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John Baez
Works at Centre for Quantum Technologies
Attended Massachusetts Institute of Technology
Lives in Riverside, California
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John Baez

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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

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
Ian Petersen's profile photoBill Kemp's profile photoWilliam Rutiser's profile photoSebastian Fernandez Alberdi's profile photo
+James Salsman I don't know if it's a linear proportion, but it's dependent on the principal quantum number because the less probability there is for the proton and antiproton to be right on top of one another, the less chance they have to annihilate. And the bigger the principal quantum number is, the less probability there is right at the origin.
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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:

At this site, apparently made by someone named 'Archery', you can see kaleidocyles with 8, 10 and 12 regular tetrahedra.
Chance Chandler (Gryphon)'s profile photoMichael Butros's profile photoMarcelo Ariel Paez Jaime's profile photoCarlos Andrés Carrizo's profile photo
Ha! Earlier tonight I enlisted my friends' kids to make these for me with paper. The 6-tetrahedron version would turn one way, but not the other, probably as a result of the irregularity in our constructions. Ping +Chris Robinson+Dev Poling​
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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
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Thank you, +Dragana Biocanin, for your great work on behalf of these wonderful creatures, and for getting other people interested in them.
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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?,

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

#spnetwork arXiv:0908.1803 #blackhole
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Lovely nice art work done by computer .., and hope it will help developed other missing link science project ., all special affects computer graphic simulation ..., makes lots of money ...  
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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:

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+John Baez hah I see
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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:
Gizmo the Sane's profile photoSeb Paquet's profile photoKaren Peck's profile photoMichael Schuh (M.)'s profile photo
+Charles Wells - the gravitational field of a circular ring of matter is discussed here:

He gives a simple argument to show that a particle in the plane of the ring will always be attracted to the closest point in the ring.  This is intuitively obvious if the particle is outside the ring.  But it's also true if it's inside the ring.  In other words: instead of being attracted to the center of the ring, a particle inside the ring will be pulled out, towards the ring.

It would be fun to study this for a more general torus, so I bet someone has. 

On my previous post about the Heawood graph, someone mentioned that a spinning torus of matter can be gravitationally stable (though probably not when it's shaped exactly like a round torus).  This calculation sounds harder.
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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

200,000 km

29,800 km

7700 km

16.26 km

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....
On Tuesday night the clocks will stand still at 23:59:60 to keep our time in sync with the universe. But does our high-speed world demand a new solution?
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+Jungshik Shin - too bad not everyone at New Scientist went to your elementary school!
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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:

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),

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,

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:

#spnetwork #saturn #prometheus #pandora #rings doi:10.1038/ngeo2508
Zack Rowan's profile photoDenice Petit's profile photocynthia davidson's profile photoДмитрий Поляков's profile photo
Yes - though those are 'classical', not 'quantum', they'd typically be lumped under 'fluid mechanics' rather than 'classical mechanics'.
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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:

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:

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:'s_theorem_%28geometry%29
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  Nothing is simple (I'm sure it used to be less complicated, a long
time ago ...)

  Using Firefox under Mavericks, accessing the link with "?dl=0" produces an
irritating dialogue, then displays but fails to animate.  Chopping
"?dl=0" loses the
dialogue, but still fails to animate.  Substituting "?dl=1" downloads
the 1.3Mb file,
but instead of in the browser, opens it in Preview which fails to animate.

  "?dl=1", clicking on downloads button, then entering "Open" dialogue
to select
the browser seems to work --- on this configuration, anyway!  Sorry, everybody!
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John Baez

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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:


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:

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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John Baez

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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:

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:
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Stephen Jay Gould had a similar arrogance as Gell-Mann. I read Gell-Mann's The Quark and the Jaguar and I went to one of his talks. Didn't get as much out of Gell-Mann as I did with Gould and Feynman, but I did gain a little bit of something. I agree with Gould's idea of that you should always analyze the previous Generation's ideas in context. Get as much as you can from them as you can (either by standing on their shoulders or their toes) for they are right and wrong. "Give me a fruitful error any time, full of seeds, bursting with its own corrections. You can keep your sterile truth for yourself." -- Vilfredo Pareto For that reason I prefer reading Euler over Gauss, but appreciate both. "Read Euler! He is the master of us all." -- Laplace And +John Baez isn't bad either.
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John Baez

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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,

• 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,

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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+Steve Esterly - while we're on corrections, Alan Schoen pointed out that the double gyroid does not consist of two minimal surfaces; it's just two surfaces that show up naturally when you write the gyroid as

f(x,y,z) = 0

and take the surfaces

f(x,y,z) = +-c

for some small c.
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I'm a mathematical physicist.
  • Centre for Quantum Technologies
    Visiting Researcher, 2011 - present
  • U.C. Riverside
    Professor, 1989 - present
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I'm trying to get mathematicians and physicists to help save the planet.
I teach at U. C. Riverside and work on mathematical physics — which I interpret broadly as ‘math that could be of interest in physics, and physics that could be of interest in math’. I’ve spent a lot of time on quantum gravity and n-categories, but now I want to work on more practical things, too.

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

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

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