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

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/ 

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

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

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

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

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

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  

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/

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/

The best part: it's solar-powered

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.