On the 4th of July, a NASA spacecraft named Juno will try to start orbiting Jupiter. It has traveled for 5 years and 2.8 billion kilometers to get there. This is going to be exciting!
Juno will try to aim its main engine towards the Sun, turn it on for 35 minutes, and slow down to 58 kilometers per second, so it can be captured by Jupiter's gravitational field. Says the lead scientist:
There's a mixture of tension and anxiety because this is such a critical maneuver and everything is riding on it. We have to get into orbit. The rocket motor has to burn at the right time, in the right direction, for just the right amount of time.
With luck, Juno will enter a highly eccentric polar orbit, and make 37 orbits lasting 14 days each. Each time it will dive down to just 4000 kilometers above Jupiter's cloud tops, closer than we've ever come! Each time it will shoot back up to a height of 2.7 million kilometers. It will map Jupiter using many instruments. The first dive is scheduled for August.
Juno will gradually be damaged by Jupiter's intense radiation, even though the main computer is encased in a 200-kilogram titanium box. After its last orbit, it will deliberately plunge to its death — so that it has no chance of contaminating the oceans of Europa.
Juno has already entered Jupiter's magnetosphere - the region of space dominated by Jupiter's powerful magnetic field. You can hear it here:
For details of Juno's trajectory, go here:
The Jupiter orbit insertion should begin at 03:18 July 5th UTC, which
is 20:18 on the 4th of July in California.
July 3, 2016
This is a view of Barth's decic surface drawn by Abdelaziz Nait Merzouk. It's a frightening shape with 345 cone-shaped singularities — the most possible for a surface described by a polynomial of degree 10.
And yet, despite its nightmarish complexity, this surface is highly symmetrical. It has the same symmetries as a regular icosahedron!
For more views of this surface, go here:
A while back I wrote a story about infinity on Google+. It featured a character who was recruited by the US government to fight in the War on Chaos. His mission was to explore larger and larger infinities.
You can see that story in my bigness collection on G+: lots of posts, each one its own little chapter.
But I keep wanting to talk about infinity — it's endlessly interesting! I keep learning more about it. Some posts here by +Refurio Anachro re-ignited my desire to write about it, and now I have. Here's the first of three articles:
The picture above shows some of the first ones we'll meet on our trip. Omega, written \(\omega\), is the first infinite ordinal: $$ \omega = \{0,1,2,3,4,5,6,7,8,9,...\} $$ Each turn of the spiral here takes you to a higher power of omega, and if you go around infinitely many times, you reach omega to the omegath power. There are many ways to visualize this ordinal, and I explain a few.
But my road trip will take you much further than that!
In this first episode, we reach an ordinal called 'epsilon nought', first discovered by Cantor. In the second episode we'll go up the Feferman–Schütte ordinal. In the third we'll reach the small Veblen ordinal and even catch a glimpse of the large Veblen ordinal.
All these are countable ordinals, and you can write computer
programs to calculate with them, so I consider them just as concrete
as the square root of 2. And yet, they're quite mind-blowing.
July 5, 2016
The 'Jupiter orbit insertion' went flawlessly, and I had fun watching it live on NASA TV. The most risky moments occurred from 11:18 to 11:53 am, here in Singapore. Success wasn't guaranteed, and indeed NASA cleverly played up the dangers, to get people interested, and to have an excuse if things didn't work.
What could possibly go wrong?
Lots."I'm confident it's going to work," Scott Bolton, Juno's principal investigator, said before the announcement Monday night that the spacecraft had arrived, "but I'll be happy when it's over and we're in orbit."
Some of the ways that this could turn into a bad day:
Juno blows up. In August 1993, NASA's instrument-packed Mars Observer spacecraft vanished. An inquiry concluded that a fuel leak caused the spacecraft to spin quickly and fall out of communication. While Juno's setup is different, there is always a chance of an explosion with rocket fuel.
The engine doesn't fire at all. The Japanese probe Akatsuki was all set to arrive at Venus in December 2010, but its engine didn't fire, and Akatsuki sailed right past Venus. Last year, Akatsuki crossed paths with Venus again, and this time, using smaller thrusters, it was able to enter orbit.
It crashes into something. Jupiter does not possess the majestic rings of Saturn, but it does have a thin of ring of debris orbiting it. Juno will pass through a region that appears clear, but that does not mean it actually is. Even a dust particle could cause significant damage, as Juno will be moving at a speed of 132,000 miles per hour relative to Jupiter.
It flies too close to Jupiter and is ripped to pieces. In one of NASA's most embarrassing failures, the Mars Climate Orbiter spacecraft, was lost in 1999 because of a mix-up between English and metric units. Climate Orbiter went far deeper into Mars. atmosphere than planned. On its first orbit, Juno is to pass within 2,900 miles of Jupiter's cloud tops, so a miscalculation could be catastrophic.
The computer crashes. On July 4 last year, the mission controllers of the New Horizons spacecraft that was about to fly by Pluto experienced some nervous moments when the spacecraft stopped talking to them. The computer on New Horizons crashed while trying to interpret some new commands and compressing some images it had taken, the electronic equivalent of walking while chewing gum.
The controllers put New Horizons back in working order within a few days, and the flyby occurred without a hitch. For Juno, the scientific instruments have been turned off for its arrival at Jupiter. "We turn off everything that is not necessary for making the event work," said Dr. Levin, the project scientist. "This is very important to get right, so you don't do anything extra."
The intense barrage of radiation at Jupiter could knock out Juno's computer, even though it is shielded in a titanium vault. Usually, when there is a glitch, a spacecraft goes into "safe mode" to await new instructions from Earth, but in this case, that would be too late to save Juno. The spacecraft has been programmed to automatically restart the engine to allow it to enter orbit.
"If that doesn't go just right, we fly past Jupiter, and of course, that's not desirable," Dr. Bolton said.
Some infinities are countable, like the number of integers. Others are uncountable, like the number of points on a line.
Uncountable infinities are hard to fully comprehend. For example, even if you think an infinity is uncountable, someone else may consider it countable! That's roughly what the Löwenheim–Skolem theorem says.
How is this possible?
Ultimately, it's because there are only a countable number of sentences in any language with finitely many letters. So, no matter how much you talk, you can never convince me that you're talking about something uncountable!
Now, if we take a really hard-ass attitude, we have to admit we can never actually write infinitely many sentences. So even countable infinities remain outside our grasp. However, we come "as close as we want", in the sense that we can keep counting $$ 0, 1, 2, 3, 4, \dots $$
and nothing seems to stop us. So, while we never actually reach the countably infinite, it's pretty easy to imagine and work with.
Thus, my favorite infinities are the countable ordinals — in particular, the computable ones. You can learn to do arithmetic with them. You can learn to visualize them just as vividly as the set of all natural numbers, which is the first countable ordinal: $$ \omega = \{0,1,2,3,4,5,6,7,8,9,\dots \} $$ For example, $$ \omega+1 = \{0,1,2,3,4,5,6,7,8,9,\dots, \omega\} $$ But as you keep trying to understand larger and larger countable ordinals, strange things happen. You discover that you're fighting your own mind.
As soon as you see a systematic way to generate a sequence of larger and larger countable ordinals, you know this sequence has a limit that's larger then all of those! And this opens the door to even larger ones....
So, this journey feels a bit like trying to outrace your car's own shadow as you drive away from the sunset: the faster you drive, the faster it shoots ahead of you. You'll never win.
On the other hand, you never need to lose. You only lose when you get tired.
And that's what I love: it becomes so obvious that the struggle to understand the infinite is a kind of mind game. But it's a game that allows clear rules and well-defined outcomes, not a disorganized mess.
In this post:
I'll take you on a tour of countable ordinals up to the Feferman–Schütte ordinal. Hop in and take a ride!
And if you don't know the Löwenheim–Skolem theorem, you've gotta learn about it. It's one of the big surprises of early 20th-century logic:
The pink and the hearts, by the way, are there to see if you
become uncomfortable. They are 'girly'.
July 7, 2016
Wow! These plastic cylinders look round — but in the mirror they look diamond-shaped. If you turn them around, they look diamond-shaped - but in the mirror they look round!
This video was made by Kokichi Sugihara, an engineer at Meiji University in Tokyo. How did he do it???
To answer this question, we should "science the hell out of it", as Matt Damon said in The Martian. Figure out how objects change appearance when you look at them in a mirror... and design an object that does this! So David Richeson, a math professor at Dickinson College in Pennsylvania, scienced the hell out of it:
The basic idea is this. The top rim of this object is not flat. More precisely, it's not horizontal: it curves up and down! This affects how it looks. If you're looking down on this object, you can make part of the top look farther away by having it be lower.
But a mirror reflects front and back. So in the mirror, part of the top looks closer if it's lower.
By cleverly taking advantage of this, we can make this object look round, but diamond-shaped in the mirror.
And if we turn it around, this effect is reversed!
Here's a bit more of the math. David Richeson gives the details, so I'll try to present just the basic idea.
Suppose you're making a video. Suppose you're looking down at an angle of 45 degrees, just as in this video. Suppose you're videotaping an object that's fairly far away.
Think about one pixel of the object's image on your camera's viewscreen.
Its height on your viewscreen depends on two things. It depends on how far up that piece of the object actually is. But it also depends on how far back that piece of the object is: how far away it is from your camera. Things farther away give higher pixels on your viewscreen.
There's a simple formula for how this works:
(It's only this simple when you're looking down at an angle of 45 degrees and the thing you're videotaping is fairly far away.)
But what if we're looking in a mirror? You may think a mirror reverses left and right, but that's wrong: it reverses front and back. So we basically get
So, you just need to craft an object for which
give two different curves: one round and one a diamond!
Now for some puzzles:
Puzzle 1. All that sounds fine: by cleverly adjusting the top rim of the object we can make it look different in a mirror. But look at the bottom of the object! What's going on there? How do you explain that?
Puzzle 2. Sometimes I know the answers to the puzzles I'm posing. Sometimes I don't. Do I know the answer to Puzzle 1, or not?
Puzzle 3. Same question for Puzzle 2.
Finally, I should admit that I simplified the formula for the mirror image pixel height. Actually we have
and thus
In other words, I ignored a constant. This constant is why the whole mirror image looks higher on your viewscreen than the original object! For more, see:
I'm back in Singapore, the land of explosive cuisine. This is the menu from our favorite Chinese restaurant. It's on Southbridge Road across from the Sri Mariamman Temple — a popular Hindu temple where they do firewalking on the holiday called Theemithi. Maybe they do it to cool down after eating here.
I hadn't known it was called the Explosion Pot Barbecue. They sell excellent barbecued fish, roast skewers of lamb with cumin, roast chives, dumplings, and other Szechuan delights. The food is a bit spicy, but I haven't seen any exploding pots, so this may be a mistranslation of something that makes more sense in Chinese.
As usual, I'm working at the Centre for Quantum Technologies and Lisa is teaching at the philosophy department at NUS. You can see her in the background ordering our food.
Meanwhile, my student Blake Pollard is in a small town in the hills of Yunnan Province in southern China, helping teach some local students science, English... and American folk songs!
This seems much more adventurous than what I'm doing. But he has a good reason for doing it. His great grandfather, Sam Pollard, was a Methodist missionary in this area — and he invented a script that is still used by the locals:
The Miao are an ethnic group that includes the Hmong, Hmub, Xong, and A-Hmao. These folks live in the borderlands of southern China, northern Vietnam, Laos, Myanmar and Thailand. The A-Hmao had a legend about how their ancestors knew a system of writing but lost it. According to this legend, the script would eventually be brought back some day. When Sam Pollard introduced his script for writing A-Hmao, it became extremely popular, and he became a kind of hero. Blake and his family visited this part of China last year. He enjoyed it a lot, so he decided to do some teaching there this summer.
Watch firewalking at the Sri Mariamman Temple:
and if you live around here, check out the Explosion Pot Barbecue!
Here you see 3 rotating rings called gimbals. Gimbals are used in gyroscopes and inertial measurement units, which are gadgets that measure the orientation of some object — like a drone, or a spacecraft. Gimbals are also used to orient thrusters on rockets.
With 3 gimbals, you can rotate the inner one to whatever orientation you want. The basic reason is that it takes 3 numbers to describe a rotation in 3 dimensional space. This is a special lucky property of the number 3.
But when two of the gimbal's axes happen to be lined up, you get gimbal lock. In other words: you lose the ability to rotate the inner gimbal a tiny bit in any way you want. The reason is that in this situation, rotating one of the two aligned gimbals has the same effect on the inner gimbal as rotating the other!
I've always found gimbal lock to be a bit mysterious: people refer to it in ominous tones without explaining it, like some incurable deadly disease. So I'm trying to demystify it here.
As the wise heads at Wikipedia point out,
The word lock is misleading: no gimbal is restrained. All three gimbals can still rotate freely about their respective axes of suspension. Nevertheless, because of the parallel orientation of two of the gimbals' axes there is no gimbal available to accommodate rotation along one axis.
Gimbal lock can actually be dangerous! When it happens, or even when it almost happens, you lose some control over what's going on.
It caused a problem when Apollo 11 was landing on the moon. This spacecraft had 3 nested gimbals on its inertial measurement unit. The engineers were aware of the gimbal lock problem but decided not to use a fourth gimbal. They wrote:
"The advantages of the redundant gimbal seem to be outweighed by the equipment simplicity, size advantages, and corresponding implied reliability of the direct three degree of freedom unit."They decided instead to trigger a warning when the system came close to gimbal lock. But it didn't work right:
"Near that point, in a closed stabilization loop, the torque motors could theoretically be commanded to flip the gimbal 180 degrees instantaneously. Instead, in the Lunar Module, the computer flashed a 'gimbal lock' warning at 70 degrees and froze the inertial measurment unit at 85 degrees."
The spacecraft had to be manually moved away from the gimbal lock position, and they had to start over from scratch, using the stars as a reference.
After the Lunar Module landed, Mike Collins aboard the Command Module joked:
"How about sending me a fourth gimbal for Christmas?"
Fun story! But ultimately, it's all about math.
Puzzle. Show that gimbal lock is inevitable with just 3 gimbals by showing that every smooth map from the 3-torus to SO(3) has at least one point where the rank of its differential drops below 3.
See what I mean? Math. This result shows not only that gimbal lock occurs with the setup shown above, but that any scheme for describing a rotation by 3 angles — or more precisely, 3 points on the circle — must suffer gimbal lock.
Here's a sketch of an answer to the puzzle: if \(X\) and \(Y\) are
smooth \(n\)-manifolds and the rank of the differential of a smooth
map \(f\colon X \to Y\) is equal to \(n\) everywhere, it's locally a
diffeomorphism. If \(X\) is compact and \(Y\) is connected you can
show this makes \(X\) into a covering space of \(Y\). So, if there
were a smooth map \(f \colon T^3 \to \mathrm{SO}(3)\) whose
differential has rank 3 everywhere, the 3-torus would be a covering
space of \(\mathrm{SO}(3)\), but this is not possible, since a
covering \(f \colon T^3 \to \mathrm{SO}(3)\) would induce an inclusion
of \(\pi_1(T^3) \cong \mathbb{Z}^3\) in \(\pi_1(\mathrm{SO}(3)) \cong
\mathbb{Z}/2\), which is impossible.
July 24, 2016
You can get electrons to behave in many strange ways in different materials. They act like various kinds of particles... but they're not truly fundamental particles, so they're called quasiparticles.
For example, the spin, charge and position of electrons can move in completely independent ways.
Imagine an audience at a football game holding up signs, and then creating a wave by wiggling their signs. This wave can move even even if the people stand still!
Similarly, we can have electrons more or less standing still, with their spins lined up. Then their spins can wiggle a bit, and this wiggle can move through the material, even though the electrons don't move. This wave of altered spin can act like a particle! It's called a spinon.
You can also imagine a hole in a dense crowd of people, moving along as if it were an entity of its own. When this happens with electrons it's called a holon, or more commonly just a hole. A hole acts like a particle with positive charge, since electrons have negative charge.
Since holes have positive charge and electrons have negative charge, they attract. Sometimes they orbit each other for long enough that this combined thing acts like a particle of its own! This kind of quasiparticle is called an exciton.
There are also other quasiparticles. If you're a student who wants to do particle physics, please switch to studying quasiparticles! The math is almost the same, and you don't need huge particle accelerators to make cool new discoveries. Some are even useful.
One of the most fundamental things about a quasiparticle, or for that matter an ordinary particle, is its energy. Its energy depends on its momentum. The relation between them is called the dispersion relation. This says a lot about how the quasiparticle acts.
Right next door to the Centre for Quantum Technologies where I'm working in Singapore there's a lab that studies graphene — a crystal made of carbon that's just one atom thick. When you've got a very thin film like this, a quasiparticle inside it acts like it's living in a 2-dimensional world! Since it can't go up and down, only 2 components of its momentum can be nonzero.
The picture above shows a graph of energy as a function of momentum for a new kind of quasiparticle they're studying. They haven't made it in the lab yet; they've just shown it's possible. It involves a 3-dimensional material, not a thin sheet, so there are really 3 components of momentum, \(k_x, k_y\) and \(k_z\). But only two are shown in this picture.
The three colored sheets show that 3 different energies are possible for each momentum — except momentum zero, where all three sheet meet, and also a line of momenta where two sheets meet.
If we only had the green and blue sheets, that would be the dispersion relation for a massless particle. People already know how to make massless quasiparticles with graphene.
The new thing is the yellow sheet! This will make very strange things happen, I'm sure.
I got interested in these new quasiparticles thanks to this article pointed out by Rasha Kamel:
But I got the picture from here:
Here's the abstract, for you physics nerds out there:
Abstract. Topological metals and semi-metals (TMs) have recently drawn significant interest. These materials give rise to condensed matter realizations of many important concepts in high-energy physics, leading to wide-ranging protected properties in transport and spectroscopic experiments. The most studied TMs, i.e., Weyl and Dirac semi-metals, feature quasiparticles that are direct analogues of the textbook elementary particles. Moreover, the TMs known so far can be characterized based on the dimensionality of the band crossing. While Weyl and Dirac semimetals feature zero-dimensional points, the band crossing of nodal-line semimetals forms a one-dimensional closed loop. In this paper, we identify a TM which breaks the above paradigms. Firstly, the TM features triply-degenerate band crossing in a symmorphic lattice, hence realizing emergent fermionic quasiparticles not present in quantum field theory. Secondly, the band crossing is neither 0D nor 1D. Instead, it consists of two isolated triply-degenerate nodes interconnected by multi-segments of lines with two-fold degeneracy. We present materials candidates. We further show that triply-degenerate band crossings in symmorphic crystals give rise to a Landau level spectrum distinct from the known TMs, suggesting novel magneto-transport responses. Our results open the door for realizing new topological phenomena and fermions including transport anomalies and spectroscopic responses in metallic crystals with nontrivial topology beyond the Weyl/Dirac paradigm.
Weirdly, I had learned the word 'symmorphic' just yesterday. Greg
Egan were working on crystals, and he explained that a crystal
is symmorphic if it contains a point where every symmetry of the
crystal consists of a symmetry fixing this point followed by a
translation. It was important for our work to notice that a diamond
is not symmorphic.
July 26, 2016
Just kidding! There's nothing devilish about the pentagram here. It's what scientists saw when they shot X-rays through a tiny piece of a meteorite found in the far northeast of Russia.
No ordinary crystal can produce this pattern - it takes a quasicrystal, where the atoms are packed in a way that never quite repeats. Scientists have made lots of quasicrystals in the lab, but only two have been found in nature, both in meteorites!
This is the second one. It contains a mineral called icosahedrite, made of aluminum, copper and iron. It's only stable at high temperatures and pressures, so it must have formed in a collision. It's been slowly decaying ever since, but very slowly. It could be billions of years old.
To see how this mineral could have formed, scientists simulated the collision between two asteroids in their lab. They took thin slices of minerals found in the Khatyrka meteorite and sandwiched them together in a gadget that looks like a a steel hockey puck. They attached it to the muzzle of a four-meter-long gun and blasted it with a projectile moving nearly one kilometer per second! Yup. Icosahedrite. For details and more pictures, see:
In June 2015, after a two-year upgrade, the Large Hadron Collider turned on again. In its first run it had discovered the Higgs boson, a particle 133 times heavier than the proton — and the main missing piece of the Standard Model. When the collider restarted, with a lot more energy, everyone was hoping to see something new.
In December 2015, two separate detectors saw something: pairs of photons, seemingly emitted by the decay of a brand new particle 6 times heavier than the Higgs boson.
But was it for real? Maybe it was just a random fluctuation: noise, rather than a true signal.
It seemed unlikely to be just chance. Combining the data from both detectors, the chance of coincidentally seeing a bump this big at this location in the photon spectrum was one in 100 thousand.
But in particle physics that's not good enough. Physicists are looking for lots of different things in these big experiments, so rare coincidences do happen. To feel safe, they want to push the chance down to one in 3 million. That's called a '5 sigma event'.
So they looked harder.
Meanwhile, theoretical physicists wrote 500 papers trying to explain this so-called diphoton bump. It turned out to be easy to make up theories that have a particle of the right sort. Not so easy, though, to make a convincingly elegant theory.
New data have come in. The bump is gone.
Theorists are bummed. On his blog, a particle physicist named Adam Falkowski wrote:
The loss of the 750 GeV diphoton resonance is a big blow to the particle physics community. We are currently going through the 5 stages of grief, everyone at their own pace, as can be seen e.g. in this comments section. Nevertheless, it may already be a good moment to revisit the story one last time, so as to understand what went wrong.In the recent years, physics beyond the Standard Model has seen 2 other flops of comparable impact: the faster-than-light neutrinos in OPERA, and the cosmic microwave background tensor fluctuations in BICEP. Much as the diphoton signal, both of the above triggered a binge of theoretical explanations, followed by a massive hangover. There was one big difference, however: the OPERA and BICEP signals were due to embarrassing errors on the experimentalists' side. This doesn't seem to be the case for the diphoton bump at the Large Hadron Collider. Some may wonder whether the Standard Model background may have been slightly underestimated, or whether one experiment may have been biased by the result of the other... But, most likely, the 750 GeV bump was just due to a random fluctuation of the background at this particular energy. Regrettably, the resulting mess cannot be blamed on experimentalists, who were in fact downplaying the anomaly in their official communications. This time it's the theorists who have some explaining to do.
For more, see Adam Falkowski's blog. He goes by the name of 'Jester':
By now we have to admit it's quite possible that the Large Hadron Collider will not see any new physics not predicted by the Standard Model. Unfortunately, this triumph of the Standard Model would leave a lot of big questions unanswered... for now.
The video above explains the diphoton bump in simple terms. It was
made back in the early optimistic days.
July 31, 2016
In South Dakota, in a town named Lead, there was a gold mine. Now it's abandoned. But at the bottom of this mine, more than a mile underground, there sits a one-meter-tall, 12-sided container. It contains 370 kilograms of a noble gas chilled to liquid form. Liquid xenon!
It's called the Large Underground Xenon experiment, or LUX. It's been looking for particles that could explain dark matter. If such a particle interacts with a xenon atom, LUX can detect it.
Of course, we also need to distinguish these particles from other things. A mile of rock helps block cosmic rays. But xenon, a gas at room temperatures, chilled to liquid form, is a great choice here. For one thing, it's self-shielding! Xenon is so dense that gamma rays and neutrons don't usually get through more than a few centimeters of the stuff. But it's perfectly transparent to ordinary light... so if a dark matter particle hits an atom of xenon in the middle of the tank, LUX will see a flash of light. It can also detect electrons that shoot out from the collision.
Four other experiments had reported hints of dark matter particles about 5 times heavier than a proton. But LUX is much more sensitive!
The LUX team, with over a hundred scientists, has been looking for dark matter since 2014. Ten days ago they announced their results: no dark matter particles seen.
This "non-discovery" is actually an important discovery. The most popular theory of dark matter — that it consists of weakly interacting massive particles — has taken a serious hit.
We now know that if these hypothetical particles, affectionately called WIMPs, are responsible for most of the dark matter and have a mass between 1/5 and 1000 times the mass of a proton, they must be very, very unwilling to interact with ordinary matter.
There's no rule saying particles have to interact with ordinary matter. So, we can't rule out such WIMPs, but they're looking less plausible. People are getting more interested in other theories:
Theory 3) is getting a lot of attention, because the gravitational wave detector called LIGO is now able to detect black hole collisions! It's seen two collisions so far, and the first one involved black holes that seem quite strange, not like the ones we know. They might be primordial black holes, left over from the early Universe. Perhaps dark matter consists of primordial black holes!
More on that later. For now, try these. The new announcement from the LUX team is here:
For how the LUX detector works, read this nice article:
For a nice intro to the new LUX results, on a website that requires you to look at ads, try this:
For primordial black holes as dark matter, try this:
The picture above comes from here: