Time Crystals

When water freezes and forms a crystal, it creates a periodic pattern in space. Could there be something that crystallizes to form a periodic pattern in time? Frank Wilczek, who won the Nobel Prize for helping explain why quarks and gluons trapped inside a proton or neutron act like freely moving particles when you examine them very close up, dreamt up this idea and called it a time crystal:

• Frank Wilczek, Classical time crystals.

• Frank Wilczek, Quantum time crystals.

‘Time crystals’ sound like something from Greg Egan’s Orthogonal trilogy, set in a universe where there’s no fundamental distinction between time and space. But Wilczek wanted to realize these in our universe.

Of course, it’s easy to make a system that behaves in an approximately periodic way while it slowly runs down: that’s how a clock works: tick tock, tick tock, tick tock… But a system that keeps ‘ticking away’ without using up any resource or running down would be a strange new thing. There’s no telling what weird stuff we might do with it.

It’s also interesting because physicists love symmetry. In ordinary physics there are two very important symmetries: spatial translation symmetry, and time translation symmetry. Spatial translation symmetry says that if you move an experiment any amount to the left or right, it works the same way. Time translation symmetry says that if you do an experiment any amount of time earlier or later, it works the same way.

Crystals are fascinating because they ‘spontaneously break’ spatial translation symmetry. Take a liquid, cool it until it freezes, and it forms a crystal which does not look the same if you move it any amount to the right or left. It only looks the same if you move it certain discrete steps to the right or left!

The idea of a ‘time crystal’ is that it’s a system that spontaneously breaks time translation symmetry.

Given how much physicists have studied time translation symmetry and spontaneous symmetry breaking, it’s sort of shocking that nobody before 2012 wrote about this possibility. Or maybe someone did—but I haven’t heard about it.

It takes real creativity to think of an idea so radical yet so simple. But Wilczek is famously creative. For example, he came up with anyons: a new kind of particle, neither boson nor fermion, that lives in a universe where space is 2-dimensional. And now we can make those in the lab.

Unfortunately, Wilczek didn’t know how to make a time crystal. But now a team including Xiang Zhang (seated) and Tongcang Li (standing) at U.C. Berkeley have a plan for how to do it.

Actually they propose a ring-shaped system that’s periodic in time and also in space, as shown in the picture. They call it a space-time crystal:

Here we propose a space-time crystal of trapped ions and a method to realize it experimentally by confining ions in a ring-shaped trapping potential with a static magnetic field. The ions spontaneously form a spatial ring crystal due to Coulomb repulsion. This ion crystal can rotate persistently at the lowest quantum energy state in magnetic fields with fractional fluxes. The persistent rotation of trapped ions produces the temporal order, leading to the formation of a space-time crystal. We show that these space-time crystals are robust for direct experimental observation. The proposed space-time crystals of trapped ions provide a new dimension for exploring many-body physics and emerging properties of matter.

The new paper is here:

• Tongcang Li, Zhe-Xuan Gong, Zhang-Qi Yin, H. T. Quan, Xiaobo Yin, Peng Zhang, L.-M. Duan and Xiang Zhang, Space-time crystals of trapped ions.

Alas, the press release put out by Lawrence Berkeley National Laboratory is very misleading. It describes the space-time crystal as a clock that

will theoretically persist even after the rest of our universe reaches entropy, thermodynamic equilibrium or “heat-death”.

NO!

First of all, ‘reaching entropy’ doesn’t mean anything. More importantly, as time goes by and things fall apart, this space-time crystal, assuming anyone can actually make it, will also fall apart.

I know what the person talking to the reporter was trying to say: the cool thing about this setup is that it gives a system that’s truly time-periodic, not gradually using up some resource and running down like an ordinary clock. But nonphysicist readers, seeing an article entitled ‘A Clock that Will Last Forever’, may be fooled into thinking this setup is, umm, a clock that will last forever. It’s not.

If this setup were the whole universe, it might keep ticking away forever. But in fact it’s just a small, carefully crafted portion of our universe, and it interacts with the rest of our universe, so it will gradually fall apart when everything else does… or in fact much sooner: as soon as the scientists running it turn off the experiment.

Classifying space-time crystals

What could we do with space-time crystals? It’s way too early to tell, at least for me. But since I’m a mathematician, I’d be happy to classify them. Over on Google+, William Rutiser asked if there are 4d analogs of the 3d crystallographic groups. And the answer is yes! Mathematicians with too much time on their hands have classified the analogues of crystallographic groups in 4 dimensions:

• Space group: classification in small dimensions, Wikipedia.

In general these groups are called space groups (see the article for the definition). In 1 dimension there are just two, namely the symmetry groups of this:

— o — o — o — o — o — o —

and this:

— > — > — > — > — > — > —

In 2 dimensions there are 17 and they’re called wallpaper groups. In 3 dimensions there are 230 and they are called crystallographic groups. In 4 dimensions there are 4894, in 5 dimensions there are… hey, Wikipedia leaves this space blank in their table!… and in 6 dimensions there are 28,934,974.

So, there is in principle quite a large subject to study here, if people can figure out how to build a variety of space-time crystals.

There’s already book on this, if you’re interested:

• Harold Brown, Rolf Bulow, Joachim Neubuser, Hans Wondratschek and Hans Zassenhaus, Crystallographic Groups of Four-Dimensional Space, Wiley Monographs in Crystallography, 1978.

A Course on Quantum Techniques for Stochastic Mechanics

Jacob Biamonte and I have come out with a draft of a book!

• A course on quantum techniques for stochastic mechanics.

It’s based on the first 24 network theory posts on this blog. It owes a lot to everyone here, and the acknowledgements just scratch the surface of that indebtedness. At some later time I’d like to go through the posts and find the top twenty people who need to be thanked. But I’m leaving Singapore on Friday, going back to California to teach at U.C. Riverside, so I’ve been rushing to get something out before then.

If you see typos or other problems, please let us know!
We’ve reorganized the original blog articles and polished them up a bit, but we plan to do more before publishing these notes as a book.

I’m looking forward to teaching a seminar called Mathematics of the Environment when I get back to U.C. Riverside, and with luck I’ll put some notes from that on the blog here. I will also be trying to round up a team of grad students to work on network theory.

The next big topics in the network theory series will be electrical circuits and Bayesian networks. I’m beginning to see how these fit together with stochastic Petri nets in a unified framework, but I’ll need to talk and write about it to fill in all the details.

You can get a sense of what this course is about by reading this:

Foreword

This course is about a curious relation between two ways of describing situations that change randomly with the passage of time. The old way is probability theory and the new way is quantum theory

Quantum theory is based, not on probabilities, but on amplitudes. We can use amplitudes to compute probabilities. However, the relation between them is nonlinear: we take the absolute value of an amplitude and square it to get a probability. It thus seems odd to treat amplitudes as directly analogous to probabilities. Nonetheless, if we do this, some good things happen. In particular, we can take techniques devised in quantum theory and apply them to probability theory. This gives new insights into old problems.

There is, in fact, a subject eager to be born, which is mathematically very much like quantum mechanics, but which features probabilities in the same equations where quantum mechanics features amplitudes. We call this subject stochastic mechanics

Plan of the course

In Section 1 we introduce the basic object of study here: a ‘stochastic Petri net’. A stochastic Petri net describes in a very general way how collections of things of different kinds can randomly interact and turn into other things. If we consider large numbers of things, we obtain a simplified deterministic model called the ‘rate equation’, discussed in Section 2. More fundamental, however, is the ‘master equation’, introduced in Section 3. This describes how the probability of having various numbers of things of various kinds changes with time.

In Section 4 we consider a very simple stochastic Petri net and notice that in this case, we can solve the master equation using techniques taken from quantum mechanics. In Section 5 we sketch how to generalize this: for any stochastic Petri net, we can write down an operator called a ‘Hamiltonian’ built from ‘creation and annihilation operators’, which describes the rate of change of the probability of having various numbers of things. In Section 6 we illustrate this with an example taken from population biology. In this example the rate equation is just the logistic equation, one of the simplest models in population biology. The master equation describes reproduction and competition of organisms in a stochastic way.

In Section 7 we sketch how time evolution as described by the master equation can be written as a sum over Feynman diagrams. We do not develop this in detail, but illustrate it with a predator–prey model from population biology. In the process, we give a slicker way of writing down the Hamiltonian for any stochastic Petri net.

In Section 8 we enter into a main theme of this course: the study of equilibrium solutions of the master and rate equations. We present the Anderson–Craciun–Kurtz theorem, which shows how to get equilibrium solutions of the master equation from equilibrium solutions of the rate equation, at least if a certain technical condition holds. Brendan Fong has translated Anderson, Craciun and Kurtz’s original proof into the language of annihilation and creation operators, and we give Fong’s proof here. In this language, it turns out that the equilibrium solutions are mathematically just like ‘coherent states’ in quantum mechanics.

In Section 9 we give an example of the Anderson–Craciun–Kurtz theorem coming from a simple reversible reaction in chemistry. This example leads to a puzzle that is resolved by discovering that the presence of ‘conserved quantities’—quantities that do not change with time—let us construct many equilibrium solutions of the rate equation other than those given by the Anderson–Craciun–Kurtz theorem.

Conserved quantities are very important in quantum mechanics, and they are related to symmetries by a result called Noether’s theorem. In Section 10 we describe a version of Noether’s theorem for stochastic mechanics, which we proved with the help of Brendan Fong. This applies, not just to systems described by stochastic Petri nets, but a much more general class of processes called ‘Markov processes’. In the analogy to quantum mechanics, Markov processes are analogous to arbitrary quantum systems whose time evolution is given by a Hamiltonian. Stochastic Petri nets are analogous to a special case of these: the case where the Hamiltonian is built from annihilation and creation operators. In Section 11 we state the analogy between quantum mechanics and stochastic mechanics more precisely, and with more attention to mathematical rigor. This allows us to set the quantum and stochastic versions of Noether’s theorem side by side and compare them in Section 12.

In Section 13 we take a break from the heavy abstractions and look at a fun example from chemistry, in which a highly symmetrical molecule randomly hops between states. These states can be seen as vertices of a graph, with the transitions as edges. In this particular example we get a famous graph with 20 vertices and 30 edges, called the ‘Desargues graph’.

In Section 14 we note that the Hamiltonian in this example is a ‘graph Laplacian’, and, following a computation done by Greg Egan, we work out the eigenvectors and eigenvalues of this Hamiltonian explicitly. One reason graph Laplacians are interesting is that we can use them as Hamiltonians to describe time evolution in both stochastic and quantum mechanics. Operators with this special property are called ‘Dirichlet operators’, and we discuss them in Section 15. As we explain, they also describe electrical circuits made of resistors. Thus, in a peculiar way, the intersection of quantum mechanics and stochastic mechanics is the study of electrical circuits made of resistors!

In Section 16, we study the eigenvectors and eigenvalues of an arbitrary Dirichlet operator. We introduce a famous result called the Perron–Frobenius theorem for this purpose. However, we also see that the Perron–Frobenius theorem is important for understanding the equilibria of Markov processes. This becomes important later when we prove the ‘deficiency zero theorem’.

We introduce the deficiency zero theorem in Section 17. This result, proved by the chemists Feinberg, Horn and Jackson, gives equilibrium solutions for the rate equation for a large class of stochastic Petri nets. Moreover, these equilibria obey the extra condition that lets us apply the Anderson–Craciun–Kurtz theorem and obtain equlibrium solutions of the master equations as well. However, the deficiency zero theorem is best stated, not in terms of stochastic Petri nets, but in terms of another, equivalent, formalism: ‘chemical reaction networks’. So, we explain chemical reaction networks here, and use them heavily throughout the rest of the course. However, because they are applicable to such a large range of problems, we call them simply ‘reaction networks’. Like stochastic Petri nets, they describe how collections of things of different kinds randomly interact and turn into other things.

In Section 18 we consider a simple example of the deficiency zero theorem taken from chemistry: a diatomic gas. In Section 19 we apply the Anderson–Craciun–Kurtz theorem to the same example.

In Section 20 we begin the final phase of the course: proving the deficiency zero theorem, or at least a portion of it. In this section we discuss the concept of ‘deficiency’, which had been introduced before, but not really explained: the definition that makes the deficiency easy to compute is not the one that says what this concept really means. In Section 21 we show how to rewrite the rate equation of a stochastic Petri net—or equivalently, of a reaction network—in terms of a Markov process. This is surprising because the rate equation is nonlinear, while the equation describing a Markov process is linear in the probabilities involved. The trick is to use a nonlinear operation called ‘matrix exponentiation’. In Section 22 we study equilibria for Markov processes. Then, finally, in Section 23, we use these equilbria to obtain equilibrium solutions of the rate equation, completing our treatment of the deficiency zero theorem.

Rolling Circles and Balls (Part 3)

In Part 1 and Part 2 we looked at the delightful curves you get by rolling one circle on another. Now let’s see what happens when you roll one circle inside another!

Four times as big

If you roll a circle inside a circle that’s 4 times as big, we get an astroid:

Puzzle 1. How many times does the rolling circle turn as it rolls all the way around?

By the way: don’t confuse an astroid with an asteroid. They both got their names because someone thought they looked like stars, but that’s where resemblance ends!

You can get an astroid using this funny parody of the equation for a circle:

Or, if you don’t like equations, you can get a quarter of an astroid by letting a ladder slide down a wall and taking a time-lapse photo!

In other words, you get a whole astroid by taking the envelope of all line segments of length 1 going from some point on the x axis to some point on the y axis!

Three times as big

If the fixed circle is just 3 times as big as the one rolling inside it, we get an deltoid:

Puzzle 2. Now how many times does the rolling circle turn as it rolls all the way around?

By the way: it looks like we’re back to naming curves after body parts… but we’re not: both this curve and the muscle called a deltoid got their names because they look like the Greek letter delta:

Puzzle 3. Did the Greek letter delta get that name because it was shaped like a river delta, or was it the other way around?

As you might almost expect by now, if you’ve been reading this whole series, there are weird relations between the deltoid and the astroid.

For example: take a deltoid and shine parallel rays of light at it from any direction. Then the envelope of these rays is an astroid!

We summarize this by saying that the astroid is a catacaustic of the deltoid. This picture is by Xah Lee, who has also made a nice movie of what happens as you rotate the light source:

• Xah Lee, Deltoid catacaustic movie.

I don’t completelly understand the rays going through the deltoid, in either the picture or the movie. It looks like those rays are getting refracted, but that would be a diacaustic, not a catacaustic. ,I think they’re formed by continuing reflected rays to straight lines that go through the deltoid. If you didn’t do that you wouldn’t get a whole astroid, just part of one.

Anyway, at the very least you get part of an astroid, which you can complete to a whole one. And then, as you rotate the light source, the astroid you get rolls around the deltoid in a pleasant manner! This is nicely illustrated here:

• Deltoid catacaustic, Wolfram Mathworld.

You can also get a deltoid from a deltoid! Draw all the osculating circles of the deltoid—that is, circles that match the deltoid’s curvature as well as its slope at the points they touch. The centers of these circles lie on another, larger deltoid:

We summarize this by saying that the evolute of a deltoid is another deltoid.

There are also fancier ways to get deltoids if you know more math. For example, the set of traces of matrices lying in the group SU(3) forms a filled-in deltoid in the complex plane!

This raises a question which unfortunately I’m too lazy to answer myself. So, I’ll just pose it as a puzzle:

Puzzle 4. Is the set of traces of matrices lying in SU(4) a filled-in astroid? In simpler terms, consider the values of that we can get from complex numbers with and Do these values form a filled-in astroid?

Twice as big

But now comes the climax of today’s story: what happens when we let a circle roll inside a circle that’s exactly twice as big?

Now you can see the rolling circle turn around once as it rolls all the way around the big one.

More excitingly, if we track a point on the rolling circle, it traces out a straight line! Can you see why? Later I’ll give a few proofs.

This gadget is called a Tusi couple. You could use it to convert a rolling motion into a vibrating one using some gears. Greg Egan made a nice animation showing how:

The Tusi couple is named after the Persian astronomer and mathematician Nasir al-Din al-Tusi, who discovered it around 1247, when he wrote a commentary on Ptomely’s Almagest, an important astronomical text:

He wrote:

If two coplanar circles, the diameter of one of which is equal to half the diameter of the other, are taken to be internally tangent at a point, and if a point is taken on the smaller circle—and let it be at the point of tangency—and if the two circles move with simple motions in opposite direction in such a way that the motion of the smaller is twice that of the larger so the smaller completes two rotations for each rotation of the larger, then that point will be seen to move on the diameter of the larger circle that initially passes through the point of tangency, oscillating between the endpoints.

I don’t quite understand why he was interested in this, but it has something to do with using epicycles to build linear motion out of circular motion. It also has something to do with the apparent motion of planets between the Earth and the Sun.

Later Copernicus also studied the Tusi couple. He proved that the moving point really did trace out a straight line:

However, many suspect that this was not a true rediscovery: al-Tusi had also proved this, and some aspects of Copernicus’ proof seem too similar to al-Tusi’s to be coincidence:

• George Saaliba, Whose science is Arabic science in Renaissance Europe?, Section 2: Arabic/Islamic science and the Renaissance science in Italy.

• I. N. Veselovsky, Copernicus and Nasir al-Din al-Tusi, Journal for the History of Astronomy 4 (1973), 128–130.

In fact, the Tusi couple goes back way before al-Tusi. It was known to Proclus back around 450 AD! Apparently he wrote about it in his Commentary on the First Book of Euclid. Proclus is mainly famous as a philosopher: people think of him as bringing neo-Platonism to its most refined heights. Given that, it’s no surprise that he also liked math. Indeed, he said:

Wherever there is number, there is beauty.

And of course this is what I’ve been trying to show you throughout this series.

Why the Tusi couple works

So, here are three proofs that a Tusi couple really does trace out a straight line. I posed this as a puzzle on Google+, and here are my favorite three answers. I like them because they’re very different. Since people have thought about Tusi couples since 450 AD, I doubt any of these proofs are original to the people I’m mentioning here! Still, they deserve credit.

The first, due to Omar Antolín Camarena, is in the style of traditional Euclidean geometry.

Let O be the center of the big circle. Let A be the position of the traced point at the instant t when it’s the point of tangency between the circles, let B be its position at some future time t′ and let C be the point of tangency at that same time t′. Let X be the center of the small circle at that future time t′.

We want to prove A, B and O lie on a line. To do this, it suffices to show that the angle AOC equals the angle BOC:

Want: ∠BOC = ∠AOC

The arc AC of the big circle has the same length as the arc BC of the small circle, since they are both the distance rolled between times t and t′. But the big circle is twice as, so the angle BXC on the little circle must be twice the angle AOC on the big circle:

Know: ∠BXC = 2 ∠AOC

But it’s a famous fact in Euclidean geometry that the angle BXC is twice the angle BOC:

Know: ∠BXC = 2 ∠BOC

From the two equations we know, the one we want follows!

Here’s the proof of that ‘famous fact’, in case you forgot it:

The second proof is due to Greg Egan. He distilled it down to a moving picture:

It’s a ‘proof without words’, so you may need to think a while to see how it shows that the Tusi couple traces out a straight line.

The third proof is due to Boris Borcic. It uses complex numbers. Let the big circle be the unit circle in the complex plane. Then the point of contact between the rolling circle and the big one is:

so the center of the rolling circle is:

Since the rolling circle turns around once clockwise as it rolls around the big one, the point whose motion we’re tracking here:

is equal to:

So, this point moves up and down along a vertical line, and its height equals the height of the point of contact,

But the same sort of argument shows that if we track the motion of the opposite point on the rolling circle, it
equals:

So, this opposite point moves back and forth along a horizontal straight line… and its horizontal coordinate equals that of the point of contact!

You can see all this clearly in the animation Borcic made:

The point of contact, the tip of red arrowhead, is The two opposite points on the rolling circle are and So, the rectangle here illustrates the fact that

It’s interesting that this famous formula is hiding in the math of the Tusi couple! But it shouldn’t be surprising, because the Tusi couple is all about building linear motion out of circular motion… or conversely, decomposing a circular motion into linear motions.

Credits

Few of these pictures were made by me, and none of the animations. For most, you can see who created them by clicking on them. The animation of the astroid and deltoid as envelopes come from here:

• Envelope, Math Images Project.

where they were made available under a GNU Free Documentation License. Here’s another animation from there:

This is a way of creating a deltoid as the envelope of some lines called ‘Wallace-Simson lines’. The definition of these lines is so baroque I didn’t dare tell it to you it earlier. But if you’re so bored you’re actually reading these credits, you might enjoy this.

Any triangle can be circumscribed by a circle. If we take any point on this circle, say M, we can drop perpendicular
lines from it to the triangle’s three sides, and get three points, say P, Q and R:

Amazingly, these points lie on a line:

This is the Wallace–Simson line of M. If we move the point M around the circle, we get lots of Wallace–Simson lines… and the envelope of these lines is a deltoid!

By the way: the Wallace–Simson line is named after William Wallace, who wrote about it, and Robert Simson, who didn’t. Don’t confuse it with the Wallace line! That was discovered by Alfred Russel Wallace.

Symmetry and the Fourth Dimension (Part 7)

Lately I’ve been showing you what happens if you start with a Platonic solid and start chopping off its corners, more and more, until you get the dual Platonic solid. There’s just one I haven’t done yet: the tetrahedron.

This is the simplest Platonic solid, so why did I wait and do this it last?

Because the tetrahedron is self-dual. Remember, the Coxeter diagram of this shape looks like this:

V—3—E—3—F

And remember what this diagram means. It means the tetrahedron has

• 3 vertices and 3 edges touching each face,
• 3 edges and 3 faces touching each vertex.

When we take the dual of a solid, we

• replace vertices by faces;
• replace edges by edges;
• replace faces by vertices.

So when we do this to the tetrahedron, we get back a tetrahedron! This ‘self-duality’ is reflected in the symmetry of its Coxeter diagram. If we switch the letters V and F, we get the same thing back—drawn backwards, but that doesn’t matter.

This self-duality also means that when we take a tetrahedron and keep cutting off the corners more and more deeply, we wind up where we started. And in fact, when we reach the halfway point of this process, we start retracing our steps… going backwards!

Let’s see how it goes. Remember, we have a system of diagrams for drawing the most important solids we meet along the way.

Tetrahedron: •—3—o—3—o

We start with the tetrahedron:

Truncated tetrahedron: •—3—•—3—o

Then we get the truncated tetrahedron:

Octahedron: o—3—•—3—o

Then, halfway through, we get a shape we’ve seen before! It’s our friend the octahedron:

Like the other ‘halfway through’ shapes we’ve seen—the cuboctahedron and icosidodecahedron—every edge of the octahedron lies on a great circle’s worth of edges:

        

Puzzle 1. Why does it always work this way?

I don’t actually know!

Truncated tetrahedron: o—3—•—3—•

Then we get back to the truncated tetrahedron:

Tetrahedron: o—3—o—3—•

At the end, we get back where we started… the tetrahedron:

Where are we?

We’ve begun to explore the three great families of semiregular polyhedra:

• the tetrahedron family shown here,

• the cube/octahedron family shown in Part 5,

• and the dodecahedron/icosahedron family shown in Part 6.

We’ve seen that for each family, we have a Coxeter complex, which is summarized by a Coxeter diagram. By coloring the dots in this diagram either white or black, we get different polyhedra in our family.

Our goal is to explore how this works in 4 dimensions. It’s very similar, but much more rich! We can still use Coxeter diagrams, but they’ll have four dots, so there will be more ways to label them, and we’ll get more shapes. And, of course, we’ll have the fun of learning to visualize 4-dimensional shapes!

But before we can explore the 4d story, there’s a hole in the story so far, that I need to fill.

Puzzle 2. Can you guess what it is?

Maybe you’ll see it if you look over our results so far.

The tetrahedron family

Here are the shapes related to the tetrahedron. It has some repeats, because the tetrahedron is its own dual! It also repeats some shapes we’ll see in other families.

tetrahedron		•—3—o—3—o
truncated tetrahedron		•—3—•—3—o
octahedron		o—3—•—3—o
truncated tetrahedron		o—3—•—3—•
tetrahedron		o—3—o—3—•

And here’s the Coxeter complex that runs the show:

This has one right triangle for each element in the group that acts as symmetries of all these shapes. This group has 24 elements, and it’s called the tetrahedral finite reflection group, or A₃. So, we can also call this collection of polyhedra the A₃ family.

The cube/octahedron family

Here are the shapes related to the cube and the octahedron:

cube		•—4—o—3—o
truncated cube		•—4—•—3—o
cuboctahedron		o—4—•—3—o
truncated octahedron		o—4—•—3—•
octahedron		o—4—o—3—•

And here’s the Coxeter complex:

Again, this has one right triangle for each element in the group that acts as symmetries of all these shapes. This group has 48 elements, and it’s called the octahedral finite reflection group, or B₃. So, we can call this collection of polyhedra the B₃ family.

The dodecahedron/icosahedron family

And here are the shapes related to dodecahedron and icosahedron:

dodecahedron		•—5—o—3—o
truncated dodecahedron		•—5—•—3—o
icosidodecahedron		o—5—•—3—o
truncated icosahedron		o—5—•—3—•
icosahedron		o—5—o—3—•

And here’s the Coxeter complex:

Yet again, this has one right triangle for each element in the group that acts as symmetries of all these shapes. This group has 120 elements, and it’s called the icosahedral finite reflection group, or H₃. So, we can call this collection of polyhedra the H₃ family.

Melting Arctic Sea Ice

I’ve been quiet about global warming lately because I’ve decided that people won’t pay much attention until I present some ideas for what to do. But I don’t want you to think I’ve simply stopped paying attention. As you’ve probably heard, the area of the Arctic sea ice hit a new record low this year:

This graph was made using data from the National Snow and Ice Data Center. Lots of other data confirm this; you can see it here.

Here is how the minimum area of Arctic sea has been dropping, based on data from Cryosphere Today:

The volume is dropping even faster, as estimated by PIOMAS, the Pan-Arctic Ice Ocean Modeling and Assimilation System:

The rapid decline has taken a lot of experts by surprise. Neven Acropolis, who keeps a hawk’s eye on these matters at the Arctic Sea Ice Blog, writes:

Basically, I’m at a loss for words, and not just because my jaw has dropped and won’t go back up as long as I’m looking at the graphs. I’m also at a loss—and I have already said it a couple of times this year—because I just don’t know what to expect any longer. I had a very steep learning curve in the past two years. We all did. But it feels as if everything I’ve learned has become obsolete. As if you’ve learned to play the guitar a bit in two years’ time, and then all of a sudden have to play a xylophone. Will trend lines go even lower, or will the remaining ice pack with its edges so close to the North Pole start to freeze up?

Basically I have nothing to offer right now except short posts when yet another of those record dominoes has fallen. Hopefully I can come up with some useful post-melting season analysis when I return from a two-week holiday.

I’m at a loss at this loss. The 2007 record that stunned everyone, gets shattered without 2007 weather conditions. The ice is thin. PIOMAS was/is right.

The big question, of course, is how this should affect what we do. David Spratt put it this way:

The 2007 IPPC report suggested that by 2100 Arctic sea-ice would likely exist in summer, though at a much reduced extent. Because many of the Arctic’s climate system tipping points are significantly related to the loss of sea-ice, the implication was that the world had some reasonable time to eliminate greenhouse emissions, and still be on time to “save the Arctic”. The 2007 IPCC-framed goal of reducing emissions 25 to 40 per cent by 2020 and 80 per cent by 2050 would “do the job” for the Arctic.

But the physical world didn’t agree. By 2006, scientist Richard Alley had observed that the Arctic was already melting “100 years ahead of schedule”. But the Arctic is not melting 100 years ahead of schedule: the climate system appears to be more sensitive to perturbations than anticipated, with observations showing many climate change impacts happening more quickly and at lower temperatures that projected, of which the Arctic is a prime example.

Politically, we are 100 years behind where we need to be on emissions reductions.

Or carbon sequestration. Or geoengineering. Or preparing to live in a hotter world.

Rolling Circles and Balls (Part 2)

Last time we rolled a circle on another circle the same size, and looked at the curve traced out by a point on the rolling circle:

It’s called a cardioid.

But suppose we roll a circle on another circle that’s twice as big. Then we get a nephroid:

Puzzle 1. How many times does the small circle rotate as it rolls all the way around the big one here?

By the way, the name ‘cardioid’ comes from the Greek word for ‘heart’. The name ‘nephroid’ comes from the Greek word for a less noble organ: the kidney! But the Greeks didn’t talk about cardioids or nephroids—these names were invented in modern times.

Here are my 7 favorite ways to get a nephroid:

1) The way just described: roll a circle on a circle twice as big, and track the path of a point.

2) Alternatively, take a circle that’s one and a half times big as another, fit it around that smaller one, roll it around, and let one of its points trace out a curve. Again you get a nephroid!

3) Take a semicircle, point it upwards, shine parallel rays of light straight down at it, and let those rays reflect off it. The envelope of the reflected rays will be half of a nephroid:

This was discovered by Huygens in 1678, in his work on light. He was really big on the study of curves.

As I mentioned last time, a catacaustic is a curve formed as the envelope of rays emanating from a specified point and reflecting off a given curve. We can stretch the rules a bit and let that point be a ‘point at infinity’. Then the rays will be parallel. So, we’re seeing that the nephroid is a catacaustic of a circle.

Last time we saw the cardioid is also a catacaustic of a circle, but with light emanating from a point on the circle. It’s neat that the cardioid and nephroid both show up as catacaustics of the circle. But it’s just the beginning of the fun…

4) The nephroid is the catacaustic of the cardioid, if we let the light emanate from the cardioid’s cusp!

This was discovered by Jacques Bernoulli in 1692.

5) Let two points move around a circle, starting at the same place, but with one moving 3 times as fast as the other. At each moment connect them with a line. The envelope of these lines is a nephroid!

Last time we saw that if we replace the number 3 by 2 here, we get a cardioid. So, this is yet another way these two curves are related!

6) Draw a circle in blue and draw its diameter as a vertical line. Then draw all the circles that have their center somewhere on that blue circle, and are tangent to that vertical line. You get a nephroid:

The red circle here has the red dot as its center, and it’s tangent to a point on the vertical line. Here’s a nice animation of the process, made available by the Math Images Project under a GNU Free Documentation License:

7) Finally, here’s how to draw a nephroid starting with a nephroid! Draw all the osculating circles of the nephroid—that is, circles that match the nephroid’s curvature as well as its slope at the points they touch. The centers of these circles give another nephroid:

This trick is an example of an ‘evolute’. The evolute of a curve is the set of centers of the osculating circles of that curve. Last time we saw the evolute of a cardioid is another cardioid. Now we’re seeing the nephroid shares this property!

Apparently the same is true for all curves formed by rolling one circle on another. These curves are called epicycloids. In a sense, these are the mathematical leftovers of the theory of epicycles in astronomy.

It would be nice if some of the funny relations we’ve been seeing between the cardioid and the nephroid generalize to relations between the epicycloid with k cusps and the one with k+1 cusps. But I don’t know if that’s true.

It would also be nice if the epicycloids with more and more cusps were named after increasingly disgusting organs of the body. But in fact, I don’t know any special names for them once we reach k = 3.

Puzzle 2. Use one of the 7 constructions above to get an equation for the nephroid. What is the simplest equation you can find for this curve?

Next time

Most of the pictures above are from Wikicommons, but the picture of the nephroid as a catacaustic of the cardioid is from Xah Lee’s wonderful website on plane curves. As usual, you can click on the pictures and get more informatino.

My ultimate goal is to tell you some amazing things about what happens when you roll one ball on another that’s exactly 3 times as big. These things have nothing to do with plane curves, actually. But I’ve been taking many detours, and next time I’ll talk about some curves formed by rolling one circle inside another!

Right now, thought I need a break. I need to stop thinking about all these curves. I think I’ll get a cup of coffee.

Rolling Circles and Balls (Part 1)

For over a decade I’ve been struggling with certain math puzzle, first with the help of James Dolan and later also with John Huerta. It’s about something amazing that happens when you roll a ball on another ball that’s exactly 3 times as big. John Huerta and I just finished a paper about it, and I’d like to explain that here.

But I’d like to ease into it slowly, so I’ll start by talking about what happens when you roll a circle on another circle that’s the exact same size:

Can you see how the rolling circle rotates twice as it rolls around the fixed circle once? Do you understand why?

The heart-shaped curve traced out by any point on the rolling circle is called a cardioid. In her latest video Vi Hart pretends to complain about parabolas while actually telling us quite a lot about them, and much else too:

Naturally, with her last name, she prefers the cardioid. She describes various ways to draw this curve: for example, by turning the hated parabola inside out. Here are my 6 favorite ways:

1) The one we’ve seen already: roll a circle on another circle the same size, and track the motion of a point on the rolling circle:

2) Take a parabola and ‘turn it inside out’, replacing each point with polar coordinates (r, θ) by a point with coordinates (1/r, θ). As long as your parabola doesn’t contain the origin, you get a cardioid:

This ‘turning inside out’ trick is called conformal inversion.

3) Draw all circles whose centers are points on a fixed circle, and which contain a specified point on that circle:

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Here’s a nice animation of this process, made available by the Math Images Project under a GNU Free Documentation License:

4) Let light rays emanate from one point on a circle and reflect off all other points on that circle. Draw all these reflected rays, and you’ll see a cardioid:

If you draw all light rays that reflect off some curve, the curve they snuggle up against (their so-called envelope) is called a catacaustic.

5) Draw 36 equally spaced points on a circle numbered 0 to 35, and draw a line between each point n and the point 2n modulo 36. You’ll see a cardioid, approximately:

But there’s nothing special about the number 36. If you take more evenly spaced points, you get a better approximation to a cardioid. You get a perfect cardioid if you connect each point (1, θ) to the point (1, 2θ) with a line, and take the envelope of these lines.

6) Finally, here’s how to draw a cardioid starting with a cardioid! Draw all the osculating circles of the cardioid—that is, circles match the cardioid’s curvature as well as its slope at the points they touch. The centers of these circles give another cardioid:

This picture has some distracting lines on it; just look at the big and the little cardioid, and the circles. This trick is an example of an ‘evolute’. The evolute of a curve is the set of centers of the osculating circles of that curve.

All the pictures above are from Xah Lee’s wonderful website or the Wikipedia article on cardioids. Click on the picture to see where it came from and get more information.

I ❤ cardioids!

Next time we’ll see what happens when we roll a circle inside a circle that’s exactly twice as big.

Constructions on curves

We’ve seen a few constructions on curves:

• A roulette is the curve traced out by a point attached to a given curve as it rolls without slipping along a second curve.

We rolled a circle on a circle and got a cardioid, but you could roll a parabola on another parabola:

This gives a curve called the cissoid of Diocles, which in some coordinate system (not the one shown) is given by this cubic equation:

• A catacaustic is the envelope of rays emanating from a specified point (perhaps a point at infinite distance, which produces parallel rays) and reflecting off a given curve.

We’ve obtained the cardioid as a catacaustic of the circle. Supposedly if you take the cissoid of Diocles and form its catacaustic using rays emanating from its ‘focus’, you get a cardioid! This would be a seventh way to get a cardioid, but I don’t understand it, even though it’s described on Wolfram Mathworld. I don’t even know what the ‘focus’ of a cissoid of Diocles is. Can you help?

• The evolute of a curve is the curve formed by the centers of its tangent circles.

We’ve seen that the cardioid is its own evolute. The evolute of an ellipse looks like this:

It’s called an astroid, and it’s given by an equation of this form:

If you take the tangent circles of the black points on the ellipse above, their centers are the sharp pointy ‘cusps’ of the astroid.

Order from chaos?

Some other famous ways to construct new curves from old ones include the involute, the isoptic, and the pedal. I could describe them… but I won’t. You get the picture: there’s a zoo of curves and constructions on curves, and lots of relations between these constructions. It’s all very beautiful, but also a bit of a mess.

It seems that all these constructions, and their relations, should be studied more systematically in algebraic geometry. It may seem like a somewhat musty and old-fashioned branch of algebraic geometry, but surely there’s a way to make it new and fresh using modern math. Has someone done this?