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## John Baez

Occupation: I'm a mathematical physicist. (Centre for Quantum Technologies)

Location: Riverside, California

Followers: 57,290

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Cream of the Crop: 11/05/2011

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### Most comments: 140

2016-02-11 16:01:44 (140 comments; 75 reshares; 283 +1s)

**Gravitational waves**

The rumors are true: LIGO has seen gravitational waves! Based on the details of the signal detected, the LIGO team estimates that 1.3 billion years ago. two black holes spiralled into each other and collided. One was 29 times the mass of the Sun, the other 36 times. When they merged, 3 times the mass of the Sun was converted directly to energy and released as gravitational waves.

For a very short time, this event produced over 10 times more power than all the stars in the Universe!

We knew these things happened. We just weren't good enough at detecting gravitational waves to see them - until now.

I'll open comments on this breaking news item so we can all learn more. LIGO now has a page on this event, which is called **GW150914** because it was seen on September 14th, 2015:

... more »

### Most reshares: 94

2016-03-24 19:27:37 (42 comments; 94 reshares; 297 +1s)

**Famous math problem solved!**

Ten days ago, Maryna Viazovska showed how to pack spheres in 8 dimensions as tightly as possible. In this arrangement the spheres occupy about 25.367% of the space. That looks like a strange number - but it's actually a wonderful number, as shown here.

People had guessed the answer to this problem for a long time. If you try to get as many equal-sized spheres to touch a sphere in 8 dimensions, there's exactly one way to do it - unlike in 3 dimensions, where there's a lot of wiggle room! And if you keep doing this, on and on, you're forced into a unique arrangement, called the **E8 lattice**. So this pattern is an obvious candidate for the densest sphere packing in 8 dimensions. But none of this proves it's the best!

In 2001, Henry Cohn and Noam Elkies showed that no sphere packing in 8 dimensions could be more... more »

### Most plusones: 297

2016-03-24 19:27:37 (42 comments; 94 reshares; 297 +1s)

**Famous math problem solved!**

Ten days ago, Maryna Viazovska showed how to pack spheres in 8 dimensions as tightly as possible. In this arrangement the spheres occupy about 25.367% of the space. That looks like a strange number - but it's actually a wonderful number, as shown here.

People had guessed the answer to this problem for a long time. If you try to get as many equal-sized spheres to touch a sphere in 8 dimensions, there's exactly one way to do it - unlike in 3 dimensions, where there's a lot of wiggle room! And if you keep doing this, on and on, you're forced into a unique arrangement, called the **E8 lattice**. So this pattern is an obvious candidate for the densest sphere packing in 8 dimensions. But none of this proves it's the best!

In 2001, Henry Cohn and Noam Elkies showed that no sphere packing in 8 dimensions could be more... more »

Latest 50 posts

2016-04-30 16:23:22 (24 comments; 4 reshares; 107 +1s)

**The Tagish Lake meteorite**

On January 18, 2000, at 8:43 in the morning, a meteor hit the Earth's atmosphere over Canada and exploded with the energy of a 1.7 kiloton bomb. Luckily this happened over a sparsely populated part of British Columbia.

It was over 50 tons in mass when it hit the air, but 97% of it vaporized. Just about a ton reached the Earth. It landed on **Tagish Lake**, which was frozen at the time. Local inhabitants said the air smelled like sulfur.

Only about 10 kilograms was found and collected. Except for a gray crust, the pieces look like charcoal briquettes.

And here is where things get interesting.

Analysis of the Tagish Lake fragments show they're very primitive. They contain dust granules that may be from the original cloud of material that created our Solar System and Sun! They also have alot of of ... more »

2016-04-28 17:57:51 (59 comments; 33 reshares; 156 +1s)

**Bad physics**

You may have heard of the "EmDrive", a gadget that supposedly provides thrust by bouncing microwaves around in a metal can. It's sort of like trying to power a spaceship by having the crew play ping-pong.

Now there's a new "theoretical explanation" of this quite possibly nonexistent effect. It appeared on the arXiv in an unpublished paper by someone named Michael E. McCulloch. It's completely flaky, and normally I'd ignore it, but for some reason the normally respectable mag Technology Review decided to mention it. So people are starting to talk about it, not realizing how goofy it actually is!

McCulloch talks a lot about the **Unruh effect**, so you should learn a bit about that. It's never been detected, but most physicists believe in it, because it's a consequence of special relativity andquant... more »

2016-04-25 17:04:08 (21 comments; 2 reshares; 54 +1s)

**Flying over Antarctica**

There are lots of flights that go near the North Pole. When you fly from California to Europe, for example, that's an efficient route! Are there flights that go near the South Pole? If not, why not?

A friend of mine asked this question, and I promised I'd try to get an answer. When she flew from Argentina to New Zealand she took a very long route. Why, she wondered, don't airplanes take a southerly route? Is the weather too bad?

My guess is that maybe there's not enough demand to fly from South America to New Zealand for there to be direct flights. Or from South America to South Africa, or Madagascar.

But I haven't even checked! Maybe there are such flights!

Does anyone here know about this?

(Yes, I could look it up on Google. I thought a conversation would be morefun. If you... more »

2016-04-23 15:08:55 (19 comments; 15 reshares; 102 +1s)

**Points at infinity**

Math tells us three of the saddest love stories:

1) of parallel lines, who will never meet.

2) of tangent lines, who were together once, then parted forever.

3) and of asymptotes, who come closer and closer, but can never truly be together.

But mathematicians invented projective geometry to provide a happy ending to the first story. In this kind of geometry, parallel lines do meet - not in ordinary space, but at new points, called "points at infinity".

The **Barth sextic** is an amazing surface with 65 points that look like the place where two cones meet - the most possible for a surface described using polynomials of degree 6. But in the usual picture of this surface, which emphasizes its symmetry, 15 of these points lie at infinity.

In this picture by +Abdelaziz Nait Merzouk, the Barthsexti... more »

2016-04-21 17:22:18 (47 comments; 76 reshares; 140 +1s)

**"And then we wept."**

The chatter of gossip distracts us from the really big story: the **Anthropocene**, the new geological era we are bringing about. Pay attention for a minute. Most of the Great Barrier Reef, the world's largest coral reef system, now looks like a ghostly graveyard.

Most corals are colonies of tiny genetically identical animals called **polyps**. Over centuries, their skeletons build up reefs, which are havens for many kinds of sea life. Some polyps catch their own food using stingers. But most get their food by symbiosis! They cooperate with algae called **zooxanthellae**. These algae get energy from the sun's light. They actually live inside the polyps, and provide them with food. Most of the color of a coral reef comes from these zooxanthellae.

When a polyp is stressed, the zooxanthellae livinginside it m... more »

2016-04-16 15:26:20 (10 comments; 4 reshares; 61 +1s)

**Barth sextic**

Some mathematical objects look almost scary, like alien artifacts. The **Barth sextic**, drawn here by +Craig Kaplan, is one.

In school you learned to solve quadratic equations. Then come cubics, then quartics, then quintics. Then come sextics, which are more sexy, and then come septics, which are downright stinky.

A **sextic surface** is a surface defined by a polynomial equation of degree 6. The Barth sextic is the one with the biggest possible number of **ordinary double points**, meaning points where it looks like a cone. It has 65 of them!

Even better, it has the symmetries of a dodecahedron! 20 of the double points lie at the vertices of a regular dodecahedron, and 30 lie at the midpoints of the edges of another regular dodecahedron.**Puzzle:** where are the rest? I honestly don't know.

For ... more »

2016-04-14 20:21:18 (66 comments; 16 reshares; 80 +1s)

**The inaccessible infinite**

In math there are infinite numbers called **cardinals**, which say how big sets are. Some are small. Some are big. Some are infinite. Some are so infinitely big that they're **inaccessible** - very roughly, you can't reach them using operations you can define in terms of smaller cardinals.

An inaccessible cardinal is so big that if it exists, we can't prove that using the standard axioms of set theory!

The reason why is pretty interesting. Assume there's an inaccessible cardinal K. If we restrict attention to sets that we can build up using fewer than K operations, we get a whole lot of sets. Indeed, we get a set of sets that does not contain every set, but which is big enough that it's "just as good" for all practical purposes.

We call such a set a **Grothendieckuniverse... more »**

**
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2016-04-12 15:15:31 (8 comments; 14 reshares; 85 +1s)

**The crystal that nature forgot: the triamond**

Carbon can form diamonds, and the geometry of the diamond crystal is stunningly beautiful. But there's another crystal, called the **triamond**, that is just as beautiful. It was discovered by mathematicians, but it doesn't seem to exist in nature.

In a triamond, each carbon atom would be 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. And if we draw these bond planes for all the atoms in the triamond, they come in four kinds, parallel to the faces of a regular tetrahedron!

The triamond is extremely symmetrical. But it comes in left- and right-handed forms, unlike a diamond.

In a diamond, the smallestrings ... more »

2016-04-04 15:51:03 (42 comments; 30 reshares; 119 +1s)

**Computing the uncomputable**

Last month the logician +Joel David Hamkins proved a surprising result: you can compute uncomputable functions!

Of course there's a catch, but it's still interesting.

Alan Turing showed that a simple kind of computer, now called a **Turing machine**, can calculate a lot of functions. In fact we believe Turing machines can calculate anything you can calculate with any fancier sort of computer. So we say a function is **computable** if you can calculate it with some Turing machine.

Some functions are computable, others aren't. That's a fundamental fact.

But there's a loophole.

We think we know what the natural numbers are:

0, 1, 2, 3, ...

and how to add and multiply them. We know a bunch of axioms that describe this sort of arithmetic: the **Peanoaxiom... more »**

**
**2016-04-03 22:05:26 (0 comments; 36 reshares; 131 +1s)

**The right to bear arms**

As you know, a lot of conservatives in the US support the right to bear arms. It's in the Bill of Rights, after all:**"A well regulated militia being necessary to the security of a free state, the right of the people to keep and bear arms shall not be infringed."**

The idea is basically that if enough of us good guys are armed, criminals and the government won't dare mess with us.

In this they are in complete agreement with the Black Panthers, a revolutionary black separatist organization founded in the 1960s by Huey P. Newton. Later it became less active, but in 1989 the New Black Panther Party was formed in South Dallas, a predominantly black part of Dallas, Texas. They helped set up the Huey P. Long Gun Club, "uniting five local black and brown paramilitary organizations under a singleban... more »

2016-04-01 02:21:14 (36 comments; 14 reshares; 70 +1s)

**A new polyhedron**

The **rectified truncated icosahedron** is a surprising new polyhedron discovered by +Craig Kaplan. It has 60 equilateral triangles, 12 regular pentagons and 20 regular hexagons as faces.

It came as a shock because it's a brand-new **Johnson solid** - a convex polyhedron whose faces are all regular polygons.

Johnson solids are named after Norman Johnson, who in 1966 published a list of 92 such solids. He conjectured that this list was complete, but did not prove it.

In 1969, Victor Zalgaller proved that Johnson’s list was complete, using the fact that there are only 92 elements in the periodic table.

It thus came as a huge shock to the mathematical community when Craig Kaplan, a computer scientist at the University of Waterloo, discovered an additional Johnson solid!

At the time, he was compiling acoll... more »

2016-03-30 02:53:22 (0 comments; 5 reshares; 52 +1s)

**Republican delegate battles**

The Republican primaries continue to shock and appall. But some of you, especially outside the US, may not have noticed the politics going on behind the scenes.

Trump defeated Ted Cruz in Louisiana's March 5 primary - he got 3.6% more votes - but Cruz may receive up to 10 more delegates from that state than Trump! The reason is that there are 5 'unbound' delegates who can do what they want, who are expected to back Cruz. And there are 5 more won by Rubio, who has since dropped out, and these too may back Cruz.

Trump is threatening to sue, but it's not clear he has any grounds:

http://www.npr.org/2016/03/29/472253183/trump-threatens-lawsuit-over-louisiana-delegates

In South Carolina, Trump won all the delegates. They're pledged to vote for Trump at the first ballot in the Republican... more »

2016-03-28 13:50:16 (7 comments; 15 reshares; 59 +1s)

**Probability theory and the golden ratio**

Traditional Tom and Liberal Lisa are dating:**Tom:** I plan to keep having kids until I get two sons in a row.**Lisa:** What?! That’s absurd. Why?**Tom:** I want two to run my store when I get old.**Lisa:** Even ignoring your insulting assumption that only boys can manage your shop, why in the world do you need two in a row?**Tom:** From my own childhood, I’ve learned there’s a special bond between sons who are next to each other in age. They play together, they grow up together… they can run my shop together.**Lisa:** Hmm. Well, then maybe I should have children until I have a girl followed directly by a boy!**Tom:** What?!**Lisa:** Well, I’ve observed that something special happens when a boy has an older sister, with nointerveni... more »

2016-03-27 21:58:17 (4 comments; 14 reshares; 88 +1s)

**Higher-dimensional commutative laws**

This is +Scott Carter's picture of the two sides of the **Zamolodchikov tetrahedron equation**, a tongue-twisting and brain-bending equation that shows up in topology.

My blog article explains it, with pictures. But in simple terms, the idea is this. When you think of the commutative law

xy = yx

as a process rather than an equation, it's the process of switching two things: in this case, the letters x and y. You can draw this process using two strings that switch places: that is, a very simple "braid" like this:

\ /

/

/ \

It turns out that this braid obeys an equation of its own, the **Yang-Baxter equation**. This is easy to explain with pictures, but it's hard to draw pictures here, so visit my blog article.

If you then think of the... more »

2016-03-25 16:10:49 (41 comments; 6 reshares; 67 +1s)

**Carbon emissions flat**

About a year ago, the International Energy Agency announced some important news. Although the global GDP grew by 3.4% in 2014, greenhouse gas emissions due to energy use did not increase! We spewed 32.3 gigatonnes of carbon dioxide into the atmosphere by burning stuff to produce energy—just as we had in 2013.

Of course, leveling off is not good enough. Since carbon dioxide stays in the atmosphere essentially ‘forever’, we need to essentially quit burning stuff. You can’t stop a clogged sink from overflowing by levelling off the rate at which you pour in water. You have to turn off the faucet!

But still, it’s a promising start.

And now the International Energy Agency is saying the same thing about 2015. While the global GDP grew 3.1% in 2015, we spewed just 32.1 billion gigatonnes of CO2 into the air by burning stuff to makeenergy. S... more »

2016-03-24 19:27:37 (42 comments; 94 reshares; 297 +1s)

**Famous math problem solved!**

Ten days ago, Maryna Viazovska showed how to pack spheres in 8 dimensions as tightly as possible. In this arrangement the spheres occupy about 25.367% of the space. That looks like a strange number - but it's actually a wonderful number, as shown here.

People had guessed the answer to this problem for a long time. If you try to get as many equal-sized spheres to touch a sphere in 8 dimensions, there's exactly one way to do it - unlike in 3 dimensions, where there's a lot of wiggle room! And if you keep doing this, on and on, you're forced into a unique arrangement, called the **E8 lattice**. So this pattern is an obvious candidate for the densest sphere packing in 8 dimensions. But none of this proves it's the best!

In 2001, Henry Cohn and Noam Elkies showed that no sphere packing in 8 dimensions could be more... more »

2016-03-23 16:06:09 (22 comments; 7 reshares; 53 +1s)

**A mathematical mystery**

I'm asking for your help in solving a mathermatical mystery.

The blue curve is a **catenary** - you get a catenary when you hang a chain. The red curve is called a **tractrix**.

The tractrix is the "involute" of the catenary. In other words, you get it by attaching one end of a taut string to the catenary and tracing out the path of the string’s free end as you wind the string onto the catenary.

That's a long sentence, but the picture explains it.

There are a couple of confusing things about this picture if you’re just starting to learn about involutes. First, Sam Derbyshire, who made this picture, cleverly moved the end of the string attached to the catenary at the instant the other end hit the catenary! That allowed him to continue the involute past the moment it hits the catenary.

<... more »

2016-03-22 00:33:07 (14 comments; 28 reshares; 116 +1s)

**SUPERNOVA GOES BANG**

Here you can see the brilliant flash of a supernova as its core blasts through its surface. This is a cartoon made by NASA based on observations of a red supergiant star that exploded in 2011. It has been sped up by a factor of 240.

When this star ran out of fuel for nuclear fusion, its core cooled down a bit. That made the pressure go down - so the core collapsed under the force of gravity.

It stopped only when it became a huge ball of neutrons: a neutron star. Most of the energy was carried out instantly, in the form of invisible neutrinos.

Only later could we start to see things happen. A shock wave rushed upward through the star! First it broke through the star’s surface in the form of finger-like plasma jets. 20 minutes later, the full fury of the shock wave reached the surface - and the doomed star exploded in a flash ofl... more »

2016-03-20 22:04:57 (24 comments; 16 reshares; 115 +1s)

**Music of the primes**

I often hear there's no formula for prime numbers. But Riemann came up with something just as good: a formula for the **prime counting function**.

This function, called π(x), counts how many prime numbers there are less than x, where x is any number you want. It keeps climbing like a staircase, and it has a step at each prime. You can see it in this gif.

Riemann's formula is complicated, but it lets us compute the prime counting function using a sum of wiggly functions. These wiggly functions vibrate at different frequencies. Poetically, you could say they reveal the secret music of the primes.

The frequencies of these wiggly functions depend on where the Riemann zeta function equals zero.

So, Riemann's formula turns the problem of counting primes less than some number into another problem: finding... more »

2016-03-19 17:10:15 (38 comments; 35 reshares; 154 +1s)

**Scared of big numbers? Don't read this!**

People love **twin primes** - primes separated by two, like 11 and 13. Nobody knows if there are infinitely many. There probably are. There are certainly lots.

But a while back, a computer search showed that among numbers less than a trillion, most common distance between successive primes is 6.

It seems that this trend goes on for quite a while longer…

... but in 1999, three mathematicians discovered that at some point, the number 6 ceases to be the most common gap between successive primes!

When does this change happen? It seems to happen around here:

17,427,000,000,000,000,000,000,000,000,000

At about this point, the most common gap between consecutive primes switches from 6 to 30. They didn't prove this, but their argument has convinced the experts, and theyc... more »

2016-03-17 19:56:30 (16 comments; 3 reshares; 37 +1s)

**Solve for 💜**

i is the square root of minus one. It takes a bit of work to wrap ones head around i to the ith power.

Here's how you figure it out. i is e to the power of iπ/2, since multiplying by i implements a quarter turn rotation, that is, a rotation by π/2. So,

i^i = (e^ iπ/2)^i = e^(i · iπ/2) = e^(-π/2) = 0.20787957...

Sorta strange. But now: **Puzzle:** can you solve this equation for 💜?

i^i = 💜^(sqrt(-💜/2))

But my real puzzle is about the album cover this equation appears on!

Is there any way to get a good electronic copy of this album for less than $50? There's one CD of it for sale on Amazon for $50.

It's called This Crazy Paradise and it's by Pyewackett. It's a cool album! It was made in 1986. It's an unusual blend of thecutting-edge elec... more »

2016-03-16 16:27:54 (17 comments; 7 reshares; 66 +1s)

**From crackpots to climate change**

Here's part two of my interview on Physics Forums. I talk about the early days of the internet, before the world-web caught on. First we started discussing physics on "usenet newsgroups" like sci.physics - but then a flood of crackpots invaded those newgroups! Alexander Abian claimed all the world’s ills would be cured if we blew up the Moon. Archimedes Plutonium claimed the Universe is a giant plutonium atom.

That's what led me to create the Crackpot Index. But spending lots of time on newsgroups was still worthwhile, and it led me to start writing "This Week's Finds", which has been called the world's first blog, in 1993.

I also talk about my physics and math heroes, what discoveries I'm most looking forward to, and why I switched to thinking about environmental problems.

more »

2016-03-15 17:19:07 (11 comments; 19 reshares; 90 +1s)

**Me**

Here's the first part of an interview. I used it as an excuse to say what I've been doing all these years. I also talk about my uncle Albert Baez, who got me interested in physics in the first place - and what I'm working on right now.

I hope it's interesting even if you care more about math and physics. There's a lot here about quantum gravity, category theory and some of my hobbies, like the octonions. But I hope it's pretty easy to read! If you have questions, fire away.

For example: what are the octonions?

They're a number system where you can add, subtract, multiply and divide. Such number systems only exist in 1, 2, 4, and 8 dimensions: you’ve got the real numbers, which form a line, the complex numbers, which form a plane, the quaternions, which are 4-dimensional, and the octonions, which are8... more »

2016-03-15 01:42:55 (16 comments; 25 reshares; 122 +1s)

**Whoa! The primes are acting weird!**

What percent of primes end in a 7? I mean when you write them out in base ten.

Well, if you look at the first hundred million primes, the answer is 25.000401%. That looks suspiciously close to 1/4. And that makes sense, because there are just 4 digits that a prime can end in, unless it's really small: 1, 3, 7 and 9.

So, you might think the endings of prime numbers are random, or very close to it. But 3 days ago two mathematicians shocked the world with a paper that asked some other questions, like this:**If you have a prime that ends in a 7, what's the probability that the ****next**** prime ends in a 7?**

I would still expect the answer to be close to 25%. But these mathematicians, Robert Oliver and Kannan Soundarajan, actually looked!

And they found that among the first... more »

2016-03-13 16:20:37 (80 comments; 47 reshares; 225 +1s)

**Revenge of the humans**

After losing the first three, Lee Sedol won his 4th game against the program AlphaGo!

Lee was playing white, which for go means taking the second move. So, he was on the defensive at first, unlike the previous game, where he played black.

After the first two hours of play, commenter Michael Redmond called the contest "a very dangerous fight.” Lee Sedol likes aggressive play, and he seemed to be in a better position than last time.

But after another 20 minutes, Redmond felt that AlphaGo had the edge. Even worse, Lee Sedol had been taking a long time on his moves, so had only about 25 minutes left on his play clock, nearly an hour less than AlphaGo. Once your clock runs out, you need to make each move in less than a minute!

At this point, AlphaGo started to play less aggressively. Maybe it thought it wasb... more »

2016-03-12 18:27:22 (48 comments; 35 reshares; 113 +1s)

**Computer beats human at go**

As you probably know, the computer program AlphaGo is consistently beating the excellent Korean player Lee Sedol.

But what would it feel like to watch one of these games, if you're good at go? David Ormerod explains:**It was the first time we’d seen AlphaGo forced to manage a weak group within its opponent’s sphere of influence. Perhaps this would prove to be a weakness?****This, however, was where things began to get scary.****Usually developing a large sphere of influence and enticing your opponent to invade it is a good strategy, because it creates a situation where you have numerical advantage and can attack severely.****In military texts, this is sometimes referred to as ‘force ratio’.****The intention in Go though is not to kill, but to consolidate territoryand gai... more »**

**
**2016-03-11 17:51:40 (53 comments; 33 reshares; 113 +1s)

**A course on category theory**

I just finished teaching a course on category theory, and you can get notes here:

http://math.ucr.edu/home/baez/qg-winter2016/

My goal was to explain the basics leading up to a little taste of topos theory. Next quarter I'll start explaining n-categories.

Here's how the course went:

Week 1 (Jan. 5 and 7) - The definition of a category. Some familiar categories. Various kinds of categories, including monoids, groupoids, groups, preorders, equivalence relations and posets. The definition of a functor. Doing mathematics inside a category: isomorphisms, monomorphisms and epimorphisms.

Week 2 (Jan. 12 and 14) - Doing mathematics inside a category: an isomorphism is a monomorphism and epimorphism, but not necessarily conversely. Products. Any object isomorphic to a product can also be a product.... more »

2016-03-09 18:11:12 (2 comments; 0 reshares; 58 +1s)

**George Martin, 1926-2016**

The Beatles' psychedelic music blew me away, but only later did I learn how much was due to the "fifth Beatle": George Martin. His idea of using the studio as an instrument was revolutionary and wonderful.

From the New York Times article by Allan Kozinn:**Always intent on expanding the Beatles’ horizons, Mr. Martin began chipping away at the group’s resistance to using orchestral musicians on its recordings in early 1965. While recording the “Help!” album that year, he brought in flutists for the simple adornment that enlivens Lennon’s “You’ve Got to Hide Your Love Away,” and he convinced Mr. McCartney, against his initial resistance, that “Yesterday” should be accompanied by a string quartet.****A year later, during the recording of the album “Revolver,” Mr. Martin no longer had tocajole: The Beatles pre... more »**

**
**2016-03-06 17:58:24 (13 comments; 3 reshares; 47 +1s)

**The capricornoid**

The capricornoid, shown here, got its name from the zodiac symbol for Capricorn. I don't get that. But it's a cool curve! It crosses itself in two different ways, giving a a **tacnode** at the bottom and a **crunode** at top.

To understand those weird words, check out my blog article! These days, a crunode is usually called an **ordinary double point**.**Puzzle:** Can you guess the equation that describes the capricornoid?

I knew it in Cartesian coordinates, but +Kram Einsnulldreizwei found a very simple formula in polar coordinates, based on a more complicated one by +jesse mckeown. If you give up on the puzzle, you can find these answers in the blog comments.

2016-03-03 21:18:16 (8 comments; 5 reshares; 55 +1s)

**It's hot**

In the last couple of months, global temperatures have spiked to a record-breaking high! This is true not only for the surface temperatures measured by the Goddard Institute of Space Studies - shown here - but also for the temperatures further up, measured by satellite and collected by the University of Alabama at Huntsville.

These temperature records disagree in some fascinating and controversial ways, but they're both shooting up to new record highs. See my blog post for more graphs!

2016-03-02 16:46:09 (0 comments; 30 reshares; 114 +1s)

**Clebsch surface**

This is a smooth cubic surface, drawn by Greg Egan. It's most symmetrically described in five dimensions, using the equations

v + w + x + y + z = 0

v³ + w³ + x³ + y³ + z³ = 0

The first equation lets us express one variable in terms of the other four, getting us down to four dimensions. Then we can **projectivize**, counting two solutions as the same if one is a multiple of the other. That gets us down to three dimensions. The actual equation in three dimensions looks a bit messy.

This surface is famous because while every smooth cubic has 27 lines on it, for this one all lines can be seen in the real picture drawn here: we don't need to look at the complex version of the surface.

Felix Klein, with his deep love of symmetry, noticed that you can build this surface starting from the icosahedron! Learnhow ... more »

2016-03-01 15:00:28 (0 comments; 3 reshares; 26 +1s)

**What is the value of the Universe, in dollars?**

In 1997, a group of scientists tried to estimate the value of the "ecosystem services" that the whole Earth provides to humanity each year - in dollars. They published a paper on this in Nature.

In 2004, a judge named Richard A. Posner estimated the cost of humanity going extinct - in dollars. He did this as part of a cost-benefit analysis of the Relativistic Heavy Ion Collider, a device that had a small chance of converting the entire Earth into "strange matter".

The next thing is to calculate the price of the entire Universe - in dollars.

2016-02-26 19:49:53 (0 comments; 8 reshares; 37 +1s)

**Arctic melting - 2016**

The graph shows global sea ice area for Februaries of various years, from 2006 to 2016. We are setting a record low!

More importantly, since about 10 February, the extent of Arctic sea ice has been noticeably below any of the last 30 years. Why? It's easy to guess. The Arctic is experiencing record-breaking temperatures: about 4° C higher than the 1951–1980 average.

For details, go here:

https://johncarlosbaez.wordpress.com/2016/02/26/arctic-melting-2016/

2016-02-26 14:34:17 (23 comments; 15 reshares; 60 +1s)

**The case for optimism on climate change**

Check out this TED talk by Al Gore! Some quotes:**Let's look at the atmosphere. This is a depiction of what we used to think of as the normal distribution of temperatures. The white represents normal temperature days; 1951-1980 are arbitrarily chosen. The blue are cooler than average days, the red are warmer than average days. But the entire curve has moved to the right in the 1980s. And you'll see in the lower right-hand corner the appearance of statistically significant numbers of extremely hot days. In the 90s, the curve shifted further. And in the last 10 years, you see the extremely hot days are now more numerous than the cooler than average days. In fact, they are 150 times more common on the surface of the earth than they were just 30 years ago.****So we're having record-breaking temperatures. Fourteen... more »**

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**2016-02-25 21:22:04 (4 comments; 1 reshares; 44 +1s)

**Beware of Bentham**

Bentham Science Publishers runs over 230 open access journals. They claim these journals are peer reviewed. However, one of their journals accepted a paper that starts like this:**Compact symmetries and compilers have garnered tremendous interest from both futurists and biologists in the last several years. The flaw of this type of solution, however, is that DHTs can be made empathic, large-scale, and extensible. Along these same lines, the drawback of this type of approach, however, is that active networks and SMPs can agree to fix this riddle. The construction of voice-over-IP would profoundly degrade Internet QoS.**

The paper was submitted by two nonexistent authors from the Center for Research in Applied Phrenology (CRAP). It was accepted in just one day! Or at least it would have been accepted if the authors had sent $800 to a post... more »

2016-02-25 14:55:56 (0 comments; 17 reshares; 85 +1s)

Remember that gravitational wave they detected at LIGO? It's getting more interesting!

2016-02-23 15:50:41 (34 comments; 9 reshares; 80 +1s)

**My favorite number**

My favorite number is 24, but it acquires some of its magic powers through those of the number 12.

If you're near the University of Waterloo at 3:30 pm this Friday, you can see my talk about this at the William G. Davis Centre in room DC 1302. If you come half an hour earlier, you can also have tea and cookies!

I'll explain Euler's wacky argument claiming to show that the infinite sum 1 + 2 + 3 + 4 + ... adds up to -1/12. This was before the mathematician Abel declared that "divergent series are the invention of the devil". You can get anything from a divergent series: it takes good taste to get something useful, as Euler did.

Euler's formula can now be understood rigorously in terms of the Riemann zeta function, and in fact it's the reason why bosonic strings work best in 26=24+2 dimensions. I'll... more »

2016-02-22 19:12:58 (11 comments; 2 reshares; 52 +1s)

**The Harmonograph**

If you're near Waterloo, Canada, please come see Anita Chowdry and me this Friday at 7:30 pm. We're giving a joint math/art lecture - followed by a demonstration... and a party!

Here's the idea:

A **harmonograph** is a drawing machine powered by pendulums. It was first invented in the 1840s - the heyday of the industrial revolution, whose sensibilities are now celebrated by the steampunk movement.

In this presentation, artist Anita Chowdry will recount her fascinating journey into this era, culminating in her creation of a two-meter high harmonograph crafted from brass and steel: “The Iron Genie”.

Then, using computer simulations, I'll explore the underlying mathematics of the harmonograph, taking you on a trip into the fourth dimension and beyond. As time passes, the motion of the harmonograph traces out acurv... more »

2016-02-18 21:50:42 (57 comments; 17 reshares; 108 +1s)

**Inside every boring gray cube...**

... there's a colorful dodecahedron yearning to unfold!**Puzzle 1:** When we fold the dodecahedron back to a cube, does it fit together snugly, or is there some empty space left? What percent of the cube is filled?

This image was created by Hermann Serras, and you can see it here:

http://cage.ugent.be/~hs/polyhedra/dodeca.html

#geometry

2016-02-17 16:12:25 (34 comments; 20 reshares; 79 +1s)

**The trouble with QED**

If you're trying to understand charged particles and radiation in a way that takes special relativity and quantum mechanics into account, you need **QED**.

That stands for **quantum electrodynamics**. Feynman, Schwinger and Tomonaga invented this theory - with lots of help - around 1948. In QED we often compute answers to physics problems as power series in the fine structure constant

α ≈ 1/137.036

This number says how strong the electric force is. For example, if you have an electron orbiting a proton, on average it's moving about 1/137.036 times the speed of light.

We can compute lots of things using QED. A great example is the magnetic field produced by an electron. The electron is a charged spinning particle, so it has a magnetic field in addition to its electric field. How strong is thisma... more »

2016-02-16 15:43:00 (17 comments; 32 reshares; 164 +1s)

**27 lines on a cubic**

Here's the best thing I know about the number 27. Any smooth surface described by a cubic equation has 27 lines on it!

In general this is true only for complex surfaces, where we look at complex solutions of the cubic equation. For these it takes two complex numbers (rather than two real ones) to say where you are. So they're kind of hard to visualize.

But for this particular surface, drawn by Greg Egan, we can see all 27 lines even if we only look at the real surface formed by the real solutions.

This is the tip of an iceberg I'm just starting to drill down into. For more, visit:

http://blogs.ams.org/visualinsight/2016/02/15/27-lines-on-a-cubic-surface/

#geometry

2016-02-14 19:44:03 (0 comments; 18 reshares; 71 +1s)

**Zebroids versus quaggas**

This is a **zebroid** - a hybrid of a horse and a zebra.

A **quagga** was a naturally occurring kind of zebra without stripes on its back half. Alas, quaggas went extinct in 1883. But now the Quagga Project is trying to bring them back! Read the whole story on my blog:

https://johncarlosbaez.wordpress.com/2016/02/13/the-quagga/

#biology

2016-02-11 16:01:44 (140 comments; 75 reshares; 283 +1s)

**Gravitational waves**

The rumors are true: LIGO has seen gravitational waves! Based on the details of the signal detected, the LIGO team estimates that 1.3 billion years ago. two black holes spiralled into each other and collided. One was 29 times the mass of the Sun, the other 36 times. When they merged, 3 times the mass of the Sun was converted directly to energy and released as gravitational waves.

For a very short time, this event produced over 10 times more power than all the stars in the Universe!

We knew these things happened. We just weren't good enough at detecting gravitational waves to see them - until now.

I'll open comments on this breaking news item so we can all learn more. LIGO now has a page on this event, which is called **GW150914** because it was seen on September 14th, 2015:

... more »

2016-02-07 18:39:57 (0 comments; 17 reshares; 106 +1s)

**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... more »

2016-02-05 16:36:52 (0 comments; 42 reshares; 120 +1s)

**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 alsosay... more »

2016-02-03 22:25:30 (0 comments; 13 reshares; 119 +1s)

**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:

http://www.slate.com/blogs/bad_astronomy/2016/01/16/optical_phenomenon_draws_a_map_of_a_city_in_the_sky.html

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

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

more »

2016-02-02 15:56:39 (0 comments; 21 reshares; 111 +1s)

**Renormalization**

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... more »

2016-02-01 17:00:45 (0 comments; 13 reshares; 66 +1s)

**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:

http://blogs.ams.org/visualinsight/2016/02/01/hoffman-singleton-graph/

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 isbuilt usi... more »

2016-01-31 18:52:34 (0 comments; 23 reshares; 107 +1s)

**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 oft... more »

2016-01-30 02:56:19 (0 comments; 27 reshares; 96 +1s)

**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:

https://johncarlosbaez.wordpress.com/2016/01/30/among-the-bone-eaters/

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

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