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John Baez
Works at Centre for Quantum Technologies
Attended Massachusetts Institute of Technology
Lives in Riverside, California
55,640 followers|39,031,199 views


John Baez

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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.,_Copenhagen
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I used to bike past this statue every morning and got distracted by it many times.
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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:

I'll be staying in downtown Warsaw in the Polonia Palace Hotel.  Anything good to do around there?
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Higher-dimensional rewiring would be a better punchline no? :)
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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.

The picture here, of terraced rice fields in Bali, is from here:
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+Alma Ionescu - wow, great news!  I'd heard of that lawsuit, but I didn't expect this outcome.
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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:

For more on Bell numbers, try this:

and for more on partitions of sets, try this:

As usual in math, the story only stops when you get tired!
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Any literature on operads and search trees?  I don't know anything quite like that.  Operads are a special case of Joyal's species, which are very fundamental to combinatorics, and I think you might enjoy the book Combinatorial Species and Tree-like Structures.  On the other hand, the recursive definitions of tree-like structures that show up in this book also appear in computer science as recursive definitions of datatypes.  But I don't know anyone who has put all the pieces together.  Maybe that's for you to do!
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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!)

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+James Salsman - Why don't you post something about the topic you're interested in discussing, and I'll say something there if I can?  (I don't really know if I have much to say about that topic, even though it's important.)
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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: )
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+John Baez : Following your website since 2010. Infact I recommended it to a lot of people. :) 
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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,

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:

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:

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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Ya know, even with the Internet, we're still resolving difference with mayhem. Especially in the Middle East -- which is tragic, as so much math&science emanated from there in the first place. Enantiodromia strikes again.
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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,

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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+Benoit Hudson - yes, we're pushing the world outside the ice age cycle that's been going on for over 20 million years, so more frozen regions are thawing.
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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,

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:
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+Philip Gibbs - no idea!  I can't see an obvious nontrivial way to "double cover" the set of k-element subsets of the n-element set, so there could be some special magic involving S5 and S6 at work in the Petersen/dodecahedron case.
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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

#spnetwork doi:10.1371/journal.pbio.1002168 #information #bigness  
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How interested are humans, really, in communicating with other intelligent species?

On a societal level, there's an astonishing absence of effort to find the limits, and go as far is as possible to go in communicating with chimps, dolphins, etc. There is no 'space race' level of funding.

But on an individual level, certain people dedicate their lives to interaction with one or very few individuals of another species -- and many people have pets with whom they have varying levels of social/emotional mutual understanding and connection. (I remember from anthro classes that pets are one of relatively few human cultural universals.)

Often, our interest in communication seems to be primarily about what other animals can give us or do for us, and people who are interested in mutual understanding for its own sake with other species are deemed 'weird' (cat ladies and the like).
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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:

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:

The picture is from here:

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+John Baez I can respect that:)
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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,

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

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+Samuel Mimram - that would be incredibly nice.   I recently discovered there's a surprisingly large amount of work taking varieties (roughly Lawvere theories with a specified set of generating morphisms) and finding the least number of relations required for a presentation.  But I don't know any homological interpretation.  Please try to find one!

There's a good introduction here:

• Walter Taylor, Equational logic,

A theory where only n relations are needed is called an n-based theory.
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I'm a mathematical physicist.
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I'm trying to get mathematicians and physicists to help save the planet.
I teach at U. C. Riverside and work on mathematical physics — which I interpret broadly as ‘math that could be of interest in physics, and physics that could be of interest in math’. I’ve spent a lot of time on quantum gravity and n-categories, but now I want to work on more practical things, too.

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

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

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