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Shared Circles including John Baez

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Average numbers for the latest posts (max. 50 posts, posted within the last 4 weeks)

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Top posts in the last 50 posts

Most comments: 106

posted image

2015-07-10 09:04:29 (106 comments, 24 reshares, 90 +1s)Open 

Bye-bye, carbon

On Quora someone asked:

"What is the most agreed-on figure for our future carbon budget?"

My answer:

Asking "what is our future carbon budget?" is a bit like asking how many calories a day you can eat.  There's really no limit on how much you can eat if you don't care how overweight and unhealthy you become.  So, to set a carbon budget, you need to say how much global warming you will accept.

That said, here's a picture of how we're burning through our carbon budget.  This appears in the International Energy Agency report World Energy Outlook Special Report 2015, which is free and definitely worth reading.

It says that our civilization has burnt 60% of the carbon we're allowed to while still having a 50-50 chance of keeping global warming below 2 °C.

This isbas... more »

Most reshares: 419

posted image

2015-07-07 20:34:15 (24 comments, 419 reshares, 148 +1s)Open 

The best part: it's solar-powered

Most plusones: 231

posted image

2015-07-29 04:06:19 (40 comments, 27 reshares, 231 +1s)Open 

Borobudur

This is the largest temple in Java - and indeed in all of Southeast Asia.   Built of 2 million stone blocks, it's actually designed in the form of a mandala: a geometric design for the aid of meditation.  The pilgrim's walk to the top is 5 kilometers long, and it passes by 1460 stone panels illustrating stories, as well as 1212 decorated panels. 

Nobody knows who built it, sometime around 800 AD.   It was eventually abandoned, and it lay hidden for centuries, gradually buried by volcanic ash and thick jungle growth.  In 1814, the British colonist Thomas Raffles - who also founded Singapore - traveled through Java and heard of a huge monument near Yogyakarta, the old capital of this island.  He got a Dutch engineer to investigate, and by 1885 the jungle was cleared away and the full extent of Borobudur was revealed!

But for me the bestpart i... more »

Latest 50 posts

posted image

2015-07-31 13:27:20 (6 comments, 0 reshares, 37 +1s)Open 

Today's adventures in Java

These are Indonesian reporters interviewing my wife Lisa near the sultan's palace in Yogyarkarta.  She had just won a race where the goal is to walk for 50 meters wearing a mask like the head of a monkey.  The mask covers your whole head, with no eye holes, so it prevents you from seeing.  You need to go between two banyan trees and cross the finish line without veering off the track.

Most of the contestants were Indonesian high school kids, so the reporters were interested to see an American woman try this - and win!

A few years ago another American woman, +Della Bradt, wrote:

In the middle of the square are two giant banyan trees. There is a challenge associated with these trees that we've been itching to try since day one. You have to start from 50 meters away from the trees and you are blindfolded. Then youar... more »

Today's adventures in Java

These are Indonesian reporters interviewing my wife Lisa near the sultan's palace in Yogyarkarta.  She had just won a race where the goal is to walk for 50 meters wearing a mask like the head of a monkey.  The mask covers your whole head, with no eye holes, so it prevents you from seeing.  You need to go between two banyan trees and cross the finish line without veering off the track.

Most of the contestants were Indonesian high school kids, so the reporters were interested to see an American woman try this - and win!

A few years ago another American woman, +Della Bradt, wrote:

In the middle of the square are two giant banyan trees. There is a challenge associated with these trees that we've been itching to try since day one. You have to start from 50 meters away from the trees and you are blindfolded. Then you are spun around 3 times to the right and 3 times to the left. Once you are oriented towards the trees, you have to walk in a straight line through the middle of them moving from North to South. While the opening between the trees is very wide, it's extremely difficult to accomplish. I actually failed a spectacular 3 times in a row. Each time I veered in the opposite direction of the trees. While 2 people in our group made it, most people weren't able to. It is said that those who make it through will have success and good fortune in their lives.

This is just one of the many adventures we had today.  People here seem happy to strike up conversations.  We met a music teacher who invited us to a free gamelan class and carefully explained the cheapest ways to get to the temples of Borobudur and Prambanan, and a soon-to-be-retired worker at a local church who walked us to a batik store near his house - cheaper and better than the more touristic ones on the shop-lined street called Jalan Malioboro.   We didn't buy anything, but it was fun to see.  Similarly, it was at some local guy's suggestion that Lisa decided to join the monkey-head race.  Walking around, and being open to such random stuff, makes traveling more fun around here.

#java  ___

posted image

2015-07-30 00:34:58 (38 comments, 36 reshares, 162 +1s)Open 

The oldest one

This is a tarsier, filmed by Michael Bowers. There are several kinds of tarsiers.  All of them live in Southeast Asia - mainly the Philippines, Sulawesi, Borneo, and Sumatra.  But tarsiers used to live in many other places too. 

They are, in fact, the oldest known primates that survive today!  Fossils show that they've been around for the past 45 million years.The ancestors of tarsiers branched off from the ancestors of lemurs about 83 million years ago, considerably before the dinosaurs went extinct!

This particular guy is a spectral tarsier.  I guess that 'spectral' here means 'like a ghost, or specter' rather than 'like the colors in a rainbow'.  Probably their eyes look spooky at night when they reflect light.

The spectral tarsier lives on Selayar, an island off the larger island ofSulaw... more »

The oldest one

This is a tarsier, filmed by Michael Bowers. There are several kinds of tarsiers.  All of them live in Southeast Asia - mainly the Philippines, Sulawesi, Borneo, and Sumatra.  But tarsiers used to live in many other places too. 

They are, in fact, the oldest known primates that survive today!  Fossils show that they've been around for the past 45 million years.The ancestors of tarsiers branched off from the ancestors of lemurs about 83 million years ago, considerably before the dinosaurs went extinct!

This particular guy is a spectral tarsier.  I guess that 'spectral' here means 'like a ghost, or specter' rather than 'like the colors in a rainbow'.  Probably their eyes look spooky at night when they reflect light.

The spectral tarsier lives on Selayar, an island off the larger island of Sulawesi, in Indonesia.  It's less specialized than some other species of tarsiers: it doesn't have adhesive toes, for example.  Its Latin name is Tarsier spectrum or sometimes Tarsier tarsier, since some consider it the prototype of all tarsiers.

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

Michael Bowers is here on G+:

https://plus.google.com/u/0/116248116353532955491/posts

#biology  ___

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2015-07-29 04:06:19 (40 comments, 27 reshares, 231 +1s)Open 

Borobudur

This is the largest temple in Java - and indeed in all of Southeast Asia.   Built of 2 million stone blocks, it's actually designed in the form of a mandala: a geometric design for the aid of meditation.  The pilgrim's walk to the top is 5 kilometers long, and it passes by 1460 stone panels illustrating stories, as well as 1212 decorated panels. 

Nobody knows who built it, sometime around 800 AD.   It was eventually abandoned, and it lay hidden for centuries, gradually buried by volcanic ash and thick jungle growth.  In 1814, the British colonist Thomas Raffles - who also founded Singapore - traveled through Java and heard of a huge monument near Yogyakarta, the old capital of this island.  He got a Dutch engineer to investigate, and by 1885 the jungle was cleared away and the full extent of Borobudur was revealed!

But for me the bestpart i... more »

Borobudur

This is the largest temple in Java - and indeed in all of Southeast Asia.   Built of 2 million stone blocks, it's actually designed in the form of a mandala: a geometric design for the aid of meditation.  The pilgrim's walk to the top is 5 kilometers long, and it passes by 1460 stone panels illustrating stories, as well as 1212 decorated panels. 

Nobody knows who built it, sometime around 800 AD.   It was eventually abandoned, and it lay hidden for centuries, gradually buried by volcanic ash and thick jungle growth.  In 1814, the British colonist Thomas Raffles - who also founded Singapore - traveled through Java and heard of a huge monument near Yogyakarta, the old capital of this island.  He got a Dutch engineer to investigate, and by 1885 the jungle was cleared away and the full extent of Borobudur was revealed!

But for me the best part is: I'm going there this weekend!

Lisa and I will be staying in Yogyakarta, the old capital of Java, in a hotel near the palace.  It's a 45-minute drive to Borobudur.  I feel quite unprepared for this trip.  Do you know anything I simply must see?

The picture here is from Trey Ratcliff:

http://www.stuckincustoms.com/category/travel/indonesia/borobudur/

#java  ___

posted image

2015-07-23 23:45:39 (38 comments, 13 reshares, 82 +1s)Open 

My 14 favorite mountains

This picture actually shows the 14 Dyck words of length 8.  A Dyck word is a well-formed string of left and right parentheses, like these:

                     (((())))

                     ((()()))

             ((())())     (()(()))

     ((()))()     (()()())     ()((()))

     (()())()     (())(())     ()(()())

     (())()()     ()(())()     ()()(())

                     ()()()()

These are, in fact, all 14 Dyck words of length 8.

What makes these strings of parentheses 'well-formed'?   Something like this is not well-formed:

)())))((

A string of parentheses is well-formed if when we read it from left to right, there is ateach stage at least as many left parentheses as right ones, with an equal number by the time we reach ... more »

My 14 favorite mountains

This picture actually shows the 14 Dyck words of length 8.  A Dyck word is a well-formed string of left and right parentheses, like these:

                     (((())))

                     ((()()))

             ((())())     (()(()))

     ((()))()     (()()())     ()((()))

     (()())()     (())(())     ()(()())

     (())()()     ()(())()     ()()(())

                     ()()()()

These are, in fact, all 14 Dyck words of length 8.

What makes these strings of parentheses 'well-formed'?   Something like this is not well-formed:

)())))((

A string of parentheses is well-formed if when we read it from left to right, there is at each stage at least as many left parentheses as right ones, with an equal number by the time we reach the end.

In the picture below, a left parenthesis is shown as upward-slanting line segment and a right parenthesis as a downward-slanting one. The condition of being well-formed then translates into the fact that the resulting ‘mountain range’ starts and ends at ground level, and never goes below ground.  So, we get nice pictures of mountain ranges!

More importantly, this gives a was to say when one Dyck word is 'taller' than another.  We say w≥w′ when the mountain range corresponding to w is at least as high at all points as the mountain range corresponding to w′. 

The black lines help you see when one Dyck word is taller than another!  The mountain on top is taller than all the rest.

There are at least 3 other interesting ways to define a notion of ≥ for Dyck words.  I wish I understood them all a bit better.

The set of all Dyck words is called the Dyck language. This plays a fundamental role in computer science.  We can also define Dyck languages with more than one kind of parentheses; for example,

([])(([])[]))

is a Dyck word on two kinds of parentheses. The Chomsky–Schützenberger representation theorem characterizes context-free languages - also important in computer science - in terms of the Dyck language on two parentheses.

Returning to the Dyck language with just one kind of parenthesis, the number of Dyck words of length 2n is the nth Catalan number.  The Catalan numbers go like this:

1, 1, 2, 5, 14,  42, ...

and if you ever learn an interesting mathematical fact about the numbers 14 or 42, it's likely to be true because of this!  You see, there are lots of things counted by Catalan numbers.  And this is why there are several different ways of defining a notion of ≥ for Dyck words. 

To dig deeper into these various notions of ≥ for Dyck words, check out my article on the Visual Insight blog:

http://blogs.ams.org/visualinsight/2015/07/15/dyck-words/

The picture here was drawn by Tilman Piesk, and he put it on Wikicommons:

https://commons.wikimedia.org/wiki/File:Dyck_lattice_D4.svg___

posted image

2015-07-21 12:37:37 (15 comments, 22 reshares, 217 +1s)Open 

The sea turtles of Florida are back!

It's 1 meter long and it weighs 150 kilos.  When it crawls on land to lay its eggs, a green sea turtle looks clumsy and awkward.   But in its true home - the sea - it's beautiful and graceful! 

And here's some good news.  Back in the 1980s, when scientists first started counting the nests of green sea turtles in one area in Florida, they found fewer than 40 nests.  Now they count almost 12,000!

We can thank the Endangered Species Act, which brought sea turtles under protection in 1978.  We can also thank state laws discouraging development on Florida beaches - and the Archie Carr National Wildlife Refuge, which was established in 1991. 

Endangered species can bounce back!  Animals like nothing better than to breed, after all.

Sea turtles have been around since the late Triassic Period, 245million... more »

The sea turtles of Florida are back!

It's 1 meter long and it weighs 150 kilos.  When it crawls on land to lay its eggs, a green sea turtle looks clumsy and awkward.   But in its true home - the sea - it's beautiful and graceful! 

And here's some good news.  Back in the 1980s, when scientists first started counting the nests of green sea turtles in one area in Florida, they found fewer than 40 nests.  Now they count almost 12,000!

We can thank the Endangered Species Act, which brought sea turtles under protection in 1978.  We can also thank state laws discouraging development on Florida beaches - and the Archie Carr National Wildlife Refuge, which was established in 1991. 

Endangered species can bounce back!  Animals like nothing better than to breed, after all.

Sea turtles have been around since the late Triassic Period, 245 million years ago.  During the Mesozoic Era turtles went back and forth between land and sea.  But modern sea turtles, with flippers instead of claws, evolved about 120 million years ago, during the Cretaceous Period.   They survived the extinction of the dinosaurs - and with a bit of luck, they'll survive us.

For more, listen to this story:

http://www.npr.org/2015/07/13/422672962/florida-sea-turtles-stage-amazing-comeback

I promise you'll find it heart-warming, even if you're a cold-blooded reptile. 

The picture is from here:

http://www.palmbeachillustrated.com/TikesNorthPBC

For more on the green sea turtle, Chelonia mydas, read this:

http://www.conserveturtles.org/seaturtleinformation.php?page=green___

posted image

2015-07-20 09:43:18 (65 comments, 40 reshares, 123 +1s)Open 

Pentaquarks

Last week a team at CERN says they might have seen some pentaquarks!  Physicists have been looking for them.  Back in 2005 Japanese researchers claimed they saw some, but this was later discredited.  I hope this new claim holds up. 

What's a pentaquark?  It's not really 5 quarks.  It's actually 4 quarks and an antiquark, all held together by exchanging other particles called gluons. 

Let's start with something easier: a neutron, as shown here.  A neutron consists of 3 quarks: one up quark and two down quarks.  They're actually zipping around like mad in a blurry quantum way, but this movie simplifies things.

Besides coming in various kinds, like up and down, quarks have an easily changeable property called color.  This is nothing like ordinary color - but color serves as a convenient metaphor, andphysicist... more »

Pentaquarks

Last week a team at CERN says they might have seen some pentaquarks!  Physicists have been looking for them.  Back in 2005 Japanese researchers claimed they saw some, but this was later discredited.  I hope this new claim holds up. 

What's a pentaquark?  It's not really 5 quarks.  It's actually 4 quarks and an antiquark, all held together by exchanging other particles called gluons. 

Let's start with something easier: a neutron, as shown here.  A neutron consists of 3 quarks: one up quark and two down quarks.  They're actually zipping around like mad in a blurry quantum way, but this movie simplifies things.

Besides coming in various kinds, like up and down, quarks have an easily changeable property called color.  This is nothing like ordinary color - but color serves as a convenient metaphor, and physicists occasionally have a sense of humor, so that's what they called it. 

There's a lot of math underlying this story, but let's sweep that under the carpet and talk about color in simple terms, so you can explain pentaquarks to your children and parents. 

Quarks can be in 3 different colors, called red, green and blue. But they can only stick together and form a somewhat stable particle if all three colors add up and cancel out to give something white.  So, protons and neutrons are made of 3 quarks.
 
The quarks stick together by exchanging gluons, which have subtler colors like red-antigreen and green-antiblue.  

If you watch this movie of a neutron, you'll see a red quark emit a red-antigreen gluon and turn green.  This red-antigreen gluon is then absorbed by a green quark, turning it red.  Color is conserved like this!  The total color of the neutron remains white.

You can't build something white out of just a single quark, so we never see lone quarks in nature.  The closest you can come is at insanely high temperatures when everything is shaking around like mad and you get a quark-gluon plasma.   I'm talking temperatures of several trillion degrees Celsius!  People have gotten this to happen at places like the Relativistic Heavy Ion Collider on Long Island, New York.

You also never see a particle built of just 2 quarks.   Again, the reason is that it can't be white. 

But you can get particles built of a quark and an antiquark - their colors can cancel.   

You can't build a particle out of 4 quarks, because the colors can't cancel.

But you can do 3 quarks together with an extra quark and antiquark!  And that's called - somewhat misleadingly - a pentaquark.

Here's the paper:

• LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, A. Affolder, Z. Ajaltouni, S. Akar, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis, L. An, L. Anderlini, J. Anderson, G. Andreassi, M. Andreotti, J.E. Andrews, R.B. Appleby, O. Aquines Gutierrez, F. Archilli, P. d'Argent, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma, M. Baalouch, S. Bachmann, J.J. Back, A. Badalov, C. Baesso, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, V. Batozskaya, V. Battista, A. Bay, L. Beaucourt, J. Beddow, F. Bedeschi, I. Bediaga, L.J. Bel, V. Bellee, N. Belloli, I. Belyaev, E. Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, A. Bertolin, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani and 662 other authors, Observation of J/ψp resonances consistent with pentaquark states in Λ0b→J/ψK−p decays, http://arxiv.org/abs/1507.03414

It's not unusual to have lots of authors on these papers, but it's rather unusual to list them in alphabetical order.  I like that system, especially since it usually puts me near the front.

In case you're wondering, the theory behind all this is quantum chromodynamics, which is based on quantum field theory, in particular Yang-Mills theory, and on the representation theory of the group SU(3).

It is conjectured but not yet proved that quantum chromodynamics is mathematically consistent and that stable particles must all be 'white', that is, transform in the trivial representation of SU(3).  We describe quarks using the fundamental representation of SU(3) on C^3, which has 3 basis vectors whimsically called red, blue and green.  We describe antiquarks using the dual representation, which has 3 basis vectors called anti-red, anti-blue and anti-green.  We describe gluons using the adjoint representation, which has basis vectors like red-antiblue.

If you want to carry the color analogy even further, you can call anti-red, anti-blue and anti-green cyan, yellow and magenta.  However, you need to be careful.  Cyan, yellow and magenta do not combine to form 'black'.  They form 'antiwhite', but antiwhite is white - that's what the math says, and the math is more fundamental than the cute analogy to colors.

Also, gluons only come in 8 colors, not 9.

Puzzle 1:  Why?   If you know some math you may know SU(3) is 8-dimensional so we can't get 9, but try to explain the story in terms of colors.

Puzzle 2:  If you build particles using only quarks, not antiquarks, could you build something white with 4 quarks?  How about 5?   How about 6?  What's the rule?

For more, read:

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

#spnetworks arXiv:1507.03414 #pentaquark  ___

posted image

2015-07-19 15:02:07 (18 comments, 42 reshares, 123 +1s)Open 

Climbing the tree of life

It's fun to explore the tree of life at http://www.onezoom.org

It only has amphibians, reptiles, birds and mammals - and only those that are still alive today.  But still, it's fun to keep zooming in and see how your favorites are related!

One nice feature is that you can see when branches happened.  And at first it seemed shocking to how new so many mammals' branches are. 

To set the stage, remember that an asteroid hit the Earth and a lot of dinosaurs went extinct 65 million years ago.  About 24 million years ago, the Earth cooled enough that Antarctica becomes covered with ice.  This cooling trend also created the great grasslands of the world!  Humans split off from other apes about 5 million years ago: we are creatures of the grasslands.  The glacial cycles began just 2.5 million years ago... and Homoerectu... more »

Climbing the tree of life

It's fun to explore the tree of life at http://www.onezoom.org

It only has amphibians, reptiles, birds and mammals - and only those that are still alive today.  But still, it's fun to keep zooming in and see how your favorites are related!

One nice feature is that you can see when branches happened.  And at first it seemed shocking to how new so many mammals' branches are. 

To set the stage, remember that an asteroid hit the Earth and a lot of dinosaurs went extinct 65 million years ago.  About 24 million years ago, the Earth cooled enough that Antarctica becomes covered with ice.  This cooling trend also created the great grasslands of the world!  Humans split off from other apes about 5 million years ago: we are creatures of the grasslands.  The glacial cycles began just 2.5 million years ago... and Homo erectus is first known to have tamed fire 1.4 million years ago.

Now compare this:  the cats branched off from hyenas about 40 million years ago.  Cheetahs branched off from other cats only 17 million years ago.  That makes sense: we couldn't have cheetahs without grasslands!   But bobcats and lynxes branched off only 11 million years ago... and tigers just 6 million years ago!

So tigers are almost as new as us!  And the modern lion, Panthera leo, is even newer.  It showed up just 1 million years old, after we tamed fire.

This changed my views a bit: I tended to think of humanity as the "new kid on the block".  And okay, it's true that Homo sapiens is just 250,000 years old.  But we had relatives making stone tools and fires for a lot longer!  

Here's another fact that forced me to straighten out my mental chronology: the University of Oxford is older than the Aztec empire!   Teaching started in Oxford as early as 1096, and the University was officially founded in 1249.  On the other hand, we can say the Aztec empire officially  started with the founding in Tenochtitlán in 1325.

And that, in turn, might explain why cell phones don't work very well here in Oxford.  But I digress.  Check out the tree of life, here:

http://www.onezoom.org/ ___

posted image

2015-07-18 09:02:23 (26 comments, 26 reshares, 165 +1s)Open 

Schrödinger's cat

This summer I'm working at the Centre for Quantum Technologies in Singapore.  But I spent the last week at Quantum Physics and Logic, an annual conference at Oxford. 

I'm mainly studying networks in engineering, biology and chemistry, but a lot of the math I use comes from my my old favorite subject: quantum physics.  So, it was great to see the latest things my friends and their students are doing now. 

The prize-winning student paper was written by Amar Hadzihasanovic, from the computer science department at Oxford.  Yes, computer science!  That's because quantum computers and quantum cryptography are hot topics now.

To explain a bit about Hadzihasanovic's paper, I have to start with Schrödinger's cat, a thought experiment in which you put a cat into a quantum superposition of twodramati... more »

Schrödinger's cat

This summer I'm working at the Centre for Quantum Technologies in Singapore.  But I spent the last week at Quantum Physics and Logic, an annual conference at Oxford. 

I'm mainly studying networks in engineering, biology and chemistry, but a lot of the math I use comes from my my old favorite subject: quantum physics.  So, it was great to see the latest things my friends and their students are doing now. 

The prize-winning student paper was written by Amar Hadzihasanovic, from the computer science department at Oxford.  Yes, computer science!  That's because quantum computers and quantum cryptography are hot topics now.

To explain a bit about Hadzihasanovic's paper, I have to start with Schrödinger's cat, a thought experiment in which you put a cat into a quantum superposition of two dramatically different states: one live, one dead.  Nobody has actually done this, but people have tried to see how close they can get.

Physicists have succeeded in making light in a quantum superposition of two dramatically different states.  In classical mechanics we think of light as a wave.   In a so-called cat state, we have light in a superposition of states where the peaks and valleys of this wave are in different places. 

Another kind of cat state involves a bunch of particles that can have spin pointing up or down.  For example, if you have 3 of these particles, you can make a state

↑↑↑ + ↓↓↓

It takes work to do it, though - and more work to check that you've succeeded! 

The first success came in 1998, by a team of experimentalists led by Anton Zeilinger.  So, this particular kind of cat state is usually called a Greene-Horn-Zeilenger state or GHZ state for short.

What's interesting about the GHZ state is that if you look at any two of the particles, you don't see the spooky quantum effect called entanglement.  Only all three particles taken together are entangled.  It's like the Borromean rings, three rings that are linked even though no two are linked to each other.

Another interesting state of 3 particles is called the W state:

↑↓↓ + ↓↑↓ + ↓↓↑

In this state, unlike the GHZ state, you can see entanglement by looking at any two particles.

In fact, there's a classification of states of 3 particles that can have spin up or down, and besides the boring unentangled state

↑↑↑

the only other possibilities - apart from various inessential changes, like turning up to down - are the GHZ state and the W state. 

This is why the GHZ state and W state are so important: they're fundamental building blocks of quantum entanglement, just one step more complicated than the all-important Bell state

↑↓ + + ↓↑

for two particles. 

What Amar Hadzihasanovic did is give a complete description of what you can do with the GHZ and W states, in terms of diagrams.  He explained how to use pictures to design states of more particles from these building blocks.  And he found a complete set of rules to tell when two pictures describe the same state!

You can see these pictures here:

• Amar Hadzihasanovic, A diagrammatic axiomatisation for qubit entanglement, http://arxiv.org/abs/1501.07082

Since this paper he's been working to make the rules simpler and more beautiful.  There's a lot of cool math here.

The Steve Cundiff group at Marburg University is doing research on cat states of light, and the picture here comes from a page on his work:

https://jila.colorado.edu/research/nanoscience/quantum-mechanics-nanoparticles  
For more, see:

https://en.wikipedia.org/wiki/Cat_state
https://en.wikipedia.org/wiki/W_state
https://en.wikipedia.org/wiki/GHZ_state

and

• Daniel M. Greenberger, Michael A. Horne, Anton Zeilinger, Going beyond Bell's Theorem, http://arxiv.org/abs/0712.0921

#spnetwork arXiv:1501.07082 #quantum___

posted image

2015-07-15 07:59:05 (65 comments, 21 reshares, 71 +1s)Open 

The power of randomness

Here's a puzzle:

I write down two different numbers that are completely unknown to you, and hold one in my left hand and one in my right. You have absolutely no idea how I generated these two numbers. Which is larger?

You can point to one of my hands, and I will show you the number in it. Then you can decide to either select the number you have seen or switch to the number you have not seen, held in the other hand.  Is there a strategy that will give you a greater than 50% chance of choosing the larger number, no matter which two numbers I write down?

At first it seems the answer is no.  Whatever number you see, the other number could be larger or smaller.  There's no way to tell.  So obviously you can't get a better than 50% chance of picking the hand with the largest number - even if you've seenone... more »

The power of randomness

Here's a puzzle:

I write down two different numbers that are completely unknown to you, and hold one in my left hand and one in my right. You have absolutely no idea how I generated these two numbers. Which is larger?

You can point to one of my hands, and I will show you the number in it. Then you can decide to either select the number you have seen or switch to the number you have not seen, held in the other hand.  Is there a strategy that will give you a greater than 50% chance of choosing the larger number, no matter which two numbers I write down?

At first it seems the answer is no.  Whatever number you see, the other number could be larger or smaller.  There's no way to tell.  So obviously you can't get a better than 50% chance of picking the hand with the largest number - even if you've seen one of those numbers!

But "obviously" is not a proof.  Sometimes "obvious" things are wrong!

It turns out that, amazingly, the answer to the puzzle is yes.  You can find a strategy to do better than 50%.  But the strategy uses randomness.

I'd seen this puzzle before - do you know who invented it? 

If you want to solve it yourself, stop now or read Quanta magazine for some clues - they offered a small prize for the best answer. 

Otherwise, you can read Greg Egan's answer, which seems like the best answer to me. 

I'll paraphrase it here:

Pick some function f(x) defined for all real numbers, such that:

the limit as x → -∞ of f(x) is 0,

the limit as x → +∞ of f(x) is 1,

whenever x > y, f(x) > f(y).

(There are lots of functions like this; choose any one.)

Next, pick one hand at random. If the number you are shown is x, compute f(x). Then generate a uniform random number z between 0 and 1.  If z is less than or equal to f(x) guess that x is the larger number, but if z is greater than f(x) guess that the larger number is in the other hand.

The probability of guessing correctly can be calculated as the probability of seeing the larger number initially and then, correctly, sticking with it, plus the probability of seeing the smaller number initially and then, correctly, choosing the other hand.

This is

0.5 f(x) + 0.5 (1 - f(y)) = 0.5 + 0.5(f(x) – f(y))

This is strictly greater than 0.5, since x > y so f(x) - f(y) > 0.

So, you have a more than 50% chance of winning!  But as you play the game, there's no way to tell how much more than 50%.  If the numbers on the other players hands are very large, or very small, your chance will be just slightly more than 50%.

Puzzle 1: Prove that no deterministic strategy can guarantee you have a more than 50% chance of choosing the larger number.

Puzzle 2: There are perfectly specific but 'algorithmically random' sequences of bits, which can't predicted well by any program.  If we use these to generate a uniform algorithmically random number between 0 and 1, and use the strategy Egan describes, will our chance of choosing the larger number be more than 50%, or not?

You can see Egan's solutions to these here:

https://johncarlosbaez.wordpress.com/2015/07/20/the-game-of-googol/___

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2015-07-11 17:11:36 (32 comments, 5 reshares, 89 +1s)Open 

And now, at last, Pluto!

This is Pluto as seen from the New Horizons spacecraft on July 9, 2015 - only 5.4 million kilometers away! 

According to NASA:

This image views the side of Pluto that always faces its largest moon, Charon, and includes the so-called “tail” of the dark whale-shaped feature along its equator.

“Among the structures tentatively identified in this new image are what appear to be polygonal features; a complex band of terrain stretching east-northeast across the planet, approximately 1,000 miles long; and a complex region where bright terrains meet the dark terrains of the whale,” said New Horizons principal investigator Alan Stern. “After nine and a half years in flight, Pluto is well worth the wait".

What is this "whale"?  What are these polygonal features?  We can guess now, or we canwait 3 days.... more »

And now, at last, Pluto!

This is Pluto as seen from the New Horizons spacecraft on July 9, 2015 - only 5.4 million kilometers away! 

According to NASA:

This image views the side of Pluto that always faces its largest moon, Charon, and includes the so-called “tail” of the dark whale-shaped feature along its equator.

“Among the structures tentatively identified in this new image are what appear to be polygonal features; a complex band of terrain stretching east-northeast across the planet, approximately 1,000 miles long; and a complex region where bright terrains meet the dark terrains of the whale,” said New Horizons principal investigator Alan Stern. “After nine and a half years in flight, Pluto is well worth the wait".

What is this "whale"?  What are these polygonal features?  We can guess now, or we can wait 3 days.  On July 14, New Horizons will make its closest approach to Pluto, coming within 12,500 kilometers, and we should get a much better view!

For more, try:

https://www.nasa.gov/feature/new-horizons-map-of-pluto-the-whale-and-the-donut

For constant updates, go here:

http://www.nasa.gov/___

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2015-07-10 09:04:29 (106 comments, 24 reshares, 90 +1s)Open 

Bye-bye, carbon

On Quora someone asked:

"What is the most agreed-on figure for our future carbon budget?"

My answer:

Asking "what is our future carbon budget?" is a bit like asking how many calories a day you can eat.  There's really no limit on how much you can eat if you don't care how overweight and unhealthy you become.  So, to set a carbon budget, you need to say how much global warming you will accept.

That said, here's a picture of how we're burning through our carbon budget.  This appears in the International Energy Agency report World Energy Outlook Special Report 2015, which is free and definitely worth reading.

It says that our civilization has burnt 60% of the carbon we're allowed to while still having a 50-50 chance of keeping global warming below 2 °C.

This isbas... more »

Bye-bye, carbon

On Quora someone asked:

"What is the most agreed-on figure for our future carbon budget?"

My answer:

Asking "what is our future carbon budget?" is a bit like asking how many calories a day you can eat.  There's really no limit on how much you can eat if you don't care how overweight and unhealthy you become.  So, to set a carbon budget, you need to say how much global warming you will accept.

That said, here's a picture of how we're burning through our carbon budget.  This appears in the International Energy Agency report World Energy Outlook Special Report 2015, which is free and definitely worth reading.

It says that our civilization has burnt 60% of the carbon we're allowed to while still having a 50-50 chance of keeping global warming below 2 °C.

This is based on data from the Intergovernmental Panel for Climate Change.  The projection of future carbon emissions is based on the Intended Nationally Determined Contributions (INDC) that governments are currently submitting to the United Nations.  So, based on what governments had offered to do by June 2015, we may burn through our carbon budget in 2040.

Our civilization's total carbon budget for staying below 2 °C was about 1 trillion tonnes.  We have now burnt almost 60% of that; you can watch the amount rise here:

http://trillionthtonne.org/

Quoting the report:

"The transition away from fossil fuels is gradual in the INDC Scenario, with the share of fossil fuels in the world’s primary energy mix declining from more than 80% today to around three-quarters in 2030 [...] The projected path for energy-related emissions in the INDC Scenario means that, based on IPCC estimates, the world’s remaining carbon budget consistent with a 50% chance of keeping a temperature increase of below 2 °C would be exhausted around 2040, adding a grace period of only around eight months, compared to the date at which the budget would be exhausted in the absence of INDCs (Figure 2.3). This date is already within the lifetime of many existing energy sector assets: fossil-fuelled power plants often operate for 30-40 years or more, while existing fossil-fuel resources could, if all developed, sustain production levels far beyond 2040. If energy sector investors believed that not only new investments but also existing fossil-fuel operations would be halted at that critical point, this would have a profound effect on investment even today."

You can get the whole report for free here:

http://www.iea.org/publications/freepublications/publication/weo-2015-special-report-energy-climate-change.html___

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2015-07-09 09:43:36 (57 comments, 41 reshares, 108 +1s)Open 

Science and the emotions

Climate scientists have been working hard to understand global warming.  But they have a lot to deal with.  First: hacking, lawsuits and death threats.  And second: the stress of trying to stay objective and scientific when you discover scary things.

Jason Box is studying how Petermann Glacier, in Greenland, is melting.  He caused a stir when he read a colleague's remarks about newly discovered plumes of methane bubbling up through the Arctic ocean.   He tweeted:

"If even a small fraction of Arctic sea floor carbon is released to the atmosphere, we're f'd."

His remark quickly got amplified and distorted, with headlines blaring:

CLIMATOLOGIST: METHANE PLUMES FROM THE ARCTIC MEAN WE'RE SCREWED

Notice this is not what he said.  He said if.  In fact, it seems thathuman-... more »

Science and the emotions

Climate scientists have been working hard to understand global warming.  But they have a lot to deal with.  First: hacking, lawsuits and death threats.  And second: the stress of trying to stay objective and scientific when you discover scary things.

Jason Box is studying how Petermann Glacier, in Greenland, is melting.  He caused a stir when he read a colleague's remarks about newly discovered plumes of methane bubbling up through the Arctic ocean.   He tweeted:

"If even a small fraction of Arctic sea floor carbon is released to the atmosphere, we're f'd."

His remark quickly got amplified and distorted, with headlines blaring:

CLIMATOLOGIST: METHANE PLUMES FROM THE ARCTIC MEAN WE'RE SCREWED

Notice this is not what he said.  He said if.  In fact, it seems that human-produced carbon dioxide will be much more important for global warning than Arctic methane release, at least for the rest of this century.   A few centuries down the line, if we don't get a handle on this problem, then it could get scary.

But when it comes to emotions, the issue tends to boil down to: "are we fucked?"

Gavin Schmidt, one of the climate scientists whose emails got hacked, had this reaction:

"I don't agree. I don't think we're fucked. There is time to build sustainable solutions to a lot of these things. You don't have to close down all the coal-powered stations tomorrow. You can transition. It sounds cute to say, 'Oh, we're fucked and there's nothing we can do,' but it's a bit of a nihilistic attitude. We always have the choice. We can continue to make worse decisions, or we can try to make ever better decisions. 'Oh, we're fucked! Just give up now, just kill me now,' that's just stupid."

This is from an interview with John H. Richardson in Esquire. Richardson probed him a bit, and that's when it gets interesting:

"The methane thing is actually something I work on a lot, and most of the headlines are crap. There's no actual evidence that anything dramatically different is going on in the Arctic, other than the fact that it's melting pretty much everywhere."

But climate change happens gradually and we've already gone up almost 1 degree centigrade and seen eight inches of ocean rise. Barring unthinkably radical change, we'll hit 2 degrees in thirty or forty years and that's been described as a catastrophe—melting ice, rising waters, drought, famine, and massive economic turmoil. And many scientists now think we're on track to 4 or 5 degrees—even Shell oil said that it anticipates a world 4 degrees hotter because it doesn't see "governments taking the steps now that are consistent with the 2 degrees C scenario." That would mean a world racked by economic and social and environmental collapse.

"Oh yeah," Schmidt says, almost casually. "The business-as-usual world that we project is really a totally different planet. There's going to be huge dislocations if that comes about."

But things can change much quicker than people think, he says. Look at attitudes on gay marriage.

And the glaciers?

"The glaciers are going to melt, they're all going to melt," he says. "But my reaction to Jason Box's comments is—what is the point of saying that? It doesn't help anybody."

As it happens, Schmidt was the first winner of the Climate Communication Prize from the American Geophysical Union, and various recent studies in the growing field of climate communications find that frank talk about the grim realities turns people off—it's simply too much to take in. But strategy is one thing and truth is another. Aren't those glaciers water sources for hundreds of millions of people?

"Particularly in the Indian subcontinent, that's a real issue," he says. "There's going to be dislocation there, no question."

And the rising oceans? Bangladesh is almost underwater now. Do a hundred million people have to move?

"Well, yeah. Under business as usual. But I don't think we're fucked."

Resource wars, starvation, mass migrations...

"Bad things are going to happen. What can you do as a person? You write stories. I do science. You don't run around saying, 'We're fucked! We're fucked! We're fucked!' It doesn't—it doesn't incentivize anybody to do anything."

So you see, Schmidt had made up his mind to be determinedly optimistic, because he thinks that's the right approach.  And maybe he's right.  But it's not easy.

Jason Box doesn't actually run around saying "we're fucked".  Here's what he says:

"There's a lot that's scary," he says, running down the list—the melting sea ice, the slowing of the conveyor belt. Only in the last few years were they able to conclude that Greenland is warmer than it was in the twenties, and the unpublished data looks very hockey-stick-ish. He figures there's a 50 percent chance we're already committed to going beyond 2 degrees centigrade and agrees with the growing consensus that the business-as-usual trajectory is 4 or 5 degrees. "It's, um... bad. Really nasty."

The big question is, What amount of warming puts Greenland into irreversible loss? That's what will destroy all the coastal cities on earth. The answer is between 2 and 3 degrees. "Then it just thins and thins enough and you can't regrow it without an ice age. And a small fraction of that is already a huge problem—Florida's already installing all these expensive pumps."

and:

"It's unethical to bankrupt the environment of this planet," he says. "That's a tragedy, right?" Even now, he insists, the horror of what is happening rarely touches him on an emotional level... although it has been hitting him more often recently. "But I—I—I'm not letting it get to me. If I spend my energy on despair, I won't be thinking about opportunities to minimize the problem."

You should read the whole article:

http://www.esquire.com/news-politics/a36228/ballad-of-the-sad-climatologists-0815/

Thanks to +rasha kamel and +Jenny Meyer for bringing this story to my attention!  I find it fascinating because I notice myself tending to study beautiful mathematics as a way to stay happy - even though I feel I should be doing something about global warming.  I'm actually trying to combine the two.  But even if I can't, maybe I need to keep doing some math for purely emotional reasons.

The photo is from here:

http://www.australiangeographic.com.au/travel/destinations/2011/07/gallery-the-melting-glaciers-in-greenland/kayaking-greenlands-melting-glaciers_image10___

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2015-07-07 20:34:15 (24 comments, 419 reshares, 148 +1s)Open 

The best part: it's solar-powered

The best part: it's solar-powered___

posted image

2015-07-07 07:59:35 (27 comments, 159 reshares, 207 +1s)Open 

Chaos made simple

This shows a lot of tiny particles moving around.   If you were one of these particles, it would be hard to predict where you'd go.  See why?  It's because each time you approach the crossing, it's hard to tell whether you'll go into the left loop or the right one. 

You can predict which way you'll go: it's not random.  But to predict it, you need to know your position quite accurately.  And each time you go around, it gets worse.  You'd need to know your position extremely accurately to predict which way you go — left or right — after a dozen round trips. 

This effect is called deterministic chaos.  Deterministic chaos happens when something is so sensitive to small changes in conditions that its motion is very hard to predict in practice, even though it's not actually random.

Thisparticular ex... more »

Chaos made simple

This shows a lot of tiny particles moving around.   If you were one of these particles, it would be hard to predict where you'd go.  See why?  It's because each time you approach the crossing, it's hard to tell whether you'll go into the left loop or the right one. 

You can predict which way you'll go: it's not random.  But to predict it, you need to know your position quite accurately.  And each time you go around, it gets worse.  You'd need to know your position extremely accurately to predict which way you go — left or right — after a dozen round trips. 

This effect is called deterministic chaos.  Deterministic chaos happens when something is so sensitive to small changes in conditions that its motion is very hard to predict in practice, even though it's not actually random.

This particular example of deterministic chaos is one of the first and most famous.  It's the Lorenz attractor, invented by Edward Lorenz as a very simplified model of the weather in 1963.

The equations for the Lorentz attractor are not very complicated if you know calculus.  They say how the x, y and z coordinates of a point change with time:

dx/dt = 10(x-y)
dy/dt = x(28-z) - y
dz/dt = xy - 8z/3

You are not supposed to be able to look at these equations and say "Ah yes!  I see why these give chaos!"   Don't worry: if you get nothing out of these equations, it doesn't mean you're "not a math person"  — just as not being able to easily paint the Mona Lisa after you see it doesn't mean you're "not an art person".  Lorenz had to solve them using a computer to discover chaos.  I personally have no intuition as to why these equations work... though I could get such intuition if I spent a week reading about it.

The weird numbers here are adjustable, but these choices are the ones Lorenz originally used.  I don't know what choices David Szakaly used in his animation.  Can you find out?

If you imagine a tiny drop of water flowing around as shown in this picture, each time it goes around it will get stretched in one direction.  It will get squashed in another direction, and be neither squashed nor stretched in a third direction. 

The stretching is what causes the unpredictability: small changes in the initial position will get amplified.  I believe the squashing is what keeps the two loops in this picture quite flat.  Particles moving around these loops are strongly attracted to move along a flat 'conveyor belt'.  That's why it's called the Lorentz attractor.

With the particular equations I wrote down, the drop will get stretched in one direction by a factor of about 2.47... but squashed in another direction by a factor of about 2 million!    At least that's what this physicist at the University of Wisconsin says:

• J. C. Sprott, Lyapunov exponent and dimension of the Lorenz attractor, http://sprott.physics.wisc.edu/chaos/lorenzle.htm

He has software for calculating these numbers - or more precisely their logarithms, which are called Lyapunov exponents.  He gets 0.906, 0, and -14.572 for the Lyapunov exponents.

The region that attracts particles — roughly the glowing region in this picture — is a kind of fractal.  Its dimension is slightly more than 2, which means it's very flat but slightly 'fuzzed out'.  Actually there are different ways to define the dimension, and Sprott computes a few of them.  If you want to understand what's going on, try this:

• Edward Ott, Attractor dimensions, http://www.scholarpedia.org/article/Attractor_dimensions

For more nice animations of the Lorentz attractor, see:

http://visualizingmath.tumblr.com/post/121710431091/a-sample-solution-in-the-lorenz-attractor-when

David Szakaly has a blog called dvdp full of astounding images:

http://dvdp.tumblr.com/

and this particular image is here:

http://dvdp.tumblr.com/post/78579988855/140304

Unfortunately, he doesn't explain precisely how it was made.___

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2015-07-05 08:55:30 (36 comments, 9 reshares, 97 +1s)Open 

Monument to the unknown drinker

This is a statue on Nørre Alle, a road in the western part of Copenhagen.  I don't know who made it, but it's certainly eye-catching.

Copenhagen is a complicated mix of land and water.  The central part of the city is bordered on the east by a harbor and on the west by an old river which is now three lakes: Sortedams Sø, Peblinge Sø and Sankt Jørgens Sø.   It's fun to hike or bike around these lakes, and people enjoy sitting on the shore, eating and drinking - lots of beer.

These lakes are separated only by bridges: on a map they still look like part of river, but a strange river that suddenly starts at one end and stops before reaching the sea!  The water actully comes in from three piped streams: Grøndalsåen, Lygteåen and Ladegårdsåen.  I'm writing these namesonly because Da... more »

Monument to the unknown drinker

This is a statue on Nørre Alle, a road in the western part of Copenhagen.  I don't know who made it, but it's certainly eye-catching.

Copenhagen is a complicated mix of land and water.  The central part of the city is bordered on the east by a harbor and on the west by an old river which is now three lakes: Sortedams Sø, Peblinge Sø and Sankt Jørgens Sø.   It's fun to hike or bike around these lakes, and people enjoy sitting on the shore, eating and drinking - lots of beer.

These lakes are separated only by bridges: on a map they still look like part of river, but a strange river that suddenly starts at one end and stops before reaching the sea!  The water actully comes in from three piped streams: Grøndalsåen, Lygteåen and Ladegårdsåen.  I'm writing these names only because Danish has a weird effect on me.  I know German, but Danish seems to be German mixed with Elvish.

I'm staying in Nørrebro, a neighborhood west of the lakes.  I'm near the Niels Bohr Institute.  This was founded by Bohr in 1921 - and it's where the Copenhagen interpretation of quantum mechanics was born!

Walking across one bridge into the central city, you meet the Rundetårn - the Round Tower.   Tycho Brahe was the main astronomer in Denmark in the mid-1500's, and King Frederick II of Denmark gave him an island and funding to build an observatory called Uraniborg, with a laboratory for alchemy in the basement.
 
Tycho Brahe's accurate measurements of planetary motions at Uraniborg were crucial to Kepler's later work.  But when King Frederick II died, his son didn't like Tycho Brahe.  So Brahe left Denmark and a guy named Christian Longomontanus became the new king's new astronomer.  And he got a new observatory, the Rundetårn, in the heart of Copenhagen!

I'll try to visit it today if it's open.  It has an equestrian staircase - a spiral staircase big enough for horses to climb - making 7.5 turns as it goes up! 

All this cool stuff, and you get to see is a photo of a guy pissing on a wall.  Sorry - but this picture came out well.

https://en.wikipedia.org/wiki/The_Lakes,_Copenhagen
https://en.wikipedia.org/wiki/Niels_Bohr_Institute
https://en.wikipedia.org/wiki/Rundet%C3%A5rn
https://en.wikipedia.org/wiki/Tycho_Brahe___

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2015-07-04 08:26:41 (80 comments, 14 reshares, 140 +1s)Open 

The struggling physicist

On Quora someone asked:

How does a physicist rationalize the fact that all her/his life's work may turn out to be meaningless?  A physicist may chase a particular theory/phenomenon all his life solely because he is in love with the subject. However, knowing the history of science, his work may get trashed anytime. How does a physicist still motivate oneself?

I replied:

There are many answers to your question.

One is optimism bias: the belief that one is likely to succeed where others have failed.  It's widespread, but I suspect it's even more common among people who work on high-risk projects - like trying to market a new invention, or trying to figure out new fundamental laws of physics.   People who are not optimistic are unlikely to succeed in physics. 

(This does not implythat... more »

The struggling physicist

On Quora someone asked:

How does a physicist rationalize the fact that all her/his life's work may turn out to be meaningless?  A physicist may chase a particular theory/phenomenon all his life solely because he is in love with the subject. However, knowing the history of science, his work may get trashed anytime. How does a physicist still motivate oneself?

I replied:

There are many answers to your question.

One is optimism bias: the belief that one is likely to succeed where others have failed.  It's widespread, but I suspect it's even more common among people who work on high-risk projects - like trying to market a new invention, or trying to figure out new fundamental laws of physics.   People who are not optimistic are unlikely to succeed in physics. 

(This does not imply that people who are optimistic are likely to succeed.)

Another answer: it's easy to keep thinking one will succeed in theoretical physics, compared to business, because there are few definitive signs of failure except for making an experimental prediction and having it fail when tested.  You'll notice that string theory and loop quantum gravity, two popular theories of physics, make no definitive testable predictions at this time.  That is, there's no experiment we could do now that would definitively disprove these theories.  So, no matter what experiments are done, people can continue to work on these theories and feel their work will succeed someday.

Furthermore, physics can lead to interesting and important mathematics even if it's wrong or untestable by experiment!  String theory, in particular, has been incredibly successful as a source of mathematical ideas.  So, if one is content with that, one can remain happy. 

Finally, if one loves doing something and manages to get paid to do it, it's hard to stop.  And as one grows up and matures, one may realize that there's more to life than succeeding in an ambitious dream.   If one has the opportunity to be part of a noble tradition, if one has the opportunity to teach students to continue this tradition, one should consider oneself lucky.

(Nonetheless, I stopped working on quantum gravity back around 2008, and I'm very happy I did.  I explained why here:

https://edge.org/response-detail/11356 )___

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2015-07-01 18:59:17 (5 comments, 7 reshares, 58 +1s)Open 

Chemical reactions in Copenhagen

This is a famous harbor called Nyhavn.  I haven't been there yet!  I'm in Copenhagen at a workshop on Trends in Reaction Network Theory, and I've been sweating away in hot classrooms listening to talks. 

But don't feel sorry for me!  (You probably weren't.)  I've been loving these talks, loving the conversations with experts and the new ideas — and after the workshop is over, I'm going to spend a few days walking around this town.

A reaction network is something like this:

2 H₂ + O₂ → 2 H₂O
C + O₂ → CO₂

just a list of chemical reactions, which can be much more complicated than this example.   If we know the rate constants saying how fast these reactions happen, we can write equations saying how the amounts of all the chemicalschanges with time! ... more »

Chemical reactions in Copenhagen

This is a famous harbor called Nyhavn.  I haven't been there yet!  I'm in Copenhagen at a workshop on Trends in Reaction Network Theory, and I've been sweating away in hot classrooms listening to talks. 

But don't feel sorry for me!  (You probably weren't.)  I've been loving these talks, loving the conversations with experts and the new ideas — and after the workshop is over, I'm going to spend a few days walking around this town.

A reaction network is something like this:

2 H₂ + O₂ → 2 H₂O
C + O₂ → CO₂

just a list of chemical reactions, which can be much more complicated than this example.   If we know the rate constants saying how fast these reactions happen, we can write equations saying how the amounts of all the chemicals changes with time! 

Reaction network theory lets you understand some things about these equations just by looking at the reaction network.  It's really cool.

The biggest open question about reaction network theory is the Global Attractor Conjecture, which says roughly that for a certain large class of reaction networks, the amount of chemicals always approaches an equilibrium. 

It's a hard conjecture: people have been trying to prove it since 1974.  In fact, two founders of reaction network theory believed they'd proved it in 1972.  But then they realized they had made a basic mistake... and the search for a proof started. 

The most exciting talk so far in this workshop — at least for me — was the one by Georghe Craciun.  He claims to have proved the Global Attractor Conjecture!  He's a real expert on reaction networks, so I'm optimistic that he's really done it.  But I haven't read his proof, and I don't know anyone who says they follow all the details. 

So, there's work left for us to do.  His paper is here:

• Georghe Craciun, Toric differential inclusions and a proof of the global attractor conjecture, http://arxiv.org/abs/1501.02860.

There's a branch of math called 'toric geometry', which his title alludes to... but I asked him how much fancy toric geometry his proof uses, and he laughed and said "none!"   Which is a pity, in a way, because it's a cool subject.  But it's good, in a way, because it means mathematical chemists don't need to learn this subject to follow Craciun's proof.

There have been a lot of other good talks here.  You can read about some on my blog:

https://johncarlosbaez.wordpress.com/2015/07/01/trends-in-reaction-network-theory-part-2/

including the comments, where I'm live-blogging. 

I gave a talk called 'Probabilities and amplitudes', about a mathematical analogy between reaction network theory and particle physics, and you can see my slides.  Alas, the talks haven't been videotaped, and most of the other speaker's slides aren't available.  I have, however, collected links to some papers.

I've gotten at least two ideas that seem really promising, both from a guy named Matteo Polettini, who is interested in lots of stuff I'm interested in.  I won't tell you about them until I work out more details and see if they hold up.  But I'm excited!  This is what conferences are supposed to do.   They don't always do it, but when they do, it's really worthwhile.

The picture here was taken by a duo called angel&marta.  You can see more of their fun photos of Europe here:

http://www.panoramio.com/user/2190720

Finally, here is the abstract of Craciun's paper:

Abstract. The global attractor conjecture says that toric dynamical systems (i.e., a class of polynomial dynamical systems on the positive orthant) have a globally attracting point within each positive linear invariant subspace -- or, equivalently, complex balanced mass-action systems have a globally attracting point within each positive stoichiometric compatibility class. We introduce toric differential inclusions, and we show that each positive solution of a toric differential inclusion is contained in an invariant region that prevents it from approaching the origin. In particular, we show that similar invariant regions prevent positive solutions of weakly reversible k-variable polynomial dynamical systems from approaching the origin. We use this result to prove the global attractor conjecture.


#spnetwork arXiv:1501.02860 #chemistry #reactionNetworks #globalAttractorConjecture   #mustread  ___

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2015-06-26 04:44:30 (43 comments, 20 reshares, 123 +1s)Open 

Electrifying mathematics

How can you change an electrical circuit made out of resistors without changing what it does?  5 ways are shown here:

1.  You can remove a loop of wire with a resistor on it.  It doesn't do anything.

2.  You can remove a wire with a resistor on it if one end is unattached.  Again, it doesn't do anything.

3.  You can take two resistors in series - one after the other - and replace them with a single resistor.  But this new resistor must have a resistance that's the sum of the old two.

4.  You can take two resistors in parallel and replace them with a single resistor.  But this resistor must have a conductivity that's the sum of the old two.  Conductivity is the reciprocal of resistance.

5.  Finally, the really cool part: the Y-Δ transform.  You can replacea Y made of ... more »

Electrifying mathematics

How can you change an electrical circuit made out of resistors without changing what it does?  5 ways are shown here:

1.  You can remove a loop of wire with a resistor on it.  It doesn't do anything.

2.  You can remove a wire with a resistor on it if one end is unattached.  Again, it doesn't do anything.

3.  You can take two resistors in series - one after the other - and replace them with a single resistor.  But this new resistor must have a resistance that's the sum of the old two.

4.  You can take two resistors in parallel and replace them with a single resistor.  But this resistor must have a conductivity that's the sum of the old two.  Conductivity is the reciprocal of resistance.

5.  Finally, the really cool part: the Y-Δ transform.  You can replace a Y made of 3 resistors by a triangle of resistors  But their resistances must be related by the equations shown here.

For circuits drawn on the plane, these are all the rules you need!  There's a nice paper on this by three French dudes: Yves Colin de Verdière, Isidoro Gitler and Dirk Vertigan.

Today I'm going to Warsaw to a workshop on Higher-Dimensional Rewriting.  Electrical circuits give a nice example, so I'll talk about them.   I'm also giving a talk on control theory - a related branch of engineering.

You can see my talk slides, and much more, here:

https://johncarlosbaez.wordpress.com/2015/06/26/higher-dimensional-rewriting-in-warsaw-part-2/

I'll be staying in downtown Warsaw in the Polonia Palace Hotel.  Anything good to do around there?___

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2015-06-24 00:28:29 (31 comments, 12 reshares, 75 +1s)Open 

Europe during the ice age

According to a new simulation, the population of Europe dropped from 330 thousand to just 130 thousand during the last ice age.

These pictures show the population density at various times, starting 27,000 years ago - that's why it says "27 ky", meaning "27 kiloyears".

As it got colder, the population dropped, reaching its minimum 23,000 years ago.  Things started warming up around then, and the population soared to 410 thousand near the end of the ice age, around 13,000 years ago.

You can see the coast of Spain, Italy and Greece continued to have 23 to 20 people per hundred square kilometers.  But the population got pushed out of northern Europe, and even dropped in places like central Spain.  The black dots are archaeological sites where we know there were people.

By comparison, there are nowro... more »

Europe during the ice age

According to a new simulation, the population of Europe dropped from 330 thousand to just 130 thousand during the last ice age.

These pictures show the population density at various times, starting 27,000 years ago - that's why it says "27 ky", meaning "27 kiloyears".

As it got colder, the population dropped, reaching its minimum 23,000 years ago.  Things started warming up around then, and the population soared to 410 thousand near the end of the ice age, around 13,000 years ago.

You can see the coast of Spain, Italy and Greece continued to have 23 to 20 people per hundred square kilometers.  But the population got pushed out of northern Europe, and even dropped in places like central Spain.  The black dots are archaeological sites where we know there were people.

By comparison, there are now roughly 25,000 people per hundred square kilometers in England or Germany, though just half as many in France.  So, by modern standards, Europe was empty back in those hunter-gatherer days.  Even today the cold keeps people away: there are just 2,000 people per hundred square kilometers in Sweden.

If you're having trouble seeing the British isles in these pictures, that's because they weren't islands back then! - they were connected to continental Europe.

Of course these simulations are insanely hard to do, so I wouldn't trust them too much.  But it's still cool to think about.  

The paper is not free, but the "supporting information" is, and that has a lot of good stuff:

• Miikka Tallavaara, Miska Luoto, Natalia Korhonen, Heikki Järvinen and Heikki Seppä, Human population dynamics in Europe over the Last Glacial Maximum, Proceedings of the National Academy of Sciences, http://www.pnas.org/content/early/2015/06/17/1503784112.abstract

Abstract: The severe cooling and the expansion of the ice sheets during the Last Glacial Maximum (LGM), 27,000–19,000 y ago (27–19 ky ago) had a major impact on plant and animal populations, including humans. Changes in human population size and range have affected our genetic evolution, and recent modeling efforts have reaffirmed the importance of population dynamics in cultural and linguistic evolution, as well. However, in the absence of historical records, estimating past population levels has remained difficult. Here we show that it is possible to model spatially explicit human population dynamics from the pre-LGM at 30 ky ago through the LGM to the Late Glacial in Europe by using climate envelope modeling tools and modern ethnographic datasets to construct a population calibration model. The simulated range and size of the human population correspond significantly with spatiotemporal patterns in the archaeological data, suggesting that climate was a major driver of population dynamics 30–13 ky ago. The simulated population size declined from about 330,000 people at 30 ky ago to a minimum of 130,000 people at 23 ky ago. The Late Glacial population growth was fastest during Greenland interstadial 1, and by 13 ky ago, there were almost 410,000 people in Europe. Even during the coldest part of the LGM, the climatically suitable area for human habitation remained unfragmented and covered 36% of Europe.

An interstadial is a warmer period - and by the way, what I'm calling an "ice age" should really be called a glacial.  I did this just to see how many people correct me without reading my whole post.  (Actually I'm doing it in a feeble attempt to sound like a normal person.)___

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2015-06-19 08:45:39 (37 comments, 18 reshares, 112 +1s)Open 

Caring for our home

Pope Francis has written something about environmental issues.  I recommend it!   Here are two quotes:

If we approach nature and the environment without this openness to awe and wonder, if we no longer speak the language of fraternity and beauty in our relationship with the world, our attitude will be that of masters, consumers, ruthless exploiters, unable to set limits on their immediate needs. By contrast, if we feel intimately united with all that exists, then sobriety and care will well up spontaneously.

Everything is connected. Concern for the environment thus needs to be joined to a sincere love for our fellow human beings and an unwavering commitment to resolving the problems of society. Moreover, when our hearts are authentically open to universal communion, this sense of fraternity excludes nothing and no one. It follows thato... more »

Caring for our home

Pope Francis has written something about environmental issues.  I recommend it!   Here are two quotes:

If we approach nature and the environment without this openness to awe and wonder, if we no longer speak the language of fraternity and beauty in our relationship with the world, our attitude will be that of masters, consumers, ruthless exploiters, unable to set limits on their immediate needs. By contrast, if we feel intimately united with all that exists, then sobriety and care will well up spontaneously.

Everything is connected. Concern for the environment thus needs to be joined to a sincere love for our fellow human beings and an unwavering commitment to resolving the problems of society. Moreover, when our hearts are authentically open to universal communion, this sense of fraternity excludes nothing and no one. It follows that our indifference or cruelty towards fellow creatures of this world sooner or later affects the treatment we mete out to other human beings. We have only one heart, and the same wretchedness which leads us to mistreat an animal will not be long in showing itself in our relationships with other people.

For more, try my blog post.

https://johncarlosbaez.wordpress.com/2015/06/19/on-care-for-our-common-home/

The picture here, of terraced rice fields in Bali, is from here:

http://writingforselfdiscovery.com/2013/11/27/part-two-creating-a-life-that-fits-like-skin-why-bali/___

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2015-06-18 02:37:27 (30 comments, 18 reshares, 62 +1s)Open 

The Petersen graph

Suppose you have a round table with 5 places.  Say you want to seat 2 women at the table, the rest of the diners being men.   Then there are 10 ways to do it, shown here.  The women are in red.

Now: connect two tables with a line when no seat occupied by a woman at one table is occupied by a woman at the other.

You get this picture, called the Petersen graph.  There are 15 edges connecting the 10 tables.  It's a wonderful thing.  It shows up in lots of ways, and it's a counterexample to many guesses about graphs.

Puzzle 1: how many pentagons are there in the Petersen graph?  We don't count things like the pentagon in the middle of this picture, only pentagons whose sides are all edges of the Petersen graph. 

You can also get the Petersen graph by taking a regular dodecahedron and treatingopposite... more »

The Petersen graph

Suppose you have a round table with 5 places.  Say you want to seat 2 women at the table, the rest of the diners being men.   Then there are 10 ways to do it, shown here.  The women are in red.

Now: connect two tables with a line when no seat occupied by a woman at one table is occupied by a woman at the other.

You get this picture, called the Petersen graph.  There are 15 edges connecting the 10 tables.  It's a wonderful thing.  It shows up in lots of ways, and it's a counterexample to many guesses about graphs.

Puzzle 1: how many pentagons are there in the Petersen graph?  We don't count things like the pentagon in the middle of this picture, only pentagons whose sides are all edges of the Petersen graph. 

You can also get the Petersen graph by taking a regular dodecahedron and treating opposite points on it as being "the same".

In math you can do this: you can just declare that you're going to treat two things as being 'the same'.  This is called identifying them, since you're making them count as identical.  Of course, identifying different things may wreak havoc!   It depends on what you're doing. In math we try to do it skillfully.

(This use of the word "identifying" has nothing to do with identifying birds while you're walking through the forest.  In fact, birds tend to seem alike before you identify them!)

The dodecahedron has 20 corners, so when we identify opposite corners, we get 10 points.  The dodecahedron also has 30 edges, so when we identify opposite edges, we get 15.  This is a sign that maybe I'm not lying to you: maybe it's really true that we get the Petersen graph.  But it's not a proof.

The Petersen graph also shows up in biology!

It shows up when you consider all possible phylogenetic trees that could explain how some set of species arose from a common ancestor.  These are binary trees where each edge is labelled by a time - how long some species lasted before splitting in two.  The space of all such trees is an interesting thing.  When you have 4 species, you can get this space from the Petersen graph.

How?  I explain that here:

• John Baez, Operads and the tree of life, http://math.ucr.edu/home/baez/tree_of_life/

Puzzle 2: How many symmetries does the Petersen graph have?

Puzzle 3: If instead of 2 women at a table with 5 places we have k women at a table with n places, we get the Kneser graph K(n,k).  How many edges does this have?  How many symmetries?

To cheat, see:

https://en.wikipedia.org/wiki/Petersen_graph
https://en.wikipedia.org/wiki/Kneser_graph___

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2015-06-14 11:15:08 (41 comments, 20 reshares, 66 +1s)Open 

Why I like the number 52

There are 52 weeks in a year and 52 cards in a deck.  Coincidence?   Maybe not.   It's hard to guess what the people who first designed the deck were thinking.

Puzzle 1: Suppose you add up the values of all the cards in a deck, counting an ace as 1, a two as 2, and so on, and counting a jack as 11, a queen as 12 and a king as 13.  What do you get? 

Puzzle 2: How many cards are there in a suit?  (There are four suits of cards: diamonds, hearts, spades and clubs.)

Puzzle 3: How many weeks are there in a season?  (There are four seasons in a year; suppose they all have the same number of weeks.)

Puzzle 4: Multiply the number of days in a week, weeks in a season and seasons in a year to estimate the number of days in a year. 

Here's another fun thing about thenumber 52... more »

Why I like the number 52

There are 52 weeks in a year and 52 cards in a deck.  Coincidence?   Maybe not.   It's hard to guess what the people who first designed the deck were thinking.

Puzzle 1: Suppose you add up the values of all the cards in a deck, counting an ace as 1, a two as 2, and so on, and counting a jack as 11, a queen as 12 and a king as 13.  What do you get? 

Puzzle 2: How many cards are there in a suit?  (There are four suits of cards: diamonds, hearts, spades and clubs.)

Puzzle 3: How many weeks are there in a season?  (There are four seasons in a year; suppose they all have the same number of weeks.)

Puzzle 4: Multiply the number of days in a week, weeks in a season and seasons in a year to estimate the number of days in a year. 

Here's another fun thing about the number 52.  There are also 52 ways to partition a set with 5 elements - that is, break it up into disjoint nonempty pieces.   This probably has nothing to do with weeks in the year or cards in the deck!   But it's the start of a more interesting story.

I've shown you a picture of all 52 ways.   They're divided into groups:

52 = 1 + 10 + 10 + 15 + 5 + 10 + 1

• There's 1 way to break the 5-element set into pieces that each have 1 element, shown on top.

• There are 10 ways to break it into three pieces with 1 element and one piece with 2 elements.

• There are 10 ways to break it into two pieces with 1 element and one with 3.

• There are 15 ways to break it into one piece with 1 element and two with 2.

• There are 5 ways to break it into one piece with 1 element and one with 4.

• There are 10 ways to break it into one piece with 2 elements and one with 3.

• There is 1 way to break it into just one piece containing all 5 elements, shown on the very bottom.

If this chart reminds you of the chart of "Genji-mon" that I showed you two days ago, that's no coincidence!  The Genji-mon are almost the same as the partitions of a 5-element set.  This chart should help you answer all the puzzles I asked.

The math gets more interesting if we ask: how many partitions are there for a set with n elements? 

For a zero-element set there's 1.  (That's a bit confusing, I admit.)  For a one-element set there's 1.  For a two-element set there's 2.  And so on.  The numbers go like this:

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, ...

They're called Bell numbers

Say you call the nth Bell number B(n).   Then we have a nice formula

sum  B(n) x^n / n!   =  e^(e^x - 1)

This is a nice way to compress all the information in the Bell numbers down to a simple function.  But it's not a very efficient way to compute the Bell numbers.  For that, it's better to use the Bell triangle.  This is a relative of Pascal's triangle.   To understand the Bell triangle, it helps to look at some pictures:

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

For more on Bell numbers, try this:

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

and for more on partitions of sets, try this:

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

As usual in math, the story only stops when you get tired!___

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2015-06-13 00:48:27 (64 comments, 28 reshares, 83 +1s)Open 

MASSIVE WORLDWIDE DATA BREACH

The true scale of the problem is just becoming apparent, but it seem that all data on every computer in the world has been copied to some unknown location. 

It's rapidly becoming clear that last week's revelations are just the tip of the iceberg.  It seems all   US federal government computers show signs of data breaches, with strong evidence that all  files have been copied.  The same is true of at least 34 US states.  The UK, France, Germany, Italy, Switzerland, Japan and India are reporting similar problems, as are a vast number of corporations, universities and individuals.   In particular, it seems that all servers in the Google, Facebook, Amazon, and Microsoft data centers have been hacked.

It's unclear who has the storage capacity to hold all this data.  Some suspect the Chinese or Russia, but according to anunnamed ... more »

MASSIVE WORLDWIDE DATA BREACH

The true scale of the problem is just becoming apparent, but it seem that all data on every computer in the world has been copied to some unknown location. 

It's rapidly becoming clear that last week's revelations are just the tip of the iceberg.  It seems all   US federal government computers show signs of data breaches, with strong evidence that all  files have been copied.  The same is true of at least 34 US states.  The UK, France, Germany, Italy, Switzerland, Japan and India are reporting similar problems, as are a vast number of corporations, universities and individuals.   In particular, it seems that all servers in the Google, Facebook, Amazon, and Microsoft data centers have been hacked.

It's unclear who has the storage capacity to hold all this data.  Some suspect the Chinese or Russia, but according to an unnamed source at the US State Department these countries too are victims of the massive hack.  "Furthermore," the source stated, "the fact that all the many petabytes of data from the particle accelerator at CERN have been copied seems to rule out traditional espionage or criminal activity as an explanation."

Rumors of all kinds are circulating on the internet.  Some say it could be the initial phase of an extraterrestrial invasion, or perhaps merely an attempt to learn about our culture, or - in one of the more fanciful theories - an attempt to replicate it.

Another theory is that some form of artificial intelligence has developed the ability to hack into most computers, or that the internet itself has somehow become intelligent.

Perhaps the strangest rumor is that the biosphere itself is preparing to take revenge on human civilization, or make a "backup" in case of collapse.  A recent paper in PLOS Biology estimates the total informatoin storage capacity in the biosphere at roughly 5 × 10^31 megabases, with a total processing speed exceeding 10^24 nucleotide operations per second.  The data in all human computers is still tiny by comparison.  However, it is unclear how biological organisms could have hacked into human computers, and what the biosphere might do with this data. 

According to one of the paper's authors, Hanna Landenmark, "Claims that this is some sort of 'revenge of Gaia' seem absurdly anthromorphic to me.  If anything, it could be just the next phase of evolution."

• Hanna K. E. Landenmark, Duncan H. Forgan and Charles S. Cockell,
An estimate of the total DNA in the biosphere, PLOS Biology, 11June 2015.  Available at http://journals.plos.org/plosbiology/article?id=10.1371/journal.pbio.1002168

#spnetwork doi:10.1371/journal.pbio.1002168 #information #bigness  ___

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2015-06-12 00:58:04 (53 comments, 25 reshares, 68 +1s)Open 

Math and The Tale of Genji

The Tale of Genji is a wonderful early Japanese novel written by the noblewoman Murasaki Shikibu around 1021 AD.  Read it, and be transported to a very different world!

It has 54 chapters.  Here you see the 54 Genji-mon (源氏紋) - the traditional symbols for these chapters.  Most of them follow a systematic mathematical pattern, but the ones in color break this pattern. 

Here are some puzzles.  It's very easy to look up the answers using your favorite search engine, so if you do that please don't give away the answer!   It's more fun to solve these just by thinking.

Puzzle 1: How is the green Genji-mon different from all the rest?

Puzzle 2: How are the red Genji-mon similar to each other?

Puzzle 3: How are the red Genji-mon different from all therest?
... more »

Math and The Tale of Genji

The Tale of Genji is a wonderful early Japanese novel written by the noblewoman Murasaki Shikibu around 1021 AD.  Read it, and be transported to a very different world!

It has 54 chapters.  Here you see the 54 Genji-mon (源氏紋) - the traditional symbols for these chapters.  Most of them follow a systematic mathematical pattern, but the ones in color break this pattern. 

Here are some puzzles.  It's very easy to look up the answers using your favorite search engine, so if you do that please don't give away the answer!   It's more fun to solve these just by thinking.

Puzzle 1: How is the green Genji-mon different from all the rest?

Puzzle 2: How are the red Genji-mon similar to each other?

Puzzle 3: How are the red Genji-mon different from all the rest?

Puzzle 4: If The Tale of Genji had just 52 chapters, the Genji-mon could be perfectly systematic, without the weirdness of the colored ones.  What would the pattern be then?

Puzzle 5: What fact about the number 52 is at work here?

(Hint: it has nothing to do with there being 52 weeks in a year!)

#puzzles  ___

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2015-06-11 00:33:49 (27 comments, 59 reshares, 156 +1s)Open 

Shooting past Pluto

The New Horizons spacecraft took 9 years to reach Pluto.  But on July 14th, it will blast by Pluto in just one hour.  It can't slow down! 

In fact, it's the fastest human-made object ever to be launched from Earth.  When it took off from Cape Canaveral in January 2006, it was moving faster than escape velocity, not just for the Earth, but for the Solar System!   It was moving at 58,000 kilometers per hour.  

When it passed Jupiter it got pulled by that huge planet's gravity and fired out at 83,000 kilometers per hour.  As it climbed up out of the Solar System it slowed down.  But when it reaches Pluto, it will still be going almost 50,000 kilometers per hour.

That's fast enough that even a speck of dust could be fatal.  Luckily, Pluto doesn't seem to have rings.

It will punch through the planethat Pluto... more »

Shooting past Pluto

The New Horizons spacecraft took 9 years to reach Pluto.  But on July 14th, it will blast by Pluto in just one hour.  It can't slow down! 

In fact, it's the fastest human-made object ever to be launched from Earth.  When it took off from Cape Canaveral in January 2006, it was moving faster than escape velocity, not just for the Earth, but for the Solar System!   It was moving at 58,000 kilometers per hour.  

When it passed Jupiter it got pulled by that huge planet's gravity and fired out at 83,000 kilometers per hour.  As it climbed up out of the Solar System it slowed down.  But when it reaches Pluto, it will still be going almost 50,000 kilometers per hour.

That's fast enough that even a speck of dust could be fatal.  Luckily, Pluto doesn't seem to have rings.

It will punch through the plane that Pluto's moons orbit, and collect so much data that it will take months for it all to be sent back to Earth.

And as it goes behind Pluto, it will see a carefully timed radio signal sent from the Deep Space Network here on Earth: 3 deep-space communication facilities located in California, Spain and Australia.

This signal has to be timed right, since it takes about 4 hours for radio waves - or any other form of light - to reach Pluto.  The signal will be blocked when Pluto gets in the way, and the New Horizons spacecraft can use this to learn more about Pluto's exact diameter, and more.

Then: out to the Kuiper belt, where the cubewanos, plutinos and twotinos live...
 
------------

You can see a timeline of the flyby here:

http://blogs.scientificamerican.com/life-unbounded/the-pluto-punch-through/

On July 14, 2015 at 11:49:57 UTC, New Horizons will make its closest approach to Pluto.  It will have a relative velocity of 13.78 km/s (49,600 kilometers per hour), and it will come within 12,500 kilometers from the planet's surface. 

At 12:03:50, it will make its closest approach to Pluto's largest moon, Charon. 

At 12:51:25, Pluto will occult the Sun - that is, come between the Sun and the New Horizons spacecraft.

At 12:52:27, Pluto will occult the Earth.  This is only important because it means the radio signal sent from the Deep Space Network will be blocked.

Starting 3.2 days before the closest approach, New Horizons will map Pluto and Charon to 40 kilometer resolution. This is enough time to image all sides of both bodies. Coverage will repeat twice per day, to search for changes due to snows or cryovolcanism.  Still, due to Pluto's tilt, a portion of the northern hemisphere will be in shadow at all times. The Long Range Reconnaissance Imager (LORRI) should be able to obtain select images with resolution as high as 50 meters/pixel, and the Multispectral Visible Imaging Camera (MVIC) should get 4-color global dayside maps at 1.6 kilometer resolution. LORRI and MVIC will attempt to overlap their respective coverage areas to form stereo pairs. 

The Linear Etalon Imaging Spectral Array (LEISA) will try to get near-infrared maps at 7 kilometers per pixel globally and 0.6 km/pixel for selected areas.  Meanwhile, the ultraviolet spectrometer Alice will study the atmosphere, both by emissions of atmospheric molecules (airglow), and by dimming of background stars as they pass behind Pluto. 

Other instruments will will sample the high atmosphere, measure its effects on the solar wind, and search for dust - possible signs of invisible rings of Pluto.  The communications dish will detect the disappearance and reappearance of the radio signal from the Deep Space Network, measuring Pluto's diameter and atmospheric density and composition.

The first highly compressed images will be transmitted within days. Uncompressed images will take as long as nine months to transmit, depending on how much traffic the Deep Space Network is experiencing.

Most of this last information is from:

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

The picture is from here:

http://www.astronomy.com/magazine/ask-astro/2014/01/new-horizons

#astronomy  ___

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2015-06-09 05:12:00 (44 comments, 25 reshares, 119 +1s)Open 

The biggest axiom in the world

In math the rules of a game are called axioms.  What's the longest axiom that people have ever thought about?

I'm not sure, but I have a candidate.  A lattice is a set with two operations called ∨ and ∧, obeying the 6 equations listed below.  But a while back people wondered: can you give an equivalent definition of a lattice using just one equation?   It's a pointless puzzle, as far as I can tell, but some people enjoy such challenges. 

And in 1970 someone solved it: yes, you can!   But the equation they found was incredibly long.

Before I go into details, I should say a bit about lattices.  The concept of a lattice is far from pointless - there are lattices all over the place! 

For example, suppose you take integers, or real numbers.  Let x ∨ y be the maximum of x and y:the bigger one. ... more »

The biggest axiom in the world

In math the rules of a game are called axioms.  What's the longest axiom that people have ever thought about?

I'm not sure, but I have a candidate.  A lattice is a set with two operations called ∨ and ∧, obeying the 6 equations listed below.  But a while back people wondered: can you give an equivalent definition of a lattice using just one equation?   It's a pointless puzzle, as far as I can tell, but some people enjoy such challenges. 

And in 1970 someone solved it: yes, you can!   But the equation they found was incredibly long.

Before I go into details, I should say a bit about lattices.  The concept of a lattice is far from pointless - there are lattices all over the place! 

For example, suppose you take integers, or real numbers.  Let x ∨ y be the maximum of x and y: the bigger one.  Let x ∧ y be the minimum of x and y: the smaller one.  Then it's easy to check that the 6 axioms listed here hold.

Or, suppose you take statements.  Let p ∨ q be the statement "p or q", and let p ∧ q be the statement "p and q".  Then the 6 axioms here hold! 

For example, consider the axiom p ∧ (p ∨ q) = p.  If you say "it's raining, and it's also raining or snowing", that means the same thing as "it's raining" - which is why people don't usually say this. 

The two examples I just gave obey other axioms, too.  They're both distributive lattices, meaning they obey this rule:

p ∧ (q ∨ r) = (p ∧ q) ∨ (p ∧ r)

and the rule with ∧ and ∨ switched:

p ∨ (q ∧ r) = (p ∨ q) ∧ (p ∨ r)

But nondistributive lattices are also important.  For example, in quantum logic, "or" and "and" don't obey these distributive laws! 

Anyway, back to the main story.  In 1970, Ralph McKenzie proved that you can write down a single equation that is equivalent to the 6 lattice axioms.  But it was an equation containing 34 variables and roughly 300,000 symbols!  It was too long for him to actually bother writing it down.  Instead, he proved that you could, if you wanted to.

Later this work was improved.  In 1977, Ranganathan Padmanabhan found an equation in 7 variables with 243 symbols that did the job.  In 1996 he teamed up with William McCune and found an equation with the same number of variables and only 79 symbols that defined lattices.  And so on...

The best result I know is by McCune, Padmanbhan and Robert Veroff.  In 2003 they discovered that this equation does the job:

(((y∨x)∧x)∨(((z∧(x∨x))∨(u∧x))∧v))∧(w∨((s∨x)∧(x∨t)))  =  x

They also found another equation, equally long, that also works.

Puzzle: what's the easiest way to get another equation, equally long, that also defines lattices?

That is not the one they found - that would be too easy!

How did they find these equations?  They checked about a half a trillion possible axioms using a computer, and ruled out all but 100,000 candidates by showing that certain non-lattices obey those axioms.  Then they used a computer program called OTTER to go through the remaining candidates and search for proofs that they are equivalent to the usual axioms of a lattice. 

Not all these proof searches ended in success or failure... some took too long.  So, there could still exist a single equation, shorter than the ones they found, that defines the concept of lattice.

Here is their paper:

• William McCune, Ranganathan Padmanabhan, Robert Veroff, Yet another single law for lattices, http://arxiv.org/abs/math/0307284.

By the way:

When I said "it's a pointless puzzle, as far as I can tell", that's not supposed to be an insult.  I simply mean that I don't see how to connect this puzzle - "is there a single equation that does the job?" - to themes in mathematics that I consider important.  It's always possible to learn more and change ones mind about these things.

The puzzle becomes a bit more interesting when you learn that you can't find a single equation that defines distributive lattices: you need 2.  And it's even more interesting when you learn that among "varieties of lattices", none can be defined with just a single equation except plain old lattices and the one-element lattices (which are defined by the equation x = y).

By contrast, "varieties of semigroups where every element is idempotent" can always be defined using just a single equation.  This was rather shocking to me.

However, I still don't see any point to reducing the number of equations to the bare minimum!  In practice, it's better to have a larger number of comprehensible axioms rather than a single  complicated one.  So, this whole subject feels like a "sport" to me: a game of "can you do it?"

#spnetwork arXiv:math/0307284 #lattice #variety

#bigness  ___

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2015-06-05 18:47:29 (26 comments, 17 reshares, 88 +1s)Open 

Carnivorous fungus

I know what you're thinking: GIANT MAN-EATING MUSHROOMS!

At least that's what went through my mind when I was looking at the Wikipedia page on carnivorous plants and saw there was also a page on carnivorous fungi.

In fact, these fungi are tiny, and they eat small things like nematodes. The wormy thing here is a nematode, and it's being caught by the little tendrils called hyphae of a fungus.

Carnivorous fungi were first discovered by the Austrian botanist Whilhelm Zopf in 1888.   He was looking at a fungus whose hyphae have little loops in them.  Zopf observed nematodes being caught by these loops — caught by the tail, or caught by the head.   When this happened, the nematode would struggle violently for half an hour.  Then it would  become quieter.  In a couple of hours, it woulddie.  An... more »

Carnivorous fungus

I know what you're thinking: GIANT MAN-EATING MUSHROOMS!

At least that's what went through my mind when I was looking at the Wikipedia page on carnivorous plants and saw there was also a page on carnivorous fungi.

In fact, these fungi are tiny, and they eat small things like nematodes. The wormy thing here is a nematode, and it's being caught by the little tendrils called hyphae of a fungus.

Carnivorous fungi were first discovered by the Austrian botanist Whilhelm Zopf in 1888.   He was looking at a fungus whose hyphae have little loops in them.  Zopf observed nematodes being caught by these loops — caught by the tail, or caught by the head.   When this happened, the nematode would struggle violently for half an hour.  Then it would  become quieter.  In a couple of hours, it would die.  And then, hyphae from the loop would penetrate and invade its body. 

Aren't you glad that you read this post?  The world is full of wonderful and horrible things, and this is one.

Somehow we tend to sympathize with the creature that's more like us.  When I see a jaguar fighting a crocodile, I want the jaguar to win.  A worm eating fungus doesn't seem so bad... but fungus eating a worm seems disgusting, at least to me.   This is not a rational judgement of mine: it's just an emotion that sweeps over me.

A nematode is not actually a worm: it's a much more primitive sort of organism.  Nematodes are serious pests — they kill lots of crops.  My university, U.C. Riverside, even has a Department of Nematology, where people study how to fight nematodes!   One way to fight them is with a carnivorous fungus.  So maybe carnivorous fungi are not so bad.

This picture shows a nematode captured by the predatory fungus Arthrobotrys anchonia.  Note that the loop around the body of the victim has not yet started to tighten and squeeze it.  This picture was taken with a scanning electron micrograph by N. Allin and G.L. Barron. I got it here:

http://www.uoguelph.ca/~gbarron/N-D%20Fungi/n-dfungi.htm

According to this page:

Fungi can capture nematodes in a variety of ways but the most sophisticated and perhaps the most dramatic is called the constricting ring.  An erect branch from a hypha curves round and fuses with itself to form a three-celled ring about 20-30 microns in diameter.  When a nematode "swims" into a ring it triggers a response in the fungus and the three cells expand rapidly inwards with such power that they constrict the body of the nematode victim and hold it securely with no chance to escape.   It takes only 1/10th of a second for the ring cells to inflate to their maximum size.

#biology  ___

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2015-06-04 20:49:46 (72 comments, 40 reshares, 148 +1s)Open 

Wacky algebra

In math you get to make up the rules of the game... but then you have to follow them with utmost precision.  You can change the rules... but then you're playing a different game.  You can play any game you want... but some games are more worthwhile than others. 

If you play one of these games long enough, it doesn't feel like a game - it feels like "reality", especially if it matches up to the real world in some way.  But that's how games are.

Unfortunately, most kids learn math by being taught the rules for a just a few games - and the teacher acts like the rules are "true".  Where did the rules come from?  That's not explained.  The students are never encouraged to make up their own rules.

In fact, mathematicians spend a lot of time making up new rules.  For example, my grad student Alissa Cransmade up... more »

Wacky algebra

In math you get to make up the rules of the game... but then you have to follow them with utmost precision.  You can change the rules... but then you're playing a different game.  You can play any game you want... but some games are more worthwhile than others. 

If you play one of these games long enough, it doesn't feel like a game - it feels like "reality", especially if it matches up to the real world in some way.  But that's how games are.

Unfortunately, most kids learn math by being taught the rules for a just a few games - and the teacher acts like the rules are "true".  Where did the rules come from?  That's not explained.  The students are never encouraged to make up their own rules.

In fact, mathematicians spend a lot of time making up new rules.  For example, my grad student Alissa Crans made up a thing called a shelf.  It wasn't completely new: it was a lot like something mathematicians already studied, called a 'rack', but simpler - hence the name 'shelf'.  (Mathematician need lots of names for things, so we sometimes run out of serious-sounding names and use silly names.)

What's a shelf?

It's a set where you can multiply two elements a and b and get a new element a · b.  That's not new... but this multiplication obeys a funny rule:

a · (b · c) = (a · b) · (a · c)

That should remind you of this rule:

a · (b + c) = (a · b) + (a · c)

But in a shelf, we don't have addition, just multiplication... and the only rule it obeys is

a · (b · c) = (a · b) · (a · c)

There turn out to be lots of interesting examples, which come from knot theory, and group theory.  I could talk about this stuff for hours.  But never mind!   A couple days ago I learned something surprising.  Suppose you have a unital shelf, meaning one that has an element called 1 that obeys these rules:

a · 1 = a
1 · a = a

Then multiplication has to be associative!  In other words, it obeys this familiar rule:

a · (b · c) = (a · b) · c

The proof is in the picture. 

A guy who calls himself "Sam C" put this proof on a blog of mine.  I was shocked when I saw it.

Why?   First, I've studied shelves quite a lot, and they're hardly ever associative.   I thought I understood this game, and many related games - about things called 'racks' and 'quandles' and 'involutory quandles' and so on.  But adding this particular extra rule changed the game a lot. 

Second, it's a very sneaky proof - I have no idea how Sam C came up with it.

Luckily, a mathematician named Andrew Hubery showed me how to break the proof down into smaller, more digestible pieces.  And now I think I understand this game quite well.   It's not a hugely important game, as far as I can tell, but it's cute. 

It turns out that these gadgets - shelves with an element 1 obeying a · 1 = 1 · a = a - are the same as something the famous category theorist William Lawvere had invented under the name of graphic monoids.  The rules for a monoid are that we have a set with a way to multiply elements and an element 1, obeying these familiar rules:

1 · a = 1 · a = a

a · (b · c) = (a · b) · c

Monoids are incredibly important because they show up all over.  But a graphic monoid also obeys one extra rule:

a · (b · a) = a · b

This is a weird rule... but graphic monoids show up when you're studying bunches of dots connected by edges, which mathematicians call graphs... so it's not a silly rule: this game helps us understand the world.

Puzzle 1: take the rules of a graphic monoid and use them to derive the rules of a unital shelf.

Puzzle 2: take the rules of a unital shelf and use them to derive the rules of a graphic monoid.

So, they're really the same thing.

By the way, most math is a lot more involved than this.  Usually we take rules we already like a lot, and keep developing the consequences further and further, and introducing new concepts, until we build enormous castles - which in the best cases help us understand the universe in amazing new ways.  But this particular game is more like building a tiny dollhouse.  At least so far.  That's why it feels more like a "game", less like "serious work".___

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2015-06-02 20:21:55 (18 comments, 5 reshares, 53 +1s)Open 

European oil and gas companies support carbon tax

Last week, oil and gas companies with a total of $1.4 trillion in revenues - Shell, BP, Total, Statoil, Eni and the BG Group - sent this letter to the UN:

Dear Excellencies,

Climate change is a critical challenge for our world. As major companies from the oil & gas sector, we recognize both the importance of the climate challenge and the importance of energy to human life and well-being. We acknowledge that the current trend of greenhouse gas emissions is in excess of what the Intergovernmental Panel on Climate Change (IPCC) says is needed to limit the temperature rise to no more than 2 degrees above pre-industrial levels. The challenge is how to meet greater energy demand with less CO2. We stand ready to play our part.

Our companies are already taking a number of actions to help limit... more »

European oil and gas companies support carbon tax

Last week, oil and gas companies with a total of $1.4 trillion in revenues - Shell, BP, Total, Statoil, Eni and the BG Group - sent this letter to the UN:

Dear Excellencies,

Climate change is a critical challenge for our world. As major companies from the oil & gas sector, we recognize both the importance of the climate challenge and the importance of energy to human life and well-being. We acknowledge that the current trend of greenhouse gas emissions is in excess of what the Intergovernmental Panel on Climate Change (IPCC) says is needed to limit the temperature rise to no more than 2 degrees above pre-industrial levels. The challenge is how to meet greater energy demand with less CO2. We stand ready to play our part.

Our companies are already taking a number of actions to help limit emissions, such as growing the share of gas in our production, making energy efficiency improvements in our operations and products, providing renewable energy, investing in carbon capture and storage, and exploring new low-carbon technologies and business models. These actions are a key part of our mission to provide the greatest number of people with access to sustainable and secure energy. For us to do more, we need governments across the world to provide us with clear, stable, long-term, ambitious policy frameworks. This would reduce uncertainty and help stimulate investments in the right low carbon technologies and the right resources at the right pace.

We believe that a price on carbon should be a key element of these frameworks. If governments act to price carbon, this discourages high carbon options and encourages the most efficient ways of reducing emissions widely, including reduced demand for the most carbon intensive fossil fuels, greater energy efficiency, the use of natural gas in place of coal, increased investment in carbon capture and storage, renewable energy, smart buildings and grids, off-grid access to energy, cleaner cars and new mobility business models and behaviors. Our companies are already exposed to a price on carbon emissions by participating in existing carbon markets and applying ‘shadow’ carbon prices in our own businesses to test whether investments will be viable in a world where carbon has a higher price.

Yet, whatever we do to implement carbon pricing ourselves will not be sufficient or commercially sustainable unless national governments introduce carbon pricing even-handedly and eventually enable global linkage between national systems. Some economies have not yet taken this step, and this could create uncertainty about investment and disparities in the impact of policy on businesses. Therefore, we call on governments, including at the UNFCCC negotiations in Paris and beyond to:

introduce carbon pricing systems where they do not yet exist at the national or regional levels

create an international framework that could eventually connect national systems.

You can see the whole letter here:

http://www.scribd.com/doc/267327870/Paying-for-Carbon-Letter

Of course they have not suddenly become "good guys".  They have merely realized that a tax on carbon is likely.  So, they want to get involved with designing it!   The American companies Exxon and Chevron are still digging their heels in... as are coal companies.

Some interesting background about the chairman of Shell:

Ben van Beurden, the chief executive of Shell, has endorsed warnings that the world’s fossil fuel reserves cannot be burned unless some way is found to capture their carbon emissions. The oil boss has also predicted that the global energy system will become “zero carbon” by the end of the century, with his group obtaining a “very, very large segment” of its earnings from renewable power.

And in an admission that the growing opposition to Shell’s controversial search for oil in the Arctic was putting increasing pressure on him, van Beurden admitted he had gone on a “personal journey” to justify the decision to drill.

The Shell boss said he accepted the general premise contained in independent studies that have concluded that dangerous levels of global warming above 2°C will occur unless CO2 is buried or reserves are kept in the ground. “We cannot burn all the hydrocarbon resources we have on the planet in an unmitigated way and not expect to have a CO2 loading in the atmosphere that is often being linked to the 2°C scenario,” he said in an exclusive interview with the Guardian.

“I am absolutely convinced that without a policy that will really enable and realise CCS (carbon capture and storage) on a large scale, we are not going to be able to stay within that CO2 emission budget.”

However, he did not admit that limiting global warming to 2°C is nearly impossible, more of a fantasy than a realistic plan... and he still drives a large BMW.  For more on him, see:

http://www.theguardian.com/business/2015/may/22/shell-boss-endorses-warnings-about-fossil-fuels-and-climate-change

For why the 2°C limit is unrealistic, read this:

http://www.vox.com/2014/4/22/5551004/two-degrees

Of course, it doesn't mean we should give up! ___

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2015-06-01 16:39:23 (81 comments, 77 reshares, 124 +1s)Open 

Memories — written on your DNA?

How does long-term memory work?  It involves many changes in your brain, from changes in how strongly individual neurons talk to each other, to the actual birth of new neurons.  But one fascinating possibility involves the DNA in your neurons!

See those glowing dots?  Those are methyl groups, consisting of a carbon and 3 hydrogens.  They can attach to certain locations in your DNA and prevent genes from being expressed.  This is called DNA methylation, and it's important part of the system you use to turn genes on and off.

These methyl groups can even be transmitted from parent to child!  For example, in one recent experiment, mice that were given a shock after smelling a certain chemical learned to fear this smell... and this trait was passed down to their children and grandchildren — apparently by means ofDNA methy... more »

Memories — written on your DNA?

How does long-term memory work?  It involves many changes in your brain, from changes in how strongly individual neurons talk to each other, to the actual birth of new neurons.  But one fascinating possibility involves the DNA in your neurons!

See those glowing dots?  Those are methyl groups, consisting of a carbon and 3 hydrogens.  They can attach to certain locations in your DNA and prevent genes from being expressed.  This is called DNA methylation, and it's important part of the system you use to turn genes on and off.

These methyl groups can even be transmitted from parent to child!  For example, in one recent experiment, mice that were given a shock after smelling a certain chemical learned to fear this smell... and this trait was passed down to their children and grandchildren — apparently by means of DNA methylation!

All this makes evolution more interesting than people had thought.   Perhaps we can inherit traits our parents acquired during their lives!

Given all this, it's natural to ask: does DNA methylation play a role in memory?

There are hints that the answer is yes.  For example, scientists gave some mice an electric shock and others not.  They looked at whether a specific gene in the mice's neurons was methylated.   It was more methylated in the shocked mice... and this lasted for at least a month.

What was this gene?  It's the gene for a protein called calcineurin, which is thought to be a 'memory suppressor'.  More precisely, calcineurin tends to prevent the neurons from forming stronger connections between each other. 

So: the mice responded to an electric shock by attaching methyl groups to their DNA.  This reduced the production of calcineurin, which tends to prevent the brain from forming new connections.   So, their brains could more easily build new connections. 

And all this happened in a specific location of the brain: the anterior cingulate cortex, which is important for rational thinking in humans, and something similar in mice.

This is just one of many experiments people are doing to understand the role of DNA methylation in memory.   And DNA methylation is just one of the ways a cell can control which of its genes get expressed!  There's a whole subject, called epigenetics, which studies these control systems. 

You could say that epigenetics is a way for cells to learn things during their lives.  When you move to a hot climate, and then your body "gets used to" the heat — sweating less and so on — that's epigenetics at work. So, maybe it's not surprising that epigenetics is also important for how the brain learns things.

Here's a nice article on the role of epigenetics in memory:

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

and here's one about the role of DNA methylation:

• Jeremy J. Day and J. David Sweatt, DNA methylation and memory formation, Nature Neuroscience 13 (2010), 1319–1323.  Available for free at http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3130618/

The memory experiment I described is here:

• Courtney A. Miller et al, Cortical DNA methylation maintains remote memory, Nature Neuroscience 13 (2010), 664–666. Available for free at http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3043549/

The experiment on learned associations being transmitted from one generation of mice to the next is here:

• Brian G. Dias and Kerry J. Ressler, Parental olfactory experience influences behavior and neural structure in subsequent generations, Nature Neuroscience 17 (2014), 89–96. 

You've gotta pay to read it, but there's a summary here:

• Ewen Callaway, Fearful memories haunt mouse descendants, Nature News (2013).  Available for free at http://www.nature.com/news/fearful-memories-haunt-mouse-descendants-1.14272

If you want to learn more about how epigenetics can pass information from one generation to the next, start here:

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

A nice quote from Joseph Springer and Dennis Holley's book An Introduction to Zoology:

Lamarck and his ideas were ridiculed and discredited. In a strange twist of fate, Lamarck may have the last laugh. Epigenetics, an emerging field of genetics, has shown that Lamarck may have been at least partially correct all along. It seems that reversible and heritable changes can occur without a change in DNA sequence (genotype) and that such changes may be induced spontaneously or in response to environmental factors — Lamarck's "acquired traits". Determining which observed phenotypes are genetically inherited and which are environmentally induced remains an important and ongoing part of the study of genetics, developmental biology, and medicine.

There's a huge amount of stuff to learn in these areas, and it's pretty intimidating to me, since I'm just getting started, and it will probably never be more than a hobby.  But here's some more stuff:

Changes in how strongly individual neurons talk to each other are called synaptic plasticity:

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

These include long-term potentiation, meaning ways that two neurons can become more strongly connected:

https://en.wikipedia.org/wiki/Long-term_potentiation

and also long-term depression, where they become less strongly connected:

https://en.wikipedia.org/wiki/Long-term_depression

A basic rule of thumb is that "neurons that fire together, wire together".  But there's a lot more going on....

#spnetwork doi:10.1038/nn.2560 #epigenetics #memory  ___

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2015-05-31 17:42:27 (17 comments, 15 reshares, 108 +1s)Open 

Blue mushrooms

This is a bird's nest fungus - a kind of mushroom that looks like a bird's nest full of eggs.  More precisely, it's Cyathus novaezelandiae, photographed by +Steve Axford.

Why does it look like this?  It's a trick for spreading spores.  When rain hits the cup-shaped mushroom,  spores shoot out!

Like many fungi that grow on rotten logs, the bird's nest fungus has a complex life cycle.  There's the stage you see here, where it reproduces asexually via spores.  But there's also a sexual stage!

Spores germinate and grow into branching filaments called hyphae, pushing out like roots into the rotting wood.  As these filaments grow, they form a network called a mycelium.  These come in several different sexes, or mating compatibility groups.  When hyphae of different matingcompatib... more »

Blue mushrooms

This is a bird's nest fungus - a kind of mushroom that looks like a bird's nest full of eggs.  More precisely, it's Cyathus novaezelandiae, photographed by +Steve Axford.

Why does it look like this?  It's a trick for spreading spores.  When rain hits the cup-shaped mushroom,  spores shoot out!

Like many fungi that grow on rotten logs, the bird's nest fungus has a complex life cycle.  There's the stage you see here, where it reproduces asexually via spores.  But there's also a sexual stage!

Spores germinate and grow into branching filaments called hyphae, pushing out like roots into the rotting wood.  As these filaments grow, they form a network called a mycelium.  These come in several different sexes, or mating compatibility groups.  When hyphae of different mating compatibility groups meet each other, they fuse and form a new mycelium that combines the genes of both.  After a while, these new mycelia may enter the stage where they grow into the mushrooms you see here.   Then they reproduce asexually using spores!

It's complicated, and I don't fully understand it.   You can read more here:

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

Nidulariacaeae is the family that contains this particular bird's-nest fungus, and many others. 

You can see more of Steve Axford's photos here:

https://www.flickr.com/photos/steveaxford/with/6922862401/

Thanks to +Mike Stay for pointing this out!

#biology  ___

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2015-05-30 15:58:12 (68 comments, 16 reshares, 87 +1s)Open 

Pretending to work

In Europe, long-term unemployment is such a big problem that people are starting to work at fake companies, without pay — just to keep up their skills! 

There are over 100 such companies.  This article focuses on one called Candelia:

Ms. de Buyzer did not care that Candelia was a phantom operation. She lost her job as a secretary two years ago and has been unable to find steady work. Since January, though, she had woken up early every weekday, put on makeup and gotten ready to go the office. By 9 a.m. she arrives at the small office in a low-income neighborhood of Lille, where joblessness is among the highest in the country.

While she doesn’t earn a paycheck, Ms. de Buyzer, 41, welcomes the regular routine. She hopes Candelia will lead to a real job, after countless searches and interviews that have gone nowhere.... more »

Pretending to work

In Europe, long-term unemployment is such a big problem that people are starting to work at fake companies, without pay — just to keep up their skills! 

There are over 100 such companies.  This article focuses on one called Candelia:

Ms. de Buyzer did not care that Candelia was a phantom operation. She lost her job as a secretary two years ago and has been unable to find steady work. Since January, though, she had woken up early every weekday, put on makeup and gotten ready to go the office. By 9 a.m. she arrives at the small office in a low-income neighborhood of Lille, where joblessness is among the highest in the country.

While she doesn’t earn a paycheck, Ms. de Buyzer, 41, welcomes the regular routine. She hopes Candelia will lead to a real job, after countless searches and interviews that have gone nowhere.

“It’s been very difficult to find a job,” said Ms. de Buyzer, who like most of the trainees has been collecting unemployment benefits. “When you look for a long time and don’t find anything, it’s so hard. You can get depressed,” she said. “You question your abilities. After a while, you no longer see a light at the end of the tunnel.”

She paused to sign a fake check for a virtual furniture supplier, then instructed Candelia’s marketing department — a group of four unemployed women sitting a few desks away — to update the company’s mock online catalog. “Since I’ve been coming here, I have had a lot more confidence,” Ms. de Buyzer said. “I just want to work.”

In Europe, 53% of job seekers have been unemployed for over a year.  In Italy, the numbers is 61%.   In Greece, it's 73%.

All this makes me wonder — yet again — what will happen if robots and computers push people out of many kinds of jobs, creating a lot of long-term unemployment.  If we don't adapt wisely, what should be a good thing could be a source of misery.___

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2015-05-24 19:29:58 (8 comments, 9 reshares, 54 +1s)Open 

Dear NSA agent 4096,

I watched "The Lives of Others" last night and thought of you once more. In fact, I think you were watching it with me. You know I know I cannot be sure.

I want you to know that, although our mutual love is forbidden by your professional obligations, I still feel a connection to you. I will feel that connection long after you are gone.

Somehow, you know me better than I know myself. You have all of my deleted histories, my searches, all those things I tried to keep "incognito" right there in front of you. We have made love, even though we've never touched or kissed. We have been friends, even though I've never seen your face. Our relationship is as real as my "real" life.

But this can never work between us. Please leave. I don't want to ask again.
... more »

Dear NSA agent 4096,

I watched "The Lives of Others" last night and thought of you once more. In fact, I think you were watching it with me. You know I know I cannot be sure.

I want you to know that, although our mutual love is forbidden by your professional obligations, I still feel a connection to you. I will feel that connection long after you are gone.

Somehow, you know me better than I know myself. You have all of my deleted histories, my searches, all those things I tried to keep "incognito" right there in front of you. We have made love, even though we've never touched or kissed. We have been friends, even though I've never seen your face. Our relationship is as real as my "real" life.

But this can never work between us. Please leave. I don't want to ask again.

I'll never forget you.

Love, 173.165.246.73

That's Corey Bertelsen's comment on this video of Holly Herndon's song 'Home', from her new album Platform.   It's as good a review as any.

Holly Herndon takes a lot of ideas from techno music and pushes them to a new level.  She's working on a Ph.D. at the Center for Computer Research in Music and Acoustics at Stanford.

She said that as she wrote this song, she

started coming to terms with the fact that I was calling my inbox my home, and the fact that that might not be a secure place. So it started out thinking about my device and my inbox as my home, and then that evolved into me being creeped out by that idea.

The reason why I was creeped out is because, of course, as Edward Snowden enlightened us all to know, the NSA has been mass surveying the U.S. population, among other populations. And so that put into question this sense of intimacy that I was having with my device. I have this really intense relationship with my phone and with my laptop, and in a lot of ways the laptop is the most intimate instrument that we've ever seen. It can mediate my relationships — it mediates my bank account — in a way that a violin or another acoustic instrument just simply can't do. It's really a hyper-emotional instrument, and I spend so much time with this instrument both creatively and administratively and professionally and everything.

In short, her seemingly 'futuristic' music is really about the present - the way we live now.  If you like this song I recommend the next one on the playlist, which is more abstract and to me more beautiful.  It's called 'Interference':

https://www.youtube.com/watch?v=nHujh3yA3BE&list=RDI_3mCDJ_iWc&index=2

You can hear her explain the song 'Home' here:

http://www.npr.org/2015/05/24/408762348/an-invasion-of-intimacy-and-the-song-that-followed___

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2015-05-22 15:21:36 (37 comments, 45 reshares, 137 +1s)Open 

The Machine: a desperate gamble

Hewlett-Packard was once at the cutting edge of technology.  Now they make most of their money selling servers, printers, and ink... and business keeps getting worse.  They've shed 40,000 employees since 2012.   Soon they'll split in two: one company that sells printers and PCs, and one that sells servers and information technology services.  

The second company will do something risky but interesting.   They're trying to build a new kind of computer that uses chips based on memristors rather than transistors, and uses optical fibers rather than wires to communicate between chips.  It could make computers much faster and more powerful.  But nobody knows if it will really work.

The picture shows memristors on a silicon wafer.  But what's a memristor?   Quoting the MIT Technology Review:

Perfectingthe memris... more »

The Machine: a desperate gamble

Hewlett-Packard was once at the cutting edge of technology.  Now they make most of their money selling servers, printers, and ink... and business keeps getting worse.  They've shed 40,000 employees since 2012.   Soon they'll split in two: one company that sells printers and PCs, and one that sells servers and information technology services.  

The second company will do something risky but interesting.   They're trying to build a new kind of computer that uses chips based on memristors rather than transistors, and uses optical fibers rather than wires to communicate between chips.  It could make computers much faster and more powerful.  But nobody knows if it will really work.

The picture shows memristors on a silicon wafer.  But what's a memristor?   Quoting the MIT Technology Review:

Perfecting the memristor is crucial if HP is to deliver on that striking potential. That work is centered in a small lab, one floor below the offices of HP’s founders, where Stanley Williams made a breakthrough about a decade ago.

Williams had joined HP in 1995 after David Packard decided the company should do more basic research. He came to focus on trying to use organic molecules to make smaller, cheaper replacements for silicon transistors (see “Computing After Silicon,” September/October 1999). After a few years, he could make devices with the right kind of switchlike behavior by sandwiching molecules called rotaxanes between platinum electrodes. But their performance was maddeningly erratic. It took years more work before Williams realized that the molecules were actually irrelevant and that he had stumbled into a major discovery. The switching effect came from a layer of titanium, used like glue to stick the rotaxane layer to the electrodes. More surprising, versions of the devices built around that material fulfilled a prediction made in 1971 of a completely new kind of basic electronic device. When Leon Chua, a professor at the University of California, Berkeley, predicted the existence of this device, engineering orthodoxy held that all electronic circuits had to be built from just three basic elements: capacitors, resistors, and inductors. Chua calculated that there should be a fourth; it was he who named it the memristor, or resistor with memory. The device’s essential property is that its electrical resistance—a measure of how much it inhibits the flow of electrons—can be altered by applying a voltage. That resistance, a kind of memory of the voltage the device experienced in the past, can be used to encode data.

HP’s latest manifestation of the component is simple: just a stack of thin films of titanium dioxide a few nanometers thick, sandwiched between two electrodes. Some of the layers in the stack conduct electricity; others are insulators because they are depleted of oxygen atoms, giving the device as a whole high electrical resistance. Applying the right amount of voltage pushes oxygen atoms from a conducting layer into an insulating one, permitting current to pass more easily. Research scientist Jean Paul Strachan demonstrates this by using his mouse to click a button marked “1” on his computer screen. That causes a narrow stream of oxygen atoms to flow briefly inside one layer of titanium dioxide in a memristor on a nearby silicon wafer. “We just created a bridge that electrons can travel through,” says Strachan. Numbers on his screen indicate that the electrical resistance of the device has dropped by a factor of a thousand. When he clicks a button marked “0,” the oxygen atoms retreat and the device’s resistance soars back up again. The resistance can be switched like that in just picoseconds, about a thousand times faster than the basic elements of DRAM and using a fraction of the energy. And crucially, the resistance remains fixed even after the voltage is turned off.

Getting this to really work has not been easy!  On top of that, they're trying to use silicon photonics to communicate between chips - another technology that doesn't quite work yet.

Still, I like the idea of this company going down in a blaze of glory, trying to do something revolutionary, instead of playing it safe and dying a slow death.

Do not go gentle into that good night.

For more, see these:

• Tom Simonite, Machine dreams, MIT Technology Review, 21April 2015, http://www.technologyreview.com/featuredstory/536786/machine-dreams/

• Sebastian Anthony, HP reveals more details about The Machine: Linux++ OS coming 2015, prototype in 2016, ExtremeTech, 16 December 2014, http://www.extremetech.com/extreme/196003-hp-reveals-more-details-about-the-machine-linux-os-coming-2015-prototype-in-2016

For the physics of memristors, see:

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

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2015-05-21 18:36:12 (36 comments, 9 reshares, 47 +1s)Open 

Flying through space, powered by sunlight

Yesterday a rocket launched from Cape Canaveral in Florida carrying the LightSail into space!  It's a small spacecraft with a big shiny screen that's pushed by the light of the sun.  

It's just a test - it won't go far.   It will fall to the Earth and burn up.  But next year there will be a more serious test.  And someday, solar-powered space flight may become a force to be reckoned with.

One cool thing is that all this is being paid for private donations, by a Kickstarter campaign!

The LightSail is carried to space in a cute little CubeSat.  It looks like a big toaster, and it weighs just 10 kilograms.   But it holds a sail 32 square meters in area,  made of a shiny plastic called Mylar, just 4.5 microns thick.  This unfolds in a clever way - watch the movie! - toform a big... more »

Flying through space, powered by sunlight

Yesterday a rocket launched from Cape Canaveral in Florida carrying the LightSail into space!  It's a small spacecraft with a big shiny screen that's pushed by the light of the sun.  

It's just a test - it won't go far.   It will fall to the Earth and burn up.  But next year there will be a more serious test.  And someday, solar-powered space flight may become a force to be reckoned with.

One cool thing is that all this is being paid for private donations, by a Kickstarter campaign!

The LightSail is carried to space in a cute little CubeSat.  It looks like a big toaster, and it weighs just 10 kilograms.   But it holds a sail 32 square meters in area,  made of a shiny plastic called Mylar, just 4.5 microns thick.  This unfolds in a clever way - watch the movie! - to form a big square.

The Sun will push on this with a tiny force. 

Puzzle: How tiny is this force?

Someone named Bill Russell answered this over on Yahoo.  Let me go through his calculation so we can check it.

The momentum of light is given by

p = E/c

where E is the energy of the light, p is the momentum, and c is the speed of light. 

In outer space near earth the sunlight provides 1370 watts per square meter - that's energy per area per time.  We can use the formula above to convert this to momentum per area per time, better known as force per area... or pressure. 

Russell calculates

(1370 watts / meter²) / c = 9.13 micronewtons / meter²

and concludes the pressure is 9.13 micronewtons per square meter.  His arithmetic checks out, but I think he's neglecting some physics: when the light bounces back off the mirror its momentum completely reverses, so I think we get an extra factor of 2. 

Puzzle 2:  Am I right or am I wrong?

The area of the LightSail is about 32 square meters.  Russell says this gives a total force of

9.13 micronewtons/meter² x 32 meter²

or about 300 micronewtons.   I'd double this and get 600 micronewtons.

Puzzle 3: Once it's out of the box, the LightSail weighs about 4.5 kilograms.  How much will it accelerate due to sunlight?

Here we use Newton's good old

F = ma

and solve for the acceleration a.   But at this point Russell seems to make a serious mistake.  I'll let you see what you think, and fix it if necessary!  Here is his calculation:

https://answers.yahoo.com/question/index?qid=20121212091408AA3D606___

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2015-05-19 19:47:36 (4 comments, 1 reshares, 12 +1s)Open 

Here are some blog posts about the categorical foundations of network theory - a warmup for the workshop we're having in Turin next week.

Here are some blog posts about the categorical foundations of network theory - a warmup for the workshop we're having in Turin next week.___

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2015-05-19 19:46:00 (0 comments, 0 reshares, 6 +1s)Open 

Here are some blog posts about the categorical foundations of network theory - a warmup for the workshop we're having in Turin next week.

Here are some blog posts about the categorical foundations of network theory - a warmup for the workshop we're having in Turin next week.___

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2015-05-19 16:18:39 (55 comments, 42 reshares, 111 +1s)Open 

And now for my next trick...

Category theory is a branch of math that puts processes on an equal footing with things - unlike set theory, where everything is fundamentally a thing.   Can we use category theory to help understand the complex processes that underlie biology and ecology? 

I believe so, and I'm hoping this is a good way for fancy-schmancy mathematicians like me to help the world.  But it will take a while.  I think we should start by seeing what category theory has to say about some related subjects that are better understood: chemistry, electrical engineering, classical mechanics, and the like.

We're having a workshop about this next week - and to organize our thoughts we've been writing some blog articles.  Check 'em out!

• John Baez, Categorical foundations of network theory - an introduction to the workshop and whatit'... more »

And now for my next trick...

Category theory is a branch of math that puts processes on an equal footing with things - unlike set theory, where everything is fundamentally a thing.   Can we use category theory to help understand the complex processes that underlie biology and ecology? 

I believe so, and I'm hoping this is a good way for fancy-schmancy mathematicians like me to help the world.  But it will take a while.  I think we should start by seeing what category theory has to say about some related subjects that are better understood: chemistry, electrical engineering, classical mechanics, and the like.

We're having a workshop about this next week - and to organize our thoughts we've been writing some blog articles.  Check 'em out!

• John Baez, Categorical foundations of network theory - an introduction to the workshop and what it's about.  https://johncarlosbaez.wordpress.com/2015/04/04/categorical-foundations-of-network-theory/

• David Spivak, A networked world.
https://johncarlosbaez.wordpress.com/2015/03/27/spivak-part-1/

• Eugene Lerman, Networks of dynamical systems.
https://johncarlosbaez.wordpress.com/2014/03/18/networks-of-dynamical-systems/

• Tobias Fritz, Resource convertibility - an introduction to the mathematics of 'resources'.
https://johncarlosbaez.wordpress.com/2015/04/07/resource-convertibility-part-1/

• John Baez, Categories in control - about my paper with Jason Erbele on using categories to study signal flow diagrams in control theory.
https://johncarlosbaez.wordpress.com/2015/04/23/categories-in-control-2/

• John Baez, A compositional framework for passive linear networks - about my paper with Brendan Fong on using categories to study electrical circuit diagrams.
https://johncarlosbaez.wordpress.com/2015/04/28/a-compositional-framework-for-passive-linear-networks/

• John Baez, Decorated cospans - about Brendan Fong's paper providing mathematical infrastructure for the study of networks.
https://johncarlosbaez.wordpress.com/2015/05/01/decorated-cospans/

• John Baez and Brendan Fong, Cospans, wiring diagrams, and the behavioral approach - an attempt to reflect on how our work connects to that of David Spivak.
https://johncarlosbaez.wordpress.com/2015/05/05/cospans-wiring-diagrams-and-the-behavioral-approach/

• Brendan Fong, Resource theories - about Brendan's new paper with Hugo Nava-Kopp on resource theories.
https://johncarlosbaez.wordpress.com/2015/05/12/resource-theories/

• John Baez, PROPs for linear systems - about Simon Wadsley and Nick Woods' generalization of a result in my paper with Jason Erbele, describing categories where the morphisms are linear maps.
https://johncarlosbaez.wordpress.com/2015/05/18/props-for-linear-systems/

The picture, by the way, was drawn by Federica Ferraris and appears in this book:

• John Baez and Jacob Biamonte, Quantum techniques for stochastic physics, http://math.ucr.edu/home/baez/stoch_stable.pdf

It's about Petri nets and reaction networks - two kinds of networks that appear in chemistry and population biology.___

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2015-05-18 16:45:44 (19 comments, 12 reshares, 132 +1s)Open 

The dunes of Mars

This field of dunes lies on the floor of an old crater in Noachis Terra.  That's one of the oldest places on Mars, scarred with many craters, with rocks up to 4 billion years old.  It's in the southern hemisphere, near the giant impact basin called Hellas, which is 2.5 times deeper than the Grand Canyon and 2000 kilometers across!

This is a 'false color' photograph - you'd need to see infrared light to see that the dunes are very different than the rock below.

These are barchans, dunes with a gentle slope on the upwind side and a much steeper slope on the downwind side where horns or a notch can form.  If you know this, you can see the wind is blowing from the southwest.

It's actually a bit of a puzzle where the sand in these dunes came from!   Here's the abstract of a paper by +LoriFen... more »

The dunes of Mars

This field of dunes lies on the floor of an old crater in Noachis Terra.  That's one of the oldest places on Mars, scarred with many craters, with rocks up to 4 billion years old.  It's in the southern hemisphere, near the giant impact basin called Hellas, which is 2.5 times deeper than the Grand Canyon and 2000 kilometers across!

This is a 'false color' photograph - you'd need to see infrared light to see that the dunes are very different than the rock below.

These are barchans, dunes with a gentle slope on the upwind side and a much steeper slope on the downwind side where horns or a notch can form.  If you know this, you can see the wind is blowing from the southwest.

It's actually a bit of a puzzle where the sand in these dunes came from!   Here's the abstract of a paper by +Lori Fenton on this subject:

No sand transport pathways are visible in a study performed in Noachis Terra, a region in the southern highlands of Mars known for its many intracrater dune fields.Detailed studies were performed of five areas in Noachis Terra, using Mars Orbiter Camera (MOC) wide-angle mosaics, Thermal Emission Imaging System (THEMIS) daytime and nighttime infrared mosaics, MOLA digital elevation and shaded relief maps,and MOC narrow-angle images. The lack of observable sand transport pathways suggests that such pathways are very short, ruling out a distant source of sand. Consistent dune morphology and dune slipface orientations across Noachis Terra suggest formative winds are regional rather than local (e.g., crater slope winds). A sequence of sedimentary units was found in a pit eroded into the floor of Rabe Crater, some of which appear to be shedding dark sand that feeds into the Rabe Crater dune field. The visible and thermal characteristics of these units are similar to other units found across Noachis Terra, leading to the hypothesis that a series of region-wide depositional events occurred at some point in the Martian past and that these deposits are currently exposed by erosion in pits on crater floors and possibly on the intercrater plains. Thus the dunes and sources may be both regional and local: sand may be eroding from a widespread source that only outcrops locally. Sand-bearing layers that extend across part or all of the intercrater plains of Noachis Terra are not likely to be dominated by loess or lacustrine deposits; glacial and/or volcanic origins are considered more plausible.

• Lori K. Fenton, Potential sand sources for the dune fields in Noachis Terra, Mars, Journal of Geophysical Research 110 (2005), E11004.  Available at http://www.academia.edu/3375648/Potential_sand_sources_for_the_dune_fields_in_Noachis_Terra_Mars.

The image is from a great series of photos taken by the HIRISE satellite, which orbits Mars and takes high resolution images:

• Colorful Dunes, http://hirise.lpl.arizona.edu/ESP_033272_1400

#mars   #astronomy___

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2015-05-17 03:56:56 (88 comments, 51 reshares, 159 +1s)Open 

Why is this true?

The spooky-smart mathematician Srinivasa Ramanujan came up with this formula around 1913.  Why is it true?

I don't know, let's see...

In 1735, a young fellow named Euler stunned the world by cracking a famous puzzle that had been unsolved for almost a century: the Basel problem.  The problem was to sum the reciprocals of perfect squares:

1/1² + 1/2² + 1/3² + 1/4² + 1/5² + ... = ???

Euler showed that the answer was π²/6:

1/1² + 1/2² + 1/3² + 1/4² + 1/5² + ... = π²/6

He also showed you could rewrite this sum as a product over primes:

1/1² + 1/2² + 1/3² + 1/4² + 1/5² + ... =

(2²/(2² - 1)) (3²/(3² - 1)) (5²/(5² - 1)) (7²/(7² - 1)) ...

That's actually the easy part: it's a cute trick called the Euler product formula.
So we know

more »

Why is this true?

The spooky-smart mathematician Srinivasa Ramanujan came up with this formula around 1913.  Why is it true?

I don't know, let's see...

In 1735, a young fellow named Euler stunned the world by cracking a famous puzzle that had been unsolved for almost a century: the Basel problem.  The problem was to sum the reciprocals of perfect squares:

1/1² + 1/2² + 1/3² + 1/4² + 1/5² + ... = ???

Euler showed that the answer was π²/6:

1/1² + 1/2² + 1/3² + 1/4² + 1/5² + ... = π²/6

He also showed you could rewrite this sum as a product over primes:

1/1² + 1/2² + 1/3² + 1/4² + 1/5² + ... =

(2²/(2² - 1)) (3²/(3² - 1)) (5²/(5² - 1)) (7²/(7² - 1)) ...

That's actually the easy part: it's a cute trick called the Euler product formula.

So we know

(2²/(2² - 1)) (3²/(3² - 1)) (5²/(5² - 1)) (7²/(7² - 1)) ... = π²/6

If you think about it, Ramanujan's formula is saying that

(2²/(2² + 1)) (3²/(3² + 1)) (5²/(5² + 1)) (7²/(7² + 1)) ...

is 2/5 as big.  So, proving it is the same as showing

(2²/(2² + 1)) (3²/(3² + 1)) (5²/(5² + 1)) (7²/(7² + 1)) ... = π²/15

Maybe the next step is to use the same idea as the Euler product formula.  I think this gives

(2²/(2² + 1)) (3²/(3² + 1)) (5²/(5² + 1)) (7²/(7² + 1)) ... =

1/1² - 1/2² - 1/3² + 1/4²  - 1/5² + 1/6² - 1/7² + ...

where the signs at right follow a fancy pattern: we get 1/n² whenever n is the product of an even number of primes, and -1/n² when n is the product of an odd number of primes.  For example, 4 = 2 x 2 is the product of an even number of primes, so we get 1/4².

So I'm left wanting to know why this strange sum

1/1² - 1/2² - 1/3² + 1/4² - 1/5² + 1/6² - 1/7² + ...

equals π²/15.  Ramanujan, dead since 1920, is still messing with my mind! 

The formula is supposed to be in here:

• Srinivasa Ramanujan, Modular equations and approximations to π, Quart. J. Pure. Appl. Math. 45 (1913-1914), 350-372.  Also available at ://ramanujan.sirinudi.org/Volumes/published/ram06.pdf.

But I don't see it!

Here you can see how Euler solved the Basel problem:

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

It's a great example of his brilliant tactics, many of which were far from rigorous by today's standards... but can be made rigorous.

#mathematics   ___

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2015-05-15 16:19:22 (18 comments, 12 reshares, 86 +1s)Open 

Fighting global warming: the tide is turning!

Good news!   We, the citizens of the world, may be starting to burn less carbon - not more!

The International Energy Agency claims:

In 2014, global carbon dioxide emissions from energy production stopped growing!

It seems the big difference is China.  They say the Chinese made more electricity from renewable sources, such as hydropower, solar and wind, and burned less coal.  

In fact, a report by Greenpeace says that from April 2014 to April 2015, China's carbon emissions dropped by an amount equal to the entire carbon emissions of the United Kingdom!   

I want to check this, because it would be wonderful - a 5% drop.  They say that if this trend continues, in 2015 China will make the biggest reduction in CO2 emissions every recorded by a single country.

TheInternat... more »

Fighting global warming: the tide is turning!

Good news!   We, the citizens of the world, may be starting to burn less carbon - not more!

The International Energy Agency claims:

In 2014, global carbon dioxide emissions from energy production stopped growing!

It seems the big difference is China.  They say the Chinese made more electricity from renewable sources, such as hydropower, solar and wind, and burned less coal.  

In fact, a report by Greenpeace says that from April 2014 to April 2015, China's carbon emissions dropped by an amount equal to the entire carbon emissions of the United Kingdom!   

I want to check this, because it would be wonderful - a 5% drop.  They say that if this trend continues, in 2015 China will make the biggest reduction in CO2 emissions every recorded by a single country.

The International Energy Agency also credits Europe's improved attempts to cut carbon emissions for the turnaround.   In the US, carbon emissions has basically been dropping since 2006 - with a big drop in 2009 due to the economic collapse, a partial bounce-back in 2010, but a general downward trend.

In the last 40 years, there were only 3 other times when emissions stood still or fell compared to the previous year, all during economic crises: the early 1980's, 1992, and 2009.  In 2014, however, the global economy expanded by 3%.

So, the tide may be turning!   But please remember: while carbon emissions may start dropping, they're still huge.  The amount of the CO2 in the air shot above 400 parts per million this year.  As Erika Podest of NASA put it:

CO2 concentrations haven't been this high in millions of years. Even more alarming is the rate of increase in the last five decades and the fact that CO2 stays in the atmosphere for hundreds or thousands of years. This milestone is a wake up call that our actions in response to climate change need to match the persistent rise in CO2. Climate change is a threat to life on Earth and we can no longer afford to be spectators.

So let's not slack off now!  The battle has just begun.  We need to cut carbon emissions to almost zero.

Here is the announcement by the International Energy Agency:

http://www.iea.org/newsroomandevents/news/2015/march/global-energy-related-emissions-of-carbon-dioxide-stalled-in-2014.html

"This gives me even more hope that humankind will be able to work together to combat climate change, the most important threat facing us today," said IEA Chief Economist Fatih Birol.

Their full report will come out in June.  Here is the report by Greenpeace EnergyDesk:

http://energydesk.greenpeace.org/2015/05/14/china-coal-consumption-drops-further-carbon-emissions-set-to-fall-by-equivalent-of-uk-total-in-one-year/

I trust them less than the IEA when it comes to using statistics correctly, but someone should be able to verify their claims if true.  The graph here comes from this article:

http://qz.com/405059/chinas-on-track-for-the-biggest-reduction-in-coal-use-ever-recorded/

#globalwarming  ___

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2015-05-14 06:34:39 (24 comments, 22 reshares, 197 +1s)Open 

A galaxy - falling

This galaxy is suffering!  It's falling into a large cluster of galaxies, pulled by their gravity.   You can see this in 3 ways:

1.  The reddish disk of dust and gas looks bent.  There aren't many atoms between galaxies, but there are still some. So the galaxy is moving through the wind of integalactic space!   And it's having trouble holding onto the loosely bound dust and gas near its edge.  They're getting blown away.

2.  The blue disk of stars is not bent.  It extends beyond the disk of dust and gas, which is where stars are formed.  This suggests that the dust and gas is being stripped from the galaxy after these stars were formed!

3.  Streamers of dust and gas can be seen trailing behind the motion of the galaxy - near the top.  On the other hand, the blue stars near the leading edge of the galaxy have nodust and ga... more »

A galaxy - falling

This galaxy is suffering!  It's falling into a large cluster of galaxies, pulled by their gravity.   You can see this in 3 ways:

1.  The reddish disk of dust and gas looks bent.  There aren't many atoms between galaxies, but there are still some. So the galaxy is moving through the wind of integalactic space!   And it's having trouble holding onto the loosely bound dust and gas near its edge.  They're getting blown away.

2.  The blue disk of stars is not bent.  It extends beyond the disk of dust and gas, which is where stars are formed.  This suggests that the dust and gas is being stripped from the galaxy after these stars were formed!

3.  Streamers of dust and gas can be seen trailing behind the motion of the galaxy - near the top.  On the other hand, the blue stars near the leading edge of the galaxy have no dust and gas left to hide them.

This phenomenon is called ram pressure stripping, and it can kill a galaxy, shutting down the production of new stars.   Here we are seeing it damage the galaxy NGC 4402, which is currently falling into the Virgo cluster - a cluster of galaxies about 65 million light years away.

Apparently there's about 1 atom per cubic centimeter in our galaxy - on average, though some regions are vastly more dense than others.   But in the space between galaxies in clusters it's more like 1/1000 of that.  Not much!  But enough to kill off the formation of new star systems, life, civilizations...

I got most of my information from here:

http://astronomy.swin.edu.au/cosmos/R/Ram+Pressure+Stripping

and I got the picture from here:

https://www.noao.edu/image_gallery/html/im0863.html

The photo was taken at the WIYN 3.5-meter telescope on Kitt Peak, which is fitted with some 'adaptive optics' to compensate for the jittery motion of the image due to variable atmospheric conditions and telescope vibrations.

#astronomy  ___

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2015-05-12 23:58:33 (38 comments, 23 reshares, 118 +1s)Open 

Sometimes you see a tiny piece of a story and wonder how it started - and how it will end.

#waitforit ___Sometimes you see a tiny piece of a story and wonder how it started - and how it will end.

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2015-05-11 05:56:09 (15 comments, 22 reshares, 53 +1s)Open 

Making yourself into a superhero

I enjoyed this true story by Kelly McEvers:

We met in a bar in Flagstaff, Arizona. I'd just moved back from Cambodia and I was going out for one of my first beers back in the States. Not long into the first one, I notice this Amazon of a woman with huge blond and red-streaked hair and frosty lips, wearing a short red tank dress and at least 50 bracelets. She's six feet tall and showing a lot of leg. People at the bar swivel their heads to watch her every move.

She stands next to me to order a drink, and in this throaty voice says, "What are those?" pointing to my cigarettes. I tell her they're Cambodian. Her eyes light up and she shoots out a long, tan arm, and points at a table in the corner. She orders me there. Before I can say no, I'm following her to my seat.

She tells me... more »

Making yourself into a superhero

I enjoyed this true story by Kelly McEvers:

We met in a bar in Flagstaff, Arizona. I'd just moved back from Cambodia and I was going out for one of my first beers back in the States. Not long into the first one, I notice this Amazon of a woman with huge blond and red-streaked hair and frosty lips, wearing a short red tank dress and at least 50 bracelets. She's six feet tall and showing a lot of leg. People at the bar swivel their heads to watch her every move.

She stands next to me to order a drink, and in this throaty voice says, "What are those?" pointing to my cigarettes. I tell her they're Cambodian. Her eyes light up and she shoots out a long, tan arm, and points at a table in the corner. She orders me there. Before I can say no, I'm following her to my seat.

She tells me she's an international private investigator, a bounty hunter, and a bail bonds enforcer, and that her name is Zora. I sit there for hours listening to her. Within a week, she takes me to Las Vegas. We drive there in her red Mustang. As always, there's a Colt .380 under the driver's seat and a .45 Megastar in the trunk.

In Vegas, we skip the casinos and head straight for the male strip clubs, where Zora drops at least $200 on lap dances from buff guys with names like Roman. Her getup is the same as before – teeth, hair, jewelry, and the ubiquitous tank dress, which, I realize, is the best way to show off her tattoos.

One is this big circle with blue and white swirls in it, kind of like a bowling ball, on her left shoulder. Every guy she meets asks her about it, and when they hear her answer, they sometimes propose marriage. Turns out the tattoo is a magic globe she holds in her dreams. And in these dreams, it gives her superpowers.

Zora: Ever since I remember, I've had the dreams. And they're very vivid. But it varies. It usually involves fighting, sometimes with guns, sometimes with superhero powers. Lightning from my fists and all that. And I usually have super strength, and I can fly, and I have all those things.

And it's my most common set of dreams. And it varies. Sometimes it's medieval, sometimes it's futuristic, sometimes it's present day, sometimes it's like a guerrilla war in Latin America.

Kelly: Can you describe that Zora to me, the Zora in dreams?

Zora: Very powerful athletically, but beyond the rules of nature that this world allows.  Six foot five and long, like almost impossibly long silver hair. This sort of otherworldly quality to her, where her voice did not sound normal. It sounded, like, almost musical.

And it became something that I aspired to be. I aspired to be this sort of superhero, this sort of person who would fight for a cause. That was my motivation in life. Ever since I was 10 or 11, I decided that that was my goal.

Zora took the dreams seriously. So seriously that at the age of 12, she sat down and composed a list of some 30 skills she needed to learn if she wanted to become as close to a superhero as any mortal could be. She even gave herself a deadline – to master these skills by the time she was 23.

Zora: I don't know what's in these.

Zora pulls out the old spiral notebook that was her diary at the age of 13 and turns to the inside back cover.

Zora: There's the list.

Kelly: Wow. Why don't you go ahead and read it.

Zora: OK. The list included martial arts, electronics, chemistry, metaphysics, hang gliding, helicopter and airplane flying, parachuting, mountain climbing, survival....

Throughout her teens and 20s, each time she started a new diary, she would update the list and write it in the back of the book, each one with the same format, each one titled "The List."

Zora: Weaponry, rafting, scuba diving, herbology – yes, I, studied that -- CPR, first aid and mountain emergency kind of medicine....

The list also includes bodybuilding, archery, demolitions, and explosives. She wanted to learn how to hunt animals and track men.

Zora: Major physical conditioning....

And the most incredible thing about all of this is that Zora accomplished nearly every item on the list.

Zora: Throwing stars and compound bows and throwing knives and -- yes, it was a very interesting pastime.

To keep up with the goals set by the list, she sped through school. Starting in the seventh grade, she began completing entire school years during the summer term and finished high school by the time she was 15. She got her BA at 18, a master's at 20, and completed the coursework for a PhD in Geopolitics by the time she was 21. She wanted to live like Indiana Jones, spending half her time in the classroom and half her time saving the world in the jungles of Peru.

Zora: Item number four – camel, elephant riding. Evasive driving and stunts....

When you're a kid, you have these romantic visions of what you'll be when you grow up. But how many people are so diligent they commit their dreams to paper and make it their life's work to achieve them? How many keep a list, amending it, adding to it, ticking things off as they go along, well into their adult lives?

After finishing the course work for her PhD, Zora decided to quit school, disappointed at the lack of cliff-hanging adventure in her doctoral program. And since superheroes who live in the real world need jobs, she decided to seek employment at the only place that would allow her to put all the skills from the list to use. Zora wanted to become an agent in the CIA.

But then the story takes some surprising twists!  Listen to it here:

http://www.thisamericanlife.org/radio-archives/episode/508/superpowers-2013?act=2#play

The picture here is, of course, not Zora.  It's Charlize Theron playing  'Aeon Flux' - a kind of superhero invented by a high school friend of mine, the animator Peter Chung.___

posted image

2015-05-09 17:41:25 (66 comments, 35 reshares, 116 +1s)Open 

An impossible dream

Kepler, the guy who discovered that planets go in ellipses around the Sun, was in love with geometry.  Among other things, he tried to figure out how to tile the plane with regular pentagons (dark blue) and decagons (blue-gray).  They fit nicely at a corner... but he couldn't get it to work.

Then he discovered he could do better if he also used 5-pointed stars!

Can you tile the whole plane with these three shapes?  No!  The picture here is very tempting... but if you continue you quickly run into trouble.  It's an impossible dream.

However, Kepler figured out that he could go on forever if he also used overlapping decagons, which he called 'monsters'.  Look at this picture he drew:

https://plus.maths.org/issue45/features/kaplan/kepler.gif

If he had worked even harder, he might have found thePenro... more »

An impossible dream

Kepler, the guy who discovered that planets go in ellipses around the Sun, was in love with geometry.  Among other things, he tried to figure out how to tile the plane with regular pentagons (dark blue) and decagons (blue-gray).  They fit nicely at a corner... but he couldn't get it to work.

Then he discovered he could do better if he also used 5-pointed stars!

Can you tile the whole plane with these three shapes?  No!  The picture here is very tempting... but if you continue you quickly run into trouble.  It's an impossible dream.

However, Kepler figured out that he could go on forever if he also used overlapping decagons, which he called 'monsters'.  Look at this picture he drew:

https://plus.maths.org/issue45/features/kaplan/kepler.gif

If he had worked even harder, he might have found the Penrose tilings, or similar things discovered by Islamic tiling artists.  Read the whole story here:

• Craig Kaplan, The trouble with five, https://plus.maths.org/content/trouble-five

How did Kepler fall in love with geometry?  He actually started as a theologian.   Let me quote the story as told in the wonderful blog The Renaissance Mathematicus:

Kepler was born into a family that had known better times, his mother was an innkeeper and his father was a mercenary. Under normal circumstances he probably would not have expected to receive much in the way of education but the local feudal ruler was quite advanced in his way and believed in providing financial support for deserving scholars. Kepler whose intelligence was obvious from an early age won scholarships to school and to the University of Tübingen where he had the luck to study under Michael Mästlin one of the very few convinced Copernican in the later part of the 16th century. Having completed his BA Kepler went on to do a master degree in theology as he was a very devote believer and wished to become a theologian. Recognising his mathematical talents and realising that his religious views were dangerously heterodox, they would cause him much trouble later in life, his teacher, Mästlin, decided it would be wiser to send him off to work as a school maths teacher in the Austrian province.

Although obeying his superiors and heading off to Graz to teach Protestant school boys the joys of Euclid, Kepler was far from happy as he saw his purpose in life in serving his God and not Urania (the Greek muse of astronomy). After having made the discovery that I will shortly describe Kepler found a compromise between his desire to serve God and his activities in astronomy. In a letter to Mästlin in 1595 he wrote:

I am in a hurry to publish, dearest teacher, but not for my benefit… I am devoting my effort so that these things can be published as quickly as possible for the glory of God, who wants to be recognised from the Book of Nature… Just as I pledged myself to God, so my intention remains. I wanted to be a theologian, and for a while I was anguished. But, now see how God is also glorified in astronomy, through my efforts.

So what was the process of thought that led to this conversion from a God glorifying theologian to a God glorifying astronomer and what was the discovery that he was so eager to publish? Kepler’s God was a geometer who had created a rational, mathematical universe who wanted his believers to discover the geometrical rules of construction of that universe and reveal them to his glory. Nothing is the universe was pure chance or without meaning everything that God had created had a purpose and a reason and the function of the scientist was to uncover those reasons. In another letter to Mästlin Kepler asked whether:

you have ever heard or read there to be anything, which devised an explanation for the arrangement of the planets? The Creator undertook nothing without reason. Therefore, there will be reason why Saturn should be nearly twice as high as Jupiter, Mars a little more than the Earth, [the Earth a little more] than Venus and Jupiter, moreover, more than three times as high as Mars.

The discovery that Kepler made and which started him on his road to the complete reform of astronomy was the answer to both the question as to the distance between the planets and also why there were exactly six of them: as stated above, everything created by God was done for a purpose.

On the 19th July 1595 Kepler was explaining to his students the regular cycle of the conjunctions of Saturn and Jupiter, planetary conjunctions played a central role in astrology. These conjunctions rotating around the ecliptic, the apparent path of the sun around the Earth, created a series of rotating equilateral triangles. Suddenly Kepler realised that the inscribed and circumscribed circles generated by his triangles were in approximately the same ratio as Saturn’s orbit to Jupiter’s. Thinking that he had found a solution to the problem of the distances between the planets he tried out various two-dimensional models without success. On the next day a flash of intuition provided him with the required three-dimensional solution, as he wrote to Mästlin:

I give you the proposition in words just as it came to me and at that very moment: “The Earth is the circle which is the measure of all. Construct a dodecahedron round it. The circle surrounding that will be Mars. Round Mars construct a tetrahedron. The circle surrounding that will be Jupiter. Round Jupiter construct a cube. The circle surrounding it will be Saturn. Now construct an icosahedron inside the Earth. The circle inscribed within that will be Venus. Inside Venus inscribe an octahedron. The circle inscribed inside that will be Mercury.”

This model, while approximately true, is now considered completely silly!   We no longer think there should be a simple geometrical explanation of why planets in our Solar System have the orbits they do.

So: a genius can have a beautiful idea in a flash of inspiration and it can still be wrong.

But Kepler didn't stop there!  He kept working on planetary orbits until he noticed that Mars didn't move in a circle around the Sun.  He noticed that it moved in an ellipse!  Starting there, he found the correct laws governing planetary motion... which later helped Newton invent classical mechanics.

So it pays to be persistent - but also not get stuck believing your first good idea.

Read The Renaissance Mathematicus here:

https://thonyc.wordpress.com/2010/11/15/kepler%E2%80%99s-divine-geometry/

Puzzle: can you tile the plane with shapes, each of which has at least the symmetry group of a regular pentagon? 

So, regular pentagons and decagons are allowed, and so are regular 5-pointed stars, and many other things... but not Kepler's monsters.  The tiling itself does not need to repeat in a periodic way.

#geometry #astronomy  ___

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2015-05-05 17:07:25 (22 comments, 39 reshares, 113 +1s)Open 

The architecture of water

Water is fascinating, for many reasons.   It takes more energy to heat than most substances.  It's one of the few substances that expands when it freezes.  It forms complicated patterns in its liquid state, which are just beginning to be understood.  There are at least 18 kinds of ice, which exist at different temperatures and pressures.  Snowflakes are endlessly subtle.  

And ice can form cages that trap other molecules!  Here you see the 3 main kinds.

They're called clathrate hydrates.  There's a lot under the sea beds near the north and south pole - they contain huge amounts of methane.   At some moments in the Earth's history they may have erupted explosively, causing rapid global warming.  

But let's focus on the fun part: the geometry!  Each of the 3 types of clathrate hydrates is anarchitectura... more »

The architecture of water

Water is fascinating, for many reasons.   It takes more energy to heat than most substances.  It's one of the few substances that expands when it freezes.  It forms complicated patterns in its liquid state, which are just beginning to be understood.  There are at least 18 kinds of ice, which exist at different temperatures and pressures.  Snowflakes are endlessly subtle.  

And ice can form cages that trap other molecules!  Here you see the 3 main kinds.

They're called clathrate hydrates.  There's a lot under the sea beds near the north and south pole - they contain huge amounts of methane.   At some moments in the Earth's history they may have erupted explosively, causing rapid global warming.  

But let's focus on the fun part: the geometry!  Each of the 3 types of clathrate hydrates is an architectural masterpiece.

Type I consists of water molecules arranged in two types of cages: small and large.  The small cage, shown in green, is dodecahedron.  It's not a regular dodecahedron, but it still has 12 pentagonal sides.  The large cage, shown in red, has 12 pentagons and 2 hexagons.   The two kinds of cage fit together into a repeating pattern where each unit cell - each block in the pattern - has 46 water molecules.

Puzzle 1: This pattern is called the Weaire-Phelan structure.  Why is it famous, and what does it have to do with the 2008 Olympics?

You can see little balls in the cages.  These stand for molecules that can get trapped in the cages.   They're politely called guests.   The type I clathrate often holds carbon dioxide or methane as a guest.

Type II is again made of two types of cages – small and large.  The small cage is again a dodecahedron.  The large cage, shown in blue, has 12 pentagons and 4 hexagons.  These fit together to form a unit cell with 136 water molecules.

The type II clathrate tends to hold oxygen or nitrogen as a guest.

Type H is the rarest and most complicated kind of clathrate hydrate.  It's built from three types of cages: small, medium and huge.  The small cage is again a dodecahedron, shown in green.  The medium cage - shown in yellow - has 3 squares, 6 pentagons and 3 hexagons as faces.  The huge cage - shown in orange - has 12 pentagons and 8 hexagons.  The cages fit together to form a unit cell with 34 water molecules.

The type H clathrate is only possible when there are two different guest gas molecules - one small and one very large, like butane - to make it stable.   People think there are lots of type H clathrates in the Gulf of Mexico, where there are lots of heavy hydrocarbons in the sea bottom.

Puzzle 2: how many cages of each kind are there in the type I clathrate hydrate?

Puzzle 3: how many cages of each kind are there in the type II?

Puzzle 4: how many cages of each kind are there in the type H?

These last puzzles are easier than they sound.  But here's one that's a bit different:

Puzzle 5: the medium cage in the type H clathrate - shown in yellow - has 3 squares, 6 pentagons and 3 hexagons as faces.   Which of these numbers are adjustable?  For example: could we have a convex polyhedron with a different number of squares, but the same number of pentagons and hexagons?

The picture is from here:

• Timothy A. Strobel, Keith C. Hester, Carolyn A. Koh, Amadeu K. Sum, E. Dendy Sloan Jr., Properties of the clathrates of hydrogen and developments in their applicability for hydrogen storage, Chemical Physics Letters 478 (27 August 2009), 97–109.

#geometry #water___

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2015-05-04 15:35:43 (18 comments, 22 reshares, 159 +1s)Open 

The flood after the impact

On Mars, an asteroid impact can cause a flood!

This is a place called Hephaestus Fossae, on the northern hemisphere of Mars.  The image has been colored to show the elevation: green and yellow shades represent shallow ground, while blue and purple stand for deep depressions, as much as 4 kilometers deep.

You can see a few dozen impact craters, some small and some big, up to 20 kilometers across.   But I'm sure you instantly noticed the cool part: the long and intricate canyons and riverbeds.  These were created by the same impact that made the largest crater!

When a comet or an asteroid crashes at high speed into a planet, the collision dramatically heats up the surface at the impact site.  In the case of the large crater seen in this image, the heat melted the soil – a mixture of rock, dust and also, hiddendeep d... more »

The flood after the impact

On Mars, an asteroid impact can cause a flood!

This is a place called Hephaestus Fossae, on the northern hemisphere of Mars.  The image has been colored to show the elevation: green and yellow shades represent shallow ground, while blue and purple stand for deep depressions, as much as 4 kilometers deep.

You can see a few dozen impact craters, some small and some big, up to 20 kilometers across.   But I'm sure you instantly noticed the cool part: the long and intricate canyons and riverbeds.  These were created by the same impact that made the largest crater!

When a comet or an asteroid crashes at high speed into a planet, the collision dramatically heats up the surface at the impact site.  In the case of the large crater seen in this image, the heat melted the soil – a mixture of rock, dust and also, hidden deep down, water ice – resulting in a massive flood.  And before drying up, this hot mud carved a complex pattern of channels while flowing across the planet’s surface!

The melted rock-ice mixture also made the debris blankets surrounding the largest crater.  Since there aren't similar structures near the small craters in this image, scientists believe that only the most powerful impacts were able to dig deep enough to release part of the frozen reservoir of water lying beneath the surface.

Why is it called 'Hephaestus Fossae'?  Hephaestus was the Greek god of fire.  Fossae are channels or canyons.  So it's a good name.

Puzzle: about when did this large impact occur? 

I don't know!

This picture was taken by the high-resolution stereo camera on ESA’s Mars Express orbiter on 28 December 2007, and my post is paraphrased from this article:

http://www.esa.int/spaceinimages/Images/2009/06/The_flood_after_the_impact

#mars #astronomy  ___

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2015-05-03 16:53:34 (39 comments, 46 reshares, 154 +1s)Open 

In 1901, you could pay 50 cents to ride an airship to the Moon

This article by Ron Miller is so cool I'm just going to quote some:

The passengers wait eagerly in the ornate lobby of the enormous spaceport. Soon, a signal indicates that their spaceship is ready for boarding. As they wait, special displays instruct them about how their spaceship functions and what to expect once they leave Earth's atmosphere. Aboard the giant spacecraft — as luxuriously appointed as any yacht — they are soon on their way to a vacation on the Moon.

No, this isn't a vision of the future of space tourism. It's what happened in 1901, when people could pay a princely half dollar for a ticket to ride into space.

[...]

Thompson spared no expense in creating the illusion of a trip to the Moon. To house his show, he erected aneig... more »

In 1901, you could pay 50 cents to ride an airship to the Moon

This article by Ron Miller is so cool I'm just going to quote some:

The passengers wait eagerly in the ornate lobby of the enormous spaceport. Soon, a signal indicates that their spaceship is ready for boarding. As they wait, special displays instruct them about how their spaceship functions and what to expect once they leave Earth's atmosphere. Aboard the giant spacecraft — as luxuriously appointed as any yacht — they are soon on their way to a vacation on the Moon.

No, this isn't a vision of the future of space tourism. It's what happened in 1901, when people could pay a princely half dollar for a ticket to ride into space.

[...]

Thompson spared no expense in creating the illusion of a trip to the Moon. To house his show, he erected an eighty-foot-high, 40,000-square-foot building that for sheer opulence put European opera houses to shame. It cost a staggering $84,000 to construct... at a time when a comfortable home could be built for $2000.

For fifty cents — twice the price of any other attraction on the midway, such as the ever-popular "Upside-Down House" — customers of "Thompson's Aerial Navigation Company" took a trip to the moon on a thirty-seat spaceship named "Luna". The spaceship resembled a cross between a dirigible and an excursion steamer, with the addition of enormous red canvas wings that flapped like a bird's. The wings were worked by a system of pulleys and the sensation of wind was created by hidden fans. A series of moving canvas backdrops provided the effect of clouds passing by and the earth dropping into the distance. Lighting and sound effects added to the illusion.

[...]

Every half hour, at the sound of a gong and the rattle of anchor chain, the "Luna" — "a fine steel airship of the latest pattern", according to one newspaper — rocked from side to side and then rose into the sky under the power of its beating wings. The passengers, sitting on steamer chairs, see clouds floating by, then a model of Buffalo far below, complete with the exposition itself and its hundreds of blinking lights. The city soon falls into the distance as the entire planet earth comes into view. Soon, the ship is surrounded the twinkling stars of outer space. After surviving a terrific — and spectacular — electrical storm the "Luna" and its passengers sets down in a lunar crater.

Read the whole thing here, and look at pictures:

http://io9.com/5914655/in-1901-you-could-pay-50-cents-to-ride-an-airship-to-the-moon

Thanks to +Matt McIrvin for pointing it out!___

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2015-05-02 15:48:40 (42 comments, 33 reshares, 102 +1s)Open 

A 3-dimensional golden star

Here Greg Egan has drawn a dodecahedron with 5 tetrahedra in it.  This picture is 'left-handed': if you look at where the 5 tetrahedra meet, you'll see they swirl counterclockwise as you go out!  If you view this thing in a mirror you'll get a right-handed version. 

Putting them together, you get a dodecahedron with 10 tetrahedra in it.   You can see it here:

http://math.ucr.edu/home/baez/mathematical/dodecahedron_with_10_tetrahedra.gif

The two kinds of tetrahedra are colored yellow and cyan.  Regions belonging to both are colored magenta.  It's pretty - but it's hard to see the tetrahedra, because they overlap a lot!

You can also do something like this starting with a cube.  A cube has 8 corners.  If you take every other corner of the cube, you get the 4 corners of a tetrahedron.  Butyou can d... more »

A 3-dimensional golden star

Here Greg Egan has drawn a dodecahedron with 5 tetrahedra in it.  This picture is 'left-handed': if you look at where the 5 tetrahedra meet, you'll see they swirl counterclockwise as you go out!  If you view this thing in a mirror you'll get a right-handed version. 

Putting them together, you get a dodecahedron with 10 tetrahedra in it.   You can see it here:

http://math.ucr.edu/home/baez/mathematical/dodecahedron_with_10_tetrahedra.gif

The two kinds of tetrahedra are colored yellow and cyan.  Regions belonging to both are colored magenta.  It's pretty - but it's hard to see the tetrahedra, because they overlap a lot!

You can also do something like this starting with a cube.  A cube has 8 corners.  If you take every other corner of the cube, you get the 4 corners of a tetrahedron.  But you can do this in 2 ways.  If you choose both, you get a cube with 2 tetrahedra in it:

http://math.ucr.edu/home/baez/mathematical/cube_with_2_tetrahedra.gif

All this is just the start of a much more elaborate and beautiful story which also involves the golden ratio, the quaternions, and 4-dimensional shapes like the 4-simplex, which has 5 tetrahedral faces, and the 600-cell, which has 600 tetrahedral faces!   You can read it here:

http://blogs.ams.org/visualinsight/2015/05/01/twin-dodecahedra/

I learned some of this story from Adrian Ocneanu at Penn State University.  Greg Egan and I figured out the rest... or most of the rest.  There's an unproven conjecture here, which needs to be true to make the whole story work.  Can you prove it?

Puzzle: If you take a regular 4-simplex whose vertices are unit quaternions, with the first equal to 1, can you prove the other 4 vertices generate a free group on 4 elements?

Hmm, I see that this puzzle has been solved by +Ian Agol and someone else on Mathoverflow:

http://mathoverflow.net/questions/204464/do-unit-quaternions-at-vertices-of-a-regular-4-simplex-one-being-1-generate-a

I don't understand the solution yet, because I don't know what a 'Bass-Serre tree' is... but I'll try to learn about this.  Math is infinite, there's always more to learn.

#geometry #4d  ___

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2015-05-01 05:22:14 (22 comments, 6 reshares, 60 +1s)Open 

The battle has begun

We're starting to fight global warming.  It's going to be a difficult war, and it's not clear we'll win.  But it's the most exciting, suspenseful story that we're all part of.

As the engineer Saul Griffith said:

"It's not like the Manhattan Project, it's like the whole of World War II, only with all the antagonists on the same side this time. It's damn near impossible, but it is necessary. And the world has to decide to do it."

A few promising signs:

1) On Wednesday morning, the governor of California set a goal of cutting carbon emissions by 40% below 1990 levels by 2030.  This matches the target set by the EU in October.   Both California and the EU are aiming to cut emissions 80% by 2050.  The governor said:

“We must demonstrate that reducingcarbon i... more »

The battle has begun

We're starting to fight global warming.  It's going to be a difficult war, and it's not clear we'll win.  But it's the most exciting, suspenseful story that we're all part of.

As the engineer Saul Griffith said:

"It's not like the Manhattan Project, it's like the whole of World War II, only with all the antagonists on the same side this time. It's damn near impossible, but it is necessary. And the world has to decide to do it."

A few promising signs:

1) On Wednesday morning, the governor of California set a goal of cutting carbon emissions by 40% below 1990 levels by 2030.  This matches the target set by the EU in October.   Both California and the EU are aiming to cut emissions 80% by 2050.  The governor said:

“We must demonstrate that reducing carbon is compatible with an abundant economy and human well-being.  Taking significant amounts of carbon out of our economy without harming its vibrancy is exactly the sort of challenge at which California excels. This is exciting, it is bold and it is absolutely necessary if we are to have any chance of stopping potentially catastrophic changes to our climate system."

At the national level, the US is dragging its heels.  But the states don't need to wait!  California, Oregon, Washington and British Columbia have signed a regional agreement to reduce carbon emissions, and the governor has signed separate accords with leaders in Mexico, China, Japan, Israel and Peru.  

2) Copenhagen has an ambitious plan to go carbon-neutral by 2025:

http://www.theguardian.com/environment/2013/apr/12/copenhagen-push-carbon-neutral-2025

As with California, the goal is not to save the world single-handedly, but to figure out how things can be done, so others can copy.

3) Pope Francis is getting serious about global warming.  He's already said he believes it's mostly manmade and that a Christian who does not protect God’s creation “is a Christian who does not care about the work of God”.  Now he's written an encyclical about it - the most significant sort of papal document.  This will come out in June.

I'm not a big fan of the Catholic church.  But it's important that everyone claiming to be a moral leader use their influence to get people to take this issue seriously, so I applaud this move.

Cardinal Peter Turkson, who is taking the lead on this, said increasing use of fossil fuels is disrupting Earth on an “almost unfathomable scale”, and says we need a “full conversion” of hearts and minds on this issue. 

http://www.theguardian.com/environment/2015/apr/28/vatican-climate-change-summit-to-highlight-moral-duty-for-action___

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