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Shared Circles including John Baez

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The Google+ Collections of 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: 126

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2015-07-10 09:04:29 (126 comments, 25 reshares, 91 +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: 428

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2015-07-07 20:34:15 (24 comments, 428 reshares, 150 +1s)Open 

The best part: it's solar-powered

Most plusones: 329

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2015-08-19 10:34:44 (63 comments, 80 reshares, 329 +1s)Open 

Kaleidocycle

It's a bit surprising.  You can take 8 perfectly rigid regular tetrahedra and connect them along their edges to form a ring that you can turn inside-out!  

It's called a kaleidocycle, and you can actually do it with any even number of tetrahedra, as long as you have at least 8.  Fewer than 8, and they bump into each other.

You can see kaleidocycles with 8, 10, and 12 tetrahedra here:

http://intothecontinuum.tumblr.com/post/50873970770/an-even-number-of-at-least-8-regular-tetrahedra

and this is where I got my picture.  Who is the secret author of this post?  Why do people who create such great stuff want to hide behind pseudonyms?  I'm a showoff myself, I want people to know my name, so I have trouble understanding this, though obviously people are different. 

You can also make kaleidocycles out ofpaper:... more »

Latest 50 posts

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2015-08-30 08:12:30 (41 comments, 46 reshares, 247 +1s)Open 

Inverse cube force law

Here you see three planets.  The blue planet is moving around the Sun in a realistic way: it's going around an ellipse.

The other two are moving in and out just like the blue one, so they all stay on the same circle, which shrinks and expands.  But they're moving around this circle more slowly.   In fact the red planet isn't going around at all: it only goes in and out!

In 1687, Isaac Newton published his famous book, the Principia Mathematica.  And one thing he did here is figure out what extra force, besides gravity, would make it move like these weird other planets.

According to Newton, gravity obeys an inverse square force law.  For example, if you move a planet 3 times as far from the Sun, it feels a force that's 1/9 as strong.  Figuring this out and using this to explain the motion of planets is one reasonNewton... more »

Inverse cube force law

Here you see three planets.  The blue planet is moving around the Sun in a realistic way: it's going around an ellipse.

The other two are moving in and out just like the blue one, so they all stay on the same circle, which shrinks and expands.  But they're moving around this circle more slowly.   In fact the red planet isn't going around at all: it only goes in and out!

In 1687, Isaac Newton published his famous book, the Principia Mathematica.  And one thing he did here is figure out what extra force, besides gravity, would make it move like these weird other planets.

According to Newton, gravity obeys an inverse square force law.  For example, if you move a planet 3 times as far from the Sun, it feels a force that's 1/9 as strong.  Figuring this out and using this to explain the motion of planets is one reason Newton is so famous.

But he also showed that if you add an extra force obeying an inverse cube law, and adjust the angular momentum of your planet, you can make it planet move in and out just as it did before... but make it move around at a different rate!  

This goes to show that Newton had intelligence to spare.  He didn't just solve the important problems that made him famous, he also had lots of minor ideas that still required amazing cleverness.

And turns out this strange fact he discovered is just one of several weird and fascinating things about the inverse cube force law.  For example, the so-called 'centrifugal force' obeys an inverse cube law.  It's not a real force; it's sometimes called a 'fictious force'.  But you can still describe it using an inverse cube law, and this ultimately explains Newton's discovery.

And a quantum particle in an inverse cube force behaves in a truly bizarre way, which people have been struggling to understand for decades.  If you think you understand quantum mechanics, this may make you reconsider.

I recently learned a lot about these things while writing a chapter for a book called New Spaces for Mathematics and Physics.

Why?  Well, it's a long story.   I've decided that most of what I learned doesn't belong in this paper.  But I had to write it down before I forgot it!  So I wrote this blog article:

https://johncarlosbaez.wordpress.com/2015/08/30/the-inverse-cube-force-law/

Go there if you're curious for more. 

Puzzle: how many times does the blue planet orbit the Sun for each time the green one does?

The animation was made by ‘WillowW’ and placed on Wikicommons:

https://en.wikipedia.org/wiki/File:Newton_revolving_orbit_3rd_subharmonic_e0.6_240frames_smaller.gif___

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2015-08-28 04:21:33 (78 comments, 39 reshares, 218 +1s)Open 

Protonium

There's a game called Robocraft where you try to destroy your enemy's "protonium reactors".   These are imaginary devices powered by imaginary "protonium crystals".  But I think reality is cooler than any fantasy, so I'm not interested in that crap.  I'm interested in actual protonium!

Protonium is a blend of matter and antimatter.   It's a kind of exotic atom made of a proton and an antiproton.  A proton is positively charged, so its antiparticle, the antiproton, is negatively charged.  Opposite charges attract, so a proton and an antiproton can orbit each other.  That gives protonium.

A proton and an electron can also orbit each other, and that's called hydrogen.  But there are a few big differences between hydrogen and protonium.

First, hydrogen lasts forever, but protonium doesnot.  Whe... more »

Protonium

There's a game called Robocraft where you try to destroy your enemy's "protonium reactors".   These are imaginary devices powered by imaginary "protonium crystals".  But I think reality is cooler than any fantasy, so I'm not interested in that crap.  I'm interested in actual protonium!

Protonium is a blend of matter and antimatter.   It's a kind of exotic atom made of a proton and an antiproton.  A proton is positively charged, so its antiparticle, the antiproton, is negatively charged.  Opposite charges attract, so a proton and an antiproton can orbit each other.  That gives protonium.

A proton and an electron can also orbit each other, and that's called hydrogen.  But there are a few big differences between hydrogen and protonium.

First, hydrogen lasts forever, but protonium does not.  When they meet, the proton and antiproton annihilate each other.  How long does it take for this to happen?  It depends on how they're orbiting each other.  

In both hydrogen and protonium, various orbits are possible.  Particles are really waves, so these orbits are really different wave patterns, like different ways a trampoline can wiggle up and down.   These patterns are called orbitals.

Orbitals are labelled by numbers called quantum numbers.  If a hydrogen atom isn't spinning at all, it will be spherically symmetric.  Then you just need one number, cleverly called n, to say what its wave pattern looks like.  

The picture here shows the orbital with n = 30.   It has 30 wiggles as you go from the center outwards.  It's really 3-dimensional and round, but the picture shows a circular slice.  The height of the wave at some point says how likely you are to find the electron there.  So, the electron is most likely to be in the orange region.  It's very unlikely to be right in the middle, where the proton sits.

The same math works for protonium!  There's another big difference to keep in mind: the proton and antiproton have the same mass, so they both orbit each other.  But we can track just one of them, moving around their shared center of mass.  Then protonium works a lot like hydrogen.  You get spherically symmetric orbitals, one for each choice of that number called n.

So: if you can make protonium in a orbital where n = 30, it's unlikely for the two protons to meet each other.  Gradually your protonium will emit light and jump to orbitals with lower n, which have less energy.  And eventually the proton and antiproton will meet... and annihilate in a flash of light.

How long does this take?  For n = 30, about 1 microsecond.  And if you make protonium with n = 50, it lasts about 10 microseconds.  

That doesn't sound long, but in particle physics it counts as a pretty long time.  Probably not long enough to make protonium crystals, though!

Protonium was first made around 1989.  Around 2006 people made a lot of it using the Antiproton Decelerator at CERN.  This is just one of the many cool gadgets they keep near the Swiss-French border.  

You see, to create antimatter you need to smash particles at each other at almost the speed of light - so the antiparticles usually shoot out really fast.  It takes serious cleverness to slow them down and catch them without letting them bump into matter and annihilate.

Once they managed to do this, they caught the antiprotons in a Penning trap.  This holds charged particles using magnetic and electric fields.  Then they cooled the antiprotons - slowed them even more - by letting them interact with a cold gas of electrons.  Then they mixed in some protons.  And they got protonium - enough to really study it!  

The folks at CERN have also made antihydrogen, which is the antiparticle of an electron orbiting an antiproton.  And they've made antiprotonic helium, which is an antiproton orbiting a helium atom with one electron removed!   The antiproton acts a bit like the missing electron, except that it's 1836 times heavier, so it must orbit much closer to the helium nucleus.  

There are even wackier forms of matter in the works - or at least, in the dreams of theoretical physicists.  But that's another story for another day.

Here's the 2008 paper about protonium:

• N. Zurlo, M. Amoretti, C. Amsler, G. Bonomi, C. Carraro, C. L. Cesar, M. Charlton, M. Doser, A. Fontana, R. Funakoshi, P. Genova, R. S. Hayano, L. V. Jorgensen, A. Kellerbauer, V. Lagomarsino, R. Landua, E. Lodi Rizzini, M. Macri, N. Madsen, G. Manuzio, D. Mitchard, P. Montagna, L. G. Posada, H. Pruys, C. Regenfus, A. Rotondi, G. Testera, D. P. Van der Werf, A. Variola, L. Venturelli and Y. Yamazaki, Production of slow protonium in vacuum, Hyperfine Interactions 172 (2006), 97-105.  Available for free at http://arxiv.org/abs/0801.3193.

The child in me thinks it's really cool that there's an abbreviation for protonium, Pn, just like a normal element.

Puzzle 1: about how big is protonium in its n = 1 orbital, compared to hydrogen in its n = 1 orbital?  I've given you all the numbers you need to estimate this, though not all the necessary background in physics.  

In Puzzle 1 you're supposed to assume protonium in its n = 1 state is held together by the attraction of opposite charges, just like hydrogen.  But is that true?  If the proton and antiproton are too close, they'll interact a lot via the strong force!

Puzzle 2: The radius of hydrogen in its n = 1 state is about 50,000 femtometers, while the radius of a proton is about 1 femtometer.  Using your answer to Puzzle 1, compare the radius of protonium in its n = 1 orbital to the radius of a proton.

If protonium is a lot bigger than a proton, it's probably held together mostly in the same way as hydrogen: by the electromagnetic force.  

#spnetwork #arXiv :0801.3193 #protonium #particlePhysics___

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2015-08-27 05:33:17 (42 comments, 5 reshares, 71 +1s)Open 

A glitch in time

The magazine New Scientist does it again!  I picked up the June 27th edition, which has a story called 'Stop all the clocks'.   I saw a big graphic saying how far various things go in one second.  It looks sort of like this:

299,792 km
TRAVELLED BY A PHOTON IN A VACUUM

200,000 km
TRAVELLED BY THE SOLAR SYSTEM IN ORBIT AROUND GALACTIC CENTRE

29,800 km
TRAVELLED BY EARTH IN ORBIT AROUND THE SUN

7700 km
TRAVELLED BY THE INTERNATIONAL SPACE STATION IN ORBIT AROUND EARTH

16.26 km
TRAVELLED BY NEW HORIZONS, THE FASTEST SPACECRAFT EVER, NOW ARRIVING AT PLUTO

Everything looks very convincing when you see it written beautifully in a glossy magazine.  But then I thought: "Wait a minute!  They're sayingthe ... more »

A glitch in time

The magazine New Scientist does it again!  I picked up the June 27th edition, which has a story called 'Stop all the clocks'.   I saw a big graphic saying how far various things go in one second.  It looks sort of like this:

299,792 km
TRAVELLED BY A PHOTON IN A VACUUM

200,000 km
TRAVELLED BY THE SOLAR SYSTEM IN ORBIT AROUND GALACTIC CENTRE

29,800 km
TRAVELLED BY EARTH IN ORBIT AROUND THE SUN

7700 km
TRAVELLED BY THE INTERNATIONAL SPACE STATION IN ORBIT AROUND EARTH

16.26 km
TRAVELLED BY NEW HORIZONS, THE FASTEST SPACECRAFT EVER, NOW ARRIVING AT PLUTO

Everything looks very convincing when you see it written beautifully in a glossy magazine.  But then I thought: "Wait a minute!  They're saying the Sun is moving around the Galaxy at 2/3 the speed of light???"

I hope you realize that's obviously false.  For starters, our view of  the Milky Way would be enormously distorted by the effects of special relativity.  Next, gas in the Galaxy would be swirling around at relativistic speeds, causing enormous shock waves.  And third, the Galaxy would fly apart.

So: at least one of the numbers in this chart is incredibly far off.

It also seems odd that the International Space Station is moving at over 100 times the speed of New Horizons, "the fastest spacecraft ever".  

It's easy to look up the actual numbers online.  But:

Puzzle: just by thinking, what can you figure out about this chart?  Which numbers seem right, and which seem wrong?

Unfortunately the article is not free... but I'm including a link just in case someone has a subscription.  Have they fixed the online version yet?  Will they fix it after I make fun of them?

Nobody in the comment section has mentioned the mistakes....___

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2015-08-24 05:09:33 (29 comments, 41 reshares, 194 +1s)Open 

The collision of Prometheus and Pandora

Prometheus and Pandora were characters in a Greek myth, but now they are moons of Saturn.  They both orbit close to Saturn's F ring, zipping around this planet once every 15 hours or so.

Here you can see Prometheus carving strange slots in the F ring!  This ring is made of ice boulders, maybe up to 3 meters across.  Sometimes these chunks of ice form temporary clumps up to 10 kilometers in size.  At other times, these clumps get pulled apart.   Prometheus steals boulders from the F ring with its gravitational pull.  And each time it comes as close as possible to Saturn, it carves a new slot in the F ring.

Why does this happen?  It's complicated, and people keep learning more about it.  I'm certainly no expert!

People used to call Prometheus and Pandora shepherd moons. The idea ... more »

The collision of Prometheus and Pandora

Prometheus and Pandora were characters in a Greek myth, but now they are moons of Saturn.  They both orbit close to Saturn's F ring, zipping around this planet once every 15 hours or so.

Here you can see Prometheus carving strange slots in the F ring!  This ring is made of ice boulders, maybe up to 3 meters across.  Sometimes these chunks of ice form temporary clumps up to 10 kilometers in size.  At other times, these clumps get pulled apart.   Prometheus steals boulders from the F ring with its gravitational pull.  And each time it comes as close as possible to Saturn, it carves a new slot in the F ring.

Why does this happen?  It's complicated, and people keep learning more about it.  I'm certainly no expert!

People used to call Prometheus and Pandora shepherd moons.  The idea was that they help stabilize the F ring.  It's a cool idea.  The singer Enya even made an album with this title.

But more recent work casts doubt on this theory.  Last month Emily Lakdawalla of the Planetary Society wrote:

The most surprising thing I've learned: You know how Prometheus and Pandora are the F ring shepherds? Prometheus on the inside, and Pandora on the outside, herding the billions of tiny particles that make up the ring into place? It's not true. Pandora is not involved in controlling the F ring's tight shape.

The first paper I looked at was written by Jeff Cuzzi and seven coauthors: "Saturn's F Ring core: Calm in the midst of chaos." (Let's pause for a moment to appreciate the quality of that paper title, which is both interesting and accurate, not boring or silly.) The paper seeks to explain why the central core of Saturn's F ring is so consistently shaped, even though various things are constantly acting to perturb it. In particular, Prometheus periodically plunges into the F ring, drawing out dramatic streamers and fans. In fact, Prometheus and Pandora, far from behaving as shepherds, actually act to stir up the motions of particles in most of the region near the F ring. Furthermore, there are other bodies that Cassini has spotted in the F ring region whose behavior is so chaotic that it's been hard to follow them; these things have "violent collisional interactions with the F ring core," so, all in all, it's really difficult to explain why the core of the F ring generally looks the same as it has ever since the Voyagers passed by.

According to her account of some recent papers, the key is a kind of resonance.  Resonant frequencies shape Saturn's rings in many ways, but here the key is something called a 'Lindblad resonance'.

The orbit of Prometheus precesses.  In other words, its point of closest approach to Saturn keeps slowly moving around.  So, the period with which this moon orbits Saturn is slightly different than the period with which it moves in and out from Saturn.  A Lindblad resonance happens when a chunk of ice goes around Saturn exactly once each time Prometheus goes in and out!  Lakdawalla writes:

So: consider a moon and a ring particle orbiting Saturn. We don't care (for the moment) what the orbital periods of the moon and ring particles are; what we do care about is the "in-and-out" period of the ring particle in its orbit. You have a Lindblad resonance if, every time the moon passes by the ring particle, the ring particle happens to be on the same position in its in-and-out motion.

The full story is even more complicated than that - obviously, since it has to explain all the weird patterns in the picture here.  The F ring consists of several strands, and these even braid around each other.  But I'll let you read her blog for more:

http://www.planetary.org/blogs/emily-lakdawalla/2014/07010001-ringmoons-shepherds.html

What I really want to tell you is some other news: how the F ring was formed in the first place!

It's in an interesting place.  Any moon too close to Saturn would be broken up by tidal forces unless it was held together by forces stronger than gravity.  The Roche limit says how close is too close: it's 147,000 kilometers from the center of Saturn.  The F ring is 140,180 kilometers from the center of Saturn.  So it's just within the Roche limit.

That could be a clue.  But how did the F ring actually form?  A new paper says it was created by a collision between Prometheus and Pandora!   The authors write:

Saturn’s F ring is a narrow ring of icy particles, located 3,400 km beyond the outer edge of the main ring system. Enigmatically, the F ring is accompanied on either side by two small satellites, Prometheus and Pandora, which are called shepherd satellites. The inner regular satellites of giant planets are thought to form by the accretion of particles from an ancient massive ring and subsequent outward migration. However, the origin of a system consisting of a narrow ring and shepherd satellites remains poorly understood. Here we present N-body numerical simulations to show that a collision of two of the small satellites that are thought to accumulate near the main ring’s outer edge can produce a system similar to the F ring and its shepherd satellites. We find that if the two rubble-pile satellites have denser cores, such an impact results in only partial disruption of the satellites and the formation of a narrow ring of particles between two remnant satellites. Our simulations suggest that the seemingly unusual F ring system is a natural outcome at the final stage of the formation process of the ring–satellite system of giant planets.

If so, the F ring and these moons have been engaged in a drama for millions of years, starting with the very formation of Saturn's rings.   We missed the beginning of the show.

The paper is here, but it ain't free:

• Ryuki Hyodo and Keiji Ohtsuki , Saturn’s F ring and shepherd satellites a natural outcome of satellite system formation, Nature Geoscience (2015), http://www.nature.com/ngeo/journal/vaop/ncurrent/full/ngeo2508.html.

The other paper I mentioned is free:

• J. N. Cuzzi, A. D. Whizin, R. C. Hogan, A. R. Dobrovolskis, L. Dones, M. R. Showalter, J. E. Colwell and J. D. Scargle, Saturn’s F Ring core: Calm in the midst of chaos, http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20140011651.pdf

Abstract: The long-term stability of the narrow F Ring core has been hard to understand. Instead of acting as 'shepherds', Prometheus and Pandora together stir the vast preponderance of the region into a chaotic state, consistent with the orbits of newly discovered objects like S/2004 S 6. We show how a comb of very narrow radial locations of high stability in semimajor axis is embedded within this otherwise chaotic region. The stability of these semimajor axes relies fundamentally on the unusual combination of rapid apse precession and long synodic period which characterizes the region. This situation allows stable 'antiresonances' to fall on or very close to traditional Lindblad resonances which, under more common circumstances, are destabilizing. We present numerical integrations of tens of thousands of test particles over tens of thousands of Prometheus orbits that map out the effect. The stable antiresonance zones are most stable in a subset of the region where Prometheus first-order resonances are least cluttered by Pandora resonances. This region of optimum stability is paradoxically closer to Prometheus than a location more representative of 'torque balance', helping explain a longstanding paradox. One stable zone corresponds closely to the currently observed semimajor axis of the F Ring core. While the model helps explain the stability of the narrow F Ring core, it does not explain why the F Ring material all shares a common apse longitude; we speculate that collisional damping at the preferred semimajor axis (not included in the current simulations) may provide that final step. Essentially, we find that the F Ring core is not confined by a combination of Prometheus and Pandora, but a combination of Prometheus and precession.

Whew - complicated!  S/2004 S 6 is a weird little thing they've discovered in the F ring.  Nobody even knows if it's solid or just a clump of dust.  You can see it here:

https://en.wikipedia.org/wiki/S/2004_S_6

#spnetwork #saturn #prometheus #pandora #rings doi:10.1038/ngeo2508___

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2015-08-22 01:20:11 (18 comments, 14 reshares, 91 +1s)Open 

Colliding kaleidocycle

A while ago I showed you a kaleidocycle - a ring of 8 regular tetrahedra joined edge to edge, that you could turn inside out.  I said you could build one with any even number of tetrahedra that's at least 8.

Then somebody said he'd built one with just 6.

Here is Greg Egan's movie of what happens if you try a kaleidocycle with 6 regular tetrahedra.  They collide!   Very slightly.    So, it's not a true kaleidocyle - it's a collidocyle.

In other words: if these tetrahedra are completely rigid, they must pass through each other as they turn.  But if you made one out of paper, you might be able to force it to work, by bending the paper slightly.

Puzzle 1: give a mathematical proof that the tetrahedra here must intersect at some point, if they're completely rigid.
P... more »

Colliding kaleidocycle

A while ago I showed you a kaleidocycle - a ring of 8 regular tetrahedra joined edge to edge, that you could turn inside out.  I said you could build one with any even number of tetrahedra that's at least 8.

Then somebody said he'd built one with just 6.

Here is Greg Egan's movie of what happens if you try a kaleidocycle with 6 regular tetrahedra.  They collide!   Very slightly.    So, it's not a true kaleidocyle - it's a collidocyle.

In other words: if these tetrahedra are completely rigid, they must pass through each other as they turn.  But if you made one out of paper, you might be able to force it to work, by bending the paper slightly.

Puzzle 1: give a mathematical proof that the tetrahedra here must intersect at some point, if they're completely rigid.

Puzzle 2: what's the maximum fraction of the volume here that's contained in the intersection of at least two tetrahedra?   It looks like about 2% to me.

Greg Egan made this gif by adapting the Mathematica code at the website I showed you before:

http://intothecontinuum.tumblr.com/post/50873970770/an-even-number-of-at-least-8-regular-tetrahedra

At this site, apparently made by someone named 'Archery', you can see kaleidocyles with 8, 10 and 12 regular tetrahedra.___

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2015-08-19 10:34:44 (63 comments, 80 reshares, 329 +1s)Open 

Kaleidocycle

It's a bit surprising.  You can take 8 perfectly rigid regular tetrahedra and connect them along their edges to form a ring that you can turn inside-out!  

It's called a kaleidocycle, and you can actually do it with any even number of tetrahedra, as long as you have at least 8.  Fewer than 8, and they bump into each other.

You can see kaleidocycles with 8, 10, and 12 tetrahedra here:

http://intothecontinuum.tumblr.com/post/50873970770/an-even-number-of-at-least-8-regular-tetrahedra

and this is where I got my picture.  Who is the secret author of this post?  Why do people who create such great stuff want to hide behind pseudonyms?  I'm a showoff myself, I want people to know my name, so I have trouble understanding this, though obviously people are different. 

You can also make kaleidocycles out ofpaper:... more »

Kaleidocycle

It's a bit surprising.  You can take 8 perfectly rigid regular tetrahedra and connect them along their edges to form a ring that you can turn inside-out!  

It's called a kaleidocycle, and you can actually do it with any even number of tetrahedra, as long as you have at least 8.  Fewer than 8, and they bump into each other.

You can see kaleidocycles with 8, 10, and 12 tetrahedra here:

http://intothecontinuum.tumblr.com/post/50873970770/an-even-number-of-at-least-8-regular-tetrahedra

and this is where I got my picture.  Who is the secret author of this post?  Why do people who create such great stuff want to hide behind pseudonyms?  I'm a showoff myself, I want people to know my name, so I have trouble understanding this, though obviously people are different. 

You can also make kaleidocycles out of paper:

http://www.mathematische-basteleien.de/kaleidocycles.htm

and this website shows other kinds, too.  For example, there's a kaleidocycle that's a ring of 16 pyramids, all the same size and shape, that folds up into a perfect regular tetrahedron!  And there's another made of 16 pyramids, all some other size and shape, that folds into an octahedron!

What's all this good for?  I have no idea.  But it shows limitations of the Rigidity Theorem.   This theorem says if the faces of a convex polyhedron are made of a rigid material and the polyhedron edges are hinges, the polyhedron can't change shape at all: it's rigid.   The kaleidocycle show this isn't true for a polyhedron with a hole in it. 

Of course, having a hole is an extreme case of being nonconvex.  To be nonconvex, your polyhedron only needs to have a 'dent' in it.  And there are nonconvex polyhedra without a hole that aren't rigid!   The first of these was discovered by a guy named Connelly in 1978.  It has 18 triangular faces.

In 1997, Connelly, Sabitov and Waltz proved something really cool: the Bellows Conjecture.  This says that a polyhedron that's not rigid must keep the same volume as you flex it!

The famous mathematician Cauchy claimed to prove the Rigidity Theorem in 1813.  But there was a mistake in his proof. Nobody noticed it for a long time.   It seems mathematician named Steinitz spotted the mistake and fixed it in a 1928 paper.

Puzzle 1: what was the mistake?

Still, people often call this result "Cauchy's Theorem", which is really unfair, especially since Cauchy has other, better, theorems named after him.

Later the rigidity theorem was generalized to higher-dimensional convex polytopes.  (A polytope is a higher-dimensional version of a polyhedron, like a hypercube.) 

It's also been shown that 'generically' polyhedra are rigid, even if they're not convex.  In other words: if you take one that's not rigid, you can change its shape just a tiny bit and get one that's rigid. 

So, there are lots of variations on this theme: it's very flexible.

Puzzle 2: can you make higher-dimensional kaleidocycles out of higher-dimensional regular polytopes?  For example, a regular 5-simplex has 6 corners; if you attach 3 corners of one to 3 corners of another, and so on, maybe you can make a flexible ring.  Unfortunately this is in 5 dimensions - a 4-simplex has 5 corners, which doesn't sound so good, unless you leave one corner hanging free, in which case you can just take the movie here and imagine it as the 'bottom' of a 4d movie where each tetrahedron is the 'base' of a 4-simplex: sorta boring.

Puzzle 3: is a version of the Bellows Conjecture true in higher dimensions?

For more, check out these:

http://mathworld.wolfram.com/FlexiblePolyhedron.html

https://en.wikipedia.org/wiki/Cauchy's_theorem_%28geometry%29___

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2015-08-18 09:27:23 (6 comments, 3 reshares, 48 +1s)Open 

Circle +Dragana Biocanin and let sea turtles into your life.  You'll be a happier person. 

It may sound silly, but it's true.  I don't often recommend people to circle, but this is one.  Her pictures are great, and she manages to write about endangered species in a way that makes you happy rather than sad - because she explains how people are helping them thrive!  You can read news about world events, become bitter and glum... but one of her posts will help you recover.


Brazilian Scientists Save Sea Turtles from Extinction


While there are some people who want animals as trophies, there are others for whom the greatest reward is to care for them. The Tamar project was created in 1980 and aims to search for and subsequently save five species of sea turtles in Brazil.

The project is internationally renown as one of the most successful projects in marine conservation, serving as a model for other countries. In addition to conservation activities, Tamar conducts research on marine ecosystems to better understand the life cycle of sea turtles: pregnant turtles visit the Brazilian coast and leave their eggs buried in the sand. The puppies are born and begin a journey to the sea marked by difficulties, such as the risk of being trampled, threat of predators and disorientation caused by artificial lighting which can guide the turtles in the opposite direction to the sea.

Tamar aims to protect the feeding areas, spawning, growth and the other aspects of the turtle lifecycle in nine Brazilian states. A team of biologists, oceanographers and local fishermen, identify mothers who come to the beaches to spawn, collecting skin samples for genetic studies. If a nest is in a dangerous place, Tamar will transfer the eggs to safer areas or in incubators.

In visitation centers, guests are presented information about the biology of turtles, the threats to their survival and the importance of protecting them.

In April this year, 20 million babies sea turtles were released into the sea in 35 years of work on the Brazilian coast. The Tamar project is an example for other countries.


 By Pablo Mingot___Circle +Dragana Biocanin and let sea turtles into your life.  You'll be a happier person. 

It may sound silly, but it's true.  I don't often recommend people to circle, but this is one.  Her pictures are great, and she manages to write about endangered species in a way that makes you happy rather than sad - because she explains how people are helping them thrive!  You can read news about world events, become bitter and glum... but one of her posts will help you recover.

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2015-08-17 07:10:05 (73 comments, 42 reshares, 160 +1s)Open 

The amazing thing about the number 6

What's the saddest thing about being a mathematician?  Seeing worlds of soul-shattering beauty - but being unable to share this beauty with most people.

I actually used to have dreams about this.  In my dream, I would be hiking with friends.  Then, in the distance, I'd see some beautiful snow-capped mountains.  They were surreal, astoundingly tall - and not even very far away!   I wanted nothing more than to run over and start exploring them. 

But my friends weren't interested.  I had to either persuade them to go with me, stay with them and leave the beautiful mountains unexplored - or leave them and go climbing all alone.

At this point I'd always wake up, stuck in the dilemma.

Maybe this explains why I spend a lot of time explaining math here on G+.  The sad part is: you can take alot of ... more »

The amazing thing about the number 6

What's the saddest thing about being a mathematician?  Seeing worlds of soul-shattering beauty - but being unable to share this beauty with most people.

I actually used to have dreams about this.  In my dream, I would be hiking with friends.  Then, in the distance, I'd see some beautiful snow-capped mountains.  They were surreal, astoundingly tall - and not even very far away!   I wanted nothing more than to run over and start exploring them. 

But my friends weren't interested.  I had to either persuade them to go with me, stay with them and leave the beautiful mountains unexplored - or leave them and go climbing all alone.

At this point I'd always wake up, stuck in the dilemma.

Maybe this explains why I spend a lot of time explaining math here on G+.  The sad part is: you can take a lot of people a short distance toward the beautiful mountain peaks... or take a few people all the way up into the peaks.  You can't get everyone up to the top.

For example, I know the picture in this post is too complicated, and not flashy enough, for most people to enjoy.  But to me it's more beautiful than other pictures that will get a lot more +1s.

Any sort of mathematical gadget has a symmetry group.  The simplest sort of gadget is a finite set, like this:

{1,2,3,4}

The symmetries of a finite set are called permutations: they're the ways of rearranging its elements.   Here's a permutation of the set {1,2,3,4}:

1 |→ 4
2 |→ 1
3 |→ 3
4 |→ 2

The permutations form a 'group'.  This means we can 'multiply' two permutations, say f and g, by doing first f, and then g - and the result is another permutation, called fg.   Also, for every way of permuting things, there's some other permutation that undoes it.  For my example, this is

4 |→ 1
1 |→ 2
3 |→ 3
2 |→ 4

So, the permutations of a set form a group, called the permutation group of that set.

Now, I said every mathematical gadget has a symmetry group.  This is also is also true for permutation groups!   What's a symmetry of a permutation group? 

(This is where I may lose you.  This is where it gets interesting.  This is where I can see the mountain peaks and want to start climbing.)

A symmetry of a permutation group is a way of sending each permutation f to a new permutation F(f), obeying

F(f) F(g) = F(fg)

So, it's a way of permuting permutations - a way that is compatible with multiplying them. 

How do we get such a thing?  We can get it from a permutation of our set.  Any permutation lets us take any other permutation, and permute the numbers in the description of that permutation, and get a new permutation.  And that turns out to work!

Now for the cool part.  Every symmetry of the permutation group of the set {1,2,3,4} actually arises this way.

And this is also true for {1,2}, and {1,2,3}, and {1,2,3,4,5}, and so on.  In every case, all symmetries of the permutation group of the set come from permutations of the set.

Except for the exception.  The only exception is the number 6.

There are symmetries of the permutation group of the set {1,2,3,4,5,6} that don't come from permutations of this set! 

To understand this, you need to ponder the picture here, drawn by Greg Egan.

If you look carefully, there are 15 red dots and 15 blue ones.  Each red dot has a pair of the numbers 1,2,3,4,5,6 in it.  There are 15 ways to choose such a pair.  Each blue dot has all 6 numbers in it, partitioned into 3 pairs.  There are 15 ways to do this, too.  We draw an edge from a red dot to a blue dot if the pair of numbers in the red dot is one of the pairs in the blue dot. 

The resulting picture has a symmetry that switches the red dots and the blue dots!  And this symmetry is - somehow or other - the symmetry of the permutation group of {1,2,3,4,5,6} that's not a permutation of {1,2,3,4,5,6}.

"Somehow or other"?  That's not a very good explanation!  I've only shown that there's something special about the number 6, which gives a surprising symmetry.  It takes more work to see how this does the job.

For the full explanation, try my blog article:

http://blogs.ams.org/visualinsight/2015/08/15/tutte-coxeter-graph/

The trail gets a bit steeper at this point... but the view is great.

You may be wondering: So, what's this all good for?

And the answer is: nobody knows yet.  But this amazing fact about the number 6 is connected to many other amazing things in mathematics, like the group E8 and the Leech lattice, both of which show up in string theory.  I don't know if string theory is on the right track.  But I hope that someday, when we understand the universe better than we do now, these mysterious and beautiful mathematical structures will turn out to be important - not just curiosities, but part of why things are the way they are.

That is my hope, anyway.  So, I'm glad we have some people thinking about these things.  And besides, they're beautiful.___

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2015-08-15 00:48:08 (116 comments, 71 reshares, 281 +1s)Open 

How to make a black hole

+Brian Koberlein, one of the most consistently energetic and interesting people here on G+, recently wrote about how to make a black hole.

His recipe works like this:

INGREDIENTS: one small neutron star, one solar mass of hydrogen.

Take a neutron star 2 weighing solar masses.  Gradually add one solar mass of hydrogen gas, letting it fall to the surface of the neutron star.  Be careful: if you add too much too quickly, you'll create a huge nuclear explosion called a nova.  When your neutron star reaches 3 solar masses, it will collapse into a black hole.

This is the smallest type of black hole we see in nature.  The problem with this recipe is that we'd need to become at least a Kardashev Type II civilization, able to harness the power of an entire star, before we could carry it out.<... more »

How to make a black hole

+Brian Koberlein, one of the most consistently energetic and interesting people here on G+, recently wrote about how to make a black hole.

His recipe works like this:

INGREDIENTS: one small neutron star, one solar mass of hydrogen.

Take a neutron star 2 weighing solar masses.  Gradually add one solar mass of hydrogen gas, letting it fall to the surface of the neutron star.  Be careful: if you add too much too quickly, you'll create a huge nuclear explosion called a nova.  When your neutron star reaches 3 solar masses, it will collapse into a black hole.

This is the smallest type of black hole we see in nature.  The problem with this recipe is that we'd need to become at least a Kardashev Type II civilization, able to harness the power of an entire star, before we could carry it out.

My friend Louis Crane, a mathematician at the University of Kansas, has studied other ways to make a black hole.  It's slightly easier to make a smaller black hole - and perhaps more useful, since the Hawking radiation from a small black hole could be a good source of power.

Crane is interested in powering starships, but we could also use this power for anything else.  It's the ultimate power source: you drop matter into your black hole, and it gets turned into electromagnetic radiation!

Unfortunately, even smaller black holes are tough to make.  Say you want to make a black hole whose mass equals that of the Earth.  Then you need to crush the Earth down to the size of a marble.  The final stage of this crushing process would probably take care of itself: gravity would do the job!  But crushing a planet to half its original size is not easy.  I have no idea how to do it.

Luckily, to make power with Hawking radiation, it's best to make a much smaller black hole.  The smaller a black hole is, the more Hawking radiation it emits.  Louis Crane recommends making a black hole whose mass is a million tonnes.  This would put out 60,000 terawatts of Hawking radiation.  Right now human civilization uses only 20 terawatts of power.  So this is a healthy power source.

You have to be careful: the radiation emitted by such a black hole is incredibly intense.  And you have to keep feeding it.  You see, the smaller a black hole is, the more Hawking radiation it emits - and as it emits radiation, it shrinks!  Eventually it explodes in a blaze of glory: in the final second, it's about 1/100 as bright as the Sun.  To keep your black hole from exploding, you need to keep feeding it.  But for a black hole a million tons in mass, you don't need to rush: it will last about a century before it explodes if you don't feed it.  

Unfortunately, to make a black hole that weighs a million tonnes, you need to put a million tonnes of mass in a region 1/1000 times the diameter of a proton.

This is about the wavelength of a gamma ray.  So, if we could make gamma ray lasers, and focus them well enough, we could in theory put enough energy in a small enough region to create a million-ton black hole.  He says:

Since a nuclear laser can convert on the order of 1/1000 of its rest mass to radiation, we would need a lasing mass of about a gigatonne to produce the pulse. This should correspond to a mass of order 10 gigatonnes for the whole structure (the size of a small asteroid). Such a structure would be assembled in space near the sun by an army of robots and built out of space-based materials. It is not larger than some structures human beings have already built. The precision required to focus the collapsing electromagnetic wave would be of an order already possible using interferometric methods, but on a truly massive scale. This is clearly extremely ambitious, but we do not see it as impossible.

I'm not holding my breath, but with luck our civilization will last long enough, and do well enough, to try this someday.

For details, see:

• Louis Crane and Shawn Westmoreland, Are black hole starships possible?, http://arxiv.org/abs/0908.1803.

Here is Brian's post on how to build a black hole:

https://plus.google.com/u/0/+BrianKoberlein/posts/epaoFG9hsh4

#spnetwork arXiv:0908.1803 #blackhole___

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2015-08-14 00:37:12 (18 comments, 34 reshares, 160 +1s)Open 

The weight of history

Newton said he saw further because he stood on the shoulders of giants.  But this amazing sculpture illustrates how we're also weighed down and blinded by the prejudices of those who came before us - who were in turn blinded by their predecessors.

I usually post one photo at a time, because they get too small and most people won't bother to click and enlarge them.  I made an exception this time since it's hard to understand this sculpture from just one view.  Please click on the photos!  And it's really worthwhile looking at even more pictures, nice and big:

http://www.mymodernmet.com/profiles/blogs/do-ho-suh-karma

This sculpture is called Karma.  It was made by the Korean artist Do Ho Suh.  It looks infinitely tall, especially in the picture at right here.  But in fact it's 7 meters tall (23 feet), builtfrom 9... more »

The weight of history

Newton said he saw further because he stood on the shoulders of giants.  But this amazing sculpture illustrates how we're also weighed down and blinded by the prejudices of those who came before us - who were in turn blinded by their predecessors.

I usually post one photo at a time, because they get too small and most people won't bother to click and enlarge them.  I made an exception this time since it's hard to understand this sculpture from just one view.  Please click on the photos!  And it's really worthwhile looking at even more pictures, nice and big:

http://www.mymodernmet.com/profiles/blogs/do-ho-suh-karma

This sculpture is called Karma.  It was made by the Korean artist Do Ho Suh.  It looks infinitely tall, especially in the picture at right here.  But in fact it's 7 meters tall (23 feet), built from 98 figures of men, each one covering the eyes of the one below.  I think it looks taller because they shrink as you go further up, providing a false perspective that makes them seem to go on forever.

This sculpture can be seen at the Sydney and Walda Besthoff Sculpture Garden at the New Orleans Museum of Art, so if you're anywhere near New Orleans, check it out!  The photos were taken by by Lehmann Maupin, Alan Teo and Eric Harvey Brown, and CamWall.

As I mentioned, Newton said:

"If I have seen further, it is by standing on the shoulders of giants."

He said this in response to a letter to his competitor Robert Hooke, and some have interpreted it as a sarcastic poke at Hooke's slight build.  But in fact they were on good terms at the time, and came to dislike each other only later. 

Murray Gell-Mann, the theoretical physicist who came up with the idea of 'quarks', was definitely taking a poke at his competitors when he said

"If I have seen further, it is because I was surrounded by dwarves."

Ouch!  For more on the history, see:

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

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2015-08-12 00:58:49 (38 comments, 60 reshares, 208 +1s)Open 

The conic sections

Here you can see a plane moving though a cone.  Most of the time the plane intersects the cone in a curve.  These curves are called conic sections.  They have famous names and formulas:

Circles:   x² + y² = r² with r > 0

Ellipses:   ax² + by² = r² with a, b, r > 0

Hyperbolas:   ax² - by² = r² with a, b, r > 0

Parabolas:   y = ax²  with a > 0

I haven't given the most general formula for each kind of curve, but my formulas are enough to describe all possible shapes and sizes of these curves.  For example, if you have an upside-down parabola y = -2x² you can rotate it so it looks like y = 2x².  So, I say they have the same shape, and I don't bother listing both.

However, there are a few other cases that aren't on this list, whichare still extremely i... more »

The conic sections

Here you can see a plane moving though a cone.  Most of the time the plane intersects the cone in a curve.  These curves are called conic sections.  They have famous names and formulas:

Circles:   x² + y² = r² with r > 0

Ellipses:   ax² + by² = r² with a, b, r > 0

Hyperbolas:   ax² - by² = r² with a, b, r > 0

Parabolas:   y = ax²  with a > 0

I haven't given the most general formula for each kind of curve, but my formulas are enough to describe all possible shapes and sizes of these curves.  For example, if you have an upside-down parabola y = -2x² you can rotate it so it looks like y = 2x².  So, I say they have the same shape, and I don't bother listing both.

However, there are a few other cases that aren't on this list, which are still extremely important!   These are the other shapes you can define using equations of the form

ax² + bxy + cy² = 0

1) You can get two lines that cross.  This equation

x(y - mx) = 0

describes a vertical line together with a line of slope m. 

2) You can get a line:

x² = 0

3) You can get a point:

x² + y² = 0

Ordinary folks wouldn't call these 'curves'.  The last two special cases are especially upsetting!   But the famous mathematician Grothendieck figured out a way to improve algebraic geometry so that these cases are on the same footing as the rest. 

In particular, he made it really precise how

x² = 0

is different, in an important way, from

x = 0

The second one is an ordinary line, given by a linear equation.  The first one is a 'double line', the limit of two lines as they get closer and closer!  Watch the movie and see how we get to this 'double line', and you'll see what I mean.

People in algebraic geometry had already thought about 'double lines' and similar things, but Grothendieck's theory of schemes explained what these things really are.  Whatever it is, a double line is not just set of points in the plane - if we look at the set of points, there's no difference between the double line

x² = 0

and the single line

x = 0

The double line is something else - it's a 'scheme'.

But now it's time for breakfast, so I can't tell you what a 'scheme' actually is.  Instead, I'll just say this.  Grothendieck developed schemes, and more, as part of his attack on a very hard problem in number theory, called the Weil conjectures.  But his attack was a gentle one.  Instead of using brute force to crack this nut, he preferred to slowly 'soften' the problem by inventing new concepts.  Here's what he wrote about this:

The analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months—when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!

The gif in this post is from +Math Gif:

http://mathgifs.blogspot.sg/2014/09/the-conic-sections.html

#geometry  ___

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2015-08-11 02:16:59 (40 comments, 62 reshares, 244 +1s)Open 

Quantum technology from butterfly wings?

Some butterflies have shiny, vividly colored wings.  From different angles you see different colors.  This effect is called iridescence.  How does it work?

It turns out these butterfly wings are made of very fancy materials!  Light bounces around inside these materials in a tricky way.  Sunlight of different colors winds up reflecting off these materials in different directions.   

We're starting to understand the materials and make similar substances in the lab.   They're called photonic crystals.   They have amazing properties.   

Here at the +Centre for Quantum Technologies we have people studying exotic materials of many kinds.  Next door, there's a lab completely devoted to studying graphene: crystal sheets of carbon in which electrons can move as if they weremassless part... more »

Quantum technology from butterfly wings?

Some butterflies have shiny, vividly colored wings.  From different angles you see different colors.  This effect is called iridescence.  How does it work?

It turns out these butterfly wings are made of very fancy materials!  Light bounces around inside these materials in a tricky way.  Sunlight of different colors winds up reflecting off these materials in different directions.   

We're starting to understand the materials and make similar substances in the lab.   They're called photonic crystals.   They have amazing properties.   

Here at the +Centre for Quantum Technologies we have people studying exotic materials of many kinds.  Next door, there's a lab completely devoted to studying graphene: crystal sheets of carbon in which electrons can move as if they were massless particles!  Graphene has a lot of potential for building new technologies - that's why Singapore is pumping money into researching it.

Some physicists at MIT just showed that one of the materials in butterfly wings might act like a 3d form of graphene.  In graphene, electrons can only move easily in 2 directions.  In this new material, electrons could move in all 3 directions, acting as if they had no mass.

The pictures here show the microscopic structure of two materials found in butterfly wings.  

The picture at left is actually a sculpture made by the mathematical artist Bathsheba Grossman.  It's a piece of a gyroid - a surface with a very complicated shape, which repeats forever in 3 directions.  It's called a minimal surface because you can't shrink its area by tweaking it just a little.  It divides space into two regions.

The gyroid was discovered in 1970 by a mathematician, Alan Schoen.  It's a triply periodic minimal surface, meaning one that repeats itself in 3 different directions in space, like a crystal.  

Schoen was working for NASA, and his idea was to use the gyroid for building ultra-light, super-strong structures.  But that didn't happen.  Research doesn't move in predictable directions.

In 1983, people discovered that in some mixtures of oil and water, the oil naturally forms a gyroid.  The sheets of oil try to minimize their area, so it's not surprising that they form a minimal surface.  Something else makes this surface be a gyroid - I'm not sure what.
 
Butterfly wings are made of a hard material called chitin.  Around 2008, people discovered that the chitin in some iridescent butterfly wings is made in a gyroid pattern!  The spacing in this pattern is very small, about one wavelength of visible light.  This makes light move through this material in a complicated way, which depends on the light's color and the direction it's moving.

So: butterflies have naturally evolved a photonic crystal based on a gyroid!  

The universe is awesome, but it's not magic.  A mathematical pattern is beautiful if it's a simple solution to at least one simple problem. This is why beautiful patterns naturally bring themselves into existence: they're the simplest ways for things to happen.  Darwinian evolution helps out: it scans through trillions of possibilities and finds solutions to problems.  So, we should expect life to be packed with mathematically beautiful patterns... and it is.
 
The picture at right is a double gyroid, drawn by Gil Toombes.  This is actually two interlocking triply periodic minimal surfaces, shown in red and blue.  It turns out that while they're still growing, some butterflies have a double gyroid pattern in their wings.  This turns into a single gyroid when they grow up!  

The new research at MIT studied how an electron would move through a double gyroid pattern.  They calculated its dispersion relation: how the speed of the electron would depend on its energy and the direction it's moving.  

An ordinary particle moves faster if it has more energy.  But a massless particle, like a photon, moves at the same speed no matter what energy it has.  

The MIT team showed that an electron in a double gyroid pattern moves at a speed that doesn't depend much on its energy.  So, in some ways this electron acts like a massless particle.

But it's quite different than a photon.  It's actually more like a neutrino.  You see, unlike photons, electrons and neutrinos are spin-1/2 particles.  A massless spin-1/2 particle can have a built-in handedness, spinning in only one direction around its axis of motion.  Such a particle is called a Weyl spinor.  The MIT team showed that a electron moving through a double gyroid acts like a Weyl spinor.

Nobody has actually made electrons act like Weyl spinors.  The MIT team just found a way to do it.  Someone will actually make it happen, probably in less than a decade.  And later, someone will do amazing things with this ability.  I don't know what.  Maybe the butterflies know!

For more on gyroids in butterfly wings, see:

• K. Michielsen and D.G Stavenga, Gyroid cuticular structures in butterfly wing scales: biological photonic crystals, http://rsif.royalsocietypublishing.org/content/5/18/85

• Vinodkumar Saranathana et al, Structure, function, and self-assembly of single network gyroid (I4132) photonic crystals in butterfly wing scales, PNAS 107 (2010), 11676–11681.

The first one is free online!  For the new research at MIT, see:

• Ling Lu, Liang Fu, John D. Joannopoulos and Marin Soljačić, Weyl points and line nodes in gapless gyroid photonic crystals, http://arxiv.org/abs/1207.0478.

There's a lot of great math lurking here, most of which is too mind-blowing too explain quickly.  Let me just paraphrase the start of the paper, so at least experts can get the idea:

Two-dimensional (2d) electrons and photons at the energies and frequencies of Dirac points exhibit extraordinary features. As the best example, almost all the remarkable properties of graphene are tied to the massless Dirac fermions at its Fermi level. Topologically, Dirac cones are not only the critical points for 2d phase transitions but also the unique surface manifestation of a topologically gapped 3d bulk. In a similar way, it is expected that if a material could be found that exhibits a 3d linear dispersion relation, it would also display a wide range of interesting physics phenomena. The associated 3d linear point degeneracies are called “Weyl points”. In the past year, there have been a few studies of Weyl fermions in electronics. The associated Fermi-arc surface states, quantum Hall effect, novel transport properties and a realization of the Adler-Bell-Jackiw anomaly are also expected. However, no observation of Weyl points has been reported. Here, we present a theoretical discovery and detailed numerical investigation of frequency-isolated Weyl points in perturbed double-gyroid photonic crystals along with their complete phase diagrams and their topologically protected surface states.

Also a bit for the mathematicians:

Weyl points are topologically stable objects in the 3d Brillouin zone: they act as monopoles of Berry flux in momentum space, and hence are intimately related to the topological invariant known as the Chern number. The Chern number can be defined for a single bulk band or a set of bands, where the Chern numbers of the individual bands are summed, on any closed 2d surface in the 3d Brillouin zone. The difference of the Chern numbers defined on two surfaces, of all bands below the Weyl point frequencies, equals the sum of the chiralities of the Weyl points enclosed in between the two surfaces.

This is a mix of topology and physics jargon that may be hard for pure mathematicians to understand, but I'll be glad to translate if there's interest.

For starters, a ‘monopole of Berry flux in momentum space’ is a poetic way of talking about a twisted complex line bundle over the space of allowed energy-momenta of the electron in the double gyroid. We get a twist at every Weyl point, meaning a point where the dispersion relations look locally like those of a Weyl spinor when its energy-momentum is near zero. Near such a point, the dispersion relations are a Fourier-transformed version of the Weyl equation.

#spnetwork arXiv:1207.0478 #photonics #physics___

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2015-08-10 02:22:29 (22 comments, 13 reshares, 71 +1s)Open 

Cross your eyes and look at this

Here's a cool picture drawn by David Rowland.  Cross your eyes!   When the two images merge you'll get a nice 3d view of a doughnut with 7 hexagons drawn on it.  It works better if you make the image as big as you can.

Each hexagon touches all the others.  So, if the Earth were a doughnut divided into 7 countries this way, map-makers would need 7 colors of ink!  That's the most they could need for a doughnut-shaped Earth, though.

If the Earth were a 2-holed doughnut, we might need as many as 8 colors.  In general, for a doughnut with any number of holes, say g holes, the number is given by this wacky formula:

floor((7 + sqrt(1 + 48g)/2))

where "floor" means the largest integer less than or equal to the stuff in the parentheses. 

This formula was conjectured by Percy JohnHeawood... more »

Cross your eyes and look at this

Here's a cool picture drawn by David Rowland.  Cross your eyes!   When the two images merge you'll get a nice 3d view of a doughnut with 7 hexagons drawn on it.  It works better if you make the image as big as you can.

Each hexagon touches all the others.  So, if the Earth were a doughnut divided into 7 countries this way, map-makers would need 7 colors of ink!  That's the most they could need for a doughnut-shaped Earth, though.

If the Earth were a 2-holed doughnut, we might need as many as 8 colors.  In general, for a doughnut with any number of holes, say g holes, the number is given by this wacky formula:

floor((7 + sqrt(1 + 48g)/2))

where "floor" means the largest integer less than or equal to the stuff in the parentheses. 

This formula was conjectured by Percy John Heawood in 1890.  The map in the picture here is called the Heawood graph, and the conjecture is called the Heawood conjecture.  

The Heawood conjecture was proven by Gerhard Ringel and J. W. T. Youngs in 1968... except for the case g = 0, the case of a sphere, with no holes.   That case, the 4-color conjecture, turned out to be much harder!  But that's another story for another day!

For more, try:

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

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2015-08-08 06:25:03 (9 comments, 2 reshares, 74 +1s)Open 

Prambanan

This is the largest Hindu temple in Java.  I was overwhelmed by the massive structures - look closely at the bottom and you'll see tiny people.

How was it built?   It all started when a prince named Bandung Bondowoso fell in love with a beautiful princess named Rara Jonggrang.  He proposed marriage.  She rejected him because he had killed her father, the cruel man-eating giant King Boko.  But Bandung Bondowoso insisted.  Finally Rara Jonggrang relented and agreed to marry him... but only on one condition: he had to build her a thousand temples in one night.

The Prince entered into meditation and conjured up a multitude of demons from the earth.  With their help, he succeeded in building 999 temples.  But just as he was about to complete his task, the princess woke her palace maids and ordered the women of the village to begin pounding rice andset a fi... more »

Prambanan

This is the largest Hindu temple in Java.  I was overwhelmed by the massive structures - look closely at the bottom and you'll see tiny people.

How was it built?   It all started when a prince named Bandung Bondowoso fell in love with a beautiful princess named Rara Jonggrang.  He proposed marriage.  She rejected him because he had killed her father, the cruel man-eating giant King Boko.  But Bandung Bondowoso insisted.  Finally Rara Jonggrang relented and agreed to marry him... but only on one condition: he had to build her a thousand temples in one night.

The Prince entered into meditation and conjured up a multitude of demons from the earth.  With their help, he succeeded in building 999 temples.  But just as he was about to complete his task, the princess woke her palace maids and ordered the women of the village to begin pounding rice and set a fire in the east of the temple, to make the prince and his demons believe that the sun was about to rise.

As the cocks began to crow, fooled by the light and the typical sounds of morning, the prince's demon helpers fled back into the ground. The prince was furious!  In revenge he cursed Rara Jonggrang - and turned her to stone. She became the last and most beautiful of the thousand temples.

Well, okay - this is how the local Javanese peasants said these temples were built, long after they were actually built in 850 AD, abandoned to the jungle after a volcanic eruption in 930, and partially destroyed by a major earthquake in the 1600s.  In reality they were built by the Medang Kingdom, a Hindu–Buddhist kingdom that thrived from 732–1006 AD, whose sphere of influence reached all the way to Angkor Wat in Cambodia.

Speaking of "demons from the earth": in California we have earthquakes, but in Java you really sense the power of shifting tectonic plates!  As recently as February last year, Borobudur and Prambanan were closed due to volcanic ashes from the eruption of a volcano called Kelud, 200 kilometers away.  Four years earlier, a much closer volcano called Merapi erupted.   You can see this one towering over Yogyakarta on days when the clouds and smog aren't too thick.  But the ashes from that eruption missed Prambanan, and only hit Borobudur.  Right now yet another volcano on Java, Raung, is disrupting air travel to Bali. 

For more, see:

https://en.wikipedia.org/wiki/Prambanan
https://en.wikipedia.org/wiki/Medang_Kingdom
https://en.wikipedia.org/wiki/Kelud
https://en.wikipedia.org/wiki/Merapi
https://en.wikipedia.org/wiki/Raung

#java  ___

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2015-08-07 15:45:25 (70 comments, 50 reshares, 175 +1s)Open 

As we remember Hiroshima, let's not forget that the US strategy of mass slaughter of Japanese civilians didn't start there.  70 years ago on March 10th, even more people were killed in the firebombing of Tokyo - a city where most houses were made of paper.

279 planes flew over the city and dropped 1,665 tons of bombs.  Most were 500-pound cluster bombs, each one releasing 38 incendiary bomblets at an altitude of about 2000 feet.  These bomblets punched through the roofs of people's houses or landed on the ground and ignited 3–5 seconds later, throwing out jets of flaming, sticky napalm. 

The planes also dropped 100-pound jelled-gasoline and white phosphorus bombs that ignited upon impact.  The city's fire departments were overwhelmed, and the individual fires started by the bombs joined to create a huge conflagration that destroyed 16 square miles of the city. Over 10... more »

As we remember Hiroshima, let's not forget that the US strategy of mass slaughter of Japanese civilians didn't start there.  70 years ago on March 10th, even more people were killed in the firebombing of Tokyo - a city where most houses were made of paper.

279 planes flew over the city and dropped 1,665 tons of bombs.  Most were 500-pound cluster bombs, each one releasing 38 incendiary bomblets at an altitude of about 2000 feet.  These bomblets punched through the roofs of people's houses or landed on the ground and ignited 3–5 seconds later, throwing out jets of flaming, sticky napalm. 

The planes also dropped 100-pound jelled-gasoline and white phosphorus bombs that ignited upon impact.  The city's fire departments were overwhelmed, and the individual fires started by the bombs joined to create a huge conflagration that destroyed 16 square miles of the city.  Over 100,000 people died - nobody knows how many, and both the Japanese and Americans had reasons to underestimate the casualties.

General Curtis LeMay, who led this attack, said:

“Killing Japanese didn’t bother me very much at that time... I suppose if I had lost the war, I would have been tried as a war criminal."

Joe O'Donnell, a marine sent in after the war to document the effects of the bombing, wrote:

“The people I met, the suffering I witnessed, and the scenes of incredible devastation taken by my camera caused me to question every belief I had previously held about my so-called enemies.”

The picture shows the charred corpse of a woman in Tokyo who was carrying a child on her back.  In this style of war, cities of people, many perfectly innocent, are treated like rats to be exterminated.  Martin Middlebrook captured the horror of this in Hamburg, one of the German cities firebombed by the US:

"A thermal column of wind generated heat in excess of 1,400 degrees Fahrenheit, melting trolley windows and the asphalt in streets, the wind uprooting trees. When people crossed a street, their feet stuck in the melted asphalt; they tried to extricate themselves with their hands, only to find them stuck as well. They remained on all fours screaming. Small children lay like "fried eels" on the pavement. The firestorm sucked all the oxygen out of the city."

Let's try to avoid this, eh?  It's not necessarily easy, and I'm not saying I know how, but let's try to avoid making our world into a hell. 

References:

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

https://www.jacobinmag.com/2015/03/tokyo-firebombing-world-war-ii/

https://en.wikipedia.org/wiki/The_Battle_of_Hamburg_(book)___

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2015-08-05 11:24:09 (9 comments, 2 reshares, 77 +1s)Open 

Climbing the mandala

This is Lisa in Borobudur - the biggest temple in Java, and the biggest Buddhist temple in the world.  It's actually a giant 3-dimensional mandala - a diagram of Buddhist cosmology that serves as an aid to meditation.  

At the bottom is the outer square wall.   Then there are 5 nested square levels, each a bit smaller than the one below, connected by stairs.  Then there are 3 round levels, and finally huge circular dome on top.  

The bottom represents Kamadhatu: the world of desires.  The idea is that most sentient beings live here.  

The 5 square levels represent Rupadhatu: the world of forms, or patterns.  Beings who have burnt out their desire for continued existence live here: they see forms, but aren't drawn to them.  

The 3 round levels on top represent Arupadhatu,the formle... more »

Climbing the mandala

This is Lisa in Borobudur - the biggest temple in Java, and the biggest Buddhist temple in the world.  It's actually a giant 3-dimensional mandala - a diagram of Buddhist cosmology that serves as an aid to meditation.  

At the bottom is the outer square wall.   Then there are 5 nested square levels, each a bit smaller than the one below, connected by stairs.  Then there are 3 round levels, and finally huge circular dome on top.  

The bottom represents Kamadhatu: the world of desires.  The idea is that most sentient beings live here.  

The 5 square levels represent Rupadhatu: the world of forms, or patterns.  Beings who have burnt out their desire for continued existence live here: they see forms, but aren't drawn to them.  

The 3 round levels on top represent Arupadhatu, the formless world.  The idea is that buddhas live here: they go beyond form and experience reality at its most fundamental level, the formless ocean of nirvana.  

(You don't have to believe this stuff to find it interesting.  I disagree with the goal of completely transcending desire and forms, though I have a certain sympathy with it.)

The idea is that pilgrims should go up one level at a time, walking all the way around and meditating on the wall carvings and statues before climbing the steep stairs to the next level.  I think Lisa here is in the world of forms. 

To see the mandala design of Borobdur, go here:

https://upload.wikimedia.org/wikipedia/commons/2/27/Borobudur_Mandala.svg

To see a cross-section, go here:

https://upload.wikimedia.org/wikipedia/commons/4/49/Borobudur_Cross_Section_en.svg

#java___

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2015-08-02 12:10:51 (21 comments, 15 reshares, 77 +1s)Open 

If the Earth were doughnut-shaped...

... would it have to collapse under its own gravity, or could it keep from collapsing if it were spinning fast enough? 

That's a fun question, but the question for today is this:

... how many colors would map-makers need, to be sure that no countries sharing a border were drawn in the same color?

For our spherical Earth the answer is 4.  But for the surface of a doughnut the answer is 7 - and curiously, this is much easier to prove.  

It's even easier to prove you need at least 7 colors.  Just look at the picture here!  Imagine the square is 'wrapped around' so the left edge touches the right edge, and the top touches the bottom. Then you've got the surface of a doughnut divided into 7 hexagons. 

And note: each hexagon shares a boundary with each other hexagon!   So, youneed at ... more »

If the Earth were doughnut-shaped...

... would it have to collapse under its own gravity, or could it keep from collapsing if it were spinning fast enough? 

That's a fun question, but the question for today is this:

... how many colors would map-makers need, to be sure that no countries sharing a border were drawn in the same color?

For our spherical Earth the answer is 4.  But for the surface of a doughnut the answer is 7 - and curiously, this is much easier to prove.  

It's even easier to prove you need at least 7 colors.  Just look at the picture here!  Imagine the square is 'wrapped around' so the left edge touches the right edge, and the top touches the bottom. Then you've got the surface of a doughnut divided into 7 hexagons. 

And note: each hexagon shares a boundary with each other hexagon!   So, you need at least 7 colors to be sure no two hexagons that share a boundary are drawn in the same color.

This particular map is called the Heawood graph.  A graph is a collection of vertices and edges.  This graph has 14 vertices and 21 edges, and it's famous.  It was discovered around 1890 by Percy John Heawood, who was studying the map-coloring question.  

The Heawood graph is also related to something called the Fano plane, a gadget with 7 points and 7 lines.   These 14 points and lines  correspond to the 14 vertices of the Fano plane!  I explain how this works in my blog article here:

http://blogs.ams.org/visualinsight/2015/08/01/heawood-graph/

By the way, the picture here was drawn by +David Eppstein, who kindly put it into the public domain. ___

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2015-07-31 13:27:20 (9 comments, 0 reshares, 56 +1s)Open 

Today's adventures in Java

These are Indonesian reporters interviewing my wife Lisa near the sultan's palace in Yogyakarta.  She had just won a race where she had to walk 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 spunar... more »

Today's adventures in Java

These are Indonesian reporters interviewing my wife Lisa near the sultan's palace in Yogyakarta.  She had just won a race where she had to walk 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  ___

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2015-07-30 00:34:58 (38 comments, 45 reshares, 178 +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 (41 comments, 30 reshares, 238 +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  ___

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2015-07-23 23:45:39 (38 comments, 13 reshares, 86 +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___

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2015-07-21 12:37:37 (15 comments, 22 reshares, 219 +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___

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2015-07-20 09:43:18 (75 comments, 41 reshares, 127 +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

The animated gif was made by Qashqaiilove:

https://commons.wikimedia.org/wiki/File:Neutron_QCD_Animation.gif

#spnetworks arXiv:1507.03414 #pentaquark  ___

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2015-07-19 15:02:07 (18 comments, 44 reshares, 125 +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 I was shocked by 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 I was shocked by 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/ ___

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2015-07-18 09:02:23 (27 comments, 27 reshares, 167 +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___

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2015-07-15 07:59:05 (65 comments, 22 reshares, 73 +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 (126 comments, 25 reshares, 91 +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 (59 comments, 41 reshares, 110 +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___

posted image

2015-07-07 20:34:15 (24 comments, 428 reshares, 150 +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, 162 reshares, 214 +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.___

posted image

2015-07-05 08:55:30 (36 comments, 9 reshares, 100 +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, 60 +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, 113 +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, 21 reshares, 67 +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, 67 +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, 155 +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, 41 reshares, 149 +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, 6 reshares, 55 +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 (82 comments, 78 reshares, 125 +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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