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John Baez
Works at Centre for Quantum Technologies
Attended Massachusetts Institute of Technology
Lives in Riverside, California
57,454 followers|49,424,637 views


John Baez

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Endorsed by Putin

Putin, eager to weaken the EU and NATO, has been backing right-wing demagogues throughout Europe.   So it came as no surprise when he started complimenting Trump.  Not only is Trump would-be strongman of Putin's ilk (only less clever), he's also been threatening to break US commitments to NATO.

In December, Putin called Trump "an outstanding and talented personality".  Trump, in a rare moment of sweetness, replied:

"It is always a great honor to be so nicely complimented by a man so highly respected within his own country and beyond."

Putin?  Respected?

Putin now appears to be backing Trump even more strongly, with Russian operatives hacking into Democratic National Committee (DNC) computers and trying to embarrass them shortly before the convention.

On June 14th, the cybersecurity firm CrowdStrike, under contract with the DNC, announced in a blog post that two separate Russian intelligence groups had gained access to the DNC network.  One, called FANCY BEAR or APT 28, gained access in April. The other, COZY BEAR or APT 29, first breached the network in the summer of 2015.

You can read a more detailed analysis here:

Let me quote some:

CrowdStrike Services Inc., our Incident Response group, was called by the Democratic National Committee (DNC), the formal governing body for the US Democratic Party, to respond to a suspected breach. We deployed our IR team and technology and immediately identified two sophisticated adversaries on the network – COZY BEAR and FANCY BEAR. We’ve had lots of experience with both of these actors attempting to target our customers in the past and know them well. In fact, our team considers them some of the best adversaries out of all the numerous nation-state, criminal and hacktivist/terrorist groups we encounter on a daily basis. Their tradecraft is superb, operational security second to none and the extensive usage of ‘living-off-the-land’ techniques enables them to easily bypass many security solutions they encounter. In particular, we identified advanced methods consistent with nation-state level capabilities including deliberate targeting and ‘access management’ tradecraft – both groups were constantly going back into the environment to change out their implants, modify persistent methods, move to new Command & Control channels and perform other tasks to try to stay ahead of being detected. Both adversaries engage in extensive political and economic espionage for the benefit of the government of the Russian Federation and are believed to be closely linked to the Russian government’s powerful and highly capable intelligence services.

COZY BEAR (also referred to in some industry reports as CozyDuke or APT 29) is the adversary group that last year successfully infiltrated the unclassified networks of the White House, State Department, and US Joint Chiefs of Staff. In addition to the US government, they have targeted organizations across the Defense, Energy, Extractive, Financial, Insurance, Legal, Manufacturing Media, Think Tanks, Pharmaceutical, Research and Technology industries, along with Universities. Victims have also been observed in Western Europe, Brazil, China, Japan, Mexico, New Zealand, South Korea, Turkey and Central Asian countries. COZY BEAR’s preferred intrusion method is a broadly targeted spearphish campaign that typically includes web links to a malicious dropper. Once executed on the machine, the code will deliver one of a number of sophisticated Remote Access Tools (RATs), including AdobeARM, ATI-Agent, and MiniDionis. On many occasions, both the dropper and the payload will contain a range of techniques to ensure the sample is not being analyzed on a virtual machine, using a debugger, or located within a sandbox. They have extensive checks for the various security software that is installed on the system and their specific configurations. When specific versions are discovered that may cause issues for the RAT, it promptly exits. These actions demonstrate a well-resourced adversary with a thorough implant-testing regime that is highly attuned to slight configuration issues that may result in their detection, and which would cause them to deploy a different tool instead. The implants are highly configurable via encrypted configuration files, which allow the adversary to customize various components, including C2 servers, the list of initial tasks to carry out, persistence mechanisms, encryption keys and others. An HTTP protocol with encrypted payload is used for the Command & Control communication.

FANCY BEAR (also known as Sofacy or APT 28) is a separate Russian-based threat actor, which has been active since mid 2000s, and has been responsible for targeted intrusion campaigns against the Aerospace, Defense, Energy, Government and Media sectors. Their victims have been identified in the United States, Western Europe, Brazil, Canada, China, Georgia, Iran, Japan, Malaysia and South Korea. Extensive targeting of defense ministries and other military victims has been observed, the profile of which closely mirrors the strategic interests of the Russian government, and may indicate affiliation with Главное Разведывательное Управление (Main Intelligence Department) or GRU, Russia’s premier military intelligence service. This adversary has a wide range of implants at their disposal, which have been developed over the course of many years and include Sofacy, X-Agent, X-Tunnel, WinIDS, Foozer and DownRange droppers, and even malware for Linux, OSX, IOS, Android and Windows Phones. This group is known for its technique of registering domains that closely resemble domains of legitimate organizations they plan to target. Afterwards, they establish phishing sites on these domains that spoof the look and feel of the victim’s web-based email services in order to steal their credentials. FANCY BEAR has also been linked publicly to intrusions into the German Bundestag and France’s TV5 Monde TV station in April 2015.

At DNC, COZY BEAR intrusion has been identified going back to summer of 2015, while FANCY BEAR separately breached the network in April 2016. We have identified no collaboration between the two actors, or even an awareness of one by the other. Instead, we observed the two Russian espionage groups compromise the same systems and engage separately in the theft of identical credentials. While you would virtually never see Western intelligence agencies going after the same target without de-confliction for fear of compromising each other’s operations, in Russia this is not an uncommon scenario. “Putin’s Hydra: Inside Russia’s Intelligence Services”, a recent paper from European Council on Foreign Relations, does an excellent job outlining the highly adversarial relationship between Russia’s main intelligence services – Федеральная Служба Безопасности (FSB), the primary domestic intelligence agency but one with also significant external collection and ‘active measures’ remit, Служба Внешней Разведки (SVR), the primary foreign intelligence agency, and the aforementioned GRU. Not only do they have overlapping areas of responsibility, but also rarely share intelligence and even occasionally steal sources from each other and compromise operations. Thus, it is not surprising to see them engage in intrusions against the same victim, even when it may be a waste of resources and lead to the discovery and potential compromise of mutual operations.

You can even see some of the code that was used. Another security group, Fidelis, did an independent study confirming CrowdStriker's findings:

Of course, none of this excuses the DNC's dastardly behavior as revealed by the hacked emails.  But it's another sign of how sickening a disaster a Trump presidency would be.


Putin's compliment, and Trump's reply, is here:

Here's an article on Putin's "useful idiots" in Europe:

A quote, which contains lots of links in the original:

Prior to 2010, one would be hard-pressed to find public statements in praise of Putin by far-right leaders. Today, they are commonplace. UKIP’s Nigel Farage is a self-proclaimed fan of the Russian president. Jobbik’s head, Gabor Vona, is a frequent invited guest in Moscow. And, of course, Madame Le Pen, whose party was the beneficiary of a 9.4 million euro loan from a Russian-owned bank, is a consistent voice of support for Russian foreign policy in Ukraine and the Middle East. Even Germany, where the far right has failed to gain a foothold, is not immune to Moscow’s narrative. Supporters of PEGIDA, the increasingly popular xenophobic group whose acronym stands for “Patriotic Europeans Against the Islamization of the West,” often carry Russian flags and anti-government posters begging for Putin’s help.

John Baez

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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:

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.

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+Maximilian S. Glad you like it -- that's a fairly typical reaction for folks with a technical background. (Chapter 45 is just beautiful...)

People in love with the fantasy-romanticism of Rowling's novels (or the movies), on the other hand, tend not to like it nearly as much.
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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:

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,

and the photo is from here:

Frigatebirds are amazing! They have the largest ratio of wing area to body weight of any bird. This lets them fly very long distances while only rarely flapping their wings. They often stay …
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+Paul O'Malley - if I were an emu with a hangover I'd be dangerous too.
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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:

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:

Here's the data for the statements 1)-6):
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+John Baez I recently watched a documentary called Cowspiracy, which suggested a totally practical yet culturally impossible solution, if you have not seen it, it is well produced at least.

Frankly I think the nanotubes are a great idea no matter what, thank you for sharing that, it's obvious in hindsight.
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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:

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:

and if you live around here, check out the Explosion Pot Barbecue:,103.845405
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+John Baez learning that your wife teaches philosophy is interesting, I like to think it has helped make you into a great teacher yourself, you are very forgiving of the layman.
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Watch Juno meet Jupiter on NASA TV!   Go here:

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.

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I scarcely noted that pun.  I mean I noticed it, but I didn't suspect it was the point of that whole comment.
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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.

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,

But I got the picture from here:

• Guoqing Chang et al, New fermions on the line in topological symmorphic metals,

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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+Steve Wenner - actually they're considering a 3d material with hexagonal crystal symmetry about the z axis, but they're only drawing the energy as a function of the x and z momentum.   This material lacks

z |-> -z

symmetry, which is what allows the graph above to have a plane sloping upwards as the momentum in the z direction increases. But this material has

x |-> -x


So, my discussion in the post was inaccurate: you could say it's  abuot 2-dimensional simplified version of the real situation.  I was fooled because the paper was put out by the Centre for Advanced 2D Materials and the Graphene Research Centre here in Singapore, so I was expecting this to be a 2d material!
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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:

John Baez

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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:

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By the way, I urge y'all to stay polite.  I usually delete comments that get into the realm of ad hominem, but I was asleep since I'm over here in Singapore.  Luckily things seem to have righted themselves!
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John Baez

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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.
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+Steve Wenner - As +David Roberts said, the math I mentioned is at the bottom of my post after a lot of blank space.  It's in the form of a puzzle, which people here have now solved.  But yes, there's also a lot of math in the Wikipedia article.
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John Baez

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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:

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


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!
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How cool! I am off to make my own little model and have some fun.
I'm also mildly curious to see what the Flat Earth folks make of this! ;D
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John Baez

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♥ ♥ ♥ 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:

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:

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

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+John Baez​ Cool! Thanks. Those are great answers. Now I've more questions!
But that will be later, after I finally get some rest tonight.
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I'm a mathematical physicist.
  • Centre for Quantum Technologies
    Visiting Researcher, 2011 - present
  • U.C. Riverside
    Professor, 1989 - present
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Riverside, California
Contributor to
I teach at U. C. Riverside and work on mathematical physics — which I interpret broadly as ‘math that could be of interest in physics, and physics that could be of interest in math’. I’ve spent a lot of time on quantum gravity and n-categories, but now I want to work on more practical things, too.

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

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

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