical Enchantments essay on signed area, and thought I'd ask now for help with supporting animations.

One thing I'd like to include is a GIF loop whose first half animates

ac+ad+bd+bc=(a+b)(c+d)

with a small rectangle becoming big and whose second half animates

ac-ad... more »

2016-10-02 00:21:59 (20 comments; 9 reshares; 96 +1s; )

**The icosahedron that got away**

Iron pyrite can form crystals shaped like icosahedra. They aren't regular solids, with equilateral triangles as faces - that would violate the laws of math! Iron pyrite is a cubical crystal, and you can't make a regular icosahedron using little cubes.

These crystals are called **pseudoicosahedra**. They take advantage of how the golden ratio can be approximated using Fibonacci numbers:

1/1 = 1

3/2 = 1.5

5/3 = 1.6666...

8/5 = 1.6125

and so on, getting closer to

Φ = 1.6180339....

They call iron pyrite **fool's gold** - and it can fool you into thinking its proportions attain the golden ratio.

Recently the curator of the Museum of Evolution, Palaeontology and Mineralogy in Uppsala, Sweden, emailed me and told me that the handsome pseudoicosahedron shown herewas... more »

2016-09-30 19:52:28 (57 comments; 14 reshares; 97 +1s; )

**An infinite corridor of universes**

Einstein's equations for gravity have some amazing solutions. Some describe things we see: the Big Bang and black holes. Others don't - like white holes, wormholes, and the infinite corridor of universes shown here.

As far as we know, all real-world black holes were formed at some moment in time by collapsing matter. But it's easier to find solutions of Einstein's equations that describe an eternal black hole whose shape doesn't change with time.

A rotating eternal black hole is called a **Kerr black hole**, because this solution of Einstein's equation was first found by Roy Kerr in 1963. However, he just found part of the solution - not the whole picture here!

You see, when you solve Einstein's equations, you get a world obeying the rules of general relativity. Butsometimes, ... more »

2016-09-24 16:14:25 (30 comments; 28 reshares; 211 +1s; )

**Solar wind**

This is the **solar wind**, the stream of particles coming from the Sun. It was photographed by **STEREO**. That's the **Solar Terrestrial Relations Observatory**, a pair of satellites we put into orbit around the Sun at the same distance as the Earth, back in 2006. One is ahead of the Earth, one is behind. Together, they can make stereo movies of the Sun!

One interesting thing is that there's no sharp boundary between the 'outer atmosphere' of the Sun, called the **corona**, and the solar wind. It's all just hot gas, after all! STEREO has been studying how this gas leaves the corona and forms the solar wind. This picture is a computer-enhanced movie of that process, taken near the Sun's edge.

What's the solar wind made of? When you take hydrogen and helium and heat them up so much that theelectrons... more »

2016-09-23 19:07:29 (26 comments; 11 reshares; 73 +1s; )

**Life on the Infinite Farm**

This is a great book about infinity - for kids. For example, there's a cow named Gracie with infinitely many legs. She likes new shoes, but she wants to keep wearing all her old shoes. What does she do?

Life on the Infinite Farm is by Richard Evan Schwartz, and it's free here:

https://www.math.brown.edu/~res/farm.pdf

Later it will be published on paper by the American Mathematical Society. I really like turning the pages when I'm reading a book to a child. Is that old-fashioned? What do modern parents think?

Gracie's tale is just a retelling of the first **Hilbert Hotel** story. There's a hotel with infinitely many rooms. Unfortunately they're all full. A guest walks in. What do you do?

You move the guest in room 1 to room 2, the guest in room 2 to room 3, andso on. Now... more »

2016-09-22 06:29:24 (18 comments; 24 reshares; 120 +1s; )

**Poncelet's Porism**

If you can fit a triangle snugly between two circles, you can always slide the triangle around. The triangle may have to change shape, but it stays snug! All 3 corners keep touching the outside circle, and all 3 sides keep touching the inside circle.

That's really cool. But even better, it also works for polygons with more than 3 sides!

This amazing fact is called **Poncelet's Porism**.

A **porism** is like a theorem, but much cooler. **Poncelet** was a French engineer and mathematician who wrote a famous book on 'projective geometry' in 1822.

What's a porism, really?

Well, Euclid is famous for his Elements, but he also wrote a more advanced book called Porisms. Unfortunately that book is lost. I hear that someone checked it out from the library of Alexandria andnever ret... more »

2016-09-21 15:17:59 (40 comments; 12 reshares; 60 +1s; )

**Black hole versus white hole**

Last time I showed you a Schwarzschild black hole... but not the whole hole.

Besides the **horizon**, which is the imaginary surface that light can only go in, that picture had a mysterious "antihorizon", where light can only come out. When you look at this black hole, what you actually see is the antihorizon. The simplest thing is to assume no light is coming out of the antihorizon. Then the black hole will look black.

But I didn't say what was behind the antihorizon!

In a real-world black hole there's no antihorizon, so all this is just for fun. And even in the Schwarzschild black hole, you can never actually cross the antihorizon - unless you can go faster than light. So there's no real need to say what's behind the antihorizon. And we can just decree that no light comesout of it.

2016-09-20 18:21:39 (90 comments; 26 reshares; 76 +1s; )

**Understanding black holes**

This is a diagram of a **Schwarzschild black hole** - a non-rotating, uncharged black hole that has been around forever.

Real-world black holes are different. They aren't eternal - they were formed by collapsing matter. They're also rotating. But the Schwarzschild black hole is simple: you can write down a formula for it. So this is the one to start with, when you're studying black holes.

This is a **Penrose diagram**. It shows time as going up, and just one dimension of space going across. The key to Penrose diagrams is that light moves along diagonal lines. In these diagrams the speed of light is 1. So it moves one inch across for each inch it moves up - that is, forwards in time.

The whole universe outside the black hole is squashed to a diamond. The **singularity** is the wiggly line attop. T... more »

2016-09-18 18:48:33 (61 comments; 19 reshares; 87 +1s; )

**The mystical hexagram theorem**

The picture explains this amazing result, which was discovered by Pascal in 1639, when he was only sixteen.

Take six points on an ellipse, called A,B,C,D,E,F. Connect each point to the next by a line.

The red lines intersect in a point G.

The yellow lines intersect in a point H.

The blue lines intersect in a point K.

And then the cool part:

The points G, H and K lie on a line!

I'm teaching a course on 'algebraic groups' starting on Thursday, so I need to review a bit of the history of projective geometry. This result of Pascal, called the **Hexagrammum Mysticum Theorem**, was the first exciting theorem about projective geometry after the old work of Pappus. So I'll mention it in my course! But I don't really understand why it's true. Do you know a niceexplana... more »

2016-09-17 02:15:36 (92 comments; 22 reshares; 110 +1s; )

**Exploring black holes - with cats!**

There should be a series of videos exploring black holes with cats.

So far all we have is this gif made by +Dragana Biocanin. A cat can orbit just above the photon sphere of a non-rotating black hole, moving at almost the speed of light. It's impossible for a cat to orbit below the photon sphere. As long as it's outside the event horizon it can accelerate upwards and escape the black hole's gravitational pull. But if it crosses the event horizon, it's doomed!

The **event horizon** is an imaginary surface in spacetime that's defined by this property: once a cat crosses this surface, it can't come back without going faster than light! This property involves events in the future, so there's no guaranteed way for the cat to tell when it's crossing an event horizon.

Forexample, if... more »

2016-09-16 01:13:54 (36 comments; 21 reshares; 90 +1s; )

**Light moves around a rotating black hole**

This gif by +Leo Stein shows a photon orbiting a black hole. Since the black hole is rotating, the photon traces out a complicated path. You can play around with the options here:

https://duetosymmetry.com/tool/kerr-circular-photon-orbits/

If a black hole is not rotating, light can only orbit it on circles that lie on a special sphere: the **photon sphere**.

But if the black hole is rotating, photon orbits are more complicated! They always lie on some sphere or other — but now there's a range of spheres of different radii on which photons can move!

The cool part is how a rotating massive object — a black hole, the Sun or even the Earth — warps spacetime in a way that tends to drag objects along with its rotation. This is called **frame-dragging**.

Frame-dragging... more »

2016-09-15 03:41:43 (15 comments; 19 reshares; 167 +1s; )

**Just because someone's on crutches doesn't mean they're handicapped**

Nomads kick ass. James Dator explains:

The World Nomad Games concluded on Friday in what can only be described as the greatest week-long sporting event on the planet. The games, intended to showcase ethnic sports of Central Asia, featured things you have never heard of, athletes you’ll never learn about and sports that sound absolutely terrifying.

There were 16 sports with medals up for grabs. These are the ones that are the absolute wildest.**Cirit**

This Turkish equestrian sport involves teams of riders chasing each other and throwing javelins at each other while on horseback. Yes, seriously.**Er Enish**

It’s wrestling, except you’re on a horse. You win by pulling your opponent off their horse.**Kok-boru**

more »

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