Greg Egan has done it again! This is an infinite cubic lattice of rotating gears in 3-dimensional space... seen by someone who is falling through it and also rotating!
It's a bit bewildering. One reason is that after you turn 180 degrees, the view looks exactly the same.
He's been developing techniques for studying 'higher-dimensional gears', like 3d ball bearings that turn while touching each other, arranged on the surface of a hypersphere in 4d space:
His real tour de force was to get a setup with a ball bearing at each of the 600 vertices and 1200 edge-centres of a 4-dimensional shape called the '120-cell'. Getting this to work required solving thousands of linear equations in thousands of variables — too hard without bringing in some heavy-duty math. Check out his website for more about that! (You can enjoy the pictures without understanding the math.)
He wrote:
After the 120-cell, I thought it would be fun to see what an infinite lattice of gears looks like. For \(\mathbb{Z}^3\), it's easy to find both a basis for the solution space, and a nice subspace where the spheres all rotate with the same speed: $$ \omega(x,y,z) = (a (-1)^{y+z}, b (-1)^{x+z}, c (-1)^{x+y}) $$ Because the rotational periods are all the same, it's possible to replace the rolling contact of the spheres with a true gear action between circular gears, which are positioned at each circle of latitude on which there are points of contact. That's what the movie here shows, from a point of view that moves "down" through the lattice while also rotating its gaze.
The manul lives in the grasslands and high steppes of Central Asia: Mongolia, Tajikistan, Kyrgyzstan, Pakistan, Kazakhstan, Kashmir, and western China. It's the size of a house cat, but its stocky build and long, dense fur makes it look stout and plush. It has a shorter jaw with fewer teeth than most cats.
Mamuls spend most of the day in caves, cracks in the rock, or marmot burrows. In the late afternoon they come out and hunt. They can't run fast, so they mainly hunt by ambush or stalking. They feed largely on prey that are active during the day: gerbils, pikas, voles, partridges, and sometimes young marmots.
They're most closely related to the leopard cat, a small cat of southeast Asia which is sometimes crossbred with house cats to give beautiful cats called 'Bengals' that enjoy getting wet. Some even live here in Singapore, but I've sure never seen one! The manul and the leopard cat seem to have diverged just 5 million years ago. It always amazes me how new many mammal species are.
The manul is hard to raise in zoos, and they're listed as 'near threatened'. They are hunted for its fur in relatively large numbers in China, Mongolia, and Russia, although international trade in manul pelts has largely ceased since the late 1980s, and Mongolia is the only place where it's still legal to kill them.
The manul is also called Pallas's cat, after a naturalist who wrote about them. Its scientific name is Otocolobus manul :
I got this picture from a great list of wild cats:
Check out the species you may not know: the fishing cat, the Borneo bay cat, the flat headed cat, the Iberian lynx, and the margay!
We understand a lot about physics - and that makes the remaining mysteries even more tantalizing! For example: why are quarks so much like leptons?
Elementary particles come in two main kinds: the ones that carry forces (gauge bosons) and the ones that make up matter (quarks and leptons). There's also at least one more... but never mind! Today's puzzle is about quarks and leptons. You'll see from the chart that they look sort of similar. But why?
Maybe a lightning review of particle physics will help, in case you skipped that class in high school. Most of the matter you see is made of electrons, protons and neutrons. Protons and neutrons are made of up and down quarks, held together by the strong force. But electrons are 'leptons', which means they don't feel the strong force.
Up quarks, down quarks and electrons — those are 3 of the 4 particles in 'generation 1'. The 4th is the electron neutrino. It's also a lepton — it's doesn't feel the strong force. But it's also has no electric charge! So, it's very hard to detect — it whizzes easily through ordinary matter. But we have detected it, and we actually know a huge amount about it.
We also know that besides 'generation 1' there's a 'generation 2' and 'generation 3' of quarks and leptons. We're pretty sure there are only 3: people have done experiments that show there can't be more different kinds of neutrinos, unless they are very heavy, or different from all the rest in some other way. We have no idea why there are only 3 generations.
But our puzzle today is: why do quarks and leptons come in generations at all? So let's just think about generation 1.
We know that the up and down quark are closely connected. We also know that the electron and electron neutrino are closely connected. For example, you can collide an electron and an up quark and have them turn into an electron neutrino and a down quark! We understand this stuff very well, actually: there's a detailed mathematical theory of it, and it works great.
But there are other things that seem mysterious. The up quark has charge 2/3, the down quark has charge -1/3, the electron has charge -1 and the electron neutrino has charge 0. Quarks also come in 3 different kinds, called 'colors' - they change colors when they interact with the strong force. Leptons have no color.
Are all the 3's in the last paragraph a coincidence? It seems not. For example, if quarks came in 4 colors, but had the charges they do now, all hell would break loose! I could explain why, but that's not my goal today.
My goal is just to say this: there's a theory called the 'Pati-Salam model' that says leptons are secretly just a funny kind of quarks — a 'fourth color of quark'. This theory unifies quarks and leptons. And this theory also explains why quarks have charges like 2/3 and -1/3.
This theory has been around since 1974. It has some problems. If it didn't, we'd probably all believe it by now! It's very hard to find theories of elementary particles that fit all the data we have; if you just make up stuff, you'll almost surely run into problems. But the Pati-Salam model is pretty good, it's not completely ruled out by experiments... and last year something interesting happened.
A famous mathematician named Alain Connes has an approach to physics based on 'noncommutative geometry', which replaces our usual picture of spacetime by something that's more like algebra than geometry. His theory predicted the wrong mass for the Higgs boson — that's the extra particle I hinted at near the start of this story. But last year he came out with a new improved version, that doesn't suffer from this problem. And it turns out to be a lot like the Pati-Salam model!
What's interesting is how he gets it. In his earlier work, he laid down a bunch of mathematical axioms, and one of the simplest theories that obeys all these axioms turned out to be very similar to the Standard Model - our usual theory of particles.
But now, he and some other guys have noticed that if you drop one of the axioms, something like the Pati-Salam model is also allowed. Moreoever, you can get a Higgs boson with the right mass!
I wish I understood this better. Alas, I don't have much time for this stuff anymore! Here is his paper:
Here's the abstract of Connes' paper, which gives a flavor of what he's doing... at least if you know enough jargon:
Abstract. The assumption that space-time is a noncommutative space formed as a product of a continuous four dimensional manifold times a finite space predicts, almost uniquely, the Standard Model with all its fermions, gauge fields, Higgs field and their representations. A strong restriction on the noncommutative space results from the first order condition which came from the requirement that the Dirac operator is a differential operator of order one. Without this restriction, invariance under inner automorphisms requires the inner fluctuations of the Dirac operator to contain a quadratic piece expressed in terms of the linear part. We apply the classification of product noncommutative spaces without the first order condition and show that this leads immediately to a Pati-Salam SU(2)_{R} × SU(2)_{L} × SU(4) type model which unifies leptons and quarks in four colors. Besides the gauge fields, there are 16 fermions in the (2,2,4) representation, fundamental Higgs fields in the (2,2,1), (2,1,4) and (1,1,1+15) representations. Depending on the precise form of the order one condition or not there are additional Higgs fields which are either composite depending on the fundamental Higgs fields listed above, or are fundamental themselves. These additional Higgs fields break spontaneously the Pati-Salam symmetries at high energies to those of the Standard Model.
Say you have some dots and you want to draw a smooth curve that sorta almost goes through these dots. Then you can use a Bézier curve. Some drawing programs use this trick... and lots of fonts are drawn with the help of Bézier curves.
The math behind these curves had been known since 1912, but they were popularized by Pierre Bézier, an engineer who used them to design automobile bodies at Renault.
Can you figure out how they work just by looking at the movie? An explanation in words sounds complicated... but it's really easy as pie.
It's like you've got 3 guys running along straight racetracks. The 2 guys in back have rabbits that each chase the next guy, always heading straight toward that next guy. And the guy at the very back also has a dog that chases straight after the next guy's rabbit. Everyone starts at the same time and stops at the same time. The dog follows the red curve.
In other words:
First draw gray lines between your original dots \(P_0, P_1, P_2, P_3\).
Each green dot moves at a constant rate along a gray line. All the green dots start at the same time, and finish at the same time.
Then draw green lines connecting the green dots.
Each blue dot moves at a constant rate along a green line. All the blue dots start at the same time, and finish at the same time.
Then draw a blue line connecting the blue dots.
The black dot moves at a constant rate along this blue line. It starts at the same time as all the other dots, and finishes at the same time.
Get the pattern? Each time we do this trick, there's one fewer dot. There are 4 original dots, 3 green dots, 2 blue dots and 1 black dot. So now you're done!
The black dot traces out the Bézier curve shown in red here.
You can play this game starting with any number of dots. When you start with n dots, you get a curve described by a polynomial equation of degree n-1. So, this red curve is called a cubic Bézier curve.
Puzzle 1: show that our cubic Bézier curve is given by the equation $$ C(t) = (1-t)^3 P_0 + 3(1-t)^2 t P_1 + 3(1-t) t^2 P_2 + t^3 P_3 $$
Puzzle 2: generalize this to more dots. (Hint: binomial coefficients!)
When you've got a lot of dots, people usually break them into bunches and draw a quadratic or cubic Bézier curve through each bunch. They match up at the ends, so this works, though frankly I often think it looks kind of lame. This is called a composite Bézier curve. PostScript, Asymptote, Metafont, and SVG use composite Bézier curves made of cubic Bézier curves to drawing curved shapes.
I imagine there are lots of tricks that are 'better' than Bézier curves, but I'm not an expert! If I wanted to know more, I'd read about stuff like non-uniform rational B-splines, or NURBS:
But I just read this:
because I liked the animated gif, made by Phil Tregoning.
What's the densest way to pack spheres? Here are two equally good ways.
In fact there are infinitely many equally good ways! We start by laying spheres on the plane in a hexagonal arrangement, as tightly as we can. Then we put a second layer like this on top, with the new spheres resting in the gaps between the old ones. Then we put on a third layer. But now there are 2 really different ways to do it!
The spheres in the third layer can be directly above the spheres in the first layer - that's the picture at right. Or they can be not directly above - shown at left.
As we continue, we keep getting more choices.
One systematic choice is to make the layers alternate like ABABAB..... That's called the hexagonal close packing, and that's how crystals of magnesium work.
Another systematic choice makes every third layer be the same, like ABCABC... That's called the cubic close packing or face-centered cubic, and that's how crystals of lead work.
(Why "cubic?" Because - even though it's not obvious! - you can also get this pattern by putting a sphere at each corner and each face of a cubical lattice. Trying to visualize this in your head is a great way to build your brain power.)
There are also uncountably many unsystematic ways to choose how to put down the layers of spheres, like ABACBCAC.... You just can't use the same letter twice in a row.
In 1611, the famous astronomer Kepler conjectured that sphere packings of this sort were the densest possible. They fill up
of the space, and he claimed you can't do better.
Proving this turned out to be very, very hard. Wu-Yi Hsiang claimed to have a proof in 1993. It was 92 pages long. Experts said it had gaps (pardon the pun). Hsiang has never admitted there's a problem.
Thomas Hales claimed to have a proof in 1998. His proof took 250 pages... together with 3 gigabytes of computer programs, data and results!
The famous journal Annals of Mathematics agreed to check his proof with a board of 12 referees. In 2003, after four years of work, the referees accepted his paper. But they didn't exactly say it was correct. They said they were "99% certain" it was right - but they didn't guarantee the correctness of all of the computer calculations.
Hales wasn't happy.
He decided to do a completely rigorous proof using computer logic systems, so that automated proof-checking software could check it. He worked on it for about 10 years with a large team of people.
He announced that it was done on August 10th, 2014. You can see the announcement here:
To verify the proof, the main thing you need to do is check 23,000 complicated inequalities. Checking all these on the Microsoft Azure cloud took about 5000 processor-hours.
When it was done, Hales said:
An enormous burden has been lifted from my shoulders. I suddenly feel ten years younger!
Personally I prefer shorter proofs. But this is quite a heroic feat.
I actually wrote about this because I want to talk about packing tetrahedra. But I figured if you didn't know the more famous story of packing spheres, that would be no good.
For more, check out Hales' free book, which starts with a nice history of the Kepler problem:
For more on computer-aided proof, try this paper:
I'm leaving Singapore today. This wall painting in Chinatown, modeled after a classical Chinese painting, captures a bit of what I like about the place. It's a mix of old and new, East and West.
Last weekend, Lisa and I saw a Chinese opera — part of a free series in Hong Lim Park. Chinese opera used to be really popular in Singapore, with the stars being the equivalent of pop idols today. Now its appeal is dwindling, but there was still a big crowd — and some old guys were punching their fists in the air when the star-crossed lovers finally triumphed in the end. It was set in the Ming Dynasty, and featured an emperor who snuck out of the palace and wound up marrying a peasant girl. I enjoyed it a lot more than I expected. Why? Because big computer screens showed translations of the lyrics into English! Without that, I might have roughly followed the plot, but I wouldn't have gotten the jokes.
I'm going back to Riverside to teach. I've got a light teaching load this fall, just grad-level real analysis (the first quarter of a 3-part course) and my seminar — where I'll take the work my grad students have been doing on network theory and put it together into a nice story. I'll try to write lecture notes in the form of blog articles, but I find that fun and relaxing. So, I'll be able to put some energy into the talk I'm giving this December at NIPS, the Neural Information Processing Seminar, a big annual conference on neural networks. I want to talk about El Niqo prediction, climate networks and machine learning. But I've got a lot to learn, especially about machine learning. The Azimuth Code Project team want to carry out some computer experiments in that direction. It should be fun, as long as I resign myself to giving a talk that's just "work in progress", not completed and polished. I'll do some heavier teaching in the winter quarter, but the spring will be a non-teaching quarter. This seems awfully cushy, but my department chair noticed I'd taught too much last year — more than I'm paid to do! And in the spring, I'll be helping run two workshops. One is on information and entropy in biology, at the National Institute for Mathematical and Biological Synthesis, in Knoxville Tennessee. The other is on network theory, at the Institute for Scientific Interchange, in Torino Italy.
Then in June I'll come back to work at the CQT in Singapore!
This selfie was taken by Vermeer's maid Griet in 1665, moments after the painter completed his masterwork Girl With a Pearl Earring.
Enceladus, the sixth-largest moon of Saturn, has geysers that shoot jets of water into space! They look beautiful in these photographed by the Cassini probe. They create an invisible ring of ice crystals around Saturn: the E ring. And now they have been mapped!
There are at least 100 geysers near the south pole of Enceladus. They come from four big cracks in its rocky surface, called tiger stripes. Some of the water they shoot out falls back to the surface as snow, but about 200 kilograms per second shoots out into space. The tiger stripes also emit about 5 gigawatts of heat.
The geysers have recently been mapped by Carolyn Porco and her collaborators at NASA: