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John Baez

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

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

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

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

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

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

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

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

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 »

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 a lot of of organic chemicals, including amino acids.

It seems this rock was formed about 4.55 billion years ago.

Scientists tried to figure out where it came from.  They reconstructed its direction of motion and compared its properties with the spectra of various asteroids.  In the end, they guessed that it most likely came from 773 Irmintraud.

773 Irmintraud is a dark, reddish asteroid from the outer region of the asteroid belt.  It's about 92 kilometers in diameter.   It's just 0.034 AU away from a chaotic zone associated with one of the gaps in the asteroid belt created by a resonance with Jupiter.  So, if a chunk got knocked off, it could wind up moving chaotically and make it to Earth!

And here's what really intrigues me.  773 Irmintraud is a D-type asteroid - a very dark and rather rare sort.  One model of Solar System formation says these asteroids got dragged in from very far out in the Solar System - the Kuiper Belt, out beyond Pluto.   (Some scientists think Mars' moon Phobos is also a D-type asteroid.) 

So, this chunk of rock here may have been made out in the Kuiper Belt, over 4.5 billion years ago!

For more, see:

https://en.wikipedia.org/wiki/Tagish_Lake_(meteorite)
https://en.wikipedia.org/wiki/773_Irmintraud
https://en.wikipedia.org/wiki/D-type_asteroid

#astronomy  ___

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

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 »

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 and quantum mechanics.   When you put these theories together, they predict that an accelerating observer will see a faint glow of thermal radiation.

Why hasn't it been detected?   Because it's predicted to be very, very  weak.   Absurdly weak!

For example, suppose you accelerate at a trillion gee - a trillion times more than a falling object on Earth.  Then the theory predicts you'll see thermal radiation at a temperature of 40 billionths of a degree Celsius above absolute zero.   That's so faint nobody knows how to detect it!

What if you sit there watching someone else accelerate past you?  What will you see then? 

There are arguments about this, but whatever happens, it'll be too small to detect under most circumstances.  Chen and Tajima have proposed an experiment to accelerate a single electron at 10 septillion gee  (that is, 10^25 gee).  That might be enough for something interesting to happen.  However, the EmDrive gadget is nowhere near as intense. The version NASA built is weaker than a typical microwave oven.
 
This has not stopped McCulloch from claiming that the Unruh effect "explains" the EmDrive! 

He also claims it explains the rotations of galaxies, eliminating the need for dark matter.  He also claims that it explains the accelerating expansion of the Universe, eliminating the need for dark energy.  He also claims that it explains the Pioneer anomaly - a small mysterious acceleration that some spacecraft have encountered as they go far out into the Solar System. 

None of this makes any sense.  In fact, I can barely believe I'm even talking about it!  But it fooled the folks at Technology Review, so let me quote a bit of McCulloch's paper, and comment on it:

McCulloch (2007) has proposed a new model for inertia (MiHsC) that assumes that the inertia of an object is due to the Unruh radiation it sees when it accelerates [...]

So the inertial mass of an object is caused  by the Unruh radiation?   Okay... yup, that's certainly new.   Let me just say there's no evidence for this.

[...] radiation which is also subject to a Hubble-scale Casimir effect.

Oh, good, the Casimir effect!  As if things weren't confused enough already.  The Casimir effect is a very real thing: a force between very nearby metal plates, caused by the fact that the electric field can't easily penetrate a conductor.  It's a reasonably large force when the plates are a few nanometers apart, but it rapidly becomes weaker as you move them farther apart.   So now imagine they're as far apart as most distant galaxies we can see....

In this model only Unruh wavelengths that fit exactly into twice the Hubble diameter are allowed, so that a greater proportion of the waves are disallowed for low accelerations (which see longer Unruh waves) leading to a gradual new loss of inertia as accelerations become tiny.

The Hubble diameter is very roughly the size of the observable Universe.  Now he's saying that at rather small accelerations the Unruh effect is so tiny that the thermal radiation has wavelengths even larger than the size of the observable Universe.  That's true.  And that of course means that this effect is even more absurdly weak than in the example I gave. 

But he's also saying that something like the Casimir effect takes place, where the size of Universe plays the role of distance between the metal plates in the usual Casimir effect.   In other words, when an object accelerates fast enough that the Unruh radiation it sees fits inside the Universe, the Unruh effect "kicks in" and gives the object a kick, or makes its mass get bigger, or something.

Again, two things stand out: 1) it doesn't work like this, and 2) even if it did, the effect would be so tiny that... why are we even talking about it?  Even the pathetically weak thrusts the EmDrive supposedly creates - less than 100 micronewtons in the latest experiments - are like a thundering herd of giant elephants compared to what we're talking about here. 

The difficulty of demonstrating MiHsC on Earth is the huge size of [the Universe] in Eq. 1 which makes the effect very small unless the acceleration is tiny, as in deep space. One way to make the effect more obvious is to reduce the distance to the horizon and this is what the emdrive may be doing since the radiation within it is accelerating so fast that the Unruh waves it sees will be short enough to be limited by the cavity walls in a MiHsC-like manner.

So now it's the radiation inside the can that's "accelerating so fast" that it sees Unruh radiation... which is limited in wavelength by the size of the can... which somehow makes the whole can get a push when the Unruh radiation fits into the can.

In short, we've got a Rube Goldberg machine where all the parts involve brand new theories of physics with nothing backing them up, and all the actual effects cited are absurdly tiny.

But that's not all!   One amusing thing is that while the Unruh effect involves quantum mechanics, Planck's constant - the number that shows up in every calculation in quantum mechanics - never shows up in this paper.  So McCulloch is not actually doing anything with the Unruh effect!  Instead, he's making up brand new stuff, like this:

Normally, of course, photons are not supposed to have inertial mass in this way, but here this is assumed.

So his photons have mass - and on top of that, the mass changes with time: see his Equation 4!

Verdict: this paper is a stew of nonsense served with a hefty helping of warmed-over baloney.   And yet we see in the Daily Mail:

Have scientists cracked the secret of NASA’s 'impossible' fuel-free thruster? New theory could explain the EmDrive that may one day take man to Mars in 10 weeks___

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

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 »

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 more fun.  If you want to look it up, go ahead.)___

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

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 »

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 Barth sextic has been rotated to bring some of these points into view!  It's also been sliced so you can see inside.

You can see more of his images here:

https://plus.google.com/114982179961753756261/posts/B6zWUjNTaVr

and learn more about the Barth sextic here:

http://blogs.ams.org/visualinsight/2016/04/15/barth-sextic/___

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

"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 »

"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 living inside it may decide to leave.  This can happen when the sea water gets too hot.  Without its zooxanthellae, the polyp is transparent and the coral's white skeleton is revealed - as you see here.  We say the coral is bleached.

After they bleach, the polyps begin to starve.  If conditions return to normal fast enough, the zooxanthellae may come back.   If they don't, the coral will die.

The Great Barrier Reef, off the northeast coast of Australia, contains over 2,900 reefs and 900 islands.  It's huge: 2,300 kilometers long, with an area of about 340,000 square kilometers.  It can be seen from outer space!

With global warming, this reef has been starting to bleach.  Parts of it bleached in 1998 and again in 2002.  But this year, with a big El Niño pushing world temperatures to new record highs, is the worst.

Scientists have being flying over the Great Barrier Reef to study the damage, and divers have looked at some of the reefs in detail.  Of the 522 reefs surveyed in the northern section, over 80% are severely bleached and less than 1% are not bleached at all.    Of 226 reefs surveyed in the central section, 33% are severely bleached and 10% are not bleached.  Of 163 reefs in the southern section, 1% are severely bleached and 25% are not bleached. 

The top expert on coral reefs in Australia, Terry Hughes, wrote:

“I showed the results of aerial surveys of bleaching on the Great Barrier Reef to my students.  And then we wept.”

Some of the bleached reefs may recover.  But as oceans continue to warm, the prospects look bleak.  The last big El Niño was in 1998.  With a lot of hard followup work, scientists showed that in the end, 16% of the world’s corals died in that event. 

This year is quite a bit hotter.

So, global warming is not a problem for the future: it's a problem now.   It's not good enough to cut carbon emissions eventually.   We've got to get serious now.  

I need to recommit myself to this.  For example, I need to stop flying around to conferences.  I've cut back, but I need to do much better.  Future generations, living in the damaged world we're creating, will not have much sympathy for our excuses.___

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

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 »

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 pictures of this beautiful beast, including some rotating views, visit my blog Visual Insight:

http://blogs.ams.org/visualinsight/2016/04/15/barth-sextic/

The proof that the Barth sextic has the maximum possible number of ordinary double point uses the theory of codes!___

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

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 »

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 Grothendieck universe.   It's not the universe - we reserve that name for the collection of all sets, which is too big to be a set.  But all the usual axioms of set theory apply if we restrict attention to sets in a Grothendieck universe.  

In fact, if an inaccessible cardinal exists, we can use the resulting Grothendieck universe to prove that the usual axioms of set theory are consistent!   The reason is that the Grothendieck universe gives a "model" of the axioms - it obeys the axioms, so the axioms must be consistent.

However, Gödel's first incompleteness theorem says we can't use the axioms of set theory to prove themselves consistent.... unless they're inconsistent, in which case all bets are off.

The upshot is that we probably can't use the usual axioms of set theory to prove that it's consistent to assume there's an inaccessible cardinal.  If we could, set theory would be inconsistent!

Nonetheless, bold set theorists are fascinated by inaccessible cardinals, and even much bigger cardinals.  For starters, they love the infinite and its mysteries.   But also, if we assume these huge infinities exist, we can prove things about arithmetic that we can't prove using the standard axioms of set theory!

I gave a very rough definition of inaccessible cardinals.  It's not hard to be precise.  A cardinal X is inaccessible if you can't write it as a sum of fewer than X cardinals that are all smaller X, and if any cardinal Y is smaller than X, 2 to the Yth power is also smaller than X. 

Well, not quite.   According to this definition, 0 would be inaccessible - and so would the very smallest infinity.   Neither of these can be gotten "from below".  But we don't count these two cardinals as inaccessible.

https://ncatlab.org/nlab/show/inaccessible+cardinal

#bigness___

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

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 »

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 smallest rings of carbon atoms have 6 atoms.  A rather surprising thing about the triamond is that the smallest rings have 10 atoms!   Each atom lies in 15 of these 10-sided rings.

When I heard about the triamond, I had to figure out how it works.  So I wrote this:

https://johncarlosbaez.wordpress.com/2016/04/11/diamonds-and-triamonds/

The thing that got me excited in the first place was a description of the 'triamond graph' - the graph with carbon atoms as vertices and bonds as edges.  It's a covering space of the complete graph with 4 vertices.  It's not the universal cover, but it's the 'universal abelian cover'. 

I guess you need to know a fair amount of math to find that exciting.  But fear not - I lead up to this slowly: it's just a terse way to say a lot of fun stuff. 

And while the triamond isn't found in nature (yet), the mathematical pattern of the triamond may be.___

posted image

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

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 »

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 Peano axioms.  But these axioms don't completely capture our intuitions!  There are facts about natural numbers that most mathematicians would agree are true, but can't be proved from the Peano axioms.

Besides the natural numbers you think you know - but do you really? - there are lots of other models of arithmetic.  They all obey the Peano axioms, but they're different.  Whenever there's a question you can't settle using the Peano axioms, it's true in some model of arithmetic and false in some other model.

There's no way to decide which model of arithmetic is the right one - the so-called "standard" natural numbers.   

Hamkins showed there's a Turing machine that does something amazing.  It can compute any function from the natural numbers to the natural numbers, depending on which model of arithmetic we use. 

In particular, it can compute the uncomputable... but only in some weird "alternative universe" where the natural numbers aren't what we think they are. 

These other universes have "nonstandard" natural numbers that are bigger than the ones you understand.   A Turing machine can compute an uncomputable function... but it takes a nonstandard number of steps to do so.

So: computing the computable takes a "standard" number of steps.   Computing the uncomputable takes a little longer.

This is not a practical result.  But it shows how strange simple things like logic and the natural numbers really are.

For a better explanation, read my blog post:

https://johncarlosbaez.wordpress.com/2016/04/02/computing-the-uncomputable/

And for the actual proof, go on from there to the blog article by +Joel David Hamkins.___

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

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 »

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 single banner." 

Here you see some of their members marching in a perfectly legal manner down the streets of South Dallas.  They started doing this after the killing of Michael Brown by a policeman in Ferguson. 

From last year:

On a warm fall day in South Dallas, ten revolutionaries dressed in kaffiyehs and ski masks jog the perimeter of Dr. Martin Luther King Jr. Park bellowing "No more pigs in our community!" Military discipline is in full effect as the joggers respond to two former Army Rangers in desert-camo brimmed hats with cries of "Sir, yes, sir!" The Huey P. Newton Gun Club is holding its regular Saturday fitness-training and self-defense class. Men in Che fatigues run with weight bags and roll around on the grass, knife-fighting one another with dull machetes." I used to salute the fucking flag!" the cadets chant. "Now I use it for a rag!"

You'd think that white conservatives would applaud this "well-regulated militia", since they too are suspicious of the powers of the government.   Unfortunately they have some differences of opinion. 

For one thing, there's that white versus black business, and the right-wing versus left-wing business.  To add to the friction, the Black Panthers are connected to the Nation of Islam, a black Muslim group, while the white conservatives tend to be Christian.

It was thus not completely surprising when a gun-toting right-wing group decided to visit a Nation of Islam mosque in South Dallas.  This group has an amusingly bureaucratic name: The Bureau of American Islamic Relations.  They said:

“We cannot stand by while all these different Anti American, Arab radical Islamists team up with Nation of Islam/Black Panthers and White anti American Anarchist groups, joining together in the goal of destroying our Country and killing innocent people to gain Dominance through fear!”

So, yesterday, the so-called Bureau showed up at the Nation of Islam mosque in South Dallas.   They were openly carrying guns.

But the Huey P. Newton Gun Club expected this.  So they showed up in larger numbers, carrying more guns. 

Things became tense.  People stood around holding guns, holding signs, yelling at each other,  exercising all their constitutional freedoms like good Americans: the right of free speech, the right of assembly, the right to bear arms.

In the end, no shots were fired.  The outgunned Bureau went home. 

One of the co-founders of the Huey P. Newton Gun Club was interviewed while this was going on.  He said:

Those banditos are out of their minds if they think they're going to come to South Dallas like this.

See?  This is how the 2nd Amendment works.   For more:

https://www.rawstory.com/2016/04/armed-hate-group-backs-out-of-texas-mosque-protest-when-faced-with-gun-toting-worshipers/

http://www.vice.com/read/huey-does-dallas-0000552-v22n1___

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

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 »

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 a collection of ‘near misses’: polyhedra that come very close to being Johnson solids.  In an interview with the New York Times, he said:

When I found this one, I was impressed at how close it came to being a Johnson solid. But then I did some calculations, and I was utterly flabbergasted to discover that the faces are exactly regular! I don’t know how people overlooked it.

It turned out there was a subtle error in Zalgaller’s lengthy proof.

Or maybe not: for details see

http://blogs.ams.org/visualinsight/2016/04/01/rectified_truncated_icosahedron/___

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

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 »

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 convention in Cleveland this June. If Trump gets 1237 votes he wins. But in a second vote, delegates are free to do what they want. And in South Carolina, Republican party insiders are trying very hard to get them to switch sides if this happens:

http://www.politico.com/story/2016/03/south-carolina-delegates-convention-221253

Also, Cruz is working very hard, and it seems successfully, to get delegates onto key committees in the Republican convention:

http://www.nationalreview.com/article/433136/republican-contested-convention-favors-ted-cruz-over-donald-trump

None of this may matter in the end. Trump only needs to win 53% of the remaining delegates to reach the magic number of 1237. But it's interesting to watch the Republican party establishment squirm as this nightmarish event looms ever closer.

It could be quite close. A recent group of expert poll-watchers - see below - estimated that Trump will get 1208. Then we can expect a major drama.

No comments allowed, because people will get into political fights, and I don't visit G+ for fights, or even politics. I just thought this behind-the-scenes stuff is not getting enough attention.___

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

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 »

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 no intervening siblings. They play together, they grow up together… and maybe he learns not to be such a sexist pig!

They decide they are incompatible, so they split up and each one separately tries to find a mate who will go along with their own crazy reproductive plan.

Now for some puzzles:

Puzzle 1. If Tom carries out his plan of having children until he has two consecutive sons, and then stops, what is the expected number of children he will have?

Puzzle 2. If Lisa carries out her plan of having children until she has a daughter followed directly by a son, and then stops, what is the expected number of children she will have?

Puzzle 3: Which is greater, Tom’s expected number of children or Lisa’s? Or are they equal?

For maximum benefit, try to answer Puzzle 3 before doing the calculations required to answer Puzzles 1 or 2.

Puzzle 4: Suppose Tom and Lisa got married and agreed to have kids until both their criteria are satisfied. What would their expected number of children be?

In these puzzles, assume that each time someone has a child, they have a 50% chance of having either a daughter or a son. Also assume each event is independent: that is, the gender of any children so far has no effect on that of later ones. Also ignore twins and other tricky issues.

For answers, go here:

https://johncarlosbaez.wordpress.com/2016/03/26/probability-puzzles-part-3/

There's also a fifth puzzle here, which only makes sense after you answer some of the rest. Its answer is "the golden ratio". It turns out that Tom's reproductive plan is connected to the Fibonacci numbers!___

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

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 »

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 Yang-Baxter equation as a process of its own, that process satisfies an equation: the Zamolodchikov tetrahedron equation. This equation really wants to be drawn in 4 dimensions, but you can get away with drawing it in 3 - just as I drew that simple braid on the plane.

This goes on forever: whenever you reinterpret a equation as a process, that process can (and usually should) obey new equations of its own. As you do this, you naturally go to higher dimensions. The Zamolodchikov tetrahedron equation is a nice example of how this works!

#topology #4d

___

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

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 »

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 make energy. So these carbon emissions are flat or even slightly down from 2014!

For more, visit Azimuth:

https://johncarlosbaez.wordpress.com/2016/03/24/global-carbon-emissions-are-flat/

___

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

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 »

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 than 1.000001 times as dense than E8. Close... but no cigar.

Now Maryna Viazovska has used the same technique, but pushed it further. Now we know: nothing can beat E8 in 8 dimensions!

Viazovska is an expert on the math of "modular forms", and that's what she used to crack this problem. But when she's not working on modular forms, she writes papers on physics! Serious stuff, like "Symmetry and disorder of the vitreous vortex lattice in an overdoped BaFe_{2-x}Co_x As_2 superconductor."

After coming up with her new ideas, Viaskovska teamed up with other experts including Henry Cohn and proved that another lattice, the Leech lattice, gives the densest sphere packing in 24 dimensions.

Different dimensions have very different personalities. Dimensions 8 and 24 are special. You may have heard that string theory works best in 10 and 26 dimensions - two more than 8 and 24. That's not a coincidence.

The densest sphere packings of spheres are only known in dimensions 0, 1, 2, 3, and now 8 and 24. Good candidates are known in many other low dimensions: the problem is proving things - and in particular, ruling out the huge unruly mob of non-lattice packings.

For example, in 3 dimensions there are uncountably many non-periodic packings of spheres that are just as dense as the densest lattice packing!

In fact, the sphere packing problem is harder in 3 dimensions than 8. It was only solved earlier because it was more famous, and one man - Thomas Hales - had the nearly insane persistence required to crack it.

His original proof was 250 pages long, together with 3 gigabytes of computer programs, data and results. He subsequently verified it using a computerized proof assistant, in a project that required 12 years and many people.

By contrast, Viazovska's proof is extremely elegant. It boils down to finding a function whose Fourier transform has a simple and surprising property! For details on that, try my blog article:

https://golem.ph.utexas.edu/category/2016/03/e8_is_the_best.html

#geometry

___

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

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 »

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.

Second, it seems that the end of the string attached to the catenary is ‘at infinity’, very far up.

But where's the mathematical mystery? For that, go here:

https://johncarlosbaez.wordpress.com/2016/03/22/the-involute-of-a-cubical-parabola/

#geometry #curves___

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

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 »

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 of light!

In the end, it became a blue-hot ball of expanding plasma.

http://www.nasa.gov/feature/ames/Kepler/caught-for-the-first-time-the-early-flash-of-an-exploding-star

#astronomy___

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

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 »

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 the zeros of the Riemann zeta function!

This doesn't make the problem easier... but, it unlocks a whole new battery of tricks for understanding prime numbers! Many of the amazing things we now understand about primes are based on Riemann's idea.

It also opens up new puzzles, like the Riemann Hypothesis: a guess about where the Riemann zeta function can be zero. If someone could prove this, we'd know a lot more about prime numbers!

The animated gif here shows how the prime counting function is approximated by adding up wiggly functions, one for each of the first 500 zeros of the Riemann zeta function. So when you see something like "k = 230", you're getting an approximation that uses the first 230 zeros.

You may need to reload this picture to get the animated gif to move - it's not looped.

And that's a problem I want to solve! Is there a way to "loopify" a gif? Or can someone make a nice looped gif of something like this? If so, I could feature it on my blog Visual Insight, and credit you!

I got this gif here:

http://www.dartmouth.edu/~chance/chance_news/recent_news/chance_news_10.10y.html

and here you can see Riemann's formula. You'll see that some other functions, related to the prime counting function, have simpler formulas.

And by the way: when I'm talking about zeros of the Riemann zeta function, I only mean zeros in the critical strip, where the real part is between 0 and 1. The Riemann Hypothesis says that for all of these, the real part is exactly 1/2.
This has been checked for the first 10,000,000,000,000 zeros.

That sounds pretty convincing, but it shouldn't be. After all, the number 6 is the most common gap between consecutive primes if we look at numbers less than something like 17,427,000,000,000,000,000,000,000,000,000... but then that pattern stops!

So, you shouldn't look at a measly few examples and jump to big conclusions when it comes to primes.

#bigness___

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

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 »

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 they checked it with some other calculations.

The same argument shows that eventually 30 ceases to be the most common gap between successive primes. The number 210 takes over! It happens somewhere roughly around here:

1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

That's 10 to the 450th power.

In fact, they argued that this pattern keeps on forever. For a little while 2 is the most common prime gap, then it's 6, then it's 30, then it's 210, then it's 2310, and so on. And these numbers are very interesting. They're called primorials:

2

2⋅3=6

2⋅3⋅5= 30

2⋅3⋅5⋅7=210

2⋅3⋅5⋅7⋅11=2310

For more details, check out my blog article:

https://golem.ph.utexas.edu/category/2016/03/the_most_common_prime_gaps.html

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

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 »

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 the cutting-edge electronic rock of that day and traditional folk music. The singer, Rosie Cross, has a voice that reminds me of Maddy Prior of Steeleye Span.

When it first came out I liked the electronic aspects, but not the folk. Now I like both - and it bothers me that this unique album seems almost lost to the world!

My wife Lisa has a tape of it. She transferred the tape to mp3 using Audacity but the result was fairly bad... a lot of distortion. Part of the problem ws a bad cassette tape deck - I've got a much better one, and I should try it. But I'm afraid another part of the problem is that without a special sound card, using the 'line in' on your laptop produces crappy recordings. I'll see.

Here's a review of the band from Last.fm:

The English folkrock group Pyewackett was founded at the end of the 1970’s by Ian Blake and Bill Martin. They were a resident band and the London University college folk club.

Pyewackett played traditional folk music by the motto "pop music from the last five centuries": 15th century Italian dances, a capella harmonies, traditional songs in systems/minimalist settings, 1920’s ballads, etc.. The distinctive sound was characterised by the unusual combination of woodwinds, strings and keyboards. The voices were also a strong trademark. All these features of the Pyewackett sound are shown best on the second album the band released in 1984, The Man in the Moon Drinks Claret. It is this album that has been re-released in Music & Words’ Folk Classics series. At the time of this recording the band was formed by Ian Blake, Bill Martin, Mark Emerson, Rosie Cross and guest drummer Micky Barker. The album has been co-produced by Andrew Cronshaw.

Here's a review by Craig Harris:

One of the lesser-known of the British folk bands, Pyewackett is remembered for updating 18th century songs with modern harmonies and inventive instrumentation. While none of their four albums are easy to find, the search is worth it. The group's sense of fun and reverence for musical traditions allowed them to bring ancient tunes to life. Pyewackett took their name from an imp that a 17th century Essex woman claimed possessed her. According to legendary witch-hunter Matthew Hopkins, it was a name that "no mortal could invent."

If you want to hear a bit of Pyewackett, try this:

https://www.youtube.com/watch?v=BOIne9rTsOs

It's less electronic, but still a good bass line spices up this rendition of the traditional "Tam Lin". For the story of Tam Lin, try this:

http://tam-lin.org/versions/steel.html

Tam Lin is

a character in a legendary ballad originating from the Scottish Borders. It is also associated with a reel of the same name, also known as Glasgow Reel. The story revolves around the rescue of Tam Lin by his true love from the Queen of the Fairies. While this ballad is specific to Scotland, the motif of capturing a person by holding him through all forms of transformation is found throughout Europe in folktales. The story has been adapted into various stories, songs and films.
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2016-03-16 16:27:54 (17 comments; 7 reshares; 66 +1s)Open 

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 »

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.

It was a great chance to ponder lots of things, including the far future of the Universe.
___

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

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 »

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 are 8­-dimensional.

The octonions are the biggest, but also the weirdest. For example, multiplication of octonions violates the associative law: (xy)z is not equal to x(yz).

So the octonions sound completely crazy at first, but they turn out to have fascinating connections to string theory and other things. They’re pretty addictive, and if became a decadent wastrel I would spend even more time on them!

There’s a concept of “integer” for the octonions, and integral octonions form a lattice, a repeating pattern of points, in 8 dimensions. This is called the E8 lattice. Each point in this lattice has 240 nearest neighbors, which form a beautiful shape called the E8 root polytope.

The picture here is not me: it's the E8 root polytope. It's projected down to the plane, of course, and a bunch of points are directly in front of others, so you don't see 240 of them.
___

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

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 »

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 hundred million primes, the answer is just 17.757%. That's way off!

So if a prime ends in a 7, it seems to somehow tell the next prime "I rather you wouldn't end in a 7. I just did that."

Pardon my French, but this is fucking weird. And I'm not just saying this because I don't know enough number theory. Ken Ono is a real expert on number theory. And when he learned about this, he said:

“I was floored. I thought, ‘For sure, your program’s not working.’ "

Needless to say, it's not magic. There will be an explanation. In fact, Oliver and Soundarajan have conjectured a formula that says exactly how much of a discrepancy to expect - and they've checked it, and it seems to work. It works in every base, not just base ten. But we still need a proof that it really works.

By the way, their formula says the discrepancy gets smaller and smaller when we look at more and more primes. If we look at primes less than N, the discrepancy is on the order of log(log(N))/log(N). This goes to zero when N goes to infinity. But this discrepancy is huge compared to the discrepancy for the simpler question, "what percentage of primes ends in a given digit?" For that, the discrepancy - the difference between reality and what you'd expect if primes were random - is on the order of 1/(log(N) sqrt(N)).

Of course, what's really surprising is not this correlation between the last digits of consecutive primes, but that number theorists hadn't thought to look for it until now!

For more, read the article below, and of course the actual paper:

• Robert J. Lemke Oliver and Kannan Soundararajan, Unexpected biases in the distribution of consecutive primes, http://arxiv.org/abs/1603.03720.___

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

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 »

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 was bound to win: it tries to maximize its probability of winning, so when it thinks it's winning it becomes more conservative. Commenter Chris Garlock said “This was AlphaGo saying: ‘I think I’m ahead. I’m going to wrap this stuff up. And Lee Sedol needs to do something special, even if it doesn’t work. Otherwise, it’s just not going to be enough.”

On his 78th move, Lee did something startling.

He put a white stone directly between two of his opponent's stones, with no other white stone next to it. You can see it marked in red here. This is usually a weak type of move, since a stone that's surrounded is "dead".

I'm not good enough to understand precisely how strange this move was, or why it was actually good. At first all the commenters were baffled. And it seems to have confused AlphaGo. In the 87th move, AlphaGo placed a stone in a strange position which commentators said was "difficult to understand."

"AlphaGo yielded its own territory more while allowing its opponent to expand his own," said commentator Song Tae-gon, a Korean nine-dan professional go player. "This could be the starting point of AlphaGo's self-destruction."

Later AlphaGo placed a stone in the bottom left corner without reinforcing its territory in the center. Afterwards it seemed to recover, which Song said would be difficult for human players under such pressure. But Lee remained calm and blocked AlphaGo's attacks. The machine resigned on the 180th move.

Lee was ecstatic. "This win cannot be more joyful, because it came after three consecutive defeats. It is the single priceless win that I will not exchange for anything."

"AlphaGo seemed to feel more difficulties playing with black than white," he said. "It also revealed some kind of bug when it faced unexpected positions."

Lee has already lost the match, since AlphaGo won 3 out of the 5 games. But Lee wants to play black next time, and see if he can win that way.

You can play through the whole game here:

http://eidogo.com/#xS6Qg2A9

Even if you don't understand go, it has a certain charm.
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2016-03-12 18:27:22 (48 comments; 35 reshares; 113 +1s)Open 

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 »

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 territory and gain advantages elsewhere while the opponent struggles to defend themselves.

Lee appeared to be off to a good start with this plan, pressuring White’s invading group from all directions and forcing it to squirm uncomfortably.

But as the battle progressed, White gradually turned the tables — compounding small efficiencies here and there.

Lee seemed to be playing well, but somehow the computer was playing even better.

In forcing AlphaGo to withstand a very severe, one-sided attack, Lee revealed its hitherto undetected power.

Move after move was exchanged and it became apparent that Lee wasn’t gaining enough profit from his attack.

By move 32, it was unclear who was attacking whom, and by 48 Lee was desperately fending off White’s powerful counter-attack.

I can only speak for myself here, but as I watched the game unfold and the realization of what was happening dawned on me, I felt physically unwell.

Generally I avoid this sort of personal commentary, but this game was just so disquieting. I say this as someone who is quite interested in AI and who has been looking forward to the match since it was announced.

One of the game’s greatest virtuosos of the middle game had just been upstaged in black and white clarity.
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2016-03-11 17:51:40 (53 comments; 33 reshares; 113 +1s)Open 

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 »

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. Products are unique up to isomorphism. Coproducts. What products and coproducts are like in various familiar categories. General limits and colimits. Examples: products and coproducts, equalizers and coequalizers, pullbacks and pushouts, terminal and initial objects.

Week 3 (Jan. 19 and 21) - Equalizers and coequalizers, and what they look like in Set and other familiar categories. Pullbacks and pushouts, and what they look like in Set. Composing pullback squares.

Week 4 (Jan. 26 and 28) - Doing mathematics between categories. Faithful, full, and essentially surjective functors. Forgetful functors: what it means for a functor to forget nothing, forget properties, forget structure or forget stuff. Transformations between functors. Natural transformations. Functor categories. Natural isomorphisms. In a category with binary products, the product becomes a functor, and the commutative and associative laws hold up to natural isomorphism. Cartesian categories. In a cartesian category, the left and right unit laws also hold up to natural isomorphism. A G-set is a functor from a group G to Set. What is a natural transformation between such functors?

Week 5 (Feb. 2 and 4) - A G-set is a functor from a group G to Set, and a a natural transformation between such functors is a map of G-sets. Equivalences of categories. Adjoint functors: the rough idea. The hom-functor. Adjoint functors: the definition. Examples: the left adjoint of the forgetful functor from Grp to Set. The left adjoint of the forgetful functor from Vect to Set. The forgetful functor from Top to Set has both a left and right adjoint. If a category C has binary products, the diagonal functor from C to C × C has a right adjoint. If it has binary coproducts, the diagonal functor has a left adjoint.

Week 6 (Feb. 9 and 11) - Diagrams in a category as functors. Cones as natural transformations. The process of taking limits as a right adjoint. The process of taking colimits as a left adjoint. Left adjoints preserve colimits; right adjoints preserve limits. Examples: the 'free group' functor from sets to groups preserve coproducts, while the forgetful functor from groups to sets preserves products. The composite of left adjoints is a left adjoint; the composite of right adjoints is a right adjoint. The unit and counit of a pair of adjoint functors.

Week 7 (Feb. 16 and 18) - Adjunctions. The naturality of the isomorphism hom(Fc,d)≅hom(c,Ud) in an adjunction. Given an adjunction, we can recover this isomorphism and its inverse from the unit and counit. Toward topos theory: cartesian closed categories and subobject classifiers. The definition of cartesian closed category, or 'ccc'. Examples of cartesian cloed categories. In a cartesian closed category with coproducts, the product distributes over the coproduct, and exponentiation distributes over the product.

Week 8 (Feb. 23) - Internalization. The concept of a group in a cartesian category. Any pair of objects X,Y in a cartesian closed category has an 'internal' hom, the object Y^X as well as the usual 'external' hom, the set hom(X,Y). Evaluation and coevaluation. Internal composition. In a category with a terminal object, we can define the set of elements of any object.

Week 8 (Feb. 25) - Guest lecture by Christina Osborne on symmetric monoidal categories.

Week 9 (Mar. 1 and 3) - For any category C with a terminal object, elements define a functor elt:C→Set. If C is cartesian, this functor preserves finite products. If C is cartesian closed, elt(Y^X)≅hom(X,Y), so it converts the internal hom into the external hom. The 'name' of a morphism. Subobjects. The subobject classifier in Set. The general definition of subobject classifier in any category with finite limits. The definition of a topos. Examples of topoi, including the topos of graphs.

Week 10 (Mar. 8 and 10) - The subobject classifier in the topos of graphs. Any topos has finite colimits. Any morphism in a topos has an epi-mono factorization, which is unique up to a unique isomorphism. The 'image' of a morphism in topos. The poset Sub(X), whose elements are subobjects of an object X in a topos. The correspondence between set theory and logic: given a set X, subsets of X correspond to predicates defined for elements of X, intersection corresponds to 'and', union corresponds to 'or', the set X itself corresponds to 'true', and the empty set corresponds to 'false'. the intersection of subsets of X∈Set is their product in Sub(X), their union is their coproduct in Sub(X), the set X is the terminal object in Sub(X), and the empty set is the initial object. A lattice is a poset with finite limits and finite colimits, and a Heyting algebra is a lattice that is also cartesian closed. For any object X in any topos, Sub(X) is a Heyting algebra. If we think of these elements of Sub(X) as predicates, the exponential is 'implication'.

Where does topos theory go from here?

The notes are handwritten, by Christina Osborne. I'd consider paying someone to TeX them up! . But as you see, there are a lot of diagrams.... so you should only try it if you want to learn category theory while practicing your TikZ. And if you try it, you should let us know - it would be silly to have more than one person doing the same job, while chopping the job into parts might work well.

When I started teaching this course, I imagined that the notes might someday grow into a book I’d always dreamt of: an introduction to category theory that includes lots of examples, talks to the reader in a friendly way, and explains what’s ‘really going on’. However, while teaching the course, I noticed that Emily Riehl has written a book like this, probably better than I ever could. Even better, her book is free online and will soon be published by Dover (which sounds nice and affordable):

Emily Riehl, Category Theory in Context, 2014, http://www.math.jhu.edu/~eriehl/727/context.pdf

So, I don’t feel much urge to write that book anymore. But there might still be room for a more quirky book on category theory that only I could write. It would probably need to include not only this 'standard' material but more on applications.
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2016-03-09 18:11:12 (2 comments; 0 reshares; 58 +1s)Open 

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 »

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 to cajole: The Beatles prevailed on him to augment their recordings with arrangements for strings (on “Eleanor Rigby”), brass (on “Got to Get You Into My Life”), marching band (on “Yellow Submarine”) and solo French horn (on “For No One”), as well as a tabla player for Harrison’s Indian-influenced song “Love You To.”

It was also at least partly through Mr. Martin’s encouragement that the Beatles became increasingly interested in electronic sound. Noting their inquisitiveness about both the technical and musical sides of recording, Mr. Martin ignored the traditional barrier between performers and technicians and invited the group into the control room, where he showed them how the recording equipment at EMI’s Abbey Road studios worked. He also introduced them to unorthodox recording techniques, including toying with tape speeds and playing tapes backward.

Mr. Martin had used some of these techniques in his comedy and novelty recordings, long before he began working with the Beatles.

“When I joined EMI,” he told The New York Times in 2003, “the criterion by which recordings were judged was their faithfulness to the original. If you made a recording that was so good that you couldn’t tell the difference between the recording and the actual performance, that was the acme. And I questioned that. I thought, O.K., we’re all taking photographs of an existing event. But we don’t have to make a photograph; we can paint. And that prompted me to experiment.”

Soon the Beatles themselves became intent on searching for new sounds, and Mr. Martin created another that the group adopted in 1966 (followed by many others). During the sessions for “Rain,” Mr. Martin took part of Lennon’s lead vocal and overlaid it, running backward, over the song’s coda.

“From that moment,” Mr. Martin said, “they wanted to do everything backwards. They wanted guitars backwards and drums backwards, and everything backwards, and it became a bore.” The technique did, however, benefit “I’m Only Sleeping” (with backward guitars) and “Strawberry Fields Forever” (with backward drums).

Mr. Martin was never particularly trendy, and when the Beatles adopted the flowery fashions of psychedelia in 1966 and 1967 he continued to attend sessions in a white shirt and tie, his hair combed back in a schoolmasterly pre-Beatles style. Musically, though, he was fully in step with them. When Lennon wanted a circus sound for his “Being for the Benefit of Mr. Kite,” Mr. Martin recorded a barrel organ and, following the example of John Cage, cut the tape into small pieces and reassembled them at random. His avant-garde orchestration and spacey production techniques made “A Day in the Life” into a monumental finale for the kaleidoscopic album “Sgt. Pepper’s Lonely Hearts Club Band.”

http://www.nytimes.com/2016/03/10/arts/music/george-martin-producer-of-the-beatles-dies-at-90.html___

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

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.

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.___

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

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!

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!___

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

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 »

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! Learn how here:

http://blogs.ams.org/visualinsight/2016/03/01/clebsch-surface/

Also learn where you can buy a model of this surface! And look at an antique model of this surface in Oxford.

Puzzle: what is the first homology group of this surface? Draw a loop representing each generator. What is the intersection matrix?

___

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

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. 

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. ___

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

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/



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/

___

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

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 »

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 of the 15 of the hottest years ever measured with instruments have been in this young century. The hottest of all was last year. Last month was the 371st month in a row warmer than the 20th-century average. And for the first time, not only the warmest January, but for the first time, it was more than two degrees Fahrenheit warmer than the average. These higher temperatures are having an effect on animals, plants, people, ecosystems.

and later....

The insurance industry has certainly noticed, the losses have been mounting up. They're not under any illusions about what's happening. And the causality requires a moment of discussion. We're used to thinking of linear cause and linear effect -- one cause, one effect. This is systemic causation. As the great Kevin Trenberth says, "All storms are different now. There's so much extra energy in the atmosphere, there's so much extra water vapor. Every storm is different now." So, the same extra heat pulls the soil moisture out of the ground and causes these deeper, longer, more pervasive droughts and many of them are underway right now.

It dries out the vegetation and causes more fires in the western part of North America. There's certainly been evidence of that, a lot of them.

More lightning, as the heat energy builds up, a considerable amount of additional lightning also.

These climate-related disasters also have geopolitical consequences and create instability. The climate-related historic drought that started in Syria in 2006 destroyed 60 percent of the farms in Syria, killed 80 percent of the livestock, and drove 1.5 million climate refugees into the cities of Syria, where they collided with another 1.5 million refugees from the Iraq War. And along with other factors, that opened the gates of Hell that people are trying to close now. The US Defense Department has long warned of consequences from the climate crisis, including refugees, food and water shortages and pandemic disease.

Right now we're seeing microbial diseases from the tropics spread to the higher latitudes; the transportation revolution has had a lot to do with this. But the changing conditions change the latitudes in the areas where these microbial diseases can become endemic and change the range of the vectors, like mosquitoes and ticks that carry them. The Zika epidemic now -- we're better positioned in North America because it's still a little too cool and we have a better public health system. But when women in some regions of South and Central America are advised not to get pregnant for two years -- that's something new, that ought to get our attention. The Lancet, one of the two greatest medical journals in the world, last summer labeled this a medical emergency now. And there are many factors because of it.

This is also connected to the extinction crisis. We're in danger of losing 50 percent of all the living species on earth by the end of this century. And already, land-based plants and animals are now moving towards the poles at an average rate of 15 feet per day.

and later....

So the answer to the first question, "Must we change?" is yes, we have to change. Second question, "Can we change?" This is the exciting news! The best projections in the world 16 years ago were that by 2010, the world would be able to install 30 gigawatts of wind capacity. We beat that mark by 14 and a half times over. We see an exponential curve for wind installations now. We see the cost coming down dramatically. Some countries -- take Germany, an industrial powerhouse with a climate not that different from Vancouver's, by the way -- one day last December, got 81 percent of all its energy from renewable resources, mainly solar and wind. A lot of countries are getting more than half on an average basis.

More good news: energy storage, from batteries particularly, is now beginning to take off because the cost has been coming down very dramatically to solve the intermittency problem. With solar, the news is even more exciting! The best projections 14 years ago were that we would install one gigawatt per year by 2010. When 2010 came around, we beat that mark by 17 times over. Last year, we beat it by 58 times over. This year, we're on track to beat it 68 times over.

We're going to win this. We are going to prevail. The exponential curve on solar is even steeper and more dramatic. When I came to this stage 10 years ago, this is where it was. We have seen a revolutionary breakthrough in the emergence of these exponential curves.

And the cost has come down 10 percent per year for 30 years. And it's continuing to come down.

Now, the business community has certainly noticed this, because it's crossing the grid parity point. Cheaper solar penetration rates are beginning to rise. Grid parity is understood as that line, that threshold, below which renewable electricity is cheaper than electricity from burning fossil fuels. That threshold is a little bit like the difference between 32 degrees Fahrenheit and 33 degrees Fahrenheit, or zero and one Celsius. It's a difference of more than one degree, it's the difference between ice and water. And it's the difference between markets that are frozen up, and liquid flows of capital into new opportunities for investment. This is the biggest new business opportunity in the history of the world, and two-thirds of it is in the private sector. We are seeing an explosion of new investment. Starting in 2010, investments globally in renewable electricity generation surpassed fossils. The gap has been growing ever since. The projections for the future are even more dramatic, even though fossil energy is now still subsidized at a rate 40 times larger than renewables. And by the way, if you add the projections for nuclear on here, particularly if you assume that the work many are doing to try to break through to safer and more acceptable, more affordable forms of nuclear, this could change even more dramatically.

So is there any precedent for such a rapid adoption of a new technology? Well, there are many, but let's look at cell phones. In 1980, AT&T, then Ma Bell, commissioned McKinsey to do a global market survey of those clunky new mobile phones that appeared then. "How many can we sell by the year 2000?" they asked. McKinsey came back and said, "900,000." And sure enough, when the year 2000 arrived, they did sell 900,000 -- in the first three days. And for the balance of the year, they sold 120 times more. And now there are more cell connections than there are people in the world.

So, why were they not only wrong, but way wrong? I've asked that question myself, "Why?"

And I think the answer is in three parts. First, the cost came down much faster than anybody expected, even as the quality went up. And low-income countries, places that did not have a landline grid -- they leap-frogged to the new technology. The big expansion has been in the developing counties. So what about the electricity grids in the developing world? Well, not so hot. And in many areas, they don't exist. There are more people without any electricity at all in India than the entire population of the United States of America. So now we're getting this: solar panels on grass huts and new business models that make it affordable. Muhammad Yunus financed this one in Bangladesh with micro-credit. This is a village market. Bangladesh is now the fastest-deploying country in the world: two systems per minute on average, night and day. And we have all we need: enough energy from the Sun comes to the Earth every hour to supply the full world's energy needs for an entire year. It's actually a little bit less than an hour. So the answer to the second question, "Can we change?" is clearly "Yes." And it's an ever-firmer "yes."

Last question, "Will we change?" Paris really was a breakthrough, some of the provisions are binding and the regular reviews will matter a lot. But nations aren't waiting, they're going ahead. China has already announced that starting next year, they're adopting a nationwide cap and trade system. They will likely link up with the European Union. The United States has already been changing. All of these coal plants were proposed in the next 10 years and canceled. All of these existing coal plants were retired. All of these coal plants have had their retirement announced. All of them -- canceled. We are moving forward. Last year -- if you look at all of the investment in new electricity generation in the United States, almost three-quarters was from renewable energy, mostly wind and solar.

We are solving this crisis. The only question is: how long will it take to get there? So, it matters that a lot of people are organizing to insist on this change. Almost 400,000 people marched in New York City before the UN special session on this. Many thousands, tens of thousands, marched in cities around the world. And so, I am extremely optimistic. As I said before, we are going to win this.___

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

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 »

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 office box in the SAIF Zone - a tax-free zone in the United Arab Emirates.

This is not the only nonsense paper that Bentham Science Publishers has accepted. They're also notorious for spam where they invite people to conferences - for money.

They claim to be based in the United Arab Emirates, and that's what it says in Wikipedia. However, today I received an email that claims otherwise. I don't know if it's correct, but I think it's worth reading - at least for anyone who cares. I think I will keep the source anonymous, so they don't get in trouble.

My research team gathered facts on the business practices of Pakistan based Bentham Science Publishers.

We would appreciate if you would bring to the attention of your friends and colleagues the following information.

Bentham Science Publishers (BSP) is based in Karachi, Pakistan. It works in Karachi with the name of Information Technology Services (ITS). We found that they adopted this name because the IT services were exempted from taxes there from 2002 to January 2016.

They started organizing international conferences,in 2008 by starting another company called Eureka Conferences. Bentham Science Publishers organize the conferences but pretend that the organizer is Eureka Conferences. Actually Bentham Science staff operates Eureka and both the companies have the same leadership. In such conferences, they invite their authors and editors (many of them from China) and scientific researchers elsewhere. On the conference web page, Bentham Science is mentioned as a media partner.

The entire BSP operations, including the main activities related to conferences (invitations, registrations etc.) are carried out in Karachi. Only subscriptions and invoices are managed in the UAE.They have hired some subscription agents and consultant services in some other countries. They do not place their Pakistan address on their website and tell scientists and librarians that they are located in the UAE/Europe.

For more details, see:

http://scholarlykitchen.sspnet.org/2009/06/10/nonsense-for-dollars/

http://gunther-eysenbach.blogspot.ca/2008/03/black-sheep-among-open-access-journals.html

Here is a Richard Poynder's interview of Matthew Honan, editorial director of Bentham Science Publishers:

http://www.richardpoynder.co.uk/Honan.pdf

Honan refused to say who owns Bentham Science Publishers! That should already be enough to make you suspicious.

___

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

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

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

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

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 »

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 explain how!

Another strange equation

1^2 + 2^2 + 3^2 + … + 24^2 = 70^2

then sets up a curious link between string theory, the Leech lattice (the densest known way of packing spheres in 24 dimensions) and a group called the Monster. I'll do my best to demystify some of these issues without getting very technical.

If you can't come to my talk, you can see a video of an earlier version here:

https://www.youtube.com/watch?v=vzjbRhYjELo

and see the slides here:

http://math.ucr.edu/home/baez/numbers/24_waterloo.pdf

But I hope some of you can stop by and say hi.

___

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

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 »

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 a curve in a multi-dimensional space. The picture it draws is just the two-dimensional "shadow" of this curve.

This presentation will be enhanced by the output of a four-day workshop with University of Waterloo students at the department of Fine Arts, who will build a harmonograph with Anita Chowdry.

Everyone is welcome at this free public lecture!  It will be followed by a reception.   The show starts at 7:30 pm, Friday February 26, 2016 - and it's here:

St. Jerome's University
Siegfried Hall
290 Westmount Road North
Waterloo

You can see a map here:

https://uwaterloo.ca/events/events/bridges-lecture-harmonograph

If you can't make it, don't worry: I think the talk will be videotaped, and I'll put my slides online.   But if you can, that would be great. 

By the way, this lecture is part of the Bridges Lecture Series - a series of talks that try to bridge the gap between the arts and mathematics.  Each lecture features both an artist and a mathematician.

Also by the way: earlier on Friday I'll give a talk on My favorite number at the William G. Davis Centre in room DC 1302.  This will be at 3:30 pm, following refreshments in DC 1301 at 3:00pm.  This is a Pure Mathematics and Combinatorics & Optimization joint colloquium.  More about this later.
 ___

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

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

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___

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

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 »

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 this magnetic field?

With a truly heroic computation, physicists have used QED to compute this quantity up to order α⁵. This required computing and adding up over 13,000 integrals. If we also take other Standard Model effects into account we get agreement with experiment to roughly one part in a trillion!

This is often called the most accurate prediction of science. However, if we continue adding up terms in this power series, there is no guarantee that the answer converges. Indeed, in 1952 Freeman Dyson gave a heuristic argument that makes physicists expect that the series diverges, along with most other power series in QED!

I explain that argument in this blog article. I'm especially happy because I think I've made it a bit more precise. But it's not a proof: just an argument that something very strange must happen if the answer converges.

Currently, the consensus among physicists is that ultimately QED is inconsistent. I explain why. But again, there's no proof. We need some mathematicians to help settle these questions!___

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

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



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

___

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

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

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___

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

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 »

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:

http://www.ligo.org/science/Publication-GW150914/index.php

You can see the gravitational waveforms here:

https://pbs.twimg.com/media/Ca8dpleUAAALI2n.jpg

At left in blue is the wave detected in Livingston, Louisiana. At right in red is the wave detected in Hanford Washington. The detector in Hanford saw the wave a few milliseconds later, so it must have come from the sky in the Southern hemisphere.

#LIGO #astronomy___

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

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 »

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 http://elanormal.com/posts/28051-rumor-mill-heats-up-again-for-discovery-of-gravitational-waves 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?

#gravitywaves  ___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:

https://johncarlosbaez.wordpress.com/2016/02/07/rumors-of-gravitational-waves/

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

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 »

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.

___

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

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 »

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:

http://www.citylab.com/weather/2016/01/eura-finland-map-sky-street-lights/424287/

___

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

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 »

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 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:

https://www.physicsforums.com/insights/struggles-continuum-part-5/

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

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 »

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

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 »

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:

https://golem.ph.utexas.edu/category/2016/01/integral_octonions_part_12.html
___

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

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.

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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