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John Baez

John Baez 

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

Location: Riverside, California

Followers: 57,450

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Views: 49,390,854

Cream of the Crop: 11/05/2011

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Most comments: 153

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2016-05-30 16:00:05 (153 comments; 113 reshares; 332 +1s; )Open 

Let us read what we paid for

Imagine a business like this: you get highly trained experts to give you their research for free... and then you sell it back to them.  Of course these experts need equipment, and they need to earn a living... so you get taxpayers to foot the bill.  

And if the taxpayers want to actually read the papers they paid for?   Then you charge them a big fee!

It's not surprising that with this business model, big publishers are getting rich while libraries go broke.  Reed-Elsevier has a 37% profit margin!

But people are starting to fight back — from governments to energetic students like ‎Alexandra Elbakyan here.

On Friday, the Competitiveness Council —a gathering of European ministers of science, innovation, trade, and industry—said that by 2020, all publicly funded scientific papers published in Europeshould be ma... more »

Most reshares: 113

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2016-05-30 16:00:05 (153 comments; 113 reshares; 332 +1s; )Open 

Let us read what we paid for

Imagine a business like this: you get highly trained experts to give you their research for free... and then you sell it back to them.  Of course these experts need equipment, and they need to earn a living... so you get taxpayers to foot the bill.  

And if the taxpayers want to actually read the papers they paid for?   Then you charge them a big fee!

It's not surprising that with this business model, big publishers are getting rich while libraries go broke.  Reed-Elsevier has a 37% profit margin!

But people are starting to fight back — from governments to energetic students like ‎Alexandra Elbakyan here.

On Friday, the Competitiveness Council —a gathering of European ministers of science, innovation, trade, and industry—said that by 2020, all publicly funded scientific papers published in Europeshould be ma... more »

Most plusones: 332

posted image

2016-05-30 16:00:05 (153 comments; 113 reshares; 332 +1s; )Open 

Let us read what we paid for

Imagine a business like this: you get highly trained experts to give you their research for free... and then you sell it back to them.  Of course these experts need equipment, and they need to earn a living... so you get taxpayers to foot the bill.  

And if the taxpayers want to actually read the papers they paid for?   Then you charge them a big fee!

It's not surprising that with this business model, big publishers are getting rich while libraries go broke.  Reed-Elsevier has a 37% profit margin!

But people are starting to fight back — from governments to energetic students like ‎Alexandra Elbakyan here.

On Friday, the Competitiveness Council —a gathering of European ministers of science, innovation, trade, and industry—said that by 2020, all publicly funded scientific papers published in Europeshould be ma... more »

Latest 50 posts

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2016-07-24 04:19:04 (10 comments; 7 reshares; 60 +1s; )Open 

New kinds of quasiparticles

You can get electrons to behave in many strange ways in different materials.   They act like various kinds of particles... but they're not truly fundamental particles, so they're called quasiparticles

For example, the spin, charge and position of electrons can move in completely independent ways. 

Imagine an audience at a football game holding up signs, and then creating a wave by wiggling their signs.  This wave can move even even if the people stand still! 

Similarly, we can have electrons more or less standing still, with their spins lined up.   Then their spins can wiggle a bit, and this wiggle can move through the material, even though the electrons don't move.  This wave of altered spin can act like a particle!  It's called a spinon.

You can also imagine a hole in a densecrowd of... more »

New kinds of quasiparticles

You can get electrons to behave in many strange ways in different materials.   They act like various kinds of particles... but they're not truly fundamental particles, so they're called quasiparticles

For example, the spin, charge and position of electrons can move in completely independent ways. 

Imagine an audience at a football game holding up signs, and then creating a wave by wiggling their signs.  This wave can move even even if the people stand still! 

Similarly, we can have electrons more or less standing still, with their spins lined up.   Then their spins can wiggle a bit, and this wiggle can move through the material, even though the electrons don't move.  This wave of altered spin can act like a particle!  It's called a spinon.

You can also imagine a hole in a dense crowd of people, moving along as if it were an entity of its own.  When this happens with electrons it's called a holon, or more commonly just a hole.  A hole acts like a particle with positive charge, since electrons have negative charge. 

Since holes have positive charge and electrons have negative charge, they attract.   Sometimes they orbit each other for long enough that this combined thing acts like a particle of its own!   This kind of quasiparticle is called an exciton.

There are also other quasiparticles.  If you're a student who wants to do particle physics, please switch to studying quasiparticles!  The math is almost the same, and you don't need huge particle accelerators to make cool new discoveries.  Some are even useful.

One of the most fundamental things about a quasiparticle, or for that matter an ordinary particle, is its energy.  Its energy depends on its momentum.  The relation between them is called the dispersion relation.  This says a lot about how the quasiparticle acts.

Here in Singapore there's a lab that studies graphene - a crystal made of carbon that's just one atom thick.  When you've got a very thin film like this, a quasiparticle inside it acts like it's living in a 2-dimensional world!   Since it can't go up and down, only 2 components of its momentum can be nonzero - say the x and y components.

Right next door to the +Centre for Quantum Technologies where I'm working in Singapore, there's a lab that specializes in graphene. The picture here shows a graph of energy as a function of momentum for a new kind of quasiparticle they're studying.  They haven't made it in the lab yet; they've just shown it's possible. 

The three colored sheets show that 3 different energies are possible for each momentum - except momentum zero, where all three sheet meet, and also a line of momenta where two sheets meet.

If we only had the green and blue sheets, that would be the dispersion relation for a massless particle.  People already know how to make massless quasiparticles with graphene.

The new thing is the yellow sheet!  This will make very strange things happen, I'm sure.

I got interested in these new quasiparticles thanks to this article pointed out by +rasha kamel:

• Unconventional quasiparticles predicted in conventional crystals, ScienceDaily, https://www.sciencedaily.com/releases/2016/07/160721151219.htm

But I got the picture from here:

• Guoqing Chang et al, New fermions on the line in topological symmorphic metals, http://arxiv.org/1605.06831.

Here's the abstract, for you physics nerds out there:

Abstract. Topological metals and semi-metals (TMs) have recently drawn significant interest. These materials give rise to condensed matter realizations of many important concepts in high-energy physics, leading to wide-ranging protected properties in transport and spectroscopic experiments. The most studied TMs, i.e., Weyl and Dirac semi-metals, feature quasiparticles that are direct analogues of the textbook elementary particles. Moreover, the TMs known so far can be characterized based on the dimensionality of the band crossing. While Weyl and Dirac semimetals feature zero-dimensional points, the band crossing of nodal-line semimetals forms a one-dimensional closed loop. In this paper, we identify a TM which breaks the above paradigms. Firstly, the TM features triply-degenerate band crossing in a symmorphic lattice, hence realizing emergent fermionic quasiparticles not present in quantum field theory. Secondly, the band crossing is neither 0D nor 1D. Instead, it consists of two isolated triply-degenerate nodes interconnected by multi-segments of lines with two-fold degeneracy. We present materials candidates. We further show that triply-degenerate band crossings in symmorphic crystals give rise to a Landau level spectrum distinct from the known TMs, suggesting novel magneto-transport responses. Our results open the door for realizing new topological phenomena and fermions including transport anomalies and spectroscopic responses in metallic crystals with nontrivial topology beyond the Weyl/Dirac paradigm.

Weirdly, I had learned the word 'symmorphic' just yesterday.  Greg Egan are writing a paper on crystals, and he explained that a crystal is symmorphic if it contains a point where every symmetry of the crystal consists of a symmetry fixing this point followed by a translation.   It was important for our work to notice that a diamond is not symmorphic.

#spnetwork arXiv:1605.06831 #condensedMatter #physics  ___

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2016-07-23 02:54:41 (17 comments; 4 reshares; 35 +1s; )Open 

The quest for larger infinities

There are different kinds of bigness.   But they're connected.

There's a fascinating contest where you try to write the computer program of a certain length that would print out the largest possible integer.    This contest was actually carried out on the xkcd blog, and Eliezer Yudkowsky won.  Unless you know more about logic than he does, you won't be able to beat him.

There's another contest where you try to name the largest "computable ordinal", and that's what my post is about:

https://johncarlosbaez.wordpress.com/2016/07/07/large-countable-ordinals-part-3/

And there's another contest where you try to name the largest "cardinal".   Here we get into inaccessible cardinals, indescribable cardinals, huge cardinals, superhuge cardinals and the like. 
But th... more »

The quest for larger infinities

There are different kinds of bigness.   But they're connected.

There's a fascinating contest where you try to write the computer program of a certain length that would print out the largest possible integer.    This contest was actually carried out on the xkcd blog, and Eliezer Yudkowsky won.  Unless you know more about logic than he does, you won't be able to beat him.

There's another contest where you try to name the largest "computable ordinal", and that's what my post is about:

https://johncarlosbaez.wordpress.com/2016/07/07/large-countable-ordinals-part-3/

And there's another contest where you try to name the largest "cardinal".   Here we get into inaccessible cardinals, indescribable cardinals, huge cardinals, superhuge cardinals and the like. 

But these three contests turn out to deeply related!   There's a way to name huge integers using fast-growing functions that you can describe using large computable ordinals.  And Yudkowsky won the contest to write a program that prints out a large integer by taking advantage of a very large cardinal.

So, there's a spooky connection between large finite numbers, large computable ordinals - which are all countable, by the way - and large cardinals, which are not countable.  Many theorems point at this connection, but the full story remains obscure.  I believe when it becomes clear we'll get a whole new idea of what the infnite is all about.

As for me, I need a break.  My post takes you up to the large Veblen ordinal, a whopping large computable ordinal... but I know people have studied others that dwarf this one.  As Bilbo said:

The Road goes ever on and on
Out from the door where it began.
Now far ahead the Road has gone,
Let others follow it who can!
Let them a journey new begin,
But I at last with weary feet
Will turn towards the lighted inn,
My evening-rest and sleep to meet.

#bigness  ___

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2016-07-21 23:28:18 (0 comments; 18 reshares; 95 +1s; )Open 

Republicans for Trump

Cruz caused a stir at the Republican convention by not endorsing Trump.  But here's what other Republicans say:

“He’s a race-baiting, xenophobic religious bigot. He doesn’t represent my party. He doesn’t represent the values that the men and women who wear the uniform are fighting for.” — Senator Lindsey Graham, Republican of South Carolina

“I don’t think this guy has any more core principles than a Kardashian marriage.” — Senator Ben Sasse, Republican of Nebraska

“We saw and looked at true hate in the eyes last year in Charleston. I will not stop until we fight a man that chooses not to disavow the K.K.K. That is not a part of our party.” — Nikki Haley, Republican governor of South Carolina

“Donald Trump is a madman who must be stopped,” — Bobby Jindal, former Republican governor ofLouisiana

“I won’t vo... more »

Republicans for Trump

Cruz caused a stir at the Republican convention by not endorsing Trump.  But here's what other Republicans say:

“He’s a race-baiting, xenophobic religious bigot. He doesn’t represent my party. He doesn’t represent the values that the men and women who wear the uniform are fighting for.” — Senator Lindsey Graham, Republican of South Carolina

“I don’t think this guy has any more core principles than a Kardashian marriage.” — Senator Ben Sasse, Republican of Nebraska

“We saw and looked at true hate in the eyes last year in Charleston. I will not stop until we fight a man that chooses not to disavow the K.K.K. That is not a part of our party.” — Nikki Haley, Republican governor of South Carolina

“Donald Trump is a madman who must be stopped,” — Bobby Jindal, former Republican governor of Louisiana

“I won’t vote for Donald Trump because of who he isn’t. He isn’t a Republican. He isn’t a conservative. He isn’t a truth teller. ... I also won’t vote for Donald Trump because of who he is. A bigot. A misogynist. A fraud. A bully.” — Norm Coleman, former Republican senator from Minnesota

“To support Trump is to support a bigot. It’s really that simple.” — Stuart Stevens, chief strategist to Mitt Romney’s 2012 presidential campaign

“Donald Trump is unfit to be president. He is a dishonest demagogue who plays to our worst fears. Trump would take America on a dangerous journey.” — Meg Whitman, Hewlett-Packard Enterprise C.E.O. and former national finance co-chairwoman for Chris Christie’s presidential campaign

“I thought he was an embarrassment to my party; I think he’s an embarrassment to my country. … I can’t vote for him.” — Tom Ridge, former Republican governor of Pennsylvania and secretary of homeland security under George W. Bush

“I would not vote for Trump, clearly. If there is any, any, any other choice, a living, breathing person with a pulse, I would be there.” — Mel Martinez, former Republican senator from Florida and former chairman of the Republican National Committee

“The G.O.P., in putting Trump at the top of the ticket, is endorsing a brand of populism rooted in ignorance, prejudice, fear and isolationism. This troubles me deeply as a Republican, but it troubles me even more as an American. … Never Trump.” — Henry M. Paulson Jr., Treasury secretary under George W. Bush

“Donald Trump is a phony, a fraud. His promises are as worthless as a degree from Trump University.” — Mitt Romney, 2012 Republican nominee for president

“When you’ve got a guy favorably quoting Mussolini, I don’t care what party you’re in, I’m not voting for that guy.” — Ken Cuccinelli, president of the Senate Conservatives Fund

“Donald Trump is a scam. Evangelical voters should back away... Trump is a misogynist and philanderer. He demeans women and minorities. His preferred forms of communication are insults, obscenities and untruths.” — The Christian Post, a popular U.S. evangelical website

“A moral degenerate.” — Peter Wehner, evangelical Christian commentator who served in last three Republican administrations

“A man utterly unfit for the position by temperament, values and policy preferences … whose personal record of chicanery and wild rhetoric of bigotry, misogyny and misplaced belligerence are without parallel in the modern history of either major party.” — Eliot A. Cohen, a senior State Department official under George W. Bush

“Leaders don’t need to do research to reject Klan support. #NeverTrump” — Ken Mehlman, former chairman of the Republican National Committee

“God bless this man” — Daily Stormer, white supremacist website

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

Sources for all these quotes can be found by clicking on the links here:

http://www.nytimes.com/2016/07/21/opinion/what-republicans-really-think-about-trump.html___

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2016-07-19 02:19:16 (13 comments; 11 reshares; 68 +1s; )Open 

Into the clouds

Frigatebirds are amazing:

Ornithologist Henri Weimerskirch put satellite tags on a couple of dozen frigatebirds, as well as instruments that measured body functions such as heart rate. When the data started to come in, he could hardly believe how high the birds flew.

“First, we found, ‘Whoa, 1,500 meters. Wow. Excellent, fantastique,’ ” says Weimerskirch, who is with the National Center for Scientific Research in Paris. “And after 2,000, after 3,000, after 4,000 meters — OK, at this altitude they are in freezing conditions, especially surprising for a tropical bird.”

Four thousand meters is more than 12,000 feet, or as high as parts of the Rocky Mountains. “There is no other bird flying so high relative to the sea surface,” he says.

Weimerskirch says that kind of flying should take a hugeamount of energy.... more »

Into the clouds

Frigatebirds are amazing:

Ornithologist Henri Weimerskirch put satellite tags on a couple of dozen frigatebirds, as well as instruments that measured body functions such as heart rate. When the data started to come in, he could hardly believe how high the birds flew.

“First, we found, ‘Whoa, 1,500 meters. Wow. Excellent, fantastique,’ ” says Weimerskirch, who is with the National Center for Scientific Research in Paris. “And after 2,000, after 3,000, after 4,000 meters — OK, at this altitude they are in freezing conditions, especially surprising for a tropical bird.”

Four thousand meters is more than 12,000 feet, or as high as parts of the Rocky Mountains. “There is no other bird flying so high relative to the sea surface,” he says.

Weimerskirch says that kind of flying should take a huge amount of energy. But the instruments monitoring the birds’ heartbeats showed that the birds weren’t even working up a sweat. (They wouldn’t, actually, since birds don’t sweat, but their heart rate wasn’t going up.)

How did they do it? By flying into a cloud.

“It’s the only bird that is known to intentionally enter into a cloud,” Weimerskirch says. And not just any cloud—a fluffy, white cumulus cloud. Over the ocean, these clouds tend to form in places where warm air rises from the sea surface. The birds hitch a ride on the updraft, all the way up to the top of the cloud.

But this is far from the only amazing thing about frigatebirds!  For the full story, read this:

https://johncarlosbaez.wordpress.com/2016/07/18/frigatebirds/

You'll also learn the dark side of frigatebirds: they're kleptoparasites.

The quote is from here:

• Christopher Joyce, Nonstop flight: how the frigatebird can soar for weeks without stopping, All Things Considered, National Public Radio, 30 June 2016, http://www.npr.org/sections/thetwo-way/2016/06/30/484164544/non-stop-flight-how-the-frigatebird-can-soar-for-months-without-stopping

and the photo is from here:

https://www.pinterest.com/pin/237353842833650981/

#biology  ___

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2016-07-16 00:51:06 (25 comments; 4 reshares; 68 +1s; )Open 

A winning septic

A septic tank is a system for disposing of sewage.  A septic surface is a surface described by a polynomial equation of degree 7.

This picture by +Abdelaziz Nait Merzouk shows a septic surface discovered by Oliver Labs when he was working on his PhD thesis. 

It looks like a beautiful flower of some strange sort.  But it's famous because it's the septic with the largest known number of points that look like two cones meeting tip to tip. 

How many?  Ninety-nine!  We know that no septic can have more than 104 of these ordinary double points, as they're called.  But we don't know any with more than 99.  So this is currently one of the winners.  There are others, too, also discovered by Labs.

This surface is called the Labs septic, which reminds me of yet another meaning of the word'sep... more »

A winning septic

A septic tank is a system for disposing of sewage.  A septic surface is a surface described by a polynomial equation of degree 7.

This picture by +Abdelaziz Nait Merzouk shows a septic surface discovered by Oliver Labs when he was working on his PhD thesis. 

It looks like a beautiful flower of some strange sort.  But it's famous because it's the septic with the largest known number of points that look like two cones meeting tip to tip. 

How many?  Ninety-nine!  We know that no septic can have more than 104 of these ordinary double points, as they're called.  But we don't know any with more than 99.  So this is currently one of the winners.  There are others, too, also discovered by Labs.

This surface is called the Labs septic, which reminds me of yet another meaning of the word 'septic'.  

Sepsis occurs when harmful bacteria start to grow in tissue.  So, 'septic' also means 'infected with bacteria'... and 'Labs septic' has a strangely medical sound.   But this septic is pure and beautiful.

For more on how the Labs septic was found, and another view of it, visit my blog Visual Insight:

http://blogs.ams.org/visualinsight/2016/07/15/labs-septic/

#geometry  ___

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2016-07-12 04:25:39 (24 comments; 27 reshares; 93 +1s; )Open 

Global warming: demand the truth

After announcements that 2015 was the hottest year on record and February 2016 was the hottest month, the news station CNN aired five times more fossil fuel advertising than actual climate reporting!

So, please sign this petition to CNN.  Tell them: start reporting on climate change.   And please reshare this message.

A study by the group Media Matters showed that the American Petroleum Institute is getting more coverage than actual news about global warming.  This doesn't even include the ads from individual fossil fuel companies and the Koch brothers.

Here's some actual news, in case you hadn't heard:

1) The extent of Arctic sea ice in June was the lowest in recorded history for that month of the year: 260,000 square kilometers less than ever before!   It's on track to break allrecor... more »

Global warming: demand the truth

After announcements that 2015 was the hottest year on record and February 2016 was the hottest month, the news station CNN aired five times more fossil fuel advertising than actual climate reporting!

So, please sign this petition to CNN.  Tell them: start reporting on climate change.   And please reshare this message.

A study by the group Media Matters showed that the American Petroleum Institute is getting more coverage than actual news about global warming.  This doesn't even include the ads from individual fossil fuel companies and the Koch brothers.

Here's some actual news, in case you hadn't heard:

1) The extent of Arctic sea ice in June was the lowest in recorded history for that month of the year: 260,000 square kilometers less than ever before!   It's on track to break all records this year.

2) Recently every month from October until May has been the hottest on record worldwide.  June was the second hottest, since the El Niño is fading.

3) India recorded its hottest day ever on May 19th. The temperature in Phalodi hit 51 degrees Celsius (124 degrees Fahrenheit), and a nationwide drought has affected more than 300 million people marched on, leaving armed guards at dams, and reservoirs well below their usual levels.

4) Alaska, along with the rest of the Arctic, has experienced record-breaking heat this year.  Its average year-to-date temperature has been 5.5C above the long term average.

5) In the atmosphere, carbon dioxide has been increasing every year for decades - but this year the speed of increase is also record-breaking!   The increase for 2016 is expected to be 3.1 parts per million, up from an annual average of 2.1.

6) The Great Barrier Reef, a natural wonder and world heritage site, recently experienced its worst ever coral bleaching event.  An aerial study found that just 7% of the reef escaped bleaching. 

7) A new study in Nature argues that even despite the actions pledged in the Paris Agreement, the Earth is still on course for a temperature increase of 2.6 - 3.1C by the end of this century.  Read this:

http://www.nature.com/nature/journal/v534/n7609/full/nature18307.html

The Paris agreement is a step in the right direction, but we need to ratchet it up.  We can't afford to slack off now.  One piece of the puzzle is clear information about the crisis we're in.

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

Media Matters writes:

In Week After Hottest Year Announcement, CNN Aired Less Than One Minute Of Climate-Related Coverage And 13.5 Minutes Of Oil Industry Ads.

From January 20 to January 26, CNN morning, daytime and primetime programming included only 57 seconds of coverage about climate change or the announcement that 2015 was the hottest year on record. Over that same time period, CNN aired 13.5 minutes of American Petroleum Institute ads. The climate-related segments included one on the January 21 edition of Early Start, in which anchor Christine Romans reported that 2015 was the hottest year on record and that officials say “the planet is still warming with no apparent change in the long term global warming rate.” Additionally, CNN senior legal analyst Jeffrey Toobin briefly mentioned Republican climate science denial during a discussion of Hillary Clinton’s emails on Anderson Cooper 360, and CNN host Fareed Zakaria noted that the “The World Economic Forum said this year that the greatest global risk is the failure of climate change mitigation and adaptation,” during a Fareed Zakaria GPS segment about a study finding that humans have entered a new geological epoch known as the Anthropocene.

Following Announcement That February 2016 Was Most Unusually Hot Month Ever, CNN Aired Four Minutes Of Climate-Related Coverage And 10 Minutes Of Fossil Fuel Ads.

In the one-week period beginning March 17, when NOAA released data showing that February 2016 was the most unusually hot month ever recorded, CNN aired only four minutes of coverage about climate change or the temperature record during its morning, daytime, and primetime coverage. During that same time period, CNN aired ten minutes of American Petroleum Institute ads. On March 18, CNN anchors Christine Romans and John Berman delivered nearly-identical reports on February’s “astounding” temperature record during the 4 a.m. and 5 a.m. editions of Early Start, respectively, but neither explicitly mentioned climate change or the role fossil fuel pollution and other human activities play in driving climate change. The March 20 edition of Fareed Zakaria GPS featured an interview with astronaut Piers Sellers about his climate change advocacy, followed by a brief report about International Energy Administration (IEA) data showing a decline in carbon emissions from energy production, which Zakaria described as “some good news on the climate front” and a “welcome update in the climate battle.” Finally, on the March 20 edition of New Day Sunday, anchor Christi Paul reported that major cities around the world were participating in Earth Hour, an event meant to bring awareness to climate change, by switching off their lights.

For more details see:

http://mediamatters.org/research/2016/04/25/study-cnn-viewers-see-far-more-fossil-fuel-advertising-climate-change-reporting/209985

Here's the data for the statements 1)-6):

https://www.theguardian.com/environment/2016/jun/17/seven-climate-records-set-so-far-in-2016

https://www.theguardian.com/environment/2016/jul/07/arctic-sea-ice-crashes-to-record-low-for-june

http://www.netnewsledger.com/2016/07/05/june-2016-second-hottest-june-ever/___

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2016-07-10 15:13:26 (44 comments; 24 reshares; 124 +1s; )Open 

Gimbal lock

Here you see 3 rotating rings called gimbals.  Gimbals are used in gyroscopes and inertial measurement units, which are gadgets that measure an object's orientation - like a drone, or a spacecraft.   Gimbals are also used to orient thrusters on rockets.

With 3 gimbals, you can rotate the inner one to whatever orientation you want.  The basic reason is that it takes 3 numbers to describe a rotation in 3 dimensional space.  This is a special lucky property of the number 3. 

But when two of the gimbal's axes happen to be lined up, you get gimbal lock.   In other words: you lose the ability to rotate the inner gimbal a tiny bit in any way you want.   The reason is that in this situation, rotating one of the two aligned gimbals has the same effect on the inner gimbal as rotating the other!  

I've always foundgimbal lock... more »

Gimbal lock

Here you see 3 rotating rings called gimbals.  Gimbals are used in gyroscopes and inertial measurement units, which are gadgets that measure an object's orientation - like a drone, or a spacecraft.   Gimbals are also used to orient thrusters on rockets.

With 3 gimbals, you can rotate the inner one to whatever orientation you want.  The basic reason is that it takes 3 numbers to describe a rotation in 3 dimensional space.  This is a special lucky property of the number 3. 

But when two of the gimbal's axes happen to be lined up, you get gimbal lock.   In other words: you lose the ability to rotate the inner gimbal a tiny bit in any way you want.   The reason is that in this situation, rotating one of the two aligned gimbals has the same effect on the inner gimbal as rotating the other!  

I've always found gimbal lock to be a bit mysterious, so I'm trying to demystify it here. 

As the wise heads at Wikipedia point out,

The word lock is misleading: no gimbal is restrained. All three gimbals can still rotate freely about their respective axes of suspension. Nevertheless, because of the parallel orientation of two of the gimbals' axes there is no gimbal available to accommodate rotation along one axis.

Gimbal lock can actually be dangerous!  When it happens, or even when it almost happens, you lose some control over what's going on.

It caused a problem when Apollo 11 was landing on the moon.  This spacecraft had 3 nested gimbals on its inertial measurement unit. The engineers were aware of the gimbal lock problem but decided not to use a fourth gimbal.  They wrote:

"The advantages of the redundant gimbal seem to be outweighed by the equipment simplicity, size advantages, and corresponding implied reliability of the direct three degree of freedom unit."

They decided instead to trigger a warning when the system came close to gimbal lock.  But it didn't work right:

"Near that point, in a closed stabilization loop, the torque motors could theoretically be commanded to flip the gimbal 180 degrees instantaneously. Instead, in the Lunar Module, the computer flashed a 'gimbal lock' warning at 70 degrees and froze the inertial measurment unit at 85 degrees."

The spacecraft had to be manually moved away from the gimbal lock position, and they had to start over from scratch, using the stars as a reference.

After the Lunar Module had landed, Mike Collins aboard the Command Module joked:

"How about sending me a fourth gimbal for Christmas?"

Fun story!  But ultimately, it's all about math.  If you don't like math, stop reading here.

























Don't say I didn't warn you!

Puzzle: show that gimbal lock is inevitable with just 3 gimbals by showing that every smooth map from the 3-torus to SO(3) has at least one point where the rank of its differential drops below 3.

See what I mean?  Math.  This result shows not only that gimbal lock occurs with the setup shown here, but that any scheme of describing a rotation by 3 angles - or more precisely, 3 points on the circle - must suffer gimbal lock.

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

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2016-07-08 08:08:55 (11 comments; 1 reshares; 43 +1s; )Open 

Adventures in Asia

I'm back in Singapore, the land of explosive cuisine.  This is the menu from our favorite Chinese restaurant.  It's on Southbridge Road across from the Sri Mariamman Temple - a popular Hindu temple where they do firewalking on the holiday called Theemithi.  Maybe they do it to cool down after eating here. 

I hadn't known it was called The Explosion Pot Barbecue.  They sell excellent barbecued fish, roast skewers of lamb with cumin, roast chives, dumplings, and other Szechuan delights.  The food is a bit spicy, but I haven't seen any exploding pots, so this may be a mistranslation of something that makes more sense in Chinese. 

As usual I'm working at the +Centre for Quantum Technologies and my wife +Lisa Raphals is teaching at the philosophy department at NUS.  You can see her in the background ordering ourfood.more »

Adventures in Asia

I'm back in Singapore, the land of explosive cuisine.  This is the menu from our favorite Chinese restaurant.  It's on Southbridge Road across from the Sri Mariamman Temple - a popular Hindu temple where they do firewalking on the holiday called Theemithi.  Maybe they do it to cool down after eating here. 

I hadn't known it was called The Explosion Pot Barbecue.  They sell excellent barbecued fish, roast skewers of lamb with cumin, roast chives, dumplings, and other Szechuan delights.  The food is a bit spicy, but I haven't seen any exploding pots, so this may be a mistranslation of something that makes more sense in Chinese. 

As usual I'm working at the +Centre for Quantum Technologies and my wife +Lisa Raphals is teaching at the philosophy department at NUS.  You can see her in the background ordering our food.

Meanwhile, my student +Blake Pollard is in a small town in the hills of Yunnan Province in southern China, helping teach some local students science, English... and American folk songs!  

This seems much more adventurous than what I'm doing.  But he has a good reason for doing it.   His great grandfather, Sam Pollard, was a Methodist missionary in this area - and he invented a script that is still used by the locals:

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

The Miao are an ethnic group that includes the Hmong, Hmub, Xong, and A-Hmao.  These folks live in the borderlands of southern China, northern Vietnam, Laos, Myanmar and Thailand.  The A-Hmao had a legend about how their ancestors knew a system of writing but lost it. According to this legend, the script would eventually be brought back some day.  When Sam Pollard introduced his script for writing A-Hmao, it became extremely popular, and he became a kind of hero.  Blake and his family visited this part of China last year.  He enjoyed it a lot, so he decided to do some teaching there this summer. 

I hope to say more about both our adventures in a while...

Watch firewalking at the Sri Mariamman Temple:

https://www.youtube.com/watch?v=nxPuTKx3OEI

and if you live around here, check out the Explosion Pot Barbecue:

https://www.google.com/maps?daddr=1.282462,103.845405___

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2016-07-07 02:51:27 (35 comments; 46 reshares; 229 +1s; )Open 

Sugihara's illusion explained

Wow!  These plastic cylinders look round - but in the mirror they look diamond-shaped.  If you turn them around, they look diamond-shaped - but in the mirror they look round!

This video was made by Kokichi Sugihara, an engineer at Meiji University in Tokyo.  How did he do it???

To answer this question, we should "science the hell out of it", as Matt Damon said in The Martian.   Figure out how objects change appearance when you look at them in a mirror... and design an object that does this!

So +David Richeson scienced the hell out of it:

https://divisbyzero.com/2016/07/05/sugiharas-circlesquare-optical-illusion/

The basic idea is this.  The top rim of this object is not flat.  More precisely, it's not horizontal: it curves up and down!  This affects how it looks.  If you'relooking ... more »

Sugihara's illusion explained

Wow!  These plastic cylinders look round - but in the mirror they look diamond-shaped.  If you turn them around, they look diamond-shaped - but in the mirror they look round!

This video was made by Kokichi Sugihara, an engineer at Meiji University in Tokyo.  How did he do it???

To answer this question, we should "science the hell out of it", as Matt Damon said in The Martian.   Figure out how objects change appearance when you look at them in a mirror... and design an object that does this!

So +David Richeson scienced the hell out of it:

https://divisbyzero.com/2016/07/05/sugiharas-circlesquare-optical-illusion/

The basic idea is this.  The top rim of this object is not flat.  More precisely, it's not horizontal: it curves up and down!  This affects how it looks.  If you're looking down on this object, you can make part of the top look farther away  by having it be lower. 

But a mirror reflects front and back.   So in the mirror, part of the top looks closer  if it's lower.

By cleverly taking advantage of this, we can make this object look round, but diamond-shaped in the mirror. 

And if we turn it around, this effect is reversed!

Here's a bit more of the math.  +David Richeson gives the details, so I'll try to present just the basic idea. 

Suppose you're making a video.  Suppose you're looking down at an angle of 45 degrees, just as in this video.   Suppose you're videotaping an object that's fairly far away.

Think about one pixel of the object's image on your camera's viewscreen.

Its height on your viewscreen depends on two things.  It depends on how far up  that piece of the object actually is.  But it also depends on how far back  that piece of the object is: how far away it is from your camera.   Things farther away give higher pixels on your viewscreen.

There's a simple formula for how this works:

pixel height = actual object height + actual distance back

(It's only this simple when you're looking down at an angle of 45 degrees and the thing you're videotaping is fairly far away.)

But what if we're looking in a mirror?  You may think a mirror reverses left and right, but that's wrong: it reverses front and back.  So we basically get
 
mirror image pixel height = actual object height - actual distance back

So, you just need to craft an object for which

actual object height + actual distance back

and

actual object height - actual distance back

give two different curves: one round and one a diamond!

But now for some puzzles:

Puzzle 1.  All that sounds fine: by cleverly adjusting the top rim of the object we can make it look different in a mirror.  But look at the bottom of the object!   What's going on there?  How do you explain that?

Puzzle 2.  Sometimes I know the answers to the puzzles I'm posing.  Sometimes I don't.   Do I know the answer to Puzzle 1, or not?

Puzzle 3.  Same question for Puzzle 2.

Finally, I should admit that I simplified the formula for the mirror image pixel height.  Actually we have

mirror image distance back = constant - actual distance back

and thus

mirror image pixel height =
actual object height + mirror image distance back =
actual object height + constant - actual distance back

In other words, I ignored a constant.  This constant is why the whole mirror image looks higher on your viewscreen than the original object!___

posted image

2016-07-06 02:10:25 (31 comments; 16 reshares; 104 +1s; )Open 

♥ ♥ ♥ I love infinity ♥ ♥ ♥

Some infinities are countable, like the number of integers.  Others are uncountable, like the number of points on a line. 

Uncountable infinities are hard to fully comprehend.  For example, even if you think an infinity is uncountable, someone else may consider it countable!  That's roughly what the Löwenheim–Skolem theorem says. 

How is this possible? 

Ultimately, it's because there are only a countable number of sentences in any language with finitely many letters.  So, no matter how much you talk, you can never convince me that you're talking about something uncountable!
 
Now, if we take a really hard-ass attitude, we have to admit we can never actually write infinitely many sentences.   So even countable infinities remain outside our grasp.   However, we come "asclose as we want", i... more »

♥ ♥ ♥ I love infinity ♥ ♥ ♥

Some infinities are countable, like the number of integers.  Others are uncountable, like the number of points on a line. 

Uncountable infinities are hard to fully comprehend.  For example, even if you think an infinity is uncountable, someone else may consider it countable!  That's roughly what the Löwenheim–Skolem theorem says. 

How is this possible? 

Ultimately, it's because there are only a countable number of sentences in any language with finitely many letters.  So, no matter how much you talk, you can never convince me that you're talking about something uncountable!
 
Now, if we take a really hard-ass attitude, we have to admit we can never actually write infinitely many sentences.   So even countable infinities remain outside our grasp.   However, we come "as close as we want", in the sense that we can keep counting

0, 1, 2, 3, 4,  ...

and nothing seems to stop us.  So, while we never actually reach the countably infinite, it's pretty easy to imagine and work with. 

Thus, my favorite infinities are the countable ordinals - in particular, the computable ones.   You can learn to do arithmetic with them.  You can learn to visualize them just as vividly as the set of all natural numbers, which is the first countable ordinal:

ω = {0,1,2,3,4,5,6,7,8,9,...}

For example,

ω+1 = {0,1,2,3,4,5,6,7,8,9,..., ω}

But as you keep trying to understand larger and larger countable ordinals, strange things happen.  You discover that you're fighting your own mind.

As soon as you see a systematic way to generate a sequence of larger and larger countable ordinals, you know this sequence has a limit that’s larger then all of those! And this opens the door to even larger ones….

So, this journey feels a bit like trying to outrace your car’s own shadow as you drive away from the sunset: the faster you drive, the faster it shoots ahead of you. You’ll never win.

On the other hand, you never need  to lose.  You only lose when you get tired.

And that's what I love: it becomes so obvious that the struggle to understand the infinite is a kind of mind game.  But it's a game that allows clear rules and well-defined outcomes, not a disorganized mess.

In this post:

https://johncarlosbaez.wordpress.com/2016/07/04/large-countable-ordinals-part-2/

I'll take you on a tour of countable ordinals up to the Feferman–Schütte ordinal.  Hop in and take a ride!

And if you don't know the Löwenheim–Skolem theorem, you've gotta learn about it.  It's one of the big surprises of early 20th-century logic:

https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem

The pink and the hearts, by the way, are just to scare certain people.

#bigness  ___

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2016-07-05 00:35:02 (48 comments; 17 reshares; 155 +1s; )Open 

Watch Juno meet Jupiter on NASA TV!   Go here:

http://www.ustream.tv/nasahdtv

Here's the timeline - all these times are Eastern Daylight Time (GMT-4):

July 4th, 9:13 p.m. Start of transmission of single frequency “tones” that will provide updates on the spacecraft’s condition.

9:16 p.m. Juno begins turning away from the sun to position the engine in the right direction to slow the spacecraft for its arrival at Jupiter.

10:41 p.m. With the main antenna pointing away from the sun, Juno switches to a smaller antenna for sending the tones.

10:45 p.m. Juno adjusts itself to eliminate any wobbling.

10:56 p.m. Juno speeds up its spin rate from two rotations a minute to five rotations a minute, a process that takes about five minutes.

11:18 p.m.  The main engine beginsfiring!<... more »

Watch Juno meet Jupiter on NASA TV!   Go here:

http://www.ustream.tv/nasahdtv

Here's the timeline - all these times are Eastern Daylight Time (GMT-4):

July 4th, 9:13 p.m. Start of transmission of single frequency “tones” that will provide updates on the spacecraft’s condition.

9:16 p.m. Juno begins turning away from the sun to position the engine in the right direction to slow the spacecraft for its arrival at Jupiter.

10:41 p.m. With the main antenna pointing away from the sun, Juno switches to a smaller antenna for sending the tones.

10:45 p.m. Juno adjusts itself to eliminate any wobbling.

10:56 p.m. Juno speeds up its spin rate from two rotations a minute to five rotations a minute, a process that takes about five minutes.

11:18 p.m.  The main engine begins firing!

11:38 p.m. The spacecraft has slowed down enough to be captured into orbit around Jupiter.

11:53 p.m. The main engine shuts off, leaving Juno in the desired orbit.

11:55 p.m. The spacecraft starts slowing its spin rate back down to two revolutions per minute.

July 5th, 12:07 a.m. Juno changes direction to point its antenna back at Earth.

12:11 a.m. Juno ends the transmission of status tones and switches to its medium-gain antenna.

12:16 a.m. Juno begins transmitting detailed telemetry, although it may take 20 minutes or longer to lock into the signal.

So, the real excitement starts at 11:18 pm on July 4th if you live on the East Coast of the US.  In California this is 8:18 pm, in London it's 4:18 am on July 5th, here in Singapore it's 11:18 am on July 5th, etc.

From the New York Times:

What could possibly go wrong?

Juno blows up.  In August 1993, NASA’s instrument-packed Mars Observer spacecraft vanished. An inquiry concluded that a fuel leak caused the spacecraft to spin quickly and fall out of communication. While Juno’s setup is different, there is always a chance of an explosion with rocket fuel.

The engine doesn’t fire at all. The Japanese probe Akatsuki was all set to arrive at Venus in December 2010, but its engine didn’t fire, and Akatsuki sailed right past Venus. Last year, Akatsuki crossed paths with Venus again, and this time, using smaller thrusters, it was able to enter orbit.

It crashes into something. Jupiter does not possess the majestic rings of Saturn, but it does have a thin of ring of debris orbiting it. Juno will pass through a region that appears clear, but that does not mean it actually is. Even a dust particle could cause significant damage, as Juno will be moving at a speed of 132,000 miles per hour relative to Jupiter.

It flies too close to Jupiter and is ripped to pieces. In one of NASA’s most embarrassing failures, the Mars Climate Orbiter spacecraft, was lost in 1999 because of a mix-up between English and metric units. Climate Orbiter went far deeper into Mars’ atmosphere than planned. On its first orbit, Juno is to pass within 2,900 miles of Jupiter’s cloud tops, so a miscalculation could be catastrophic.

The computer crashes. On July 4 last year, the mission controllers of the New Horizons spacecraft that was about to fly by Pluto experienced some nervous moments when the spacecraft stopped talking to them. The computer on New Horizons crashed while trying to interpret some new commands and compressing some images it had taken, the electronic equivalent of walking while chewing gum.

The controllers put New Horizons back in working order within a few days, and the flyby occurred without a hitch. For Juno, the scientific instruments have been turned off for its arrival at Jupiter. “We turn off everything that is not necessary for making the event work,” said Dr. Levin, the project scientist. “This is very important to get right, so you don’t do anything extra.”

The intense barrage of radiation at Jupiter could knock out Juno’s computer, even though it is shielded in a titanium vault. Usually, when there is a glitch, a spacecraft goes into “safe mode” to await new instructions from Earth, but in this case, that would be too late to save Juno. The spacecraft has been programmed to automatically restart the engine to allow it to enter orbit.

“If that doesn’t go just right, we fly past Jupiter, and of course, that’s not desirable,” Dr. Bolton said.

http://www.nytimes.com/2016/07/05/science/juno-nasa-jupiter-what-to-expect.html

#astronomy  ___

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2016-07-04 03:10:41 (25 comments; 31 reshares; 134 +1s; )Open 

Driving to infinity

A long time ago, before most of you showed up on Google+, I wrote a story about infinity.  It featured a character who was recruited by the US government to fight in the War on Chaos.  His mission was to explore larger and larger infinities.  

You can see that story in my "bigness" collection - lots of posts, each one its own little chapter.

But I keep wanting to talk about infinity - it's endlessly interesting!   I keep learning more about it.  Some posts here by +Refurio Anachro re-ignited my desire to write about it, and now I have.  Here's the first of three articles:

https://johncarlosbaez.wordpress.com/2016/06/29/large-countable-ordinals-part-1/

If you read this, you'll learn about the two basic kinds of infinities discovered by Cantor: cardinals and ordinals.   Then we'llgo on a ... more »

Driving to infinity

A long time ago, before most of you showed up on Google+, I wrote a story about infinity.  It featured a character who was recruited by the US government to fight in the War on Chaos.  His mission was to explore larger and larger infinities.  

You can see that story in my "bigness" collection - lots of posts, each one its own little chapter.

But I keep wanting to talk about infinity - it's endlessly interesting!   I keep learning more about it.  Some posts here by +Refurio Anachro re-ignited my desire to write about it, and now I have.  Here's the first of three articles:

https://johncarlosbaez.wordpress.com/2016/06/29/large-countable-ordinals-part-1/

If you read this, you'll learn about the two basic kinds of infinities discovered by Cantor: cardinals and ordinals.   Then we'll go on a road trip through larger and larger ordinals.

The picture here shows some of the first ones we'll meet on our trip.  Omega, written ω, is the first infinite ordinal:

ω = {0,1,2,3,4,5,6,7,8,9,...}

Each turn of the spiral here takes you to a higher power of omega, and if you go around infinitely many times, you reach omega to the omegath power.   There are many ways to visualize this ordinal, and I explain a few. 

But my road trip will take you much further than that!

In this first episode, we reach an ordinal called epsilon nought, first discovered by Cantor.  In the second episode we'll go up the Feferman–Schütte ordinal.  In the third we'll reach the small Veblen ordinal and even catch a glimpse of the large Veblen ordinal.

All these are countable ordinals, and you can write computer programs to calculate with them, so I consider them just as concrete as the square root of 2.  And yet, they're quite mind-blowing.

#bigness  ___

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2016-07-03 08:00:09 (23 comments; 9 reshares; 64 +1s; )Open 

Scary geometry

This is a view of Barth's decic surface drawn by +Abdelaziz Nait Merzouk.  It's a frightening shape with 345 cone-shaped singularities - the most possible for a surface described by a polynomial of degree 10. 

And yet, despite its nightmarish complexity, this surface is highly symmetrical.  It has the same symmetries as a regular icosahedron!    

For more views of this surface, go here:

http://blogs.ams.org/visualinsight/2016/07/01/barth-decic/

I have no idea how Wolf Barth dreamt up this surface, along with the closely related 'Barth sextic', back in 1994.   The equations describing them feature the golden ratio... but they're complicated.  I bet there's a more conceptual way to get your hands on these surfaces.  If you know it, please tell me!

#geometry  

Scary geometry

This is a view of Barth's decic surface drawn by +Abdelaziz Nait Merzouk.  It's a frightening shape with 345 cone-shaped singularities - the most possible for a surface described by a polynomial of degree 10. 

And yet, despite its nightmarish complexity, this surface is highly symmetrical.  It has the same symmetries as a regular icosahedron!    

For more views of this surface, go here:

http://blogs.ams.org/visualinsight/2016/07/01/barth-decic/

I have no idea how Wolf Barth dreamt up this surface, along with the closely related 'Barth sextic', back in 1994.   The equations describing them feature the golden ratio... but they're complicated.  I bet there's a more conceptual way to get your hands on these surfaces.  If you know it, please tell me!

#geometry  ___

posted image

2016-07-02 00:20:03 (25 comments; 30 reshares; 119 +1s; )Open 

Jupiter orbit insertion

On the 4th of July, a NASA spacecraft named Juno will try to start orbiting Jupiter.  It has traveled for 5 years and 2.8 billion kilometers to get there.  This is going to be exciting!

Juno will try to aim its main engine towards the Sun, turn it on for 35 minutes, and slow down to 58 kilometers per second, so it can be captured by Jupiter's gravitational field.   Says the lead scientist:

“There’s a mixture of tension and anxiety because this is such a critical maneuver and everything is riding on it. We have to get into orbit. The rocket motor has to burn at the right time, in the right direction, for just the right amount of time.”

With luck, Juno will enter a highly eccentric polar orbit, and make 37 orbits lasting 14 days each.   Each time it will dive down to just 4000 kilometers above Jupiter'scloud tops... more »

Jupiter orbit insertion

On the 4th of July, a NASA spacecraft named Juno will try to start orbiting Jupiter.  It has traveled for 5 years and 2.8 billion kilometers to get there.  This is going to be exciting!

Juno will try to aim its main engine towards the Sun, turn it on for 35 minutes, and slow down to 58 kilometers per second, so it can be captured by Jupiter's gravitational field.   Says the lead scientist:

“There’s a mixture of tension and anxiety because this is such a critical maneuver and everything is riding on it. We have to get into orbit. The rocket motor has to burn at the right time, in the right direction, for just the right amount of time.”

With luck, Juno will enter a highly eccentric polar orbit, and make 37 orbits lasting 14 days each.   Each time it will dive down to just 4000 kilometers above Jupiter's cloud tops, closer than we've ever come!  Each time it will shoot back up to a height of 2.7 million kilometers.   It will map Jupiter using many instruments.  The first dive is scheduled for August.

Juno will gradually be damaged by Jupiter's intense radiation, even though the main computer is encased in a 200-kilogram titanium box.   After its last orbit, it will deliberately plunge to its death - so that it has no chance of contaminating the oceans of Europa.

Juno has already entered Jupiter's magnetosphere - the region of space dominated by Jupiter's powerful magnetic field.  You can hear it here:

http://www.cnet.com/news/the-sounds-of-juno-approaching-jupiter-are-totally-spooky/

For details of Juno's trajectory, go here:

http://spaceflight101.com/juno/juno-mission-trajectory-design/

And watch the NASA "preview" here.  It's like the preview of a science fiction movie, emphasizing the dangers rather than the potential rewards - but it's fun.

The Jupiter orbit insertion should begin at 03:18 July 5th UTC, which is 20:18 on the 4th of July in California.

#astronomy  ___

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2016-06-30 04:59:55 (23 comments; 20 reshares; 84 +1s; )Open 

Zero to the zero

What's zero to the zeroth power?  People love to argue about this question.   Wimps - that is, wisely cautious people - say that it's undefined. Bolder people take bolder views.  +David Tanzer has written a nice article about this puzzle:

https://johncarlosbaez.wordpress.com/2016/06/25/a-quirky-function/

and various readers, including the redoubtable Greg Egan, have posted beautiful animations in the comments. 

The animation here was made by one of those wonderful pseudonymous internet beings - in this case, someone named "etatoby".  It's a graph of the surface

z = x ʸ

in the uncontroversial region, namely where x > 0.   The horizontal arrows point along the x and y axes, forming the boundary of the first quadrant: the region where x > 0 and y > 0.  

If you lookcarefull... more »

Zero to the zero

What's zero to the zeroth power?  People love to argue about this question.   Wimps - that is, wisely cautious people - say that it's undefined. Bolder people take bolder views.  +David Tanzer has written a nice article about this puzzle:

https://johncarlosbaez.wordpress.com/2016/06/25/a-quirky-function/

and various readers, including the redoubtable Greg Egan, have posted beautiful animations in the comments. 

The animation here was made by one of those wonderful pseudonymous internet beings - in this case, someone named "etatoby".  It's a graph of the surface

z = x ʸ

in the uncontroversial region, namely where x > 0.   The horizontal arrows point along the x and y axes, forming the boundary of the first quadrant: the region where x > 0 and y > 0.  

If you look carefully, you can see that x ʸ = 0 when x = 0 and y > 0. 

And if you look harder, you can see that x ʸ takes a constant positive value when x > 0 and y = 0.  This is consistent with the fact that x ⁰ = 1 in this case.

But the really interesting part is that x ʸ approaches all possible positive values as x and y both get close to zero.  This is the main reason that cautious people say 0 ⁰ is undefined.

Puzzle: when we say someone is "redoubtable", we don't mean that you can doubt them again.  We mean they're formidable or awe-inspiring.  So what's the etymology of this word?

Googling is cheating. ___

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2016-06-28 22:42:52 (29 comments; 27 reshares; 100 +1s; )Open 

Metaculus - a prediction website

Metaculus is a website where you can ask about future events and predict their probabilities.  The "wisdom of crowds" says that this is a pretty reasonable way to divine the future.  But some people are better predictors than others, and this skill can be learned.  Check it out:

http://www.metaculus.com/questions/

Metaculus was set up by two professors at U.C. Santa Cruz.  Anthony Aguirre, a physicist, is a co-founder of the Foundational Questions Institute, which tries to catalyze breakthrough research in fundamental physics, and the Future of Life Institute, which studies disruptive technologies like AI.  Greg Laughlin, an astrophysicist, is an expert at predictions from the millisecond predictions relevant to high-frequency trading to the ultra-long-term stability of the solar system.

I've askedand ... more »

Metaculus - a prediction website

Metaculus is a website where you can ask about future events and predict their probabilities.  The "wisdom of crowds" says that this is a pretty reasonable way to divine the future.  But some people are better predictors than others, and this skill can be learned.  Check it out:

http://www.metaculus.com/questions/

Metaculus was set up by two professors at U.C. Santa Cruz.  Anthony Aguirre, a physicist, is a co-founder of the Foundational Questions Institute, which tries to catalyze breakthrough research in fundamental physics, and the Future of Life Institute, which studies disruptive technologies like AI.  Greg Laughlin, an astrophysicist, is an expert at predictions from the millisecond predictions relevant to high-frequency trading to the ultra-long-term stability of the solar system.

I've asked and answered a few questions there.  It's fun, and it will get more fun as more people take it seriously!   Here's some stuff from their latest report:

Dear Metaculus Users,

We recently logged our 10,000th prediction. Not quite Big Data (which will take lots more growth), but we’re making progress! With this milestone passed, it seems like a good time to share an overview of our results
.
First, the big picture. This can be summarized with a single histogram that shows the distribution of the first 10,042 predictions on our first 146 questions. Unambiguously, the three most popular predictions are 1%, 50% and 99%, with spikes of varying strength at each multiple of 5%. There’s a definite overall skew toward lower percentages. This phenomenon stems in part from the fact that the subset of provocative low-probability questions is most naturally worded in a way that the default outcome is negative, e.g., Question: Will we confirm evidence for megastructures orbiting the star KIC 8462852? (Answer: No.) The histogram also makes the point that while 99% confidence — the equivalent of complete confidence -- is very common, it’s very rare that anyone is ever 98% sure about anything. One takeaway from the pileup at 1% and 99% is that we could use more possible values there, so we plan to introduce an expanded range, from 0.1% to 99.9% soon — but as cautioned below, be careful in using it. Excluding the 1% and 99% spikes and smoothing a bit, the prediction distribution turns out to be a pretty nice gaussian, illustrating the ubiquitous effect of the law of large numbers.

The wheels of Metaculus are grinding slowly, but they grind very fine. Almost 80% of the questions that have been posed on site are still either active (open), or closed (pending resolution) We are starting, however, to get meaningful statistics on questions that have resolved to date — a collection that spans a wide range of topics (from Alpha Go to LIGO and from VIX to SpaceX). We’ve been looking at different metrics to evaluate collective predictive success. A simple approach is to chart the fraction of outcomes that actually occurred, after aggregating over all of the predictions in each percentage bin. In the limit of a very large number of optimally calibrated predictions on a very large number of questions, the result would be the straight line shown in gold on Figure 2 below. It’s clear that the optimal result compares quite well to the aggregation produced by the Metaculus user base. Error bars are 25% and 75% confidence intervals, based on bootstrap resampling of the questions. The only marginally significant departure from the optimal result comes at the low end: as a whole, the user base has been slightly biased toward pessimism, assigning a modest overabundance of low probabilities to events that actually wound up happening. In particular, the big spike in the 1% bin in Figure 1 isn’t fully warranted. (This is also somewhat true at 99%: these predictions have come true 90% of the time.) Take-away: if you’re inclined to pull the slider all the way to the left or even right, give it a second thought...

It has been demonstrated that the art of successful prediction is a skill that can be learned. Predictors get better over time, and so it’s interesting to look at the performance of the top predictors on Metaculus, as defined by users with a current score greater than 500. The histogram of predictions for the subset of top users shows some subtle differences with the histogram of all the predictions. The top predictors tend to be more equivocal. The 50% bin is still highly prominent, whereas the popularity of 1% votes is quite strongly diminished.

I recently predicted - not on Metaculus - that Hillary Clinton has a 99% chance of getting the Democratic nomination.  Maybe I should have said 98%.  But I definitely should put my prediction on Metaculus!  This could develop into a useful resource.

If you want to become a "super-forecaster", you need to learn about the Good Judgment Project.  Start here:

http://www.npr.org/sections/parallels/2014/04/02/297839429/-so-you-think-youre-smarter-than-a-cia-agent

A little taste:

For the past three years, Rich and 3,000 other average people have been quietly making probability estimates about everything from Venezuelan gas subsidies to North Korean politics as part of the Good Judgment Project, an experiment put together by three well-known psychologists and some people inside the intelligence community.

According to one report, the predictions made by the Good Judgment Project are often better even than intelligence analysts with access to classified information, and many of the people involved in the project have been astonished by its success at making accurate predictions.

Then read Philip Tetlock's books Expert Political Judgment and Superforecasting: The Art and Science of Prediction.  I haven't!   But I would like to become a super-forecaster.___

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2016-06-27 15:44:24 (20 comments; 25 reshares; 143 +1s; )Open 

Why bees are fuzzy

The fuzz on bees helps them collect pollen.  But it may also help them detect electric fields!

The surprising part - to me - is that flowers have electric fields.   And different kinds of flowers have noticeably different fields.

Gregory Sutton, a  biomechanical engineer who is studying this, says that flower petals tend to accumulate electric charge.  So, they produce an electric field just like when you rub a balloon on a woolly sweater - but smaller:

“It’s a very small electrical field, which is why we're quite astounded that bees can actually detect it,” Sutton says. “And there is different charge distribution at different locations on the petals of different species of flowers. So two flowers of the same species will have a similar electric field, whereas two flowers of a different species will have differentelectric fi... more »

Why bees are fuzzy

The fuzz on bees helps them collect pollen.  But it may also help them detect electric fields!

The surprising part - to me - is that flowers have electric fields.   And different kinds of flowers have noticeably different fields.

Gregory Sutton, a  biomechanical engineer who is studying this, says that flower petals tend to accumulate electric charge.  So, they produce an electric field just like when you rub a balloon on a woolly sweater - but smaller:

“It’s a very small electrical field, which is why we're quite astounded that bees can actually detect it,” Sutton says. “And there is different charge distribution at different locations on the petals of different species of flowers. So two flowers of the same species will have a similar electric field, whereas two flowers of a different species will have different electric fields.”

Together with biophysics researcher Erica Morley and some other scientists, Sutton did experiments to test the theory that bees use electric fields to help find food.  

They built 10 flowers with the same shape, size and smell. They put sugar water on some of the flowers and then added small static electric fields to those flowers. On the rest of the flowers, they put bitter water and no electric field. They let the bees loose among the flowers and kept moving the flowers around so the bees couldn’t learn the location of the sugar water.
 
“As they forage, they start to go to the flowers with the sugar water 80 percent of the time,” Sutton says. “So you know they've figured out the difference between the two sets of flowers. The last step is you just turn off the voltage and then check to see if they can continue telling the difference. And when we turned off the voltage, they were unable to tell the difference. And that's how we knew it was the voltage itself that they were using to tell the difference between the flowers.”

It's good that they did this last step, because otherwise I'd be unconvinced.  They also studied the mechanism that bees use to detect electric fields.  Basically, bee hairs get pulled by an electric field, and the bee can feel it:

“What we found in bees is that they're using a mechanic receptor,” Morley says. “It's not a direct coupling of this electrical signal to the sensory system. They’re using mechanical movement of hair in a very non-conductive medium. Air doesn't conduct electricity very well — it's very resistive. So these hairs have moved in response to the field, which then causes the nerve impulses from the cells at the bottom of the hair.”

I love results like this, which show the world is bigger and more interesting than I thought.   But I'm a bit suspicious too, so I hope more scientists try to replicate these experiments or poke holes in them.

The paper is open-access, so if you have questions you can read it yourself!

• Gregory P. Sutton, Dominic Clarke, Erica L. Morley and Daniel Robert, Mechanosensory hairs in bumblebees (Bombus terrestris) detect weak electric fields, Proc. Nat. Acad. Sci. 113 (2016), 7261–7265. http://www.pnas.org/content/early/2016/05/25/1601624113.full

I got my quotes from here:

http://www.pri.org/stories/2016-06-26/flowers-give-electrical-signals-bees

#spnetnwork   #bees #mechanosensory  ___

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2016-06-24 16:58:40 (85 comments; 4 reshares; 105 +1s; )Open 

Youth

A lynx kitten bounds forward, confident and focused.

I need this picture today, to cheer myself up.  I don't like the Brexit.  The very best possible interpretation I can put on it is that it's ordinary folks poking a stick in the eye of the elite, demanding more local control of government, more democracy.   Maybe the elite will wake up, stop trying to hog all the wealth, and realize that in the long run it pays to help the downtrodden.  

Maybe London will become less dominated by corrupt financiers.   Maybe Scotland will become independent and join the EU.   

I can imagine a wave of decentralization and localization being a good thing.... if  it's balanced by the right larger-scale structures, allowing plenty of free trade, free movement of people, and so on.   But I don't get any sense that the Brexiters have aconstructiv... more »

Youth

A lynx kitten bounds forward, confident and focused.

I need this picture today, to cheer myself up.  I don't like the Brexit.  The very best possible interpretation I can put on it is that it's ordinary folks poking a stick in the eye of the elite, demanding more local control of government, more democracy.   Maybe the elite will wake up, stop trying to hog all the wealth, and realize that in the long run it pays to help the downtrodden.  

Maybe London will become less dominated by corrupt financiers.   Maybe Scotland will become independent and join the EU.   

I can imagine a wave of decentralization and localization being a good thing.... if  it's balanced by the right larger-scale structures, allowing plenty of free trade, free movement of people, and so on.   But I don't get any sense that the Brexiters have a constructive vision for the future. 

Back to the theme of youth:

The young are generally bolder, less careful, less fearful.  It's got pros and cons.

75% of British people between ages 18 and 24 said they voted for Britain to stay in the EU.   For people 25-49 it was 56%.  For people 50-64 it was 44%.  For people above 65, just 39%.

So this is an interesting case.  Perhaps the old are more fearful - of refugees, of Polish plumbers, of EU bureaucrats - but in this case they were more eager to do something rash.   It's quite amazing how little is known about what will happen next!    About all we be sure about is that it will create a big mess.

Good luck, Britain!  Good luck, EU!___

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2016-06-23 15:33:57 (22 comments; 11 reshares; 77 +1s; )Open 

Superstar

This is the small stellated dodecahedron.   It's like a star made of stars.   It has 12 pentagrams, 5-pointed stars, as faces.  These stars cross over each other.  Five meet at each sharp corner.

But here's the really cool part: you should think of each pentagram as a pentagon that's been mapped into space in a very distorted way, with a 'branch point of order 2' at its center.

What does that mean?  

Stand at the center of a pentagon!   Measure the angle you see between two corners that are connected by an edge.  You'll get 2π/5.   But now stand at the center of a pentagram.   Measure the angle you see between two corners that are connected by an edge.  You get 4π/5.  Twice as big! 

So, to map a pentagon into space in a way that makes it look like a pentagram, you need to wrap ittwice around it... more »

Superstar

This is the small stellated dodecahedron.   It's like a star made of stars.   It has 12 pentagrams, 5-pointed stars, as faces.  These stars cross over each other.  Five meet at each sharp corner.

But here's the really cool part: you should think of each pentagram as a pentagon that's been mapped into space in a very distorted way, with a 'branch point of order 2' at its center.

What does that mean?  

Stand at the center of a pentagon!   Measure the angle you see between two corners that are connected by an edge.  You'll get 2π/5.   But now stand at the center of a pentagram.   Measure the angle you see between two corners that are connected by an edge.  You get 4π/5.  Twice as big! 

So, to map a pentagon into space in a way that makes it look like a pentagram, you need to wrap it twice around its central point.   That's what a branch point of order 2 is all about.

That's the cool way to think of this shape you see spinning before you.  It's a surface made of 12 pentagons, each wrapped twice around its center, with 5 meeting at each sharp corner. 

There's another way to think about this surface!   Any equation of this sort

z⁵ + pz + q = 0

has 5 solutions, or roots.   To make this true we need to bend the rules a bit.  First, we let the solutions be complex numbers...  so let p and q be complex too.  Second, we must allow for the possibility of repeated roots: when you factor z⁵ + pz + q, the same root may show up twice.

Now here's the cool part: the small stellated dodecahedron is the set of all lists of 5 numbers that are roots of some equation of this form:

z⁵+ pz + q = 0

So it's not just a pretty star-shaped thing.  It's a serious mathematical entity!    It's actually a Riemann surface, the most symmetrical Riemann surface with 4 holes!   You can build it starting from a tiling of the hyperbolic plane by pentagons.  In this tiling 5 pentagons meet at each corner - just like 4 squares meet at each corner in a square tiling of the ordinary plane.

It's all about the number 5, which has a lot of star power.  To understand more, read my blog article:

http://blogs.ams.org/visualinsight/2016/06/15/small-stellated-dodecahedron/

Most of this was discovered by Felix Klein in 1877.   He discovered lots of cool facts like this.  It's almost annoying.  I keep learning cool things about Riemann surfaces and the hyperbolic plane... and it keeps turning out they were discovered by Klein.    He found more than his fair share. 

By the way, this post and many others are now part of my "geometry" collection.   If you want to binge on beauty, go there now:

https://plus.google.com/collection/UIQgaB

But beware: next morning you may wake up in a gutter with a headache, seeing stars.

This image was created by someone named 'Cyp' and placed on Wikicommons.

#geometry  ___

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2016-06-21 17:38:01 (0 comments; 24 reshares; 116 +1s; )Open 

Broke — but self-funding

During the primaries, Trump claimed he was rich and couldn't be bought.  He said he wouldn't have a super-PAC.   Now he has a lot of super-PACs - all fighting each other.   But his campaign has very little cash!  

In May he tweeted:

Good news is that my campaign has perhaps more cash than any campaign in the history of politics.

But this was a lie.   By the end of May his campaign had less than $1.3 million.  At least, that's what he reported to the Federal Election Commission.  

That may sound like a lot if you don't know US politics.  But Clinton, by comparison, had $42 million.   Even Ben Carson - remember that guy, the nutty candidate who claimed the pyramids were built for storing grain? - had $1.7 million when he quit back in March.

So, by US standards, Trump'scampaign is brok... more »

Broke — but self-funding

During the primaries, Trump claimed he was rich and couldn't be bought.  He said he wouldn't have a super-PAC.   Now he has a lot of super-PACs - all fighting each other.   But his campaign has very little cash!  

In May he tweeted:

Good news is that my campaign has perhaps more cash than any campaign in the history of politics.

But this was a lie.   By the end of May his campaign had less than $1.3 million.  At least, that's what he reported to the Federal Election Commission.  

That may sound like a lot if you don't know US politics.  But Clinton, by comparison, had $42 million.   Even Ben Carson - remember that guy, the nutty candidate who claimed the pyramids were built for storing grain? - had $1.7 million when he quit back in March.

So, by US standards, Trump's campaign is broke.  

And he keeps putting campaign money back into his own pocket!

Throughout his campaign, up to the end of May, he has given $6.2 million of campaign funds to companies he owns.  That's roughly 10% of his campaign spending so far.    And in May this rose to almost 20%: he spent $6.7 million on his campaign, but over $1 million of that went to his own companies.

According to the Huffington Post:

The most striking expenditure in the new filings was $423,372, paid by the Trump campaign for rentals and catering at Trump’s 126-room Palm Beach, Florida, mansion, Mar-A-Lago, which Trump operates as a private club.

Other Trump-owned recipients of campaign funds include Trump Restaurants, which raked in $125,080 in rent and utilities; Trump Tower Commercial, which charged $72,800 in rent and utilities in the building that houses Trump’s campaign headquarters; the Trump National Golf Club, in Jupiter, Florida, which collected $35,845 for facilities rental and catering; and the Trump International Golf Club in West Palm Beach, Florida, which billed the campaign for $29,715, for facilities rentals and catering.

So, Trump has given a whole new meaning to the term "self-funding".  In 2000, he said:

It's very possible that I could be the first presidential candidate to run and make money on it.

It seems that Trump plans to let the Republican National Committee pay for most of his campaign.  They've got some money: they started June with $20 million in cash.  But four years ago at this time, they had more than $60 million.  Their big donors are shying away from Trump.
 
I would love to get money out of US politics.  I hadn't expected Trump to take the lead. 

Here is his May report to the Federal Election Commission:

http://docquery.fec.gov/cgi-bin/forms/C00580100/1079423/

Here is the Huffington Post article:

http://www.huffingtonpost.com/entry/trump-campaign-payments_us_5768a69ee4b0853f8bf1fe2d

Here is an article on Trump's super-PACs:

http://www.motherjones.com/politics/2016/05/donald-trump-super-pac-problem

Trump's boast that his might be the first presidential campaign to make
money:

http://www.theatlantic.com/politics/archive/2016/05/trumps-self-funding-lie/482691/

More figures from here:

http://www.wsj.com/articles/hillary-clintons-war-chest-grows-aided-by-super-pac-1466464804

http://www.msnbc.com/rachel-maddow-show/more-ways-one-the-trump-campaign-broke___

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2016-06-11 17:54:44 (17 comments; 24 reshares; 167 +1s; )Open 

Big stick insect

One of the world's largest insects lives in Australia.   It looks like a stick and it's called Ctenomorpha gargantua.   It's very hard to find, because it lives in the highest parts of the rainforests in Queensland, and it's only active at night! 

In 2014 one fell down and was found hanging on a bush.   Scientists took it to the Museum Victoria, in Melbourne.  They named it Lady Gaga-ntuan.   Now it has a daughter that's 0.56 meters long - that is, 22.2 inches long.  

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

#biology

Big stick insect

One of the world's largest insects lives in Australia.   It looks like a stick and it's called Ctenomorpha gargantua.   It's very hard to find, because it lives in the highest parts of the rainforests in Queensland, and it's only active at night! 

In 2014 one fell down and was found hanging on a bush.   Scientists took it to the Museum Victoria, in Melbourne.  They named it Lady Gaga-ntuan.   Now it has a daughter that's 0.56 meters long - that is, 22.2 inches long.  

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

#biology___

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2016-06-09 15:42:47 (46 comments; 5 reshares; 50 +1s; )Open 

Silly sounding elements

Forget Trump. We have until November to prevent scientists from naming an element oganesson

I don't have anything against Yuri Oganessian, a pioneer in the study of  highly radioactive, short-lived elements.   I just think the word "oganesson" stumbles off the tongue like a dazed, jet-lagged passenger staggering off a plane and falling down the stairway. 

And it's a noble gas!   A noble gas should sound noble.  

Neon.  Argon.  Krypton.  Xenon.  Oganesson.  

Which one does not belong?  Which name was created by somebody without a shred of poetry in their soul?

The International Union of Pure and Applied Chemistry, or IUPAC, has begun a five-month public review, ending 8 November 2016, before it names these elements:

Element 113: Nihonium (Nh)
Element 115,Moscovium (Mc)... more »

Silly sounding elements

Forget Trump. We have until November to prevent scientists from naming an element oganesson

I don't have anything against Yuri Oganessian, a pioneer in the study of  highly radioactive, short-lived elements.   I just think the word "oganesson" stumbles off the tongue like a dazed, jet-lagged passenger staggering off a plane and falling down the stairway. 

And it's a noble gas!   A noble gas should sound noble.  

Neon.  Argon.  Krypton.  Xenon.  Oganesson.  

Which one does not belong?  Which name was created by somebody without a shred of poetry in their soul?

The International Union of Pure and Applied Chemistry, or IUPAC, has begun a five-month public review, ending 8 November 2016, before it names these elements:

Element 113: Nihonium (Nh)
Element 115, Moscovium (Mc)
Element 117, Tennessine (Ts)
Element 118, Oganesson (Og)

I have no opinions except about how the names sound and look.  Nihonium and Moscovium sound okay to me.   The word "Tennessine" is awkward.   I like the state of Tennessee.  I have nothing against it having its own highly radioactive element.   I just don't like this word.  This element is a halogen, and again it's the least pretty of the bunch:

Fluorine.  Chlorine.   Bromine.   Iodine.   Astatine.   Tennessine.

But "oganesson" is worse.  IUPAC could have done better hiring that unemployed guy who used to make up names for elements on Star Trek.  The only good thing about "oganesson" is that it has such a short half-life that we'll hardly ever need to say that word. 

I would gladly accept tennessine if it would stop "oganesson" from lurching onto the periodic table.  In fact, I'd volunteer to eat the world's entire supply of tennessine.

If you agree with me, or have other opinions, write a polite letter to Dr. Lynn M. Soby, the Executive Director of IUPAC, at

secretariat@iupac.org

PS - yes, I know helium is also a noble gas.  People gave it a name suitable for a metal, not a noble gas, before they knew better.  It's too late to call it helion.___

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2016-06-07 16:57:41 (51 comments; 41 reshares; 131 +1s; )Open 

Does dark matter have dark hair?

By now there's a lot of evidence that dark matter exists, but not so much about what it is.  The most popular theories say it's some kind of particles that don't interact much with ordinary matter, except through gravity.  These particles would need to be fairly massive - as elementary particles go - so that despite having been hot and energetic shortly after the Big  Bang, they'd move slow enough to bunch up thanks to gravity.  Indeed, the bunching up of dark matter seems necessary to explain the formation of the visible galaxies!  

Searches for dark matter particles have not found much.  The DAMA experiment, a kilometer underground in Italy, seemed to detect them.  Even better, it saw more of them in the summer, when the Earth is moving faster relative to the Milky Way, than in the winter.  That's just what you'dexpect!... more »

Does dark matter have dark hair?

By now there's a lot of evidence that dark matter exists, but not so much about what it is.  The most popular theories say it's some kind of particles that don't interact much with ordinary matter, except through gravity.  These particles would need to be fairly massive - as elementary particles go - so that despite having been hot and energetic shortly after the Big  Bang, they'd move slow enough to bunch up thanks to gravity.  Indeed, the bunching up of dark matter seems necessary to explain the formation of the visible galaxies!  

Searches for dark matter particles have not found much.  The DAMA experiment, a kilometer underground in Italy, seemed to detect them.  Even better, it saw more of them in the summer, when the Earth is moving faster relative to the Milky Way, than in the winter.  That's just what you'd expect!  But other similar experiments haven't seen anything.  So most physicists doubt the DAMA results.  

Maybe dark matter is not made of massive weakly interacting particles.  Maybe it's a superfluid made of light but strongly interacting particles.  Maybe there are lot more 25-solar-mass black holes than most people think!  There are lots of theories, and I don't have time to talk about them all.  

I just want to tell you about a cool idea which assumes that dark matter is made of massive weakly interacting particles.  It's still the most popular theory, so we should take it seriously and ask: if they exist, what would these particles do?

In the early Universe they'd attract each other by gravity.  They'd bunch up, helping seed the formation of galaxies.  But after stars and planets formed, they'd pull at the dark matter, making it thicker in some places, thinner in others.  

And this is something we can simulate using computers!  After all, the relevant physics is well-understood: just Newton's law of gravity, applied to stars, planets and zillions of tiny dark matter particles.  

Gary Prezeau of NASA's Jet Propulsion Laboratory did these simulations and discovered something amazing. 

When dark matter flows past the Earth, it gets deflected and focused by the Earth's gravity.  Like light passing through a lens, it gets intensely concentrated at certain locations!

This creates long thin 'hairs' where the density of dark matter is enhanced by a factor of 10 million.   Each hair is densest at its 'root'.   At the root, the density of dark matter is about a billion times greater than average!

The hairs in this picture are not to scale: the Earth is drawn too big.   The roots of the hairs would be about a million kilometers from Earth, while the Earth's radius is only 6,400 kilometers.  

Of course we don't know dark matter particles exist.  What's cool is that if they exist, it forms such beautiful structures!  And if we could do a dark matter search in space, near one of these possible roots, we might have a better chance of finding something.   

Let me paraphrase Prezeau, because the real beauty is in the details.  From his abstract:

It is shown that compact bodies form strands of concentrated dark matter filaments henceforth simply called 'hairs'. These hairs are a consequence of the fine-grained stream structure of dark matter halos surrounding galaxies, and as such they constitute a new physical prediction of the standard model of cosmology. Using both an analytical model of planetary density and numerical simulations (a fast way of computing geodesics) with realistic planetary density inputs, dark matter streams moving through a compact body are shown to produce hugely magnified dark matter densities along the stream velocity axis going through the center of the body. Typical hair density enhancements are 10^7 for Earth and 10^8 for Jupiter. The largest enhancements occur for particles streaming through the core of the body that mostly focus at a single point called the root of the hair. For the Earth, the root is located at about 10^6 kilometers from the planetary center with a density enhancement of around 10^9 while for a gas giant like Jupiter, the root is located at around 10^5 kilometes with a enhancement of around 10^11. Beyond the root, the hair density precisely reflects the density layers of the body providing a direct probe of planetary interiors.

The mathematicians and physicists among you may enjoy even more detail.  Again, I'll paraphrase:

According to the standard model of cosmology, the velocity dispersion of cold dark matter (CDM) is expected to be greatly suppressed as the universe expands and the CDM collisionless gas cools.  In particular, for a weakly interacting mass particle with mass of 100 GeV that decoupled from normal matter when the Universe cooled to an energy of 10 MeV per particle, the velocity dispersion is only about 0.0003 meters per second.

As the Universe cools and the nonlinear effects of gravity become more prominent and galactic halos grow, the dispersion of velocities will increase somewhat, but 10 kilometers per second is an upper limit on the velocity dispersion of the resulting dark matter streams.

Dark matter starts out having a very low spread in velocities, but its location can be anywhere.  So, it forms a 3-dimensional sheet in the 6-dimensional space of position-velocity pairs, called phase space

As time passes this sheets gets bent, but it can never be broken.   When this sheet gets folded enough, we get a 'caustic where lots of different dark matter particles have almost the same position, though different velocities.  You can see a caustic by shining light into a reflective coffee cup, or shining light through a magnifying glass.  The same math applies here:

A phase-space perspective sheds additional light on the processes affecting the CDM under the influence of gravity.  When the CDM decouples from normal matter, the CDM occupies a 3-dimensional sheet in the 6-dimensional phase space since it has a tiny velocity dispersions. The process of galactic halo formation cannot tear this hypersurface, thanks to generalization of Liouville’s theorem.  Under the influence of gravity, a particular phase space volume of the hypersurface is stretched and folded with each orbit of the CDM creating layers of fine-grained dark matter streams, each with a vanishingly small velocity dispersion. These stretches and folds also produce caustics: regions with very high CDM densities that are inversely proportional to the square root of the velocity dispersion.

Here are some more pictures:

http://www.nasa.gov/feature/jpl/earth-might-have-hairy-dark-matter

and here's the paper:

• Gary Prezeau, Dense dark matter hairs spreading out from Earth, Jupiter and other compact bodies, http://arxiv.org/abs/1507.07009.

#spnetwork arXiv:1507.07009 #astronomy  ___

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2016-06-04 16:57:18 (0 comments; 59 reshares; 204 +1s; )Open 

Thin-skinned

By now you've probably heard: Trump said he'd given $1 million of his own money to veterans groups, but he actually hadn't.   His campaign manager, too, falsely claimed he had given this money. 

4 months later, the Washington Post and other papers started investigating.  They contacted Trump and asked what was up.  On May 24th, feeling the heat, he broke down and handed over the million bucks.

Other donors had also given money to the Donald J. Trump Foundation on the promise that Trump would then give it to veterans.  And he did - after  he was caught.  The Associated Press found that many of his checks were dated May 24, after  the Washington Post story came out.

That's bad enough, but the really interesting part is the temper tantrum that Trump threw at a press conference where he publicly announced that he'dfinally ... more »

Thin-skinned

By now you've probably heard: Trump said he'd given $1 million of his own money to veterans groups, but he actually hadn't.   His campaign manager, too, falsely claimed he had given this money. 

4 months later, the Washington Post and other papers started investigating.  They contacted Trump and asked what was up.  On May 24th, feeling the heat, he broke down and handed over the million bucks.

Other donors had also given money to the Donald J. Trump Foundation on the promise that Trump would then give it to veterans.  And he did - after  he was caught.  The Associated Press found that many of his checks were dated May 24, after  the Washington Post story came out.

That's bad enough, but the really interesting part is the temper tantrum that Trump threw at a press conference where he publicly announced that he'd finally given the promised money.

He blasted the media for making him “look bad” by insisting that he account for $6 million.  He called them "dishonest" and "not good people", without giving any example of dishonesty.  And he personally attacked ABC reporter Tom Llamas.

“I’m not looking for credit,” Trump insisted, contrary to all appearances. “But what I don’t want is when I raise millions of dollars have people say — like this sleazy guy over here from ABC. He’s a sleaze in my book.”

"Why am I a sleaze?" Llamas shot back.

“You’re a sleaze!” Trump shouted. “Because you know the facts and you know the facts well.”   Llamas, you see, had just asked a question about this issue.  He also had a history of asking Trump tough questions about his anti-immigrant rhetoric. 

It's shocking for a US presidential candidate to act this way.  This is what a Chicago gangster or tinpot dictator would do.

“Is this what it’s going to be like covering you if you’re president?” one reporter asked.

Trump’s reply: “Yeah, it is. I’m going to continue to attack the press.”

And indeed, Trump has said that if he becomes president, he will "open up" the libel laws to make it easier to sue people who say things he doesn't like.  This is exactly  what dictators do.

In 2005, Timothy O'Brien wrote a book TrumpNation: The Art Of Being The Donald.   He raised questions about Trump's claims of vast wealth.  Trump promptly sued O'Brien for $5 billion.  It was the largest libel lawsuit in U.S. history.  Maybe Trump was trying to gain the wealth he didn't actually have.   But the lawsuit was dismissed, because of course it's not libel to report that someone is not as rich as they claim.

It's tough to take public criticism.  Hillary Clinton knows this well.   But Clinton is not thin-skinned like Trump.   I don't want a president who throws hissy fits at press conferences, yells at reporters, and threatens writers with lawsuits.

The Washington Post article about Trump on May 24:

https://www.washingtonpost.com/news/post-politics/wp/2016/05/24/four-months-later-donald-trump-says-he-gave-1-million-to-veterans-group/

A video of Trump's press conference:

https://www.youtube.com/watch?v=eJ5kNEL0KzI

On Trump wanting to "open up" libel laws, and suing O'Brien:

http://www.npr.org/2016/03/24/471762310/donald-trump-wants-to-open-up-libel-laws-so-he-can-sue-news-outlets___

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2016-06-03 17:02:01 (19 comments; 14 reshares; 95 +1s; )Open 

A mathematical mystery - part 3

Especially before the fall of the USSR, the best Russian mathematicians would often meet and discuss their work at seminars. 

Gelfand's seminar in Moscow was especially famous, since he would stop speakers any time they said something unclear.   In fact, sometimes he'd appoint an audience member to play the role of arbiter: if this guy in the audience doesn't understand it, the speaker has to explain it better!  

As a result, the seminar would often go on until late at night, even after the building was locked up.  But everyone learned a lot of math.

With such exhaustive seminars, publishing proofs sometimes became a mere afterthought.  You'll often see short papers from this era making important claims with just a tiny sketch of an argument to back them up.

That annoyed Westernmathem... more »

A mathematical mystery - part 3

Especially before the fall of the USSR, the best Russian mathematicians would often meet and discuss their work at seminars. 

Gelfand's seminar in Moscow was especially famous, since he would stop speakers any time they said something unclear.   In fact, sometimes he'd appoint an audience member to play the role of arbiter: if this guy in the audience doesn't understand it, the speaker has to explain it better!  

As a result, the seminar would often go on until late at night, even after the building was locked up.  But everyone learned a lot of math.

With such exhaustive seminars, publishing proofs sometimes became a mere afterthought.  You'll often see short papers from this era making important claims with just a tiny sketch of an argument to back them up.

That annoyed Western mathematicians.  And I've bumped into a few mysteries that I'm having trouble with, thanks to these short Russian papers without clear proofs.  Here is one.

This image by Greg Egan shows the set of points (a,b,c) for which the quintic

x^5 + ax^4 + bx^2 + c

has repeated roots... with the plane c = 0 removed.  You'll notice this surface crosses over itself in a cool way, creating lines of sharp cusps

Vladimir Arnol'd, who ran one of these famous seminars, says that one O. V. Lyashko studied this surface in 1982 with the help of a computer - a very primitive computer by our standards, I'm sure.  And he says Lyashko proved this surface looks the same as another surface defined using the icosahedron. 

Arnol'd doesn't mention removing the plane c = 0, so his claim is technically wrong.  But if you remove that plane, it looks right!   So I'd like to see a proof that these surfaces are the same (after a smooth change of coordinates).   The icosahedron and the quintic equation are connected in many ways, so there should be a nice explanation.  But I don't know it!

For more details on this surface, see my Visual Insight  blog post:

http://blogs.ams.org/visualinsight/2016/06/01/discriminant-of-restricted-quintic/

You'll also see the other surface, defined using the icosahedron.  And you can read a full explanation of that other surface here:

http://blogs.ams.org/visualinsight/2016/05/15/discriminant-of-the-icosahedral-group/

As I explain, the same surface shows up in yet another disguise - but again, I don't know a proof!   If you make progress on these mysteries, let me know!

The icosahedron is connected to some of the most fascinating symmetrical structures in the mathematical universe, such as E8 and the Golay code.   I'm trying to get to the bottom of this, so every clue helps.

Here is a longer description of Gelfand's seminar, as told by Simon Gindikin:

The Gelfand seminar was always an important event in the very vivid mathematical life in Moscow, and, doubtless, one of its leading centers. A considerable number of the best Moscow mathematicians participated in it at one time or another. Mathematicians from other cities used all possible pretexts to visit it. I recall how a group of Leningrad students agreed to take turns to come to Moscow on Mondays (the day of the seminar, to which other events were linked), and then would retell their friends what they had heard there. There were several excellent and very popular seminars in Moscow, but nevertheless the Gelfand seminar was always an event.

I would like to point out that, on the other hand, the seminar was very important in Gelfand's own personal mathematical life. Many of us witnessed how strongly his activities were focused on the seminar. When, in the early fifties, at the peak of antisemitism, Gelfand was chased out of Moscow University, he applied all his efforts to seminar. The absence of Gelfand at the seminar, even because of illness, was always something out of the ordinary.

One cannot avoid mentioning that the general attitude to the seminar was far from unanimous. Criticism mainly concerned its style, which was rather unusual for a scientific seminar. It was a kind of a theater with a unique stage director playing the leading role in the performance and organizing the supporting cast, most of whom had the highest qualifications. I use this metaphor with the utmost seriousness, without any intention to mean that the seminar was some sort of a spectacle. Gelfand had chosen the hardest and most dangerous genre: to demonstrate in public how he understood mathematics. It was an open lesson in the grasping of mathematics by one of the most amazing mathematicians of our time. This role could be only be played under the most favorable conditions: the genre dictates the rules of the game, which are not always very convenient for the listeners. This means, for example, that the leader follows only his own intuition in the final choice of the topics of the talks, interrupts them with comments and questions (a privilege not granted to other participants) [....] All this is done with extraordinary generosity, a true passion for mathematics.

Let me recall some of the stage director's strategems. An important feature were improvisations of various kinds. The course of the seminar could change dramatically at any moment. Another important mise en scene involved the "trial listener" game, in which one of the participants (this could be a student as well as a professor) was instructed to keep informing the seminar of his understanding of the talk, and whenever that information was negative, that part of the report would be repeated. A well-qualified trial listener could usually feel when the head of the seminar wanted an occasion for such a repetition. Also, Gelfand himself had the faculty of being "unable to understand" in situations when everyone around was sure that everything is clear. What extraordinary vistas were opened to the listeners, and sometimes even to the mathematician giving the talk, by this ability not to understand. Gelfand liked that old story of the professor complaining about his students: "Fantastically stupid students - five times I repeat proof, already I understand it myself, and still they don't get it."

It has remained beyond my understanding how Gelfand could manage all that physically for so many hours. Formally the seminar was supposed to begin at 6 pm, but usually started with an hour's delays. I am convinced that the free conversations before the actual beginning of the seminar were part of the scenario. The seminar would continue without any break until 10 or 10:30 (I have heard that before my time it was even later). The end of the seminar was in constant conflict with the rules and regulations of Moscow State University. Usually at 10 pm the cleaning woman would make her appearance, wishing to close the proceedings to do her job. After the seminar, people wishing to talk to Gelfand would hang around. The elevator would be turned off, and one would have to find the right staircase, so as not to find oneself stuck in front of a locked door, which meant walking back up to the 14th (where else but in Russia is the locking of doors so popular!). The next riddle was to find the only open exit from the building. Then the last problem (of different levels of difficulty for different participants) - how to get home on public transportation, at that time in the process of closing up. Seeing Gelfand home, the last mathematical conversations would conclude the seminar's ritual. Moscow at night was still safe and life seemed so unbelievably beautiful!

http://www.math.rutgers.edu/home/gelfand/

#geometry  ___

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2016-05-30 16:00:05 (153 comments; 113 reshares; 332 +1s; )Open 

Let us read what we paid for

Imagine a business like this: you get highly trained experts to give you their research for free... and then you sell it back to them.  Of course these experts need equipment, and they need to earn a living... so you get taxpayers to foot the bill.  

And if the taxpayers want to actually read the papers they paid for?   Then you charge them a big fee!

It's not surprising that with this business model, big publishers are getting rich while libraries go broke.  Reed-Elsevier has a 37% profit margin!

But people are starting to fight back — from governments to energetic students like ‎Alexandra Elbakyan here.

On Friday, the Competitiveness Council —a gathering of European ministers of science, innovation, trade, and industry—said that by 2020, all publicly funded scientific papers published in Europeshould be ma... more »

Let us read what we paid for

Imagine a business like this: you get highly trained experts to give you their research for free... and then you sell it back to them.  Of course these experts need equipment, and they need to earn a living... so you get taxpayers to foot the bill.  

And if the taxpayers want to actually read the papers they paid for?   Then you charge them a big fee!

It's not surprising that with this business model, big publishers are getting rich while libraries go broke.  Reed-Elsevier has a 37% profit margin!

But people are starting to fight back — from governments to energetic students like ‎Alexandra Elbakyan here.

On Friday, the Competitiveness Council —a gathering of European ministers of science, innovation, trade, and industry—said that by 2020, all publicly funded scientific papers published in Europe should be made immediately free for everyone to read. 

This will start a big fight, and it may take longer than 2020.   But Alexandra Elbakyan isn't waiting around.

In 2011, as a computer science grad student in Kazakhstan, she got sick of paying big fees to read science papers.  She set up SciHub, a pirate website that steals papers from the publishers and sets them free.

SciHub now has 51,000,000 papers in its database.  In October 2015, Elsevier sued them.  In November, their domain name was shut down.  But they popped up somewhere else.  By February, people were downloading 200,000 papers per day.   Even scientists with paid access to the publisher's databases are starting to use SciHub, because it's easier to use.

Clearly piracy is the not the ultimate solution. Elbakyan now lives in an undisclosed location, to avoid being extradited.  But she gave the world a much-needed kick in the butt.   The old business model of get smart people to work for free and sell the product back to them is on its way out.

For more, read:

John Bohannon, Who's downloading pirated papers? Everyone, Science, 28 April 2016, http://www.sciencemag.org/news/2016/04/whos-downloading-pirated-papers-everyone

and especially the SciHub Twitter feed:

https://twitter.com/Sci_Hub

Also read this:

Martin Enserink, In dramatic statement, European leaders call for ‘immediate’ open access to all scientific papers by 2020, Science,
27 May 2016, http://www.sciencemag.org/news/2016/05/dramatic-statement-european-leaders-call-immediate-open-access-all-scientific-papers

The key word here is immediate - right now the US lets the journals sit on publicly funded papers for a year.  The Dutch government is really pushing this!  Congratulations to them!

#openaccess  ___

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2016-05-27 15:01:48 (76 comments; 36 reshares; 169 +1s; )Open 

The world's most long-winded proof

In the 1980s, the famous mathematician Ronald Graham asked if it's possible to color each positive integer either red or blue, so that no triple of integers a, b and c obeying Pythagoras’ famous equation:

a² + b² = c²

all have the same color.  He offered a prize of $100.

Now it's been solved!  The answer is no.  You can do it for numbers up to 7824, and a solution is shown in this picture.  But you can't do it for numbers up to 7825.

To prove this, you could try all the ways of coloring these numbers and show that nothing works.  Unfortunately that would require trying

3 628 407 622 680 653 855 043 364 707 128 616 108 257 615 873 380 491 654 672 530 751 098 578 199 115 261 452 571 373 352 277 580 182 512 704 196 704 700 964 418 214 007 274 963 650 268 320 833 348 358055 727 80... more »

The world's most long-winded proof

In the 1980s, the famous mathematician Ronald Graham asked if it's possible to color each positive integer either red or blue, so that no triple of integers a, b and c obeying Pythagoras’ famous equation:

a² + b² = c²

all have the same color.  He offered a prize of $100.

Now it's been solved!  The answer is no.  You can do it for numbers up to 7824, and a solution is shown in this picture.  But you can't do it for numbers up to 7825.

To prove this, you could try all the ways of coloring these numbers and show that nothing works.  Unfortunately that would require trying

3 628 407 622 680 653 855 043 364 707 128 616 108 257 615 873 380 491 654 672 530 751 098 578 199 115 261 452 571 373 352 277 580 182 512 704 196 704 700 964 418 214 007 274 963 650 268 320 833 348 358 055 727 804 748 748 967 798 143 944 388 089 113 386 055 677 702 185 975 201 206 538 492 976 737 189 116 792 750 750 283 863 541 981 894 609 646 155 018 176 099 812 920 819 928 564 304 241 881 419 294 737 371 051 103 347 331 571 936 595 489 437 811 657 956 513 586 177 418 898 046 973 204 724 260 409 472 142 274 035 658 308 994 441 030 207 341 876 595 402 406 132 471 499 889 421 272 469 466 743 202 089 120 267 254 720 539 682 163 304 267 299 158 378 822 985 523 936 240 090 542 261 895 398 063 218 866 065 556 920 106 107 895 261 677 168 544 299 103 259 221 237 129 781 775 846 127 529 160 382 322 984 799 874 720 389 723 262 131 960 763 480 055 015 082 441 821 085 319 372 482 391 253 730 679 304 024 117 656 777 104 250 811 316 994 036 885 016 048 251 200 639 797 871 184 847 323 365 327 890 924 193 402 500 160 273 667 451 747 479 728 733 677 070 215 164 678 820 411 258 921 014 893 185 210 250 670 250 411 512 184 115 164 962 089 724 089 514 186 480 233 860 912 060 039 568 930 065 326 456 428 286 693 446 250 498 886 166 303 662 106 974 996 363 841 314 102 740 092 468 317 856 149 533 746 611 128 406 657 663 556 901 416 145 644 927 496 655 933 158 468 143 482 484 006 372 447 906 612 292 829 541 260 496 970 290 197 465 492 579 693 769 880 105 128 657 628 937 735 039 288 299 048 235 836 690 797 324 513 502 829 134 531 163 352 342 497 313 541 253 617 660 116 325 236 428 177 219 201 276 485 618 928 152 536 082 354 773 892 775 152 956 930 865 700 141 446 169 861 011 718 781 238 307 958 494 122 828 500 438 409 758 341 331 326 359 243 206 743 136 842 911 727 359 310 997 123 441 791 745 020 539 221 575 643 687 646 417 117 456 946 996 365 628 976 457 655 208 423 130 822 936 961 822 716 117 367 694 165 267 852 307 626 092 080 279 836 122 376 918 659 101 107 919 099 514 855 113 769 846 184 593 342 248 535 927 407 152 514 690 465 246 338 232 121 308 958 440 135 194 441 048 499 639 516 303 692 332 532 864 631 075 547 542 841 539 848 320 583 307 785 982 596 093 517 564 724 398 774 449 380 877 817 714 717 298 596 139 689 573 570 820 356 836 562 548 742 103 826 628 952 649 445 195 215 299 968 571 218 175 989 131 452 226 726 280 771 962 970 811 426 993 797 429 280 745 007 389 078 784 134 703 325 573 686 508 850 839 302 112 856 558 329 106 490 855 990 906 295 808 952 377 118 908 425 653 871 786 066 073 831 252 442 345 238 678 271 662 351 535 236 004 206 289 778 489 301 259 384 752 840 495 042 455 478 916 057 156 112 873 606 371 350 264 102 687 648 074 992 121 706 972 612 854 704 154 657 041 404 145 923 642 777 084 367 960 280 878 796 437 947 008 894 044 010 821 287 362 106 232 574 741 311 032 906 880 293 520 619 953 280 544 651 789 897 413 312 253 724 012 410 831 696 803 510 617 000 147 747 294 278 502 175 823 823 024 255 652 077 422 574 922 776 413 427 073 317 197 412 284 579 070 292 042 084 295 513 948 442 461 828 389 757 279 712 121 164 692 705 105 851 647 684 562 196 098 398 773 163 469 604 125 793 092 370 432

possibilities.  But recently, three mathematicians cleverly figured out how to eliminate most of the options.  That left fewer than a trillion to check!  

So they spent 2 days on a supercomputer, running 800 processors in parallel, and checked all the options.  None worked.   They verified their solution on another computer.

This is the world's biggest proof: it's 200 terabytes long!  That's about equal to all the digitized text held by the US Library of Congress.  There's also a 68-gigabyte digital signature - sort of a proof that a proof exists - if you want to skim it.

It's interesting that these 200 terabytes were used to solve a yes-or-no question, whose answer takes a single bit to state: no.

I'm not sure breaking the world's record for the longest proof is something to be proud of.  Mathematicians prize short, elegant proofs.   I bet a shorter proof of this result will eventually be found.

Still, it's fun that we can do such things.   Here's a story about the proof:

http://www.nature.com/news/two-hundred-terabyte-maths-proof-is-largest-ever-1.19990

and here's the actual paper:

• Marijn J. H. Heule, Oliver Kullmann and Victor W. Marek, Solving and verifying the Boolean Pythagorean triples problem via cube-and-conquer, http://arxiv.org/abs/1605.00723.

The cube-and-conquer paradigm is a "hybrid SAT method for hard problems, employing both look-ahead and CDCL solvers"... whatever that means.  It would be interesting to learn about this.  But it's time for breakfast!

Anyone who makes a joke about Fermat's remark:

"I have discovered a truly marvellous proof of this, which this margin is too narrow to contain."

loses 10 points, for not reading my whole post.

#bigness  ___

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2016-05-26 14:41:52 (1 comments; 66 reshares; 237 +1s; )Open 

I don't like posting about politics.  I prefer math.  Math is the beautiful game of truth.  Politics is the ugly game of lies. 

But I woke up in the middle of the night and realized that if I didn't say the obvious, I might regret it:

We've got to fight Trump with everything we've got.

The US is dangerously close to electing a buffoon and would-be dictator: our very own Berlusconi, our very own Putin.  Elizabeth Warren explains it clearly, so please reshare this video:

Unfortunately, if you’ve been watching the presidential race, you know that we need to stand up now more than ever. Just yesterday, it came out that Donald Trump had said back in 2007 that he was “excited” for the real estate market to crash because, quote, “I’ve always made more money in bad markets than in good markets.” That’s right. The rest of uswere horrified by ... more »

I don't like posting about politics.  I prefer math.  Math is the beautiful game of truth.  Politics is the ugly game of lies. 

But I woke up in the middle of the night and realized that if I didn't say the obvious, I might regret it:

We've got to fight Trump with everything we've got.

The US is dangerously close to electing a buffoon and would-be dictator: our very own Berlusconi, our very own Putin.  Elizabeth Warren explains it clearly, so please reshare this video:

Unfortunately, if you’ve been watching the presidential race, you know that we need to stand up now more than ever. Just yesterday, it came out that Donald Trump had said back in 2007 that he was “excited” for the real estate market to crash because, quote, “I’ve always made more money in bad markets than in good markets.” That’s right. The rest of us were horrified by the 2008 financial crisis, by what happened to the millions of families like Mr. Estrada’s that were forced out of their homes. But Donald Trump was drooling over the idea of a housing meltdown — because it meant he could buy up a bunch more property on the cheap.

What kind of a man does that? Root for people to get thrown out on the street? Root for people to lose their jobs? Root for people to lose their pensions? Root for two little girls in Clark County, Nevada, to end up living in a van? What kind of a man does that? I’ll tell you exactly what kind — a man who cares about no one but himself. A small, insecure moneygrubber who doesn’t care who gets hurt, so long as he makes some money off it. What kind of man does that? A man who will NEVER be President of the United States.

Sometimes Trump claims he is tough on Wall Street – tough on the guys who cheated people like Mr. Estrada. I’m sure you’ve heard him say that. But now he’s singing a very different song. Last week, he said that the new Dodd-Frank financial regulations have, and I’m quoting here, “made it impossible for bankers to function” and he will put out a new plan soon that “will be close to dismantling Dodd-Frank.” Donald Trump is worried about helping poor little Wall Street? Let me find the world’s smallest violin to play a sad, sad song.

Can Donald Trump even name three things that Dodd-Frank does? Seriously, someone ask him. But this much he should know: If he’s so tough on Wall Street, he should be cheering on Dodd-Frank’s capital and leverage requirements that have made big banks less likely to fail. If he’s so tough on Wall Street, he should be cheering on Dodd-Frank’s living wills process, which is helping push big banks to become safer. If he’s so tough on Wall Street, he should be cheering on the Consumer Financial Protection Bureau, which has already returned over $11 billion to families who were cheated.

He SHOULD be, but he’s not. Now that he’s sewn up the Republican nomination, Donald Trump is dropping the pretense. Now he’s kissing the fannies of poor, misunderstood Wall Street bankers. But the American people are a whole lot smarter than Donald Trump thinks they are. The American people are NOT looking for a bait and switch. They are NOT looking for a man so desperate for power he will say and do anything to get elected. Take the hint, Donald: the time for letting big banks call all the shots in Washington is coming to an end.

And I want to make just one last point about Donald Trump that won’t fit into a Twitter war. One last point that sums up what Donald Trump is all about – his taxes.

We don’t know what Trump pays in taxes because he is the first Presidential nominee in 40 years to refuse to disclose his tax returns. Maybe he’s just a lousy businessman who doesn’t want you to find out that he’s worth a lot less money than he claims.

But we know one thing: the last time his taxes were made public, Donald Trump paid nothing — zero. Zero taxes before, and for all we know he’s paying zero taxes today. And he’s proud of it. Two weeks ago he said he’s more than happy to dodge taxes because he doesn’t want to throw his money “down the drain.”

Trump likes being a billionaire, and doesn’t think the rules that apply to everyone else should apply to him. But let’s be clear: Donald Trump didn’t get rich on his own. His businesses rely on the roads and bridges the rest of us paid for. His businesses rely on workers the rest of us paid to educate and on police-forces and fire fighters who protect all of us and the rest of us pay to support. Donald Trump and his businesses are protected by a world-class military that defends us abroad and keeps us safe at home and that the rest of us pay to support. When anyone builds something terrific, they should get to keep a big hunk of it. But they should also pay a fair share forward so the next kid and the next kid and the next kid who come along gets their chance to build something too. That’s how we build a future that works for everyone.

And that goes double for Donald Trump, because he didn’t even get rich by building something terrific. He inherited a fortune from his father, and kept it going by scamming people, declaring bankruptcy, and skipping out on what he owed.

Nurses, teachers, and dockworkers pay their fair share for all the services that keep Trump’s businesses going. Programmers and engineers and small business owners pay their fair share to support our military who show courage and sacrifice every single day. Donald Trump thinks supporting them is throwing money “down the drain.”

I say we just throw Donald Trump down the drain.

Let’s face it: Donald Trump cares about exactly one thing — Donald Trump. It’s time for some accountability because these statements disqualify Donald Trump from ever becoming President. The free ride is over.

I am going to disable comments because I don't see a need for discussion.  +Elizabeth Hahn gets the last word.___

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2016-05-25 17:46:39 (10 comments; 9 reshares; 64 +1s; )Open 

A windy day

This is a tornado that occurred in south-central Oklahoma on May 9th.   It's impressive what moist air can do when it's not in thermal equilibrium.

At first I couldn't remember where I found this video.  But Chris Greene found it:

https://www.youtube.com/watch?v=K1R_N_pysRs

Watch the whole thing!  At the end you'll see a car driving dangerously close to an oncoming tornado.  I wonder how that worked out.

A windy day

This is a tornado that occurred in south-central Oklahoma on May 9th.   It's impressive what moist air can do when it's not in thermal equilibrium.

At first I couldn't remember where I found this video.  But Chris Greene found it:

https://www.youtube.com/watch?v=K1R_N_pysRs

Watch the whole thing!  At the end you'll see a car driving dangerously close to an oncoming tornado.  I wonder how that worked out.___

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2016-05-24 14:36:24 (26 comments; 27 reshares; 121 +1s; )Open 

Logic hacking

In mathematics, unlike ordinary life, the boundary between the knowable and the unknowable is a precisely defined thing.   But finding it isn't easy.  Its exact location could itself  be unknowable.  But we don't even know that! 

This month, a bunch of 'logic hackers' have stepped up to the plate and made a lot of progress.  They've sharpened our estimate of where this boundary lies.  How?   By writing shorter and shorter computer programs for which it's unknowable whether these programs run forever, or stop.

A Turing machine is a simple kind of computer whose inner workings have N different states, for some number N = 1,2,3,...

The Busy Beaver Game is to look for the Turing machine with N states that runs as long as possible before stopping.  Machines that never stop are not allowed in thisgame. ... more »

Logic hacking

In mathematics, unlike ordinary life, the boundary between the knowable and the unknowable is a precisely defined thing.   But finding it isn't easy.  Its exact location could itself  be unknowable.  But we don't even know that! 

This month, a bunch of 'logic hackers' have stepped up to the plate and made a lot of progress.  They've sharpened our estimate of where this boundary lies.  How?   By writing shorter and shorter computer programs for which it's unknowable whether these programs run forever, or stop.

A Turing machine is a simple kind of computer whose inner workings have N different states, for some number N = 1,2,3,...

The Busy Beaver Game is to look for the Turing machine with N states that runs as long as possible before stopping.  Machines that never stop are not allowed in this game. 

We know the winner of the Busy Beaver Game for N = 1,2,3 and 4.  Already for N = 5, the winner is unknown.  The best known contestant is a machine that runs for 47,176,870 steps before stopping.  There are 43 machines that might or might not stop - we don't know. 

When N is large enough, the winner of the Busy Beaver Game is unknowable. 

More precisely, if you use the ordinary axioms of mathematics, it's impossible to prove that any particular machine with N states is the winner of the Busy Beaver Game... as long as those axioms are consistent.

How big must N be, before we hit this wall?

We don't know. 

But earlier this month, Adam Yedidia and Scott Aaronson showed that it's 7910 or less. 

And by now, thanks to a group of logic hackers like Stefan O’Rear, we know it's 1919 or less. 

So, the unknowable kicks in - the winner of the Busy Beaver Game for N-state Turing machines becomes unknowable using ordinary math - somewhere between N = 5 and N = 1919. 

The story of how we got here is is fascinating, and you can read about it on my blog post:

https://johncarlosbaez.wordpress.com/2016/05/21/the-busy-beaver-game/

Anything that I didn't make clear here, should be explained there.  If it ain't clear there, ask me!

#bigness  ___

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2016-05-23 05:25:18 (21 comments; 18 reshares; 125 +1s; )Open 

Mountain biking is fun, but this is ridiculous!

This is Danny MacAskill on the Inaccessible Pinnacle on the Isle of Skye.

He is a great mountain biker,  but he had to carry the bike up the last part of this scary peak.

The Isle of Skye is an island off the west coast of Scotland.   It's the largest of the Inner Hebrides, and the most northerly of the large islands in this group.    In the center of this island is a mountain range called the Cuillin, and the Inaccessible Pinnacle sits among these.

Skye has been occupied since Mesolithic times, and it appears in Norse poetry, for example in this romantic line:

"The hunger battle-birds were filled in Skye with blood of foemen killed."

Almost a third of the inhabitants still speak Gaelic, and apart from a few bigger towns, thepopul... more »

Mountain biking is fun, but this is ridiculous!

This is Danny MacAskill on the Inaccessible Pinnacle on the Isle of Skye.

He is a great mountain biker,  but he had to carry the bike up the last part of this scary peak.

The Isle of Skye is an island off the west coast of Scotland.   It's the largest of the Inner Hebrides, and the most northerly of the large islands in this group.    In the center of this island is a mountain range called the Cuillin, and the Inaccessible Pinnacle sits among these.

Skye has been occupied since Mesolithic times, and it appears in Norse poetry, for example in this romantic line:

"The hunger battle-birds were filled in Skye with blood of foemen killed."

Almost a third of the inhabitants still speak Gaelic, and apart from a few bigger towns, the population lives in crofting townships scattered around the coastline.  "Crofting"?  Yeah, a croft is a small farm with a wall around it.  

The only distillery on the Isle of Skye is the Talisker Distillery, which makes a rather famous single malt Scotch whisky.  It's in a village on the south shore.

I've always been fascinated by the Inner Hebrides and the even more exotic-sounding Outer Hebrides.  I'm annoyed at how all my visits to the British Isles have only taken me to the lofty centers of academe, not places like this.  I don't know much about them, but anything remote appeals to me: inaccessible pinnacles, inaccessible cardinals, the Taklamakan desert, the underground oceans of Europa....

Danny Macaskill is actually from the Isle of Skye!  You can see his whole journey along the Cuillin Ridgeline here:

https://www.youtube.com/watch?v=xQ_IQS3VKjA

Pretty impressive!  Beautiful scenery, too!___

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2016-05-19 03:04:53 (13 comments; 36 reshares; 148 +1s; )Open 

Gently down the stream - life is but a dream

A sea otter and her sleeping pup float downstream. 

For the whole adorable video see:

https://www.youtube.com/watch?v=de6uTMEiZf0

#biology  

Gently down the stream - life is but a dream

A sea otter and her sleeping pup float downstream. 

For the whole adorable video see:

https://www.youtube.com/watch?v=de6uTMEiZf0

#biology  ___

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2016-05-16 14:25:59 (17 comments; 30 reshares; 113 +1s; )Open 

A mathematical mystery - part 2

Russian mathematicians have discovered a mysterious connection between waves and a shape called the icosahedron.   You can see it in this image by Marshall Hampton.   It shows a wave going around an obstacle. 

The obstacle, in red, has an inflection point in the middle.  In other words, it curves down  on one side of this point, but curves up  on the other side. 

The blue curve below the red curve is the front of a wave.  As this wave front moves past the inflection point, something interesting happens.  It crosses over itself, and two sharp points form!

Together with some friends, the famous Russian mathematician Vladimir Arnol'd discovered that these two sharp points are different.  One is more pointy than the other!   The top one is a cusp of order 3/2, meaning it's described by an equationlike
<... more »

A mathematical mystery - part 2

Russian mathematicians have discovered a mysterious connection between waves and a shape called the icosahedron.   You can see it in this image by Marshall Hampton.   It shows a wave going around an obstacle. 

The obstacle, in red, has an inflection point in the middle.  In other words, it curves down  on one side of this point, but curves up  on the other side. 

The blue curve below the red curve is the front of a wave.  As this wave front moves past the inflection point, something interesting happens.  It crosses over itself, and two sharp points form!

Together with some friends, the famous Russian mathematician Vladimir Arnol'd discovered that these two sharp points are different.  One is more pointy than the other!   The top one is a cusp of order 3/2, meaning it's described by an equation like

y³ = x²

The bottom one is a cusp of order 5/2, meaning it's described by an equation like

y⁵ = x²

More precisely, you can get these equations after you do a change of coordinates. 

Arnol'd and friends also discovered something else.  The blue curve, as it moves along, traces out a surface that you can also get starting from an icosahedron!   As Arnol'd wrote:

Thus the propagation of the waves is controlled by an icosahedron hidden at the inflection point of the boundary. This icosahedron is hidden, and it is difficult to find it even if its existence is known.

Here's a hint of how it works.  An icosahedron has 5 triangles meeting at each corner.  The "5" here gives the cusp of order 5/2, while the "3" hiding in the word "triangle" gives the cusp of order 3/2!

You can read a much better explanation here:

http://blogs.ams.org/visualinsight/2016/05/01/involutes-of-a-cubical-parabola/

and here:

http://blogs.ams.org/visualinsight/2016/05/15/discriminant-of-the-icosahedral-group/

But it's still mysterious to me! 

First of all, it seems the proof was never written up.  Secondly, there's the question of "what does it all really mean?"  I don't think anyone knows. 

So I plan to get to the bottom of this....

#geometry  ___

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2016-05-14 18:31:34 (50 comments; 55 reshares; 135 +1s; )Open 

Global warming pause?

Umm, not really.  You can see how the Earth is heating up rapidly.   This great image was made by Ed Hawkins, a climate scientist at the University of Reading in the United Kingdom. 

He points out some features:

1877-78: strong El Nino event warms global temperatures

1880s-1910: small cooling, partially due to volcanic eruptions

1910-1940s: warming, partially due to recovery from volcanic eruptions, small increase in solar ouput and natural variability

1950s-1970s: fairly flat temperatures as cooling sulphate aerosols mask the greenhouse gas warming

1980-now: strong warming, with temperatures pushed higher in 1998 and 2016 due to strong El Nino events

He used temperature data from January 1850 – March 2016.  The numbers give the temperature above theaverag... more »

Global warming pause?

Umm, not really.  You can see how the Earth is heating up rapidly.   This great image was made by Ed Hawkins, a climate scientist at the University of Reading in the United Kingdom. 

He points out some features:

1877-78: strong El Nino event warms global temperatures

1880s-1910: small cooling, partially due to volcanic eruptions

1910-1940s: warming, partially due to recovery from volcanic eruptions, small increase in solar ouput and natural variability

1950s-1970s: fairly flat temperatures as cooling sulphate aerosols mask the greenhouse gas warming

1980-now: strong warming, with temperatures pushed higher in 1998 and 2016 due to strong El Nino events

He used temperature data from January 1850 – March 2016.  The numbers give the temperature above the average of 1850-1900.    The temperatures are from a British data set called HadCRUT4.4.  You can get that data here:

http://www.metoffice.gov.uk/hadobs/hadcrut4/

For more details, read this article on The Guardian:

http://www.theguardian.com/environment/2016/may/10/see-earths-temperature-spiral-toward-2c-rise-graphic

and check out Hawkin's website and blog:

http://www.climate-lab-book.ac.uk/2016/spiralling-global-temperatures/

Thanks to +rasha kamel for pointing this out.

#climate  ___

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2016-05-13 04:07:47 (63 comments; 9 reshares; 77 +1s; )Open 

Fecht Yeah

Kevin Kelly has claimed that "tools never die" - that any tool ever made is still being made somewhere.   There are interesting arguments about this online.  You can find videos on how to make stone hand axes.  You can find instructions on how make a calcium oxide light - the old-fashioned "limelight" used in theaters until it was replaced by electric arc lamp in the 1890s.  

And you can certainly buy a longsword.  That's a sword with a long double-edged blade and a cross-shaped handle, as shown here.  They reached their height of popularity from 1350 to 1550.  But people still fight with them - mainly for fun.

In fact, this weekend on Staten Island there's a course for women who want to fight with longswords!   And there's a tournament, too!  It's called Fecht Yeah, and it's probably not too lateto register... more »

Fecht Yeah

Kevin Kelly has claimed that "tools never die" - that any tool ever made is still being made somewhere.   There are interesting arguments about this online.  You can find videos on how to make stone hand axes.  You can find instructions on how make a calcium oxide light - the old-fashioned "limelight" used in theaters until it was replaced by electric arc lamp in the 1890s.  

And you can certainly buy a longsword.  That's a sword with a long double-edged blade and a cross-shaped handle, as shown here.  They reached their height of popularity from 1350 to 1550.  But people still fight with them - mainly for fun.

In fact, this weekend on Staten Island there's a course for women who want to fight with longswords!   And there's a tournament, too!  It's called Fecht Yeah, and it's probably not too late to register.  Bring your weapon.

It's part of the Historical European Martial Arts movement, or HEMA.  Here's the ad:

A weekend of training, learning, and collaboration for women who study HEMA and other sword arts.

This is an event for women of all skill levels with varied interests to come together and develop their skills. Workshops for beginners will be available. Free from tournament pressure and the constraints of classes, we have the ability to workshop teaching methods, rulesets, and learning strategies with other dedicated practitioners.

We will have laurel tournaments in longsword, sword and buckler, rapier, and saber. Prizes will be modest. Attend to learn, not win.

I'm an absurdly nonviolent guy, who will pick up a spider and take it outside rather than squash it.   But I admire skills like sword-fighting, and I'm glad people are keeping those skills alive.   Why?  I'm not completely sure.  I could theorize about it, but never mind.

Check out this video of German longsword fighting:

https://www.youtube.com/watch?v=5zueF4Mu2uM

Register here:

http://www.hema.events/aboutfy/

As you might expect, female swordfighters get flack from some male ones.  There's a nice article about Fecht Yeah here, and it get into that a bit:

http://www.villagevoice.com/news/fecht-club-new-yorks-women-warriors-kick-ass-8601021

Tiby Kantorowitz, one of the women running Fecht Yeah, treats swordfighting as a spiritual exercise:

"It's the flip side to yoga.   It's easy to Zen out with twinkly music, incense, and soft light. But can I maintain the same equanimity when there's some six-foot guy" — she's four-ten — "with a sword who's trying to brain me?"

The woman in this picture is Laura McBride, and she was photographed by Brad Trent.

For Kevin Kelly's claim, try:

http://www.npr.org/templates/transcript/transcript.php?storyId=133188723

KRULWICH: And then he made this ridiculous bet. He said: I bet you can't find any tool, any machine - go back to any century you like - that still isn't being made and made new today. So all I have to do is find a single tool that's not being made anymore, and I win.

(Soundbite of laughter)

KELLY: Yes, that's right.

KRULWICH: You're so going to lose this.

And then the show explores this....___

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2016-05-11 00:24:31 (143 comments; 36 reshares; 210 +1s; )Open 

The Earth is flat - and accelerating upwards

It's fun to read the frequently asked questions on the Flat Earth Society wiki.  First question:

Is this site a joke?

Answer: no, we're just diagonally parked in a parallel universe.  

Sorry - that's my answer, not theirs.  This site is not a joke.  But it's sure funny. 

How do you explain day/night cycles and seasons?

Day and night cycles are easily explained on a flat earth. The sun moves in circles around the North Pole. When it is over your head, it's day. When it's not, it's night. The sun acts like a spotlight and shines downward as it moves. The picture below illustrates how the sun moves and also how seasons work on a flat earth. The apparent effect of the sun rising and setting is usually explained as a perspective effect.... more »

The Earth is flat - and accelerating upwards

It's fun to read the frequently asked questions on the Flat Earth Society wiki.  First question:

Is this site a joke?

Answer: no, we're just diagonally parked in a parallel universe.  

Sorry - that's my answer, not theirs.  This site is not a joke.  But it's sure funny. 

How do you explain day/night cycles and seasons?

Day and night cycles are easily explained on a flat earth. The sun moves in circles around the North Pole. When it is over your head, it's day. When it's not, it's night. The sun acts like a spotlight and shines downward as it moves. The picture below illustrates how the sun moves and also how seasons work on a flat earth. The apparent effect of the sun rising and setting is usually explained as a perspective effect.

And all that stuff about the Moon's phases, and lunar and solar eclipses, was apparently set up just to fool us into thinking the Earth, Moon and Sun are round objects, with the Earth able to come between the Sun and Moon, and the Moon able to come between the Earth and Sun.

But what I really like is the explanation of gravity.  Wouldn't gravity pull the Earth into a round ball?   No:

The earth is constantly accelerating up at a rate of 32 feet per second squared (or 9.8 meters per second squared). This constant acceleration causes what you think of as gravity. Imagine sitting in a car that never stops speeding up. You will be forever pushed into your seat.

That's brilliant!   But wait a minute...

Objects cannot exceed the speed of light. Doesn't this mean that the Earth can't accelerate forever?

They've got an answer to that too:

Due to special relativity, this is not the case. At this point, many readers will question the validity of any answer which uses advanced, intimidating-sounding physics terms to explain a position. However, it is true. The velocity can never reach the speed of light, regardless of how long one accelerates for and the rate of the acceleration.

Fantastic!

What I like about this is that people can understand special relativity, yet not believe the Earth is round.  I had never encountered that combination.  I know more people who go the other way.

Of course there's the problem of what's powering this eternal acceleration.  But they have an answer to that too: it's the Universal Accelerator, also known as dark energy or the "aetheric wind".

Here's the site:

https://wiki.tfes.org/Frequently_Asked_Questions

Do not be angry.  Enjoy.___

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2016-05-07 17:42:16 (56 comments; 49 reshares; 172 +1s; )Open 

Weapons of math destruction

This Thursday evening, a 40-year-old man with dark, curly hair, olive skin and an exotic foreign accent boarded a plane in the United States.  It was a regional jet making a short hop from Philadelphia to nearby Syracuse.

The curly-haired man tried to keep to himself, intently scribbling mysterious symbols on a notepad he’d brought aboard. His seatmate, a blonde, 30-something woman sporting flip-flops and a red tote bag, looked him over. He was wearing navy Diesel jeans and a red Lacoste sweater – but something about him didn’t seem right.

She decided to try out some small talk.

"Is Syracuse home?"

"No," he replied curtly.

He similarly deflected further questions. He appeared laser-focused — perhaps too laser-focused — on the task at hand, those strange scribblings.
She beca... more »

Weapons of math destruction

This Thursday evening, a 40-year-old man with dark, curly hair, olive skin and an exotic foreign accent boarded a plane in the United States.  It was a regional jet making a short hop from Philadelphia to nearby Syracuse.

The curly-haired man tried to keep to himself, intently scribbling mysterious symbols on a notepad he’d brought aboard. His seatmate, a blonde, 30-something woman sporting flip-flops and a red tote bag, looked him over. He was wearing navy Diesel jeans and a red Lacoste sweater – but something about him didn’t seem right.

She decided to try out some small talk.

"Is Syracuse home?"

"No," he replied curtly.

He similarly deflected further questions. He appeared laser-focused — perhaps too laser-focused — on the task at hand, those strange scribblings.

She became suspicious.   Maybe it was code, or something in in Arabic — maybe the details of a plot to destroy the dozens of innocent lives aboard their flight!    Just to be safe, she decided, it was her duty to alert the authorities.

And so she did.  

And so the suspicious-looking man was taken off the plane, and interrogated. 

[I've paraphrased part of the article.  To hear what happened, read the rest.   Thanks to +Jenny Meyer for pointing this out.   My title is stolen from Cathy O'Neil's book, which you can obtain here:

http://www.amazon.com/Weapons-Math-Destruction-Increases-Inequality/dp/0553418815

It's about the abuses of math in the financial system.  There are economists who should be in big trouble for what they do with math.]___

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2016-05-06 15:22:33 (24 comments; 17 reshares; 124 +1s; )Open 

The longest G+ post I'll ever write

Sometimes math is just downright weird.  Here's an example.

A shelf is a set with a binary operation ▷ that distributes over itself:

a ▷ (b ▷ c) = (a ▷ b) ▷ (a ▷ c)

They're important in knot theory, as you can see from the picture.  But there's a strange connection between shelves, extremely large infinities, and extremely large finite numbers!

It goes like this.  Let's write 2^n for 2 to the nth power.  For each n we can make the numbers {1,2, ...,2^n} into a shelf by defining

a ▷ 1 = a + 1  mod  2^n

So, the elements of our shelf are

1
1 ▷ 1 = 2
2 ▷ 1 = 3

and so on, until we get to

2^n ▷ 1 = 1

However, we can now calculate

1 ▷ 1
1 ▷ 2
1 ▷ 3

andso on.  You should try it yours... more »

The longest G+ post I'll ever write

Sometimes math is just downright weird.  Here's an example.

A shelf is a set with a binary operation ▷ that distributes over itself:

a ▷ (b ▷ c) = (a ▷ b) ▷ (a ▷ c)

They're important in knot theory, as you can see from the picture.  But there's a strange connection between shelves, extremely large infinities, and extremely large finite numbers!

It goes like this.  Let's write 2^n for 2 to the nth power.  For each n we can make the numbers {1,2, ...,2^n} into a shelf by defining

a ▷ 1 = a + 1  mod  2^n

So, the elements of our shelf are

1
1 ▷ 1 = 2
2 ▷ 1 = 3

and so on, until we get to

2^n ▷ 1 = 1

However, we can now calculate

1 ▷ 1
1 ▷ 2
1 ▷ 3

and so on.  You should try it yourself for some simple examples!  You'll need to use the self-distributive law.  It's quite an experience.

You'll get a list of 2^n numbers, but this list will not contain all the numbers {1, 2, ... 2^n}  Instead, it will repeat with some period P(n).

And here is where things get weird.  The numbers P(n) form this sequence:

1, 1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, ...

It may not look like it, but a logician named Richard Laver proved the numbers in this sequence approach infinity!

... if we assume an extra axiom, which goes beyond the usual axioms of set theory, but so far seems consistent! 

This axiom asserts the existence of an absurdly large infinity, called an I3 rank-into-rank cardinal..  If you read my blog post on this stuff, I'll explain what that is.  So, this is an example of how an axiom about large infinite numbers can have implications for down-to-earth math.

On the other hand, a mathematician named Randall Dougherty has proved a lower bound on how far you have to go out in this sequence to reach the number 32.

And, it's an incomprehensibly large number!  If you read my blog post, I'll explain that too:

https://johncarlosbaez.wordpress.com/2016/05/06/shelves-and-the-infinite/

So, what we've got here is a very slowly growing sequence... which is easy to define but grows so slowly that - so far, at least -mathematicians need new axioms of set theory to settle the most basic questions about it!

Like alien spores floating down from the sky, large infinite numbers can come down and contaminate the study of down-to-earth questions about ordinary finite numbers!___

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2016-05-05 06:06:53 (0 comments; 17 reshares; 94 +1s; )Open 

Canadian oil-mining town experiences global warming

The city of Fort McMurray, Canada, mainly exists thanks to petroleum mining.   And thanks to humans burning carbon, this year is the hottest on record, especially in the far north.  The climate around Fort McMurray is borderline Arctic... but on Tuesday the temperature soared over 32 Celsius (91 Fahrenheit).   Thanks to the heat and wind, a huge fire engulfed the city.   A resident said:

"It was the most terrifying feeling looking straight ahead at a wall of flames 10 times higher than us.   I was in a complete state of shock and fear.   The streets were in a panic, people were abandoning their vehicles and hitchhiking."

Most news stories aren't pointing out the connection here, or the sad irony.

http://www.cnn.com/2016/05/04/world/fort-mcmurray-fire-canada/

Canadian oil-mining town experiences global warming

The city of Fort McMurray, Canada, mainly exists thanks to petroleum mining.   And thanks to humans burning carbon, this year is the hottest on record, especially in the far north.  The climate around Fort McMurray is borderline Arctic... but on Tuesday the temperature soared over 32 Celsius (91 Fahrenheit).   Thanks to the heat and wind, a huge fire engulfed the city.   A resident said:

"It was the most terrifying feeling looking straight ahead at a wall of flames 10 times higher than us.   I was in a complete state of shock and fear.   The streets were in a panic, people were abandoning their vehicles and hitchhiking."

Most news stories aren't pointing out the connection here, or the sad irony.

http://www.cnn.com/2016/05/04/world/fort-mcmurray-fire-canada/___

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2016-04-30 16:23:22 (24 comments; 7 reshares; 130 +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 (62 comments; 35 reshares; 164 +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 was published in European Physics Letters by 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 yo... 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 was published in European Physics Letters by 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 (25 comments; 4 reshares; 55 +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; 19 reshares; 107 +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/___

posted image

2016-04-21 17:22:18 (48 comments; 79 reshares; 142 +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; 5 reshares; 62 +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!___

posted image

2016-04-14 20:21:18 (66 comments; 16 reshares; 82 +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; 15 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.___

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2016-04-04 15:51:03 (42 comments; 29 reshares; 122 +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.

#bigness  ___

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2016-04-03 22:05:26 (0 comments; 35 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 (37 comments; 14 reshares; 71 +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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