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Shared Circles including John Baez

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Most comments: 92

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

Exploring black holes - with cats!

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

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

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

Forexample, if... more »

Most reshares: 91

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

Thanksgiving

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

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

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

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

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

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

Most plusones: 412

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

Thorny devil

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

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

It's also called a thorny dragon:

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

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

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

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

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

Latest 50 posts

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

The math of networks 

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

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

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

The math of networks 

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

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

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

All three are exactly the sort of people I like to listen to - full of cool ideas.   And they're just the start of this show!   It's gonna be fun.  You can see live streaming video right now, here:

https://www.youtube.com/watch?v=F4otLUkQ-Bw

The program is here:

https://johncarlosbaez.wordpress.com/2016/11/01/compositionality-workshop/

I'm talking at 9:30 PST Tuesday.  Here are my talk slides:

http://math.ucr.edu/home/baez/networks_compositionality

I've decided to talk about some new work on 'Petri nets' with Blake Pollard.  We're using categories to study chemical reaction networks... but this is just one example of how categories can be used to study compositionality in network theory.  At the end of my talk I show a network of different examples: a network of different kinds of networks!

Abstract. To describe systems composed of interacting parts, scientists and engineers draw diagrams of networks: flow charts, Petri nets, electrical circuit diagrams, signal-flow graphs, chemical reaction networks, Feynman diagrams and the like. In principle all these different diagrams fit into a common framework: the mathematics of symmetric monoidal categories. This has been known for some time. However, the details are more challenging, and ultimately more rewarding, than this basic insight. Two complementary approaches are presentations of symmetric monoidal categories using generators and relations (which are more algebraic in flavor) and decorated cospan categories (which are more geometrical). In this talk we focus on the latter.

#networktheory #networks  ___

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

The writing is on the wall

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

Join the battle:

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

George Monbiot provides more detail:

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

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

The writing is on the wall

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

Join the battle:

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

George Monbiot provides more detail:

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

Over this time, I have watched as tobacco, coal, oil, chemicals and biotech companies have poured billions of dollars into an international misinformation machine composed of thinktanks, bloggers and fake citizens’ groups. Its purpose is to portray the interests of billionaires as the interests of the common people, to wage war against trade unions and beat down attempts to regulate business and tax the very rich. Now the people who helped run this machine are shaping the government.

I first encountered the machine when writing about climate change. The fury and loathing directed at climate scientists and campaigners seemed incomprehensible until I realised they were fake: the hatred had been paid for. The bloggers and institutes whipping up this anger were funded by oil and coal companies.

Among those I clashed with was Myron Ebell of the Competitive Enterprise Institute (CEI). The CEI calls itself a thinktank, but looks to me like a corporate lobbying group. It is not transparent about its funding, but we now know it has received $2m from ExxonMobil, more than $4m from a group called the Donors Trust (which represents various corporations and billionaires), $800,000 from groups set up by the tycoons Charles and David Koch, and substantial sums from coal, tobacco and pharmaceutical companies.

For years, Ebell and the CEI have attacked efforts to limit climate change, through lobbying, lawsuits and campaigns. An advertisement released by the institute had the punchline “Carbon dioxide: they call it pollution. We call it life.”

It has sought to eliminate funding for environmental education, lobbied against the Endangered Species Act, harried climate scientists and campaigned in favour of mountaintop removal by coal companies. In 2004, Ebell sent a memo to one of George W Bush’s staffers calling for the head of the Environmental Protection Agency to be sacked. Where is Ebell now? Oh – leading Trump’s transition team for the Environmental Protection Agency.

It's not just Ebell: Trump is hiring lots of creatures from the swamp of fake industry-funded "research institutes".  For details, and links providing evidence, go here:

https://www.theguardian.com/commentisfree/2016/nov/30/donald-trump-george-monbiot-misinformation___

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

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

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

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

Jarzynski on thermodynamics

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

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

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

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

Jarzynski on thermodynamics

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

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

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

The second law implies that the change in 'free energy' of a system is less than or equal to the amount of work done on it.  But the Jarzynski equality gives a precise equation relating these two concepts, which implies that inequality.   I won't explain it here, but it's terse and beautiful.

Last week at the +Santa Fe Institute, Jarzynski gave an incredibly clear hour-long tutorial on thermodynamics, starting with the basics and zipping forward to modern work. With his permission, you can see his slides here:

http://tinyurl.com/jarzynski

along with links to an explanation of the Jarzynski equality, and a proof.

I had a great time in Santa Fe, and this was one of the high points.
 
#physics  ___

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

Fireflies in bamboo

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

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

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

She writes:

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

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

Fireflies in bamboo

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

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

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

She writes:

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

Of course, in this phenomenon, Japanese and visiting photographers have found a gorgeous source of inspiration. Capturing the lights of the fireflies, however, can be extremely difficult. Fireflies are very sensitive to other sources of light besides themselves, meaning that camera flashes, cell phones, flashlights, and other things that photographers often need to get their equipment set up can drive the little creatures away.

The difficulty of capturing photos of the fireflies, however, hasn’t deterred the most dedicated photographers. They’ve simply adapted their strategy to account for the habits of the fireflies. Photographers often scout an area out days in advance to see where the fireflies congregate and then return very early on the day they want to shoot, setting up in daylight before the twinkling lights begin and lying in still, silent wait for hours.

You can see more of Kei Nomiyama's firefly photos here:

https://keinomiayma.smugmug.com/Firefly/

What puzzles me is this: the glowing fireflies in these photos seem more orange than what I see in the eastern US.  I'm used to firefly light being yellow-green.  So:

Puzzle 1.  Are fireflies in Japan from a different species than US fireflies?

and more importantly:

Puzzle 2. Do they use a different chemical mechanism to make light?

or more generally:

Puzzle 3. How do fireflies make light, and how do they turn the chemical reaction on and off?

#biology  ___

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

Thanksgiving

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

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

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

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

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

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

Thanksgiving

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

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

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

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

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

The extent of Arctic sea ice had shrunk to record lows, while the temperature hit new record highs for winter.   In December 2015, parts of the North Pole were covered with a lake!

A unique event?  No: this year again scientists are shocked!   Here's what I read today on phys.org:

Freakishly high temperatures in the Arctic driven by heat-packed oceans and northward winds have been reinforced by a "vicious circle" of climate change, scientists said Thursday.

Air above the Polar ice cap has been 9-12 degrees Celsius (16.2 to 21.6 degrees Fahrenheit) above average during the last four weeks, according the data from the Danish Meteorological Institute (DMI), which tracks hourly changes in Arctic weather.

And during several days last week, temperatures above the North Pole were a balmy zero degrees Celsius (32 degrees Fahrenheit), a full 20 C (36 F) above the levels typical for mid-November, said Martin Stendel, a DMI climate researcher based in Copenhagen.

"This is by far the highest recorded" in the era of satellite data, starting in 1979, he told AFP.  "What we are observing is very unusual."

At this time of year, open Arctic ocean exposed by sea ice melted away in summer should be freezing again, with thousands of square kilometres icing over every day.  But that has not been happening, at least not at the same pace, said Stendel.

"Not only was the ice not growing as it would normally, there was further melting due to warm air coming in," he explained by phone.

The US National Snow and Ice Data Center reported that sea ice extent in October was the lowest on record, some 6.4 million square kilometres (2.5 million square miles). Ice cover at the top of the globe shrank to its smallest area in 2016 — some 4.14 million sq km (1.6 million sq miles) — on September 16.

Several factors have caused the Arctic to overheat since late October, say scientists.  The most immediate are warm winds sweeping up from western Europe and off the west coast of Africa.

"The winds carrying this heat is a temporary — and fairly unprecedented — weather phenomenon," said Valerie Masson Delmotte, a scientist at the Climate and Environment Sciences Laboratory in Paris".  Only since Thursday have they abated.

A second contributor is the record-strong Pacific Ocean El Nino that tapered off earlier this year — after pumping a couple tenths of a degree of added warming into the atmosphere.

But reinforcing these periodic, if powerful, drivers is the biggest one of all: global warming, experts agreed.

Two days ago, I read this on LiveScience:

The Arctic Is a Seriously Weird Place Right Now

The sun set on the North Pole more than a month ago, not to rise again until spring. Usually that serves as a cue for sea ice to spread its frozen tentacles across the Arctic Ocean. But in the depths of the polar night, a strange thing started to happen in mid-October. Sea ice growth slowed to a crawl and even started shrinking for a bit.

Intense warmth in both the air and oceans is driving the mini-meltdown at a time when Arctic sea ice should be rapidly growing. This follows last winter, when temperatures saw a huge December spike.

Even in an age where climate change is making outliers — lowest maximum sea ice extent set two years in a row, the hottest year on record set three years in a row, global coral bleaching entering a third year — the norm, what's happening in the Arctic right now stands out for just how outlandish it is.

"I've never seen anything like it this last year and half," Mark Serreze, director of the National Snow and Ice Data Center, said.

The latest twist in the Arctic sea ice saga began in mid-October. Temperatures stayed stuck in their September range, pausing sea ice growth. By the end of the month, the Arctic was missing a chunk of ice the size of the eastern U.S.

The oddness continued into November. A large area of the Arctic saw temperatures as much as 36°F above normal, further slowing Arctic sea ice growth and even turning it around for a few days. In other words, it was so warm in the Arctic that despite the lack of sunlight, sea ice actually disappeared.

"​The ridiculously warm temperatures in the Arctic during October and November this year are off the charts over our 68 years of measurements," Jennifer Francis, a climate scientist at Rutgers University who studies the Arctic, said.

Compounding the warm air is warm water. Sea surface temperatures on the edge of the ice are also running well above normal in many places, further inhibiting sea ice growth.

Things will keep getting stranger — freakishly violent storms in the east and southeast US, droughts and fires in the west, and so on.

I'm thankful I grew up on a different planet.  I remember it fondly, and it makes me want to save what we have now.

Here's the phys.org article:

http://m.phys.org/news/2016-11-overheated-arctic-climate-vicious-circle.html

Here's the LiveScience article:

http://www.livescience.com/56954-arctic-sea-ice-record-low.html

Both of these were mentioned on +Azimuth by +rasha kamel  so make sure to add +Azimuth to your G+ feed — it'll help you keep informed.

This is Peter Gleick's tweet last February, with a graph:

https://twitter.com/PeterGleick/status/702953140853690368

Here's the video showing the Arctic sea ice minimum volume each year:

https://www.youtube.com/watch?v=9NP0L1PG9ag___

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

Hail Trump!

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

From the New York Times:

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

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

Hail Trump!

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

From the New York Times:

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

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

But now his tone changed as he began to tell the audience of more than 200 people, mostly young men, what they had been waiting to hear. He railed against Jews and, with a smile, quoted Nazi propaganda in the original German. America, he said, belonged to white people, whom he called the “children of the sun,” a race of conquerors and creators who had been marginalized but now, in the era of President-elect Donald J. Trump, were “awakening to their own identity.”

As he finished, several audience members had their arms outstretched in a Nazi salute. When Mr. Spencer, or perhaps another person standing near him at the front of the room — it was not clear who — shouted, “Heil the people! Heil victory,” the room shouted it back.

These are exultant times for the alt-right movement, which was little known until this year, when it embraced Mr. Trump’s campaign and he appeared to embrace it back. He chose as his campaign chairman Stephen K. Bannon, the media executive who ran the alt-right’s most prominent platform, Breitbart News, and then named him as a senior adviser and chief strategist.

Now the movement’s leaders hope to have, if not a seat at the table, at least the ear of the Trump White House.

While many of its racist views are well known — that President Obama is, or may as well be, of foreign birth; that the Black Lives Matter movement is another name for black race rioters; that even the American-born children of undocumented Hispanic immigrants should be deported — the alt-right has been difficult to define. Is it a name for right-wing political provocateurs in the internet era? Or is it a political movement defined by xenophobia and a dislike for political correctness?

At the conference on Saturday, Mr. Spencer, who said he had coined the term, defined the alt-right as a movement with white identity as its core idea.

“We’ve crossed the Rubicon in terms of recognition,” Mr. Spencer said at the conference, which was sponsored by his organization, the National Policy Institute.

And while much of the discourse at the conference was overtly racist and demeaning toward minorities, for much of the day the sentiments were expressed in ways that seemed intended to not sound too menacing. The focus was on how whites were marginalized and beleaguered.

One speaker, Peter Brimelow, the founder of Vdare.com, an anti-immigration website, asked why, if Hispanics had the National Council of La Raza and Jews had the Anti-Defamation League, whites were reluctant to organize for their rights. Some speakers made an effort to distance themselves from more notorious white power organizations like the Ku Klux Klan.

But as the night wore on and most reporters had gone home, the language changed.

Mr. Spencer’s after-dinner speech began with a polemic against the “mainstream media,” before he briefly paused. “Perhaps we should refer to them in the original German?” he said.

The audience immediately screamed back, “Lügenpresse,” reviving a Nazi-era word that means “lying press.”

Mr. Spencer suggested that the news media had been critical of Mr. Trump throughout the campaign in order to protect Jewish interests. He mused about the political commentators who gave Mr. Trump little chance of winning.

“One wonders if these people are people at all, or instead soulless golem,” he said, referring to a Jewish fable about the golem, a clay giant that a rabbi brings to life to protect the Jews.

Mr. Trump’s election, Mr. Spencer said, was “the victory of will,” a phrase that echoed the title of the most famous Nazi-era propaganda film. But Mr. Spencer then mentioned, with a smile, Theodor Herzl, the Zionist leader who advocated a Jewish homeland in Israel, quoting his famous pronouncement, “If we will it, it is no dream.”

The United States today, Mr. Spencer said, had been turned into “a sick, corrupted society.” But it was not supposed to be that way.

“America was, until this last generation, a white country designed for ourselves and our posterity,” Mr. Spencer thundered. “It is our creation, it is our inheritance, and it belongs to us.”

But the white race, he added, is “a race that travels forever on an upward path.”

“To be white is to be a creator, an explorer, a conqueror,” he said. More members of the audience were on their feet as Mr. Spencer described the choice facing white people as to “conquer or die.  Of other races, Mr. Spencer said: “We don’t exploit other groups, we don’t gain anything from their presence. They need us, and not the other way around.”

The ties between the alt-right movement and the Trump team are difficult to define, even by members of the alt-right.

Mr. Bannon was the chief executive of Breitbart, an online news organization that has fed the lie that Mr. Obama is a Kenyan-born Muslim. As recently as last year, Breitbart published an op-ed article urging that “every tree, every rooftop, every picket fence, every telegraph pole in the South should be festooned with the Confederate battle flag.”

Mr. Bannon told Mother Jones this year that Breitbart was now “the platform for the alt-right.”

And soon Bannon will be the Trump's "senior counselor" in the White House.  

I quoted a lot, but not all, of this article by Joseph Goldstein:

http://www.nytimes.com/2016/11/21/us/alt-right-salutes-donald-trump.html___

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

Completely integrable billiards

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

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

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

Puzzle 1.  What shape is this 'forbidden region', and why? 

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

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

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

Completely integrable billiards

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

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

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

Puzzle 1.  What shape is this 'forbidden region', and why? 

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

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

This illustrates two very different kinds of dynamical systems.  The 'completely integrable' systems, like the elliptical billiards, do very predictable things.   The 'ergodic' ones seem random. 

With some math, we can make these ideas precise.  I'll be quick: a system whose motion is described by Hamiltonian mechanics is completely integrable if it has the maximum number of conserved quantities.   It's ergodic if it has the minimum number.   All sorts of in-between cases are also possible!

For a particle moving around in n dimensions the maximum number of conserved quantities is n.   More precisely, we can write every conservated quantity as a function of n such quantities.  The minimum number is 1, since energy is always conserved.

So, for a billiard ball, the maximum number is 2 - and that's what we have for the elliptical billiard ball table.   One of them is the energy, or if you prefer, the speed of the billiard ball.  

Puzzle 2. What's the other? 

This is related to puzzle 1, since it's this extra conserved quantity that sometimes forbids the billiard ball from entering certain regions in the ellipse.

For a lot more about the Bunimovitch stadium and ergodicity, see:

http://blogs.ams.org/visualinsight/2016/11/15/bunimovich-stadium/

(If you looked at it before: I've added more since then.)   For more on complete integrability versus ergodicity, try these:

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

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

#physics  ___

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

Mammal-like reptiles

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

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

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

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

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

Mammal-like reptiles

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

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

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

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

These particular guys are called Pristerognathus vanderbyli.  This picture is from Wikicommons and was apparently made by Участник:ДиБгд:

https://commons.wikimedia.org/wiki/File:Pristeroognathus_DB.jpg

The first dinosaurs showed up around 240 million years ago - and they only became common after the great Triassic-Jurassic extinction, 200 million years ago.   Therapsids started around 275 million years ago.  Some of them evolved into mammals 225 million years ago, and all the non-mammalian ones went extinct by the early Cretaceous, 100 million years ago.   Most dinosaurs went extinct at the end of the Cretaceous, 65 million years ago.  Some, however, are still sold as food at many grocery stores.

#biology___

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

Fighting climate change in the Trump era

There's some good news and some bad news. 

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

1) Kill Obama’s Clean Power Plan
2) Withdraw from the Paris Climate Agreement
3) Dismantle US environmental rules around coal power
4) Weaken fuel economy standards for cars and trucks
5) Open up new public lands to oil and gas drilling
6) Scale back federal support for wind and solar power
7) Limit the Environmental Protection Agency
8) Reverse the White House’s climate guidance to federal agencies
9) Make the Supreme Court hostile to environmental regulation
10) Pack the executive branch with carbon dioxide lovers

These things aren'tinevit... more »

Fighting climate change in the Trump era

There's some good news and some bad news. 

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

1) Kill Obama’s Clean Power Plan
2) Withdraw from the Paris Climate Agreement
3) Dismantle US environmental rules around coal power
4) Weaken fuel economy standards for cars and trucks
5) Open up new public lands to oil and gas drilling
6) Scale back federal support for wind and solar power
7) Limit the Environmental Protection Agency
8) Reverse the White House’s climate guidance to federal agencies
9) Make the Supreme Court hostile to environmental regulation
10) Pack the executive branch with carbon dioxide lovers

These things aren't inevitable - we can fight them.  So this article is worth reading.  You'll learn a lot about the battle to come.

So what's the good news?

California, Connecticut, Minnesota, New Hampshire, New York, Oregon, Rhode Island, Vermont and Washington have signed onto a spinoff of the Paris Climate Agreement.  It's called the Under 2 Memorandum of Understanding, or Under 2 MOU for short.   

"Under 2" stands for two goals:

under 2 degrees Celsius of global warming, and
under 2 tonnes of carbon dioxide emitted per person per year.   

These states have agreed to cut greenhouse gas emissions to 80-95% below 1990 levels by 2050.  We've also agreed to share technology and scientific research, expand use of zero-emission vehicles, etc., etc.

And the Under 2 MOU Coalition includes more than just US states:

http://under2mou.org/coalition/

It includes regions and cities around the world!   I'll list them, starting with ones near the US.  If you go to the link you can find out exactly what each of these 'sub-national entities' are promising to do.   If you're into politics, maybe you can help get your local region to join.  Or maybe you have a friend who is good at politics!

For more details, go here:

https://johncarlosbaez.wordpress.com/2016/11/24/under2-coalition/

Okay, here's the list:

UNITED STATES
Austin
California
Connecticut
Los Angeles
Massachusetts
Minnesota
New Hampshire
New York City
New York State
Oakland City
Oregon
Portland City
Rhode Island
Sacramento
San Francisco
Seattle
Vermont
Washington

CANADA
British Columbia
Northwest Territories
Ontario
Québec
Vancouver

MEXICO
Baja California
Chiapas
Hidalgo
Jalisco
Mexico City
Mexico State
Michoacán 
Quintana Roo
Tabasco 
Yucatán

BRAZIL
Acre
Amazonas
Mato Grosso
Pernambuco
Rondônia
São Paulo City
São Paulo State
Tocantins

CHILE
Santiago

COLOMBIA
Guainía
Guaviare

PERU
Loreto
San Martín
Ucayali

AUSTRIA
Lower Austria

FRANCE
Alsace
Aquitaine
Auvergne-Rhône-Alpes
Bas-Rhin
Midi-Pyrénées
Pays de la Loire

GERMANY
Baden-Württemberg
Bavaria
Hesse
North Rhine-Westphalia
Schleswig-Holstein
Thuringia

HUNGARY
Budapest

ITALY
Abruzzo
Basilicata
Emilia-Romagna
Lombardy
Piedmont
Sardinia
Veneto

THE NETHERLANDS
Drenthe
North Brabant
North Holland
South Holland

PORTUGAL
Azores
Madeira

SPAIN
Andalusia
Basque Country
Catalonia
Navarra

SWEDEN
Jämtland Härjedalen

SWITZERLAND
Basel-Landschaft
Basel-Stadt

UNITED KINGDOM
Bristol
Greater Manchester
Scotland
Wales

AUSTRALIA
Australian Capital Territory (ACT)
South Australia

CHINA
Alliance of Peaking Pioneer Cities (represents 23 cities)
Jiangsu Province
Sichuan
Zhenjiang City

INDIA
Telangana

INDONESIA
East Kalimantan
South Sumatra
West Kalimantan

JAPAN
Gifu

NEPAL
Kathmandu Valley

KENYA
Laikipia County

IVORY COAST
Assemblée des Régions de Côte d’Ivoire (represents 33 subnationals)

NIGERIA
Cross River State

MOZAMBIQUE
Nampula

SENEGAL
Guédiawaye

There's more good news, but I'll dole it out a little at a time, since you'll probably keep wanting more.___

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

What is a random string of bits?

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

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

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

What is a random string of bits?

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

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

The precise statement of this result is a bit more complicated, because what you can prove depends on what system of math you use, but still: there are limits on how much information you can prove any message contains!

So what should we do?  If you want to know, check out the slides of my talk at the Santa Fe Institute workshop on Statistical Mechanics, Information Processing and Biology:

http://tinyurl.com/alg-thermo

Also make sure to watch the movie of an alien planet... a movie that contains just 4 kilobytes of information.

In my talk, I started out by drawing a Turing machine on the whiteboard.  That's not in these slides.  But if you know what a Turing machine is, you may be able to understand my explanation of recursive functions, the Church-Turing thesis, Kolomogorov complexity, the relation between Kolmogorov complexity and Shannon entropy, the uncomputability of Kolmogorov complexity, the complexity barrier, Levin’s computable version of complexity, and finally my work with Mike Stay on algorithmic thermodynamics.

In short, lots of information about information!

#information #thermodynamics #informationtheory  ___

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

Chaotic billiards

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

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

It's also an example of ergodicity:  for almost every choice of initial conditions, the trajectory of a ball will have an equal chance of visiting each tiny little region.  

Puzzle: why did I say "almost" every choice?  Can you find some exceptions?

 Check out more ofPhillipe ... more »

Chaotic billiards

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

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

It's also an example of ergodicity:  for almost every choice of initial conditions, the trajectory of a ball will have an equal chance of visiting each tiny little region.  

Puzzle: why did I say "almost" every choice?  Can you find some exceptions?

 Check out more of Phillipe Roux's animations here:

https://plus.google.com/+philipperoux/posts/fkbbjvca78J

For the precise definition of ergodicity, and the history of this billiard problem, read my Visual Insight post:

http://blogs.ams.org/visualinsight/2016/11/15/bunimovich-stadium/

This shape is called the Bunimovich stadium, after the Russian mathematician Leonid Bunimovich.

I can't get +phillipe roux to work right now - G+ is only offering me other Phillipe Rouxs.

#physics #geometry  ___

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

The math of networks

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

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

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

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

Eventually a video will be available.

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

The math of networks

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

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

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

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

Eventually a video will be available.

The main new thing here is work with Blake Pollard on a monoidal category where the morphisms are open Petri nets.  This allows us to describe ‘open’ systems of chemical reactions, where chemical flow in and out. Composing morphisms in this category then corresponds to combining these open systems to form bigger ones.

Right now I'm sitting in on a workshop called "Circumventing Turing's Achilles Heel".   The idea:

Much of the extraordinary success of the computer industry over the last half-century is because the vast majority of computing machines we’ve built are general purpose computers. Subject to the limitations of finite memory, time, and processor speed, general-purpose computers are (near) Turing complete; that is, capable of computing anything that is computable. But it’s exactly that same strategy that makes our computer and network systems so vulnerable to attack: If an outsider can gain control of your general-purpose system, then s/he can in principle use it for whatever purposes s/he is clever enough to trick your system into executing, precisely because the system is (near) Turing complete.

This Working Group will explore strategies for retaining the hardware and software advantages of general purpose computers, while denying those same general purpose capabilities to outside attackers. The focus will be on systems and applications that, by definition, require routine access from the open internet (e.g., webservers, online banking and other financial systems, etc.).

It's a lively place!

#networks  ___

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

California — heading toward a greener future

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

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

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

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

California — heading toward a greener future

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

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

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

But as Californians, we will also stay true to our basic principles. We will protect the precious rights of our people and continue to confront the existential threat of our time — devastating climate change.

E PLURIBUS UNUM.

After Trump won the electoral vote, some have been calling for California to secede.  The Yes California movement wants to put a secession referendum on the ballot.  But a more reasonable thing is for California to pursue its own course within the US. 

We're already different.   We just passed a referendum committing the state to fight the Citizens United ruling.  We also legalized marijuana and banned free plastic bags at grocery stores.  Plastic bags may seem like a small thing, but they wreak havoc in the oceans. 

More importantly, in September we passed a law saying our state will cut carbon emissions 40% below 1990 levels by 2030.    We were already legally committed to hit 1990 levels by 2020.   And we've got a cap-and-trade system for carbon emissions!

California is not alone: we're part of the Pacific Coast Collaborative.   This organization represents 54 million people — and it's got officials at the international climate change meeting that's happening in Marrakesh now:

The world's climate leaders are gathering for the twenty-second Conference of Parties (COP22) in Marrakesh, Morocco from November 7-18. Representatives from the Pacific Coast Collaborative (PCC), a partnership between California, Oregon, Washington and British Columbia, will join world leaders as we seek to build thriving, sustainable, and low-carbon economies.

Last year in Paris, states, provinces, cities, and other subnational jurisdictions grabbed headlines for their commitments to deep emissions reductions through the Under2MOU, agreeing to limit greenhouse gas emissions to 2 tons per capita, 80-95% below a 1990 benchmark, by 2050. In 2016, the Pacific Coast Collaborative affirmed its commitment to meeting these targets by signing a new Climate Leadership Action Plan and forging a partnership with the leading sustainable cities on the West Coast — Los Angeles, Oakland, Portland, San Francisco, Seattle, and Vancouver — through the Pacific North America Climate Leadership Agreement.

The PCC heads to Marrakesh with the intention of meeting with other subnational leaders to share information about how our region's states and province are working together to drive dramatic transformations in the building, electricity, and transportation sectors while demonstrating that this transformation can happen alongside regional economic and job growth. The PCC is also promoting the recently launched International Alliance to Combat Ocean Acidification (OA Alliance) at COP 22. The OA Alliance will join the PCC and other nations and subnationals, including states, provinces, cities tribes, and nongovernmental members to advance our understanding of ocean acidification, reduce its causes, and protect coastal economies and marine ecosystems. The PCC invites other governments and organizations to join us in taking action by joining this new alliance.

Now that the US has fallen into the hands of Trump, it's all the more important to take action locally.  I need to learn more about the Pacific Coast Collaborative and what a lowly mathematician can do to help  out. 

Here is Jerry Brown's announcement:

https://www.gov.ca.gov/news.php?id=19598

Here's some interesting news about the California secession movement, and how it's gotten fired up thanks to Trump:

http://www.nytimes.com/2016/11/10/us/california-today-secession-trump.html

#sustainability___

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

The loser won

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

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

The loser won

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

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

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

This man must be stopped

Trump has said on Twitter that:

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

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

Myron Ebell has said:

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

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

This man must be stopped

Trump has said on Twitter that:

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

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

Myron Ebell has said:

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

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

In fact, there is no question that most people prefer less severe winters.

After running an organization devoted to eliminating protection for endangered species, he switched to heading the Global Warming and International Environmental Policy project at an institute funded by Exxon.  His job was to sow doubt and create confusion about climate change.

But he burst into fame in 2002.  That's when he helped Bush's "council on environmental policy" water down a key report on global warming.  He was caught by Greenpeace, and a scandal erupted.  

He also tried to get the head of the Environmental Protection Agency fired.  Back then it was Christine Todd Whitman.   In a secret memo to Philip Cooney, head of Bush's anti-environmental council, Ebell wrote:

It seems to me that the folks at the EPA are the obvious fall guys, and we would only hope that the fall guy (or gal) should be as high up as possible. I have done several interviews and have stressed that the president needs to get everyone rowing in the same direction. Perhaps tomorrow we will call for [Christine Todd Whitman] to be fired. I know that that doesn't sound like much help, but it seems to me that our only leverage to push you in the right direction is to drive a wedge between the President and those in the Administration who think they are serving the president's best interests by publishing this rubbish.

"This rubbish" was a report put out by the EPA warning people of the dangers of climate change.

So, get ready: this guy will be working full-time to cause trouble!  If you want to protest his selection, you can sign this petition:

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

Here's a good Scientific American article to get you up to speed:

https://www.scientificamerican.com/article/trump-picks-top-climate-skeptic-to-lead-epa-transition/

Here's Myron Ebell on Wikipedia:

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

Here's Myron Ebell rewriting scientific reports:

https://www.theguardian.com/environment/2005/jun/09/science.environment

Myron Ebell saying global warming is, on the whole, a good thing:

http://www.forbes.com/forbes/2006/1225/038.html

Here's Trump's claim that climate change is a notion invented by the Chinese:

http://www.theverge.com/2016/9/26/13067918/donald-trump-presidential-debate-2016-climate-change-hoax

#savetheplanet___

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

___

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

Learn advanced math at a ski resort - for free!

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

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

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

Learn advanced math at a ski resort - for free!

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

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

This workshop will bring together advanced graduate students and postdocs with some background in algebraic topology, category theory, mathematical logic, or computer science, with the goal of learning how these areas come together in homotopy type theory, and working together to prove new results.  You'll need basic knowledge of one of these areas to be a successful participant.  But you don't need to be an expert on all of them!

For information on how to register and a list of sample topics that participants may work on, go here:

 http://www.ams.org/programs/research-communities/2017MRC-1

Everyone accepted into the program will get financial support: room and board at the Snowbird Resort and up to $650 towards airfare.  The application deadline is March 1st, 2017,  but it's better if  you register as soon as possible.   Most of the positions are allocated to U.S. citizens and people at U.S. institutions, but a smaller number will be open to international participants.

If you have any questions, contact one of the organizers: +Dan Christensen, +Chris Kapulkin, +Dan Licata, +Emily Riehl, and +Michael Shulman.   These are great folks, and if you're a young mathematician interested in this area this workshop will be a wonderful opportunity.

If you just want to learn a bit more about homotopy type theory, I suggest this article:

• Álvaro Pelayo and Michael A. Warren, Homotopy type theory and Voevodsky's univalent foundations, https://arxiv.org/abs/1210.5658

and the free book:

• Homotopy Type Theory, https://homotopytypetheory.org/book/

#HOTT  ___

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

Predicting the election: a bit of information

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

The obvious question is: who is right? 

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

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

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

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

Predicting the election: a bit of information

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

The obvious question is: who is right? 

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

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

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

If you have technical ideas about why Nate Silver and Sam Wang get such different results, I'd be happy to hear them.  Silver keeps saying that outcomes for different states are highly correlated, so that if the polls are wrong enough to change the outcome in one state, it's more likely to happen in others.  It's true that if the states were completely uncorrelated, we could be extremely sure about the election by now.  But what exactly is Silver's model of correlation?  And what is Wang's?  And is this enough to explain the difference?

But if you want to discuss politics, don't do it here.   There are enough other places to do that - like, the whole fucking internet.   Any word endorsing or criticizing a candidate will be enough to get your comment deleted.

Personally I just have a tiny contribution to make here.   I can only answer this question:

How much information would you get if you suddenly learned that Hillary Clinton had a 64.2% chance of being elected?  Or a 99% chance?

This is what the concept of relative information is good for.  It's relative, because how much information you get depends on what you believe beforehand.

If you thought that Clinton had a probability q of winning, and you learn that Clinton has a probability p of winning, you have gained this much information:

p log(p/q) + (1-p) log((1-p)/(1-q))

For example, suppose you started out thinking the candidates each have a 50% of winning.   That's a reasonable assumption if you just came from Mars and are completely ignorant about the situation.  Then q = 1/2, so the formula above becomes

p log(2p) + (1-p) log(2(1-p))

If we do the logarithms here in base 2, we are measuring the information in bits.   Then we can simplify the formula and get

1 + p log(p) + (1-p) log(1-p)

For example, suppose you learn that yes, Clinton indeed has a 50% chance of winning!  Then p = 1/2 and the formula above gives 0.  I will spare you the calculation, but this makes sense: you have gained no information.  You thought Clinton had a 50% chance of winning, and you learned that's right, so you learned nothing new.

Or, suppose you read Nate Silver's blog and discover that Clinton has a 64.2% chance of winning.    Then you've gotten this many bits of information:

1 + 0.642 log(0.642) + (1 - 0.642) log(1 - 0.642)

That's about 0.06 bits of information!  Not much! 

And I think it's very fascinating that such a smart guy could analyze so much polling data about the election and only feel able to extract 0.06 bits of information about this all-important question: who will win? 

It shows an amazing lack of confidence.  But that may be a good thing.  I wish I knew.

If you believe Sam Wang's blog, on the other hand, you'll discover that Clinton has an over 99% chance of winning.  If we say it's 99%, then you've gotten this many bits of information:

1 + 0.99 log(0.99) + (1 - 0.99) log(1 - 0.99)

That's about 0.92 bits.  In other words, almost the complete answer to the question.

All this illustrates the power, and value, of a single bit of information.  Claude Shannon realized early on that if you have one bit of information that other people don't know, and you can get them to bet on it, you can double your money - on average. 

Similarly, an investor who gets 1/5 a bit of such information can expect to multiply her money by 2 to the 1/5 power, which is almost 1.15.  So, one fifth a bit of 'actionable information' per year is enough to make a 15% annual return!

For more on relative information, see this:

https://en.wikipedia.org/wiki/Kullback-Leibler_divergence

It goes by a lot of other names, like Kullback-Leibler divergence - but I find that name hopelessly obscure.   For Nate Silver's election predictions, go here:

http://projects.fivethirtyeight.com/2016-election-forecast/

It's changed a bit while I was writing this post!  For Sam Wang's, go here:

http://election.princeton.edu/todays-electoral-vote-histogram/

#information #informationtheory___

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

Escudero nonic

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

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

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

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

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

Escudero nonic

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

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

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

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

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

Escudero came up with this polynomial equation in a very clever way, which I don't completely understand.  It involves Chebyschev polynomials and arrangements of lines related to substitution tilings.  You can read about it in his paper:

• Juan García Escudero, A construction of algebraic surfaces with many real nodes, Annali di Matematica 195 (2016), 571–583, available at https://arxiv.org/abs/1107.3401.

#geometry  ___

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

Thorny devil

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

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

It's also called a thorny dragon:

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

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

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

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

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

Thorny devil

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

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

It's also called a thorny dragon:

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

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

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

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

Researchers discovered long ago that because its mouth has evolved to eat ants, it cannot sip or even lick water from a source — instead it has to rely on other means. Prior research had found that the lizard has tiny folds on its skin that overlap, creating tube-like structures capable of carrying water — the tubes all lead to the back of the mouth. It was noted that setting the lizard in a small bucket of water caused the tubes to fill and the lizard to start swallowing. But what has remained a mystery is how such a technique could work in the desert, where there are rarely puddles to stand in. To solve the mystery, the researchers captured some specimens and took them back to their lab for study.

In the lab, the researchers first tried allowing the lizards to stand on sand that had been wetted — this resulted in some water being drawn into the tubes, but not enough to get the lizard to start swallowing, which meant it wasn't enough. The answer, it turned out, was the lizard's habit of pushing sand onto its back — this caused any water from recent rains or even from dew to move slowly downward, due to gravity. Eventually, it would reach the skin, where it would be sucked into the tubes like a child with a straw. At some point, the tubes would fill and the lizard would swallow it.

The researchers note that such a drinking technique is likely merely supplementary — most of the water they get comes from the ants they eat; thus, using skin for drinking would likely only occur during extreme draught conditions.

I got this picture from a website with a lot of great photos of thorny devils:

http://www.factzoo.com/reptiles/lizards/thorny-devil-lizard.html

#biology  ___

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2016-11-01 15:35:06 (0 comments; 13 reshares; 84 +1s; )Open 

A good read!  Check it out!

Today's topic of study: Hashtables

My view of hash tables is heavily influenced by an oral doctoral exam question my boyfriend in college had: "why or why not would you use a random number generator to dither an image?"

There's a whole lot of info packed in that question -- what is "dithering" for one, but also what randomness actually means. And I'll give you a hint on that last one: what does a random distribution look like in the Fourier Domain? Yeah, I'll explain that one too.

Just real quick, dithering is a computer graphics term. Best way to explain it: you know how images in newspapers have dots to make different shades of gray? That's a dither pattern. For newspapers, the dots are larger and larger for darker areas.

Color printers also use dither patterns. If you look closely at a printout on a color printer, you will see the tiny dots. These aren't the same as a newspaper dither, where the dots gets larger and larger. There are more and more of them, but they are more scattered for each pixel, not clumped like for a newspaper.

How you dither can drastically affect the quality of the image.

How you dither is tied to the method of distributing the ink.

If I may take an aside, I worked at a company in the late 80's that was building a color printer. It was one of the very first in the world. It was a dye sublimation printer -- meaning the dye sublimated (or sunk down into) the paper. The colors, because of this, were very smooth and rich. The prints were amazing! My boyfriend back then (um, a different one :-> ) was writing the drivers for the printer, and got me a job there too. I worked on their dither patterns. Wow, did I learn a lot!

The obvious one would be to place, say, a dot in the center of the pixel for value 1, then maybe a dot in the center, then a dot in the upper right for value 2, and those plus lower left for 3, and so forth. Actually, the upper right and lower left, in a field of the same color, would touch, so you have to think a lot about how the pattern wrapped on itself as well.

But the bottom line is, the obvious idea was to place the dots as far away as possible.

The prints looked terrible by this method! What was going wrong!

As it turned out, the process of printing was a static charge process. There was a drum that would get charged by the different dots, then roll into a powdered ink, then roll the ink onto the paper. Well, a single dot charge was not quite enough to pick up any ink -- two dots next to each other did much better. And so this "well-placed dots" algorithm mean for values 0-50% they didn't print anything at all! So a dither pattern that more resembled a newspaper dither worked much better. (In the end, I combined the two a bit, by placing dots far away, but always in pairs.)

I had a lot of fun on that project and wrote a little program to allow me to develop my own dither patterns -- you were presented a square of 8x8 and would click the dots in the order they should appear for darker and darker colors. I made all kinds of silly dither patterns: spirals, cats, and stuff like that. :)

A quick aside on dithering: I learned only recently that pointilism -- the method of painting using dots of paint -- has a special quality that the paintings look more bright than regular painting. I learned that this is because blank parts of the canvas are left white, and so the paintings have more light reflecting off of them. They really are physically brighter than paintings that cover the whole canvas!

From that, I noticed that today's dither patterns tend not to be the same for all the color fields. The cyan, magenta and yellow inks do not overlap where they are printed. This takes advantage of the same effect: white paper is reflected back, and makes the image look brighter. Kind of amazing that it works that way!

Ok, so that's dithering.

Now, I was stuck in the idea of an 8x8 grid to replace each pixel, but I learned about error diffusion in dithering. This is a method whereby you print your dot, then, as you move along printing your image, you keep track of how long it has been since you printed that dot, accumulate that amount, then print again when it's about time for another dot. This does not produce a grid-like pattern. It is more blended across the whole image. Error diffusion is very cool.

I can now tell you this joke:

My friend, Mark said: "I don't tan. I have freckles. When I'm in the sun, I just get more freckles."

Me: "Ah. You have a dithered tan!" Yuk yuk. :)

And...so why wouldn't a random number be just great for generating a dither pattern?

Let's talk about the Fourier Domain.

The fourier domain is a transformation of a signal into a graph of its component frequencies. Sounds technical, but it's not really. If you have a sine wave, just one wave, in the fourier domain, it's a single spike, a single frequency. If you have a wavy sine wave, that's two frequencies, and they show as two spikes in the Fourier Domain. And so forth.

I learned about the fourier domain by doing a FFT (fast fourier transform) on images in Advanced Computer Graphics class. (Boy, what a great class!!!)

(Note: I didn't really understand the math itself at the time -- I'd never even heard of the fourier domain anyway. I just implemented the algorithm and got the correct answers. I am hungering to go back and re-write that program, now that all the info has sunk in, and I understand it better.)

(Super geek info: We got two images back: one for the frequencies, and one for the phase -- where in the sine wave the frequency was in the image, up or down or in the middle, etc. As a homework exercise, we were asked to swap the frequency image with another image and undo the transform, and to do the same with the phase image. The shocker was that swapping the frequency image did nothing, but the phase image caused the donor image to be the result! The phase has far more to do with an image than its frequencies! Wow did that blow my mind!!! Let me know if you want me to elaborate on this in another post. But it's pretty geeky...)

But for random numbers, what do they look like in the fourier domain?

They are a flat line, all at the same amplitude. In short, they contain all frequencies.

For the lay person, basically, they clump. They clump big and small and all sizes in between. They clump at every scale.

And random clumpiness doesn't look very good in an image.

A more technical way to put it is: randomness is not well distributed. This is an important thing to understand: for some things, you want them all nicely separated apart from each other.

Like birds on a wire. Well distributed.

For a hash table, you kinda want that too. You want each hash to be nicely far from the others. You kinda want all of them to be nicely separated from all the others.

Well distributed.

A hash table is quite an amazing sorting and retrieval method.

Imagine first that you have a bunch of PO boxes. You put each name on the box in alphabetical order. But now you have to put a letter in "Bea's" mailbox. Which box is hers? You have to search through all the names to find Bea. That takes a long time.

How about this idea: take all the letters, assign numbers 1-26, then add all of Bea's name letters together and give her that box! Super easy!

That is a hash.

It is O(1) -- wow! All you do to store something is hash the key, and voila! In a single operation, you have the place to put that thing! Getting the letters out is the same -- one operation. That's very very fast.

So our hash algorithm is adding all the letters in a name together, and ba-da-bing! you go right to the box and get your stuff! No searching or slogging through all the boxes for the right name. In one move, you've got your stuff!

Hmm...but what happens when you have "Abe" and "Bea"? They have the same letters. When you add them, they crash. They will point to the same place in your hash table. Abe might get Bea's stuff, and vice versa.

Well, you can simply put "Bea" on the first one, then move to the next one and put "Abe" and when you pop over in your one-operation, then you scan for a match.

Hmm. Of course, this is great when there are few crashes, but your O(1) operation can become O(n) pretty quick, if your hash algorithm isn't...dun dun duuun!!! Well distributed.

So a nice well distributed hash algorithm is important.

The link below has an amazing discussion of various hashing algorithms. The super amazing thing is someone actually did a study of the various ones, and published their results of how well distributed each was. Sadly, the writer easily interchanges the words "random" and "well distributed" and those are absolutely not the same thing.

But they show some just amazing visualizations of how well distributed each algorithm is. Just look at the purdy rainbow images, and it should all be clear. (But then I really prefer things to be explained visually like that.)


Ok that's my lesson for today! I'm going to start playing around with well distributed hash algorithms today!

___A good read!  Check it out!

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

Higher Structures

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

Check it out:

https://journals.mq.edu.au/index.php/higher_structures

Focus and Scope

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

Peer Review Process

Articles appearingin... more »

Higher Structures

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

Check it out:

https://journals.mq.edu.au/index.php/higher_structures

Focus and Scope

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

Peer Review Process

Articles appearing in the journal have been carefully and critically refereed under the responsibility of members of the Editorial Board. Only papers judged to be both significant and excellent are accepted for publication.

Open Access Policy

This journal provides immediate open access to its content on the principle that making research freely available to the public supports a greater global exchange of knowledge. Accepted articles will be licensed under a Creative Commons Attribution 4.0 International License.

Archiving

This journal utilizes the LOCKSS system to create a distributed archiving system among participating libraries and permits those libraries to create permanent archives of the journal for purposes of preservation and restoration.

What really makes a journal is its editors, and this one has the best.  The managing editor is Mark Weber, the editors are Michael Batanin, Ralph Kaufmann and Martin Markl, and the editorial board is a star-studded gallery of experts on higher categories, operads, stacks and the like:

Clemens Berger, Université Nice-Sophia Antipolis

Vladimir Dotsenko, Trinity College Dublin, the University of Dublin

Tobias Dyckerhoff, Hausdorff Center for Mathematics

Benoit Fresse, Université de Lille

Richard Garner, Macquarie University

André Henriques, Universiteit Utrecht

Joachim Kock, Universitat Autònoma de Barcelona

Stephen Lack, Macquarie University

Andrey Lazarev, Lancaster University

Muriel Livernet, Université Paris Diderot

Michael Makkai, McGill University

Yuri Manin, Max Planck Institute for Mathematics

Ieke Moerdijk, Universiteit Utrecht

Amnon Neeman, Australian National University

Maria Ofelia Ronco, Universidad de Talca

Jiří Rosický, Masaryk University

James Stasheff, University of Pennsylvania

Ross Street, Macquarie University

Bertrand Toën, Université de Toulouse

Boris Tsygan, Northwestern University

Bruno Vallette, Université Paris 13

Michel Van den Bergh, Universiteit Hasselt

Alexander Voronov, University of Minnesota

The picture here is by +Scott Carter.  It shows the Zamolodchikov tetrahedron equation, the 4d analogue of the famous third Reidemeister move from knot theory.   To understand this equation better, see:

http://blogs.ams.org/visualinsight/2016/03/15/zamolodchikov-tetrahedron-equation/

#mathematics #openaccess  ___

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

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

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

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

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

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

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

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

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

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

But then suppose you discover that no, the ball moves upward at a constant rate.   Does that mean there's no mystery left?  No!  Gravity should be pulling the ball back down.  If it keeps rising at a constant rate, there still must be some extra force at work!

The actual paper is clear about this, but some of the science journalists seem confused.

The old story is shown in the picture here.  Time goes up the page, and the picture shows a patch of space expanding as time goes on.  Thanks to surveys of distant supernovas, we thought the expansion was just starting to accelerate.  That could be explained by dark energy - a hypothetical field with positive energy density but negative pressure.  Positive energy density makes space want to contract, but negative pressure makes it want to expand - and in the case of dark energy, the negative pressure wins. 

Why?   Einstein saw that without postulating an "aether" that defines a fixed frame of reference, it was possible for space to have exactly 3 times as much negative pressure as positive energy density, in suitable units.  The number 3 is not adjustable here - it comes from having 3 dimensions of space!  And so, once the ordinary matter in the universe is spread thinly enough, dark energy dominates - and negative pressure wins, making the expansion accelerate!

More recently astronomers have gotten good data on the redshift and brightness of 740 supernovae, over ten times the original sample.  The authors of this new paper claim that the new data fits a model where the universe is expanding at a constant rate:

• J. T. Nielsen, A. Guffanti, S. Sarkar. Marginal evidence for cosmic acceleration from Type Ia supernovae, Scientific Reports 6 (2016), 35596.  Freely available at http://www.nature.com/articles/srep35596

There's a lot of statistical analysis in here, and someone should check it.   However, they admit that there is other, more indirect evidence for dark matter.  So, they don't claim to have disposed of dark matter - even if they're doing the statistics right.

Moreover, if  the universe is expanding at a constant rate, the attractive force of gravity - which tends to slow down the expansion - must be counteracted somehow.  The author mention a couple of theories.  But both these theories seem iffy to me.  So, even if this paper is basically right, it's just the beginning of a big argument. 

And that's just fine.  That's how science makes progress.

#astronomy #spnetwork doi:10.1038/srep35596___

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

Category theory - an advanced online course!

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

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

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

------------
In ea... more »

Category theory - an advanced online course!

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

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

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

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

In early 2014, the n-Category Café hosted the Kan Extension Seminar, a graduate reading course in category theory modeled after Daniel Kan’s eponymous reading course in algebraic topology at MIT. My experience with the seminar, described here:

http://www.ams.org/notices/201411/rnoti-p1357.pdf

was overwhelmingly positive, so I’m delighted to announce that we’re back. Alexander Campbell, Brendan Fong, and I are organizing Kan II in early 2017 and we’re currently soliciting applications for seminar participants.

Our plan is to read the following eight papers between mid January and mid May:

* Hyland and Power, The category theoretic understanding of universal algebra: Lawvere theories and monads

* Freyd, Algebra valued functors in general and tensor products in particular

* Beck, Distributive laws

* Kelly, On the operads of J.P. May

* Kelly, Structures defined by finite limits in the enriched context, I

* Kelly, On Clubs and data-type constructors

* Lack and Rosicky, Notions of Lawvere theory

* Berger, Mellies, and Weber, Monads with arities and their associated theories

We are seeking eight participants who will read and engage with all of the papers as well as prepare an oral presentation on one of them, followed by a blog post to be published on the n-Category Café. The course will conclude with a series of short public expository lectures given, by those able to attend, on July 16 in conjunction with the 2017 International Category Theory Conference at the University of British Columbia in Vancouver.

More details, including information about how to apply, can be found on the seminar website:

http://www.math.jhu.edu/~eriehl/kanII

or by contacting any of the three of us. Applications are due November 30th. I hope you’ll help us spread the word by passing this message along to those who might be interested.

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

The picture here shows a decomposition of the 2-dimensional associahedron, which is the space of 4-ary operations in the associahedron operad.   The 2d associahedron is also called the Stasheff pentagon, because it was discovered by the topologist Jim Stasheff in his work on topology. 

For details see:

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

#mathematics #categories___

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

Activity diagrams

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

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

For the whole story, go here:
https://... more »

Activity diagrams

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

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

For the whole story, go here:

https://johncarlosbaez.wordpress.com/2016/10/18/complex-adaptive-system-design-part-2/

#networks  ___

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

Mini Saturn

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

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

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

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

Mini Saturn

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

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

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

A while ago I told you about a 'super Saturn' - an object in another solar system with rings almost a thousand times bigger than Saturn.  Chariklo, on the other hand, is a 'mini Saturn'.  Its rings are just 800 kilometers across - just 0.3% the size of Saturn's F ring.

These rings are narrow and dense.  One is about 6 kilometers wide and the other - which you can barely see in this artist's picture - is just 3 kilometers wide.   They're separated by a 9-kilometer gap.

How did they get there?  Some smaller objects - probably made of ice - must have collided and broken apart.   But they must have collided not too fast, or they would have shot all over instead of forming neat rings.

The rings are probably not very stable, unless Chariklo has one or more moons to stabilize them.  Saturn has such moons, called shepherd moons.

The second largest centaur, called Chiron, may also have rings.

Puzzle 1: who was Chariklo in Greek mythology?

Puzzle 2: who was Chiron?

Chariklo's full name is 10199 Chariklo:

https://en.wikipedia.org/wiki/10199_Chariklo

Its rings are tentatively named Oiapoque and Chuí, after two rivers in Brazil:

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

They were discovered in 2013.  How come nobody told me? 

Centaurs are lots of fun if you like celestial mechanics:

https://en.wikipedia.org/wiki/Centaur_(minor_planet)

I can't resist quoting a bit:

Because the centaurs are not protected by orbital resonances, their orbits are unstable within a timescale of 10^6–10^7 years. For example, 55576 Amycus is in an unstable orbit near the 3:4 resonance of Uranus.  Dynamical studies of their orbits indicate that being a centaur is probably an intermediate orbital state of objects transitioning from the Kuiper belt to the Jupiter family of short-period comets. Objects may be perturbed from the Kuiper belt, whereupon they become Neptune-crossing and interact gravitationally with that planet. They then become classed as centaurs, but their orbits are chaotic, evolving relatively rapidly as the centaur makes repeated close approaches to one or more of the outer planets. Some centaurs will evolve into Jupiter-crossing orbits whereupon their perihelia may become reduced into the inner Solar System and they may be reclassified as active comets in the Jupiter family if they display cometary activity. Centaurs will thus ultimately collide with the Sun or a planet or else they may be ejected into interstellar space after a close approach to one of the planets, particularly Jupiter.

The picture here was created by Nick Risinger for ESO, the European Southern Observatory in Chile.  They reported their discovery here:

https://www.eso.org/public/usa/news/eso1410/

#astronomy___

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

Trump's Russian connections

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

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

Trump's Russian connections

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

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

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

Triamond

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

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

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

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

Triamond

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

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

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

If we discount the difference between single and double bonds, the triamond is highly symmetrical. There’s a symmetry carrying any atom and any of its bonds to any other atom and any of its bonds. However, the triamond has an inherent handedness, or chirality. It comes in two mirror-image forms.

Some chemists have argued that the triamond should be metastable at room temperature and pressure: that is, it should last for a while but eventually turn to graphite. Diamonds are also considered metastable, though I’ve never seen anyone pull an old diamond ring from their jewelry cabinet and discover to their shock that it’s turned to graphite. The big difference is that diamonds are formed naturally under high pressure — while triamonds, it seems, are not.

It's the mathematics behind the triamond that really interests me.  It's a topological crystal that can be constructed starting from the complete graph on four vertices.  For the details of how this works, see my Visual Insight blog article:

http://blogs.ams.org/visualinsight/2016/10/15/laves-graph/

The picture here was made by Greg Egan.___

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

Alien machinery

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

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

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

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

Alien machinery

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

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

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

Egan thought about it a lot and did a lot of computations and discovered that this unit-distance embedding is flexible.  And here it is, flexing!

For Pegg and Egan's work, go here:

http://math.stackexchange.com/questions/1484002/is-unit-mcgee-rigid

What's the practical use of all this?  Mainly, it's a practice problem in structural rigidity: the study of whether a structure is flexible or rigid.  This is important in engineering:

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

A structure is infinitesimally flexible if, roughly, we can bend it a teeny weeny bit.  As the name suggests, infinitesimal rigidity can be determined by using calculus to take the derivative of all the edge lengths as a function of all the vertex positions and then using linear algebra to see in which directions this derivative is zero.  This is easy in principle, though complicated when you have 24 vertices and 36 edges.

Puzzle 1: with a minimum of explicit computation, prove that any unit-distance embedding of the McGee graph is infinitesimally flexible.

Infinitesimal flexibility is a necessary but not sufficient condition for true flexibility.

Puzzle 2: find a unit-distance embedding of a graph that is infinitesimally flexible but not flexible.

So, Egan had to do more work to show Pegg's unit-distance embedding of the McGee graph was actually flexible.  There is probably a high-powered theoretical way to do this, and it's probably not even very complicated, but I don't know it.   Do you?

For my Visual Insight post on the McGee graph, go here:

http://blogs.ams.org/visualinsight/2015/09/15/mcgee-graph/

By the way, I don't like the phrase 'unit-distance embedding' - we're not really embedding the McGee graph in the plane, because we're letting the edges cross.   The word 'immersion' would be better. 

#geometry  ___

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

Super Saturn

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

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

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

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

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

Super Saturn

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

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

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

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

One theory is that the rings are spinning in a retrograde way — in other words, backwards.   If you have a planet moving clockwise around a star, and its rings are turning counterclockwise, this helps keep them from getting pulled apart.   You can see a simulation here:

http://www.nytimes.com/2016/10/14/science/exoplanet-rings-saturn-j1407b.html

However, it's not obvious why the rings would turn backwards.

There's no sharp boundary between a very large planet and a very small star.    If it produces heat using nuclear fusion, it's considered a star... but there are some funny borderline cases.

Stars about 13 times heavier than Jupiter get hot enough to fuse deuterium — but they quickly fizzle out, since that isotope of hydrogen is rare.   Stars about 65 times heavier than Jupiter can also fuse lithium... but then fizzle out.  So, these things are called brown dwarfs.   Stars over 80 times heavier than Jupiter can actually fuse hydrogen, so they light up and form very small red dwarfs.

The atmosphere of a hot brown dwarf is similar to a sunspot — a cold spot on our Sun.   It contains molecular hydrogen, carbon monoxide and water vapor. This is called a class M brown dwarf.

But after they run out of fuel, they cool down. The cooler class L brown dwarfs have clouds!

But the even more chilly class T brown dwarfs do not. Why not?

Here's a popular theory: the clouds may rain down, with material moving deeper into the star!  People seem to be seeing this in Luhman 16B, a brown dwarf 7 light years from us.  It's half covered by huge clouds. These clouds are hot — 1200 °C — so they’re probably made of sand, iron or salts.  But some of them have been seen to disappear!

Finally, as brown dwarfs cool below 300 °C, astronomers expect that ice clouds start to form: first water ice, and eventually ammonia ice. These are called class Y brown dwarfs.

Wouldn’t that be neat to see? A star with icy clouds!  And maybe it could have huge rings, too!

For more on J1407b, try Wikipedia:

https://en.wikipedia.org/wiki/1SWASP_J140747.93-394542.6

The picture below is an artist's impression by Ron Miller.

#astronomy___

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

Gas, Solid, Liquid, Darkness

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

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

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

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

https://en.wikipedia.org/wiki/Weddell_Sea
Da... more »

Gas, Solid, Liquid, Darkness

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

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

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

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

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

David Burdeny took this photo in 2007 and called it 'Mercator's Projection'.  It appeared as part of a series of photos from Antarctica and Greenland.  As a fan of the Earth's desolate regions, I like these a lot:

http://davidburdeny.com/photographs/north-south/1

You can see more of his work here:

https://www.instagram.com/david_burdeny/

Most is not as thrilling to me, but there are some stunning images of tulip fields, which look don't look like tulip fields.

Thanks to +Jenny Winder for pointing this out.  If you don't have her circled yet, it's not too late!

#photography #art  ___

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

The golden ratio

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

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

Φ = (1 + sqrt(5))/2 = 1.6180339...

I tried other options, like this:

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

and this:

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

Eventually I figured out a nice formula for all expressions like these.  I was very proud of it.

Puzzle 1: what's the formula?

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

The golden ratio

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

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

Φ = (1 + sqrt(5))/2 = 1.6180339...

I tried other options, like this:

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

and this:

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

Eventually I figured out a nice formula for all expressions like these.  I was very proud of it.

Puzzle 1: what's the formula?

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

1 + 2/(3 + 4/(1 + 2/(3 + 4/(1 + 2/(3 + ....

It was silly of me not to notice this.

Puzzle 2: how do you get this formula?

If you're an expert at math, and these puzzles seem obvious to you, please wait and give others a chance!  

This animated gif was made by LucasVB,  a math enthusiast, physics student and educational animator:

https://twitter.com/lucasvb

By the way, I'm on Twitter too now:

https://twitter.com/johncarlosbaez

so if that's your thing, you can get one dose of math or science each day from me there.  I can't really explain stuff in 140 characters, so my tweets always contain links to other things I've written.___

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

McGee graph

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

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

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

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

McGee graph

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

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

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

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

http://blogs.ams.org/visualinsight/2015/09/15/mcgee-graph/

Who is მამუკა ჯიბლაძე?  It's Mamuka Jibladze, writing in Georgian script.   There are lots of good category theorists from Georgia - not the southern state in the US, the country between Russia and Turkey.  Mamuka Jibladze is one.

Here is Jibladze's website:

http://www.rmi.ge/~jib/

Here is Jibladze's post on MathOverflow:

http://mathoverflow.net/a/218435/2893

#geometry  ___

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

Vortex versus antivortex

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

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

For a better explanation, read my article:

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

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

Vortex versus antivortex

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

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

For a better explanation, read my article:

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

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

#physics  ___

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

Nobel Prize in Physics Goes to Another Weird Thing Nobody Understands

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

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

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

And for more details, read his blog:

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

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

Nobel Prize in Physics Goes to Another Weird Thing Nobody Understands

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

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

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

And for more details, read his blog:

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

He doesn't explain topological phase transitions, but he describes how vortices - swirls of spin in a magnetic material, like in his picture here - can act like particles.  That's the first step toward understanding what Kosterlitz and Thouless discovered.

#physics  ___

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

Diamonds are forever?

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

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

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

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

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

After unsuccessfuly trying tocreate a... more »

Diamonds are forever?

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

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

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

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

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

After unsuccessfuly trying to create a slogan for De Beers which would perfectly express of the qualities of a diamond mingled with romance, there was finally a stroke of genius in 1947. One of Ayer’s young copyrighters, Frances Gerety, was working late one night to the point of exhaustion. Finally in desperation, she put her head down on the table and pleaded for help. Just before she left work that night she scribbled the words “a diamond is forever” on a piece of paper and the rest is history. This may have had a simple start, but the result was America’s most famous advertising slogan and today over 90% of American’s recognize it.

Diamonds are very hard, but they don't actually last forever.  Very few things do, because forever is a very long time.  As far as we can tell, electrons are forever.   Many theories of particle physics predict that protons decay.  However, experiments indicate that they last for at least 10^34 years on average... so these theories have not been confirmed.  Protons may be forever.

Diamonds, on the other hand, are metastable at room temperature and pressure.  They are formed under high pressure, deep underground, but under ordinary conditions at the Earth's surface a diamond has more energy than the same amount of graphite!  So, it should slowly turn into graphite and release energy. 

So, diamonds are not forever.  However,  they last a very long time.

How about the abstract mathematical structure of the diamond.   Does that last forever?

Not really.   This is not a thing in spacetime, it's an abstract pattern that has the ability of being realized at any place, at any time, in any universe.  It doesn't make sense to say that this mathematical pattern lasts forever - or that it doesn't last forever.   "Lasting forever" applies to things in spacetime.

Puzzle: what is the approximate lifetime of a diamond at room temperature and pressure?

This picture of the diamond structure was made by Greg Egan back when we were working on 'topological crystals'.  The De Beers quote is from here:

http://www.thediamondauthority.org/diamonds-are-forever-the-story-behind-the-slogan/

#geometry  ___

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

Systems of systems

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

What’s a system of systems?

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

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

Metron was applying for a grantfrom ... more »

Systems of systems

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

What’s a system of systems?

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

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

Metron was applying for a grant from DARPA, the Defense Advanced Research Projects Agency, which funds a lot of cutting-edge research for the US military. It may seem surprising that DARPA is explicitly interested in using category theory to study systems of systems. But it actually shouldn’t be surprising: their mission is to try many things and find a few that work. They are willing to take risks.

Metron was applying for a grant under a DARPA program run by John S. Paschkewitz, who is interested in new paradigms and foundational approaches for the design of systems of systems.

This program is called CASCADE, short for Complex Adaptive System Composition and Design Environment.  Here’s the idea:

Complex interconnected systems are increasingly becoming part of everyday life in both military and civilian environments. In the military domain, air-dominance system-of-systems concepts, such as those being developed under DARPA’s SoSITE effort, envision manned and unmanned aircraft linked by networks that seamlessly share data and resources in real time. In civilian settings such as urban “smart cities”, critical infrastructure systems — water, power, transportation, communications and cyber — are similarly integrated within complex networks. Dynamic systems such as these promise capabilities that are greater than the mere sum of their parts, as well as enhanced resilience when challenged by adversaries or natural disasters. But they are difficult to model and cannot be systematically designed using today’s tools, which are simply not up to the task of assessing and predicting the complex interactions among system structures and behaviors that constantly change across time and space.

To overcome this challenge, DARPA has announced the Complex Adaptive System Composition and Design Environment (CASCADE) program. The goal of CASCADE is to advance and exploit novel mathematical techniques able to provide a deeper understanding of system component interactions and a unified view of system behaviors. The program also aims to develop a formal language for composing and designing complex adaptive systems.

“CASCADE aims to fundamentally change how we design systems for real-time resilient response within dynamic, unexpected environments,” said John Paschkewitz, DARPA program manager. “Existing modeling and design tools invoke static ‘playbook’ concepts that don’t adequately represent the complexity of, say, an airborne system of systems with its constantly changing variables, such as enemy jamming, bad weather, or loss of one or more aircraft. As another example, this program could inform the design of future forward-deployed military surgical capabilities by making sure the functions, structures, behaviors and constraints of the medical system — such as surgeons, helicopters, communication networks, transportation, time, and blood supply — are accurately modeled and understood.”

CASCADE could also help the Department of Defense fulfill its role of providing humanitarian assistance in response to a devastating earthquake, hurricane or other catastrophe, by developing comprehensive response models that account for the many components and interactions inherent in such missions, whether in urban or austere environs.

“We need new design and representation tools to ensure resilience of buildings, electricity, drinking water supply, healthcare, roads and sanitation when disaster strikes,” Paschkewitz said. “CASCADE could help develop models that would provide civil authorities, first responders and assisting military commanders with the sequence and timing of critical actions they need to take for saving lives and restoring critical infrastructure. In the stress following a major disaster, models that could do that would be invaluable.”

The CASCADE program seeks expertise in the following areas:

• Applied mathematics, especially in category theory, algebraic geometry and topology, and sheaf theory

• Operations research, control theory and planning, especially in stochastic and non-linear control

• Modeling and applications responsive to challenges in battlefield medicine logistics and platforms, adaptive logistics, reliability, and maintenance

• Search and rescue platforms and modeling

• Adaptive and resilient urban infrastructure

Metron already designs systems of systems used in Coast Guard search and rescue missions. Their grant proposal was to use category theory and operads to do this better. They needed an academic mathematician as part of their team: that was one of the program’s requirements. So they asked if I was interested.

I had mixed feelings.

On the one hand, I come from a line of peaceniks including Joan Baez, Mimi Fariña, their father the physicist Albert Baez, and my parents. I don’t like how the US government puts so much energy into fighting wars rather than solving our economic, social and environmental problems. It’s interesting that ‘systems of systems engineering’, as a field, is so heavily dominated by the US military. It’s an important subject that could be useful in many ways. We need it for better energy grids, better adaptation to climate change, and so on. I dream of using it to develop ‘ecotechnology’: technology that works with nature instead of trying to battle it and defeat it. But it seems the US doesn’t have the money, or the risk-taking spirit, to fund applications of category theory to those subjects.

On the other hand, I was attracted by the prospect of using category theory to design complex adaptive systems — and using it not just to tackle foundational issues, but also concrete challenges. I liked the idea of working with a team of people who are more practical than me. In this project, a big part of my job would be to write and publish papers: that’s something I can do. But Metron had other people who would try to create prototypes of software for helping the Coast Guard design search and rescue missions.

So I was torn.

In fact, because of my qualms, I’d already turned down an offer from another company that was writing a proposal for the CASCADE program. But the Metron project seemed slightly more attractive — I’m not sure why, perhaps because it was described to me in a more concrete way. And unlike that other company, Metron has a large existing body of software for evaluating complex systems, which should help me focus my theoretical ideas. The interaction between theory and practice can make theory a lot more interesting.

Something tipped the scales and I said yes. We applied for the grant, and we got it.

And so, an interesting adventure began. It will last for 3 years, and I’ll say more about it soon.

There's some interesting discussion about this on my blog:

https://johncarlosbaez.wordpress.com/2016/10/02/complex-adaptive-system-design-part-1/

I decided to simply copy my blog post from there to here, word for word.  So just go down to the comments!___

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

Mathematical Enchantments

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

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

Here's an example of what he wants:

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

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

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

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

with a small rectangle becoming big and whose second half animates 
ac-ad... more »

Mathematical Enchantments

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

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

Here's an example of what he wants:

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

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

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

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

with a small rectangle becoming big and whose second half animates 

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

with the big rectangle becoming small again. (The directed lengths a,b,c,d in the second half correspond respectively to the directed lengths a+b,b,c+d,d in the first half.)

More mathematical/visual details: Let $WXYZ denote the signed area of the oriented rectangle with boundary W->X->Y->Z->W (is there a standard symbol for this?). Then in the figure below, we have

$ABED + $DEHG + $EFJH + $BCFE = $ACJG

GHJ
DEF
ABC

One can animate the formula in a three-step way by letting the boundary of the small rectangle continuously bulge outward in successive directions (rectangle ABED becomes rectangle ABHG which becomes hexagon ABEFJG which becomes rectangle ACJG). To do the shrinking process, relabel the points as

DFE
GJH
ACB

and apply the exact same formulas. One can animate the shrinking with a three-step process involving two inward bulges and one outward bulge.

Both of the three-step procedures (the one that makes a small rectangle big and the one that makes the big rectangle small again) embody the distributive law 

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

(note the nonstandard ordering of the terms: First, Outside, Last, Inside instead of the familiar "FOIL"): set a=$AB, b=$BC, c=$AD, d=$DG, where now $XY means the directed length of XY.

Making such a GIF involves various design issues that I suspect I'd handle badly (leaving aside the technical issues involved with making good visuals). But I also think someone who knows only about animation and not about math would make choices that hide the concepts instead of highlighting them. So I'm looking to partner with someone who's strong on both.

It's possible that what I really want to have is two versions: a "gee-whiz" version that arrests people's attention, and a longer version that explains what's going on (with supporting captions, diagram-labels, and maybe sound).___

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

The icosahedron that got away

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

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

1/1 = 1
3/2 = 1.5
5/3 = 1.6666...
8/5 = 1.6125

and so on, getting closer to

Φ = 1.6180339....

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

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

The icosahedron that got away

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

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

1/1 = 1
3/2 = 1.5
5/3 = 1.6666...
8/5 = 1.6125

and so on, getting closer to

Φ = 1.6180339....

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

Recently the curator of the Museum of Evolution, Palaeontology and Mineralogy in Uppsala, Sweden, emailed me and told me that the handsome pseudoicosahedron shown here was on sale for just $30.  I am very lucky to have friends like this.  His name is Johan Kjellman.

Unfortunately, I waited a day or two.   By the time I offered to buy it, it was already sold.   :-(

I wanted to buy it from here:

http://www.trinityminerals.com/new.html

It's from the Merelani Hills near Arusha in Tanzania.  It was 2.8 x 1.4 x 1.3 centimeters in size:

A crystal of pyrite is set in calcite and graphite matrix. This find at Merelani has produced some of the most remarkable crystals in terms of crystallography. In this case the icosahedron! The luster and form are unmistakable from this locality. The back side of the crystal has damage but the display face is fine.

If you ever see a pyrite pseudoicosahedron for sale, let me know!  I'd also be happy to have a pyritohedron, which is nature's attempt to create a regular dodecahedron using little cubes.

For the math of the pseudoicosahedron and pyritohedron, go here:

http://math.ucr.edu/home/baez/golden.html

and scroll down until you start seeing pictures that look right.

#geometry___

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

An infinite corridor of universes

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

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

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

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

An infinite corridor of universes

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

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

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

You see, when you solve Einstein's equations, you get a world obeying the rules of general relativity.  But sometimes, if you're not careful, somebody else can find a bigger world that contains yours!    It's like you drew a map of the world but you forgot there was anything south of the equator.   A solution is called maximally extended if you can't make it any bigger. 

This picture shows the maximally extended Kerr solution.  It's a Penrose diagram, so moving up the page takes you forward in time, while moving to the right or left edge of the page takes you away from the black hole.  Light moves along diagonal lines.

It's a single world, but it has portions called Universe, Parallel Universe, Antiverse, and Parallel Antiverse.   Each of these is roughly like our universe, but with no Big Bang.  Each lasts forever: time is not drawn to scale.

Each universe, and each parallel universe, has a black hole in it - and also a white hole!   Each antiverse, and each parallel antiverse, has a black hole with negative mass, and also a white hole with negative mass.

Only a few of these universes and antiverses are shown here.   But there are infinitely many.  The pattern repeats forever as you continue to go up or down the picture - that is, forwards or backwards in time.  

There's also an infinite repeating sequence of black holes and white holes.    And there's more - you can see singularities drawn as wiggly lines.   But let's not worry about those yet.  There's too much to take in at once.

Let's just follow the blue curve as it goes up the page.   This describes a path you could take through space and time.

You could shoot out of a white hole at the very bottom of the picture and wind up in our universe. 

Then you could jump into the black hole. 

If you dodge the singularities, you could wind up in a new white hole!

And at this point, you have a choice.  Swerve right and you go into a new universe. Swerve left and you go into a new parallel universe.  They're different - but there's no big difference.  In this picture, you choose to enter the new universe. 

And so on!

It would be great fun if our universe were part of a grand infinite corridor of universes like this.   As far as we know, it's not.   I suspect the real universe will be even more amazing.  However, we will need much better science and technology to discover what's out there.  Right now most of us are stuck here on Earth, and we need to learn to live here.  That's a tough challenge too.

My picture is from Andrew Hamilton's wonderful website:

http://jilawww.colorado.edu/~ajsh/insidebh/penrose.html

I would like to tell you more about the Kerr black hole - but if I don't get around to it, also check out David Madore's page:

http://www.madore.org/~david/math/kerr.html

#physics___

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

Solar wind

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

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

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

Solar wind

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

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

What's the solar wind made of?   When you take hydrogen and helium and heat them up so much that the electrons get knocked off, you get a mix of electrons, hydrogen nuclei (protons), and helium nuclei (made of two protons and two neutrons).   So that's all it is.

The Sun's corona is very hot: about a million degrees Celsius.  That's hotter than the visible surface of the Sun!  Why does it get so hot?  When I last checked, this was still a bit mysterious.   But it has something to do with the Sun's powerful magnetic fields. 

When they're this hot, some electrons are moving fast enough to break free of the Sun's gravity.   Its escape velocity is 600 kilometers per second.  The protons and helium nuclei, being heavier but having the same average energy, move slower.  So, few of these reach escape velocity.

But with the negatively charged electrons leaving while the positively charged protons and helium nuclei stay behind, this means the corona builds up a positive charge!   So the electric field starts to push the protons and helium nuclei away, and some of them - the faster-moving ones - get thrown out too.  

Indeed, enough of these positively charged particles have to leave the Sun to balance out the electrons, or the Sun's electric charge would keep getting bigger.   It would eventually shoot out huge lightning bolts!  The solar wind deals with this problem in a less dramatic way - but sometimes it gets pretty dramatic.  Check out this proton storm:

http://www.spaceweather.com/images2014/08jan14/x1s2_anim.gif

When storms like this happen, the US government sends out warnings like this:

Space Weather Message Code: WATA50
Serial Number: 48
Issue Time: 2014 Jan 08 1214 UTC
WATCH: Geomagnetic Storm Category G3 Predicted
Highest Storm Level Predicted by Day:
Jan 08: None (Below G1) Jan 09: G3 (Strong) Jan 10: G3 (Strong)
THIS SUPERSEDES ANY/ALL PRIOR WATCHES IN EFFECT
Potential Impacts: Area of impact primarily poleward of 50 degrees geomagnetic latitude.
Induced Currents – Power system voltage irregularities possible, false alarms may be triggered on some protection devices.
Spacecraft – Systems may experience surface charging; increased drag on low Earth-orbit satellites and orientation problems may occur.
Navigation – Intermittent satellite navigation (GPS) problems, including loss-of-lock and increased range error may occur.
Radio – HF (high frequency) radio may be intermittent.
Aurora – Aurora may be seen as low as Pennsylvania to Iowa to Oregon.

The solar wind is really complicated, and I've just scratched the surface.  I love learning about stuff like this, surfing the web as I lie in bed sipping coffee in the morning.  Posting about it just helps organize my thoughts - when you try to explain something, you come up with more questions about it.

For more on space weather, visit this fun site:

http://www.spaceweather.com/

You can see space weather reports put out by the National Oceanic and Atmospheric Administration here:

http://www.swpc.noaa.gov/products/alerts-watches-and-warnings

For more on the solar wind, see:

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

For more on STEREO, see:

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

#physics   #astronomy  ___

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

Life on the Infinite Farm

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

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

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

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

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

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

Life on the Infinite Farm

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

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

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

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

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

You move the guest in room 1 to room 2, the guest in room 2 to room 3, and so on.  Now there's a room available!

The Hilbert Hotel stories were introduced by the great mathematician David Hilbert in a 1924 lecture, and popularized by George Gamow in his classic One Two Three... Infinity.   That book made a huge impression on me as a child: one of my first times I tasted the delights of mathematics.    

But that book is not good for children just learning to read.  Life on the Infinite Farm is.  And there's nothing that smells like "education" in this book.  It's just fun.

You can read more Hilbert Hotel stories here:

https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel

But it's probably more fun to read Gamow's One Two Three... Infinity.   He was an excellent astrophysicist who in 1942 figured out how the first elements were created - the theory of Big Bang nucleosynthesis.    He was also a coauthor of the famous Alpher-Bethe-Gamow paper on this topic, also known as the αβγ paper.    Alpher was a grad student of Gamow, and they added the famous nuclear physicist Hans Bethe as a coauthor just for fun - since 'Bethe' is pronounced like the Greek letter 'beta':

It seemed unfair to the Greek alphabet to have the article signed by Alpher and Gamow only, and so the name of Dr. Hans A. Bethe was inserted in preparing the manuscript for print. Dr. Bethe, who received a copy of the manuscript, did not object, and, as a matter of fact, was quite helpful in subsequent discussions. There was, however, a rumor that later, when the alpha, beta, gamma theory went temporarily on the rocks, Dr. Bethe seriously considered changing his name to Zacharias.

Gamow also had a real knack for explaining things in fun ways, with the help of charming pictures.   I don't do many advertisements for commercial products, but I will for this!  You can get his book for as little as $2.98 plus shipping:

https://www.amazon.com/One-Two-Three-Infinity-Speculations/dp/0486256642/

You should have read it by the time you were a teenager - but if you didn't, maybe it's not too late.

For more about Gamow, see:

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

#bigness___

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

Poncelet's Porism

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

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

This amazing fact is called Poncelet's Porism

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

What's a porism, really? 

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

Poncelet's Porism

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

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

This amazing fact is called Poncelet's Porism

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

What's a porism, really? 

Well, Euclid is famous for his Elements, but he also wrote a more advanced book called Porisms.  Unfortunately that book is lost.  I hear that someone checked it out from the library of Alexandria and never returned it.   By now the overdue fee exceeds the annual GDP of Greece, so we'll never see that book again... and we'll never know exactly what Euclid meant by 'porism'.

Wikipedia starts by saying:

A porism is a mathematical proposition or corollary. In particular, the term porism has been used to refer to a direct result of a proof, analogous to how a corollary refers to a direct result of a theorem. In modern usage, a porism is a relation that holds for an infinite range of values but only if a certain condition is assumed, for example Steiner's porism.  [...]  Note that a proposition may not have been proven, so a porism may not be a theorem, or for that matter, it may not be true.

In short: nobody knows what a porism is, but people are willing to make stuff up.

Pappus of Alexandria managed to write down a few of Euclid's porisms around 400 AD, before the book got lost.  They are quite advanced facts about geometry.  Poncelet was inspired by Pappus, so when he proved his cool result, maybe he wanted to call it a porism too.  I don't know.

A slick modern proof of Poncelet's porism uses 'elliptic curves'.  Check out David Speyer's explanation:

https://sbseminar.wordpress.com/2007/07/16/poncelets-porism/

In case you're not a mathematician, beware!  An 'elliptic curve' is jargon for a surface shaped like a doughnut.  We just call them 'elliptic curves' to keep people like you confused.

For some truly amazing connections between Poncelet's Porism and other math problems, see this paper by J. L. King:

https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/King609-628.pdf

An elementary proof of Poncelet's Porism is here:

http://user.math.uzh.ch/halbeisen/publications/pdf/poncelet.pdf

In math, 'elementary' means that we don't use fancy concepts.  It doesn't mean 'easy'. 

When I retire, I want to quit proving theorems, and prove a porism.

#geometry  ___

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

Black hole versus white hole

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

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

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

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

Black hole versus white hole

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

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

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

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

But inquiring minds want to know...  what could be behind the antihorizon?

This picture shows the answer.  This is the maximally extended Schwarzschild black hole - the biggest universe we can imagine, that contains this sort of black hole.

It's really weird.

It contains not only a black hole but also a white hole.  The wiggly lines are singularities.  Matter and light can only fall into the black hole from our universe... passing through the horizon and hitting the singularity at the top of the picture.   And they can only fall out of the white hole into our universe... shooting out of the singularity at the bottom of the picture and passing through the antihorizon.

If that weren't weird enough, there's also a parallel universe, just like ours.  

Someone from our universe and someone from the parallel universe can jump into the black hole, meet, say hi, then hit the singularity and die.    Fun!

But we can never go from our universe to the parallel universe.  :-(

Why not?   Remember, the only allowed paths for people going slower than light are paths that go more up the page than across the page - like the blue path in the picture.  To get from our universe to the parallel universe, a path would need to go more across than up.

If you could go faster than light for just a very short time, you could get from our universe to the parallel universe by zipping through the point in the very middle of the picture, where the horizon and antihorizon meet. 

Puzzle 1.  Suppose the parallel universe has stars in it more or less like ours.  You can't see it from our universe - but you could see it if you jumped into the black hole!  What would it look like?

Puzzle 2.   How would my story change if the "arrow of time" in the parallel universe pointed the other way from ours?  In other words, what if the future for them was at the bottom of the picture, rather than the top?

I should emphasize that we're playing games here, but they're games with rules.  We're not talking about the real world, but the math of this stuff is well-understood, so you can't just make stuff up.  Or you can, but it might be wrong.  These puzzles have right and wrong answers!

Unfortunately I haven't really explained things well enough, so you may need to guess  the answers instead of just figure them out.  For more info, try Andrew Hamilton's page, from which I took this picture:

http://jila.colorado.edu/~ajsh/insidebh/penrose.html

And for more, try this:

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

#physics  ___

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

Understanding black holes

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

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

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

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

Understanding black holes

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

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

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

The whole universe outside the black hole is squashed to a diamond. The singularity is the wiggly line at top.   The blue curve is the trajectory of a cat falling into the black hole.  Since it's moving slower than light, this curve must move more up than across.  So, once it crosses the diagonal line called the horizon, it is doomed to hit the singularity. 

Indeed, anyone in the region called "Black Hole" will hit the singularity.   Notice: when you're in this region, the singularity is not in front of you!  It's in your future.  Trying to avoid it is like trying to avoid tomorrow.

But what is the diagonal line called the antihorizon?   If you start in our universe, there's no way to reach the antihorizon without going faster than light.     But we can imagine things crossing it from the other direction: entering from the left  and coming in  to our universe! 

The point is that while this picture of the Schwarzschild black hole is perfectly fine, we can imagine extending it and putting it inside a larger picture.   We say it's not maximally extended

The larger picture, the maximally extended one, describes a very strange world, where things can enter our universe through the antihorizon.   But that's another story, which deserves another picture.

If we stick with the diagram here, nothing can come out of the antihorizon, so it will look black.  In fact, to anyone in the "Universe" region, it will look like a black sphere.  And that's why a Schwarzschild black hole looks like a black sphere from outside!

The weird part is that this black sphere you see, the antihorizon, is different than the sphere you can fall into, namely the horizon.

If this seem confusing, join the club.   I think I finally understand it, but nobody ever told me this - at least, not in plain English - so it took me a long time.

What could be behind the antihorizon?   If you want to peek, try Andrew Hamilton's page on Penrose diagrams, where I got this picture:

http://jila.colorado.edu/~ajsh/insidebh/penrose.html
 
I wish that Wikipedia had a really nice Penrose diagram like this!  It's very important.  They have some more complicated ones, but the most basic important ones are not drawn very nicely.  You need to think about Penrose diagrams to understand black holes and the Big Bang!

Still, their article is worth reading:

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

For more on the Schwarzschild black hole, read this:

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

#physics  ___

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

The mystical hexagram theorem

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

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

The red lines intersect in a point G.
The yellow lines intersect in a point H.
The blue lines intersect in a point K. 

And then the cool part:

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

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

The mystical hexagram theorem

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

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

The red lines intersect in a point G.
The yellow lines intersect in a point H.
The blue lines intersect in a point K. 

And then the cool part:

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

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

I'll start by reading this:

https://en.wikipedia.org/wiki/Pascal%27s_theorem

If you can manage to enable Java applets on your device - a task made ever harder by those worried for our safety - you should check out this:

http://www.cut-the-knot.org/Curriculum/Geometry/Pascal.shtml

You can move six points around a circle and see how things change.

#geometry___

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

Exploring black holes - with cats!

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

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

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

Forexample, if... more »

Exploring black holes - with cats!

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

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

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

For example, if two supermassive black holes were shooting toward our Solar System right now and collided in an hour, forming a black hole that swallowed the Earth, at some moment your cat would cross the event horizon.  That's the moment when, no matter how hard it tried, it could no longer escape.  But this moment could be happening right now, and your cat might not notice!   No alarm bells ring at this moment.

What happens inside the event horizon?

For a non-rotating black hole formed by the collapse of matter, the answer is pretty well understood - except at the 'singularity', where the laws of physics we know break down.   

Your cat will fall in, getting stretched ever thinner.    For a hypothetical non-rotating black hole with the mass of our Sun, once it crosses the event horizon it will hit the singularity in about 10 microseconds.  That's not much time!

In fact, all known black holes are heavier than our Sun.   If you double the mass of the black hole, you double the amount of time it takes to hit the singularity, and so on.  So, for a non-rotating black hole 100,000 times the mass of our Sun, it takes 1 second to hit the singularity after  crossing the horizon.

The biggest known black holes are about 30 billion times the mass of our Sun.  For a non-rotating black hole this big, it would take three and a half days for your cat to hit the singularity after it crosses the horizon!   You might want to send it in with some cat food.

But there's a catch.  Real-world black holes are always rotating!  This makes them much more complicated.  For starters, frame-dragging tends to pull you along with the black hole's rotation.

We began to see that yesterday when I showed you +Leo Stein's website about how photons orbit a black hole.  There's not just one photon sphere - there's a bunch!   

There's also a region called the ergosphere where frame-dragging becomes so strong that your cat can't stand still.   And Penrose discovered something interesting about this.

You can send a cat into the ergosphere with rockets strapped to its back.  When it shoots back out, it can carry angular momentum and energy out of the black hole!   It's a bit like how we use Jupiter to fling satellites to Pluto - except we're using the rotation rather than the motion of the black hole!  

So, we can in theory "mine" a rotating black hole, removing energy from it until it's not rotating.

Beneath the ergosphere lies the horizon.  Inside the horizon of a rotating black hole, things get even weirder.  More on that later, I hope.  But probably not with cats.

For now, try this:

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

#physics  ___

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

Light moves around a rotating black hole

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

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

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

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

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

Light moves around a rotating black hole

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

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

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

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

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

Frame-dragging was one of the last experimental predictions of general relativity to be verified, using a satellite called Gravity Probe B.    Frame-dragging was supposed to make a gyroscope precess a bit more.   This experiment was really hard.  It suffered massive delays and cost overruns.   When it was finally done, the results were not as conclusive as we'd like.   I believe in frame-dragging mainly because everything else about general relativity works great, and it's hard to make up a theory that differs in just this one prediction.

It's pretty bizarre that instead of following orbits that move in and out from the black hole - like ellipses, or something - photons can move only in orbits of constant radius, with a range of different possible radii  being allowed.   Leo Stein explains:

After you study the radial equation, you learn that the only bound photon trajectories — that is, orbits! — are those for which r=const in Boyer-Lindquist coordinates. This is why these photon orbits are sometimes called “circular” or “spherical.”

In the end, you see that for each angular momentum parameter a for the black hole, there is a one-parameter family of trajectories given by the radius r, which must be between the two limits

r₁(a) ≤ r ≤ r₂(a)

The innermost photon orbit is a prograde circle lying in the equatorial plane, and the outermost orbit is a retrograde circle lying in the equatorial plane.

Prograde means that this orbit goes around the same way the black hole is rotating; retrograde means it's moving in the opposite direction.

These orbits are all unstable.  Push the photon slightly inward and it will fall into the black hole.   Push it outward just a bit and it will fly away.  So, this stuff is mainly interesting for the math.  You won't actually find a lot of light orbiting a black hole.

For more of the math, see Leo Stein's website.  It's great!  But the most fun part is using some sliders to play with photon orbits.

For more on frame-dragging, see:

https://en.wikipedia.org/wiki/Frame-dragging

#physics  ___

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

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

Nomads kick ass.   James Dator explains:

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

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

Cirit

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

Er Enish

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

Kok-boru
more »

The World Nomad Games, Kyrgyzstan. ___Just because someone's on crutches doesn't mean they're handicapped

Nomads kick ass.   James Dator explains:

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

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

Cirit

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

Er Enish

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

Kok-boru

There’s no delicate way to explain Kok-boru. It’s horseback basketball using a goat carcass. You win by tossing the dead goat into your opponent’s well. It comes from a tradition of beating up wolves that attacked your herd of sheep and throwing a dead wolf to your friends who went wolf hunting with you.

Mas-wrestling

In this form a wrestling, athletes fight over a stick. Each wrestler is given part of the stick to hold and are seated facing each other with their feet on a plank. Whoever gets the stick wins.

Salbuurun

A three-step hunting sport involving animals.  Competitions are held in the following disciplines:

1. Burkut saluu - hunting with golden eagles. Composition of the team - 6 people: 1 leader and 5 berkutchi (hunter with eagles).

2. Dalba oynotuu - falcon flying to the lure. Composition of the team - 6 people: 1 leader and 5 Kushchu (falconer).

3. Taigan jarysh - dog racing among breeds of greyhound. Composition of the team - 6 people: 1 leader and 5 owners of dogs.

Traditional Archery

This has to be the biggest misnomer of the World Nomad Games. They say “traditional,” but really they mean on horseback and also this.

(The picture of this woman here.  Who is she?)

James Dator explains more games here:

http://www.sbnation.com/2016/9/12/12888720/world-nomad-games-burning-horseriders-dead-goat-basketball-eagle-hunting-wow

There's more here:

https://www.washingtonpost.com/news/early-lead/wp/2016/09/14/first-ever-u-s-team-plays-rugby-on-horses-with-a-decapitated-goat-at-world-nomad-games/

Not for the squeamish!  However, excellent pictures of hunters with their eagles, horse riders, etc.

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