Fun Equations of the Torus (FEniCS part 2)

The story so far: we wished to create a horseshoe-like initial condition on the torus. We needed to (1) come up with the equation for the bounding curve, and (2) distinguish between the inside and outside. One tool is perfect for the job: level sets (another common name for them is isosurfaces, and some more dimension-specific terms like level contours or level surfaces). Surfaces (or curves) of the form f(x,y,z) = c (or f(x,y) = c). In order to do this, we first start with the classic parametrization of the torus as a surface of revolution:

x = (R+r cos(u))cos(θ)
y = (R+r cos(u))sin(θ)
z = r sin(u),

with 0 ≤ u ≤ 2π and 0 ≤ θ ≤ 2π. What this is saying is that we take a circle of radius r in a vertical plane (its angle coordinate is u) and bring it out so its center is at radius R from the origin and still lying in the xy-plane (that's why the R only affects the term with r cos(u)). Then we revolve it around the z-axis, which is where the cos(θ) and sin(θ) come in. We need to come up with an equation in terms of x, y, and z.

If we square these three coordinates and add, we get

x² + y² + z² = (R+r cos(u))² + r²sin(u)² = R²+2Rr cos(u) + r².

However, r cos(u) is simply the extra distance one has to go from the radius R to the coordinates (x,y,0), that is, r cos(u) =  ρ-R, where ρ² = x² + y² is the usual polar coordinate (click to enlarge):

This gives us

x² + y² + z² = (R+r cos(u))² + r²sin(u)² = R²+2R(ρ-R)+ r² = r²+ 2Rρ – R².

Moving everything on the other side, except 2Rρ, and dividing by 2R, we finally get

(x² + y² + z² + R² – r²)/(2R) = ρ,

or

(x² + y² + z² + R² – r²)²/(4R²)= x² + y², which finally gives us

f(x,y,z) = (x² + y² + z² + R² – r²)²/(4R²) – x² – y² = 0.

This is the Cartesian equation of the torus (really, a Cartesian equation, because there are several). Of course to be really proper, we should include dependence on R and r in there too (by including that in Mathematica, I got to see how the surfaces change as I vary each parameter). The torus plus its solid interior (a donut!) is given by replacing it with a ≤ rather than a = (note also we can determine which side is inside and which side is outside by computing the differential df, also variously known as the gradient vector–it points in the direction of increasing f). To get nearby shapes, we can consider other level sets f(x,y,z) = c for c near to 0. The picture introducing this post is precisely this, for c in the interval [-1,1] with increments of 1/4. Notice that the largest two of these has no hole (look carefully at the upper center part of the picture). So it's almost nested tori.

The interesting thing about using a level set description of a family of surfaces is that nothing special happens when some topological transition (shape with a hole becoming a shape without a hole) occurs. Well, almost nothing special—the transition point is precisely when c is not a regular value of f, that is, there are one or more points on the level set for which the gradient df vanishes at p. Such ps are called critical points (and is the starting point of Morse theory). Making use of level sets and tracking various geometric dynamics (surfaces, interfaces, etc) using these concepts in numerical methods is the level set method. The trickiness, however, is finding the right level set function to use, and it can be more of an art than a science (although I don't necessarily think the strict separation of art and science is that great of a thing. But that's a story for another day).

Back to our original problem, however. We wanted to see what the cross section looks like and how it changes as we slice a bagel at different angles. This is quite simple to do in terms of level sets (which is why we went through all the trouble to derive it). We can imagine the slicing plane to be z = mx + b. The slope m is the tangent of the angle α between the slicing plane and the horizontal xy-plane. And, of course, we wish it to lie on the surface of the torus, so that we have a system of equations

z = mx + b,
f(x,y,z) = (x² + y² + z² + R² – r²)²/(4R²) – x² – y² = 0.

The key is, then, to substitute one into the other, getting

g(x,y) =  (x² + y² + (mx + b)² + R² – r²)²/(4R²) – x² – y² = 0.

If we consider this as a function of x and y only, that is, a subset of the plane, it gives the xy-projection of the 3D slice. To get the interior of the cross section, we use a ≤ instead of a = as before. The values chosen here are r = 1, R = 1.61803398874 (the Golden Ratio), m = 1/2 and b = 1/3. And that gives us our inside vs. the outside of the domain we used in the heat equation. If we had wanted to make it more circular, we could eliminate the projection by rotating the entire picture by the angle α. This makes the equations much more complicated, however.

Fun with FEniCS

Today's entry is also inspired by another talk. We had Anders Logg of Simula Research and one of the developers at The FEniCS Project come by, all the way from Norway. FEniCS is one of the main pieces of software I am learning to use. Now, I started the day not thinking I would really encounter nested tori (and don't think, of course, every post is going to be about them!). But believe it or not, they made completely unexpected side entrance. I was testing out its capabilities for time evolution (simply solving the heat equation on a square with zero Dirichlet boundary conditions) and got it running with some trivial initial data. But as another interesting test run, I figured I should try some nontrivial initial data. My basic goal was to see if I could recreate this (from Wikipedia, LOL):

Now, where was I gonna get that horseshoe?

From nested tori, of course. If you slice a torus at an angle (this can easily be tested out with a good chef's knife and a bagel), you may get completely different-looking slices depending on the angle. In the plane of the bagel slice, of course, you get an annulus... the one that you spread cream cheese on. Completely perpendicular, you will get either two circle-like curves (if the knife cuts through the hole), a figure 8 (if the knife just grazes the hole), or a single oval like curve (if the knife cuts the side). Changing the angle will make any of these configurations into an annulus. For example, the two circles will "reach out to touch each other" as the angle changes, and then the other ends start wrapping around the hole. Just before those ends join, it becomes something like a letter "C" or a horseshoe (pause the movie about halfway in and look at the red):

Anyway, supplying the initial data as 1 inside the horseshoe and 0 outside (the interpolating part of the software actually enforces continuity, albeit with some very steep slopes), we get this:

Now just how exactly did I determine what was "inside" the horseshoe and "outside" it? I used level sets. That's where the nested tori come in. But I will have to leave my dear readers in suspense, and tell you about it in another entry. But this has been quite an adventure!

Singular Value Decomposition Delights, Part 2

First off, congratulations to math homie and guest blogger Adam on his advancement to candidacy! Now back to our explorations from last week. So the goal of the previous vector bundle problem is to consider a vector bundle over the circle, represented as planes over an interval such that the fibers at one end of the interval are identified with the other end using some linear transformation T. In order to visualize this, we shall adopt the following approach: we put our base space circle in the xy-plane. arrange the fiber planes perpendicularly to circle (planes θ = constant), like so:

and then we choose a reference circle in one of these planes (say, of radius 1). Then as we move around the main circle, we consider the surface swept out as we carry that circle with us and apply the interpolated transformation

A(t) = exp(tξ) Σ^texp(–tη)

to that circle as we go. When we come around to the beginning, this circle should be deformed by the original transformation T. We do the experiment with the matrix

0.498502 0.214052
0.0433899 0.60181

which is a rotation of -2.62394 radians (-150.3°), then stretching with singular values 0.69543 and 0.41803, and finally applying a rotation of 2.24233 radians (128.5°). We start with the identity matrix, and as we move around, rotate the appropriate fraction t, times of the first angle, stretch by the second factor to the t power, and finally rotate by the same fraction of the second angle. So at time t = 1, a full revolution about the main circle, we get full operator T applied to it. This is the surface swept out:

And if we kept going, it would spiral inward more and more, which when doing a cutaway view (essentially, starting only with, say, 3/4 or 1/2 of the initial circle), we get the original picture shown in Part 1. But actually, it is more interesting to see what happens if we just unfurl it in a straight line (in the spirit of the original definition as an identification space)... and let the mapping continue past t = 1. The result is the title picture, which shows what happens when we proceed to t = 3 (which corresponds to the transformation T³). Forget donuts, let's do pasta (more comprehensively, here)!

Of course, we know that this shows our bundle is trivial—the only way to get a nontrivial bundle this way is to have a transformation with a negative singular value, in which case, since we can't raise negative numbers to real powers and stay in the reals, makes the above construction fail, as it should (since a nontrivial bundle fails to have a continuous global frame). So what good is this, really, besides making things pretty? As it turns out, such oddly behaving, yet topologically trivial bundles, do have some interesting, different geometry. It is encoded in the notion of connection—a connection will reveal nontrivialities in the geometry that topology cannot (in fact, Riemannian geometry starts out by considering a connection specifically on the tangent bundle, the Levi-Civita [LAY-vee CHEE-vee-tah] connection). This is an interesting topic for yet another part (I'm not promising it soon, however!).

An unexpectedly delightful application of the Singular Value Decomposition

We begin this entry with a bit of commentary on numerical analysis. If there's anything that I think that should be taught in math classes more, it is the Singular Value Decomposition of a matrix. Every real n × m matrix A can be factored as A = UΣV* where U and V are orthogonal, and Σ is "diagonal" (but can be rectangular). It is in fact more useful than the much more often discussed eigenvalues and eigenvectors (which often are computed for symmetric matrices where the two concepts coincide, anyway). It would probably do a great amount of good for people's intuition in linear algebra classes. For some reason, it is completely ignored in a lot of abstract linear algebra books, somehow having acquired a reputation of being "too applied," and thus would sully the purity of their most precious subject! I never heard it talked about in any upper division or graduate math class, until I'd somewhat belatedly discovered that numerics was really the best place to go to explore my interests (better late than never). But the SVD is absolutely ubiquitous in science. The first place I'd heard of it was a linguistics class, where the prof talked about some paper on latent semantic analysis. As I understood it, the SVD was essentially used to compute axes of word meanings(!!) and the information used to categorize the words. That was really a long time ago, so forgive me, linguists, if this seems a gross mischaracterization. In any case, it also is intimately connected with least squares (briefly mentioned in the last entry).

At this point, the sufficiently attention-spanned reader (all too rare these days!) is most certainly scratching their head (yes, this blog will use singular they) wondering what that has to do with the very out-sized picture. First of all, the picture generated actually is a trivial case in which the SVD isn't actually needed. But if it wasn't for the SVD, I would have never been inspired to make a picture of my findings, so it deserves the tribute. It is inspired by trying to explore the concept of orientation, and to generalize constructions like the famous Möbius Strip:

We can generalize the Möbius strip by the use of vector bundles and identification spaces. We get the topological twist in a Möbius strip by gluing the ends of the usual strip with a flip. If we allow ourselves some imagination, one way to generalize is to first imagine the Möbius strip as now being constructed from a strip that is infinitely tall ([0,1] × ℝ) to get the Möbius vector bundle. Now when we "glue" the ends together, we can still flip it (identify (0,x) with (1,-x)). Or, we can stretch it and identify, say, (0, x) to (0, 2x), the "strained cylinder" (as Roger Penrose calls it in The Road to Reality). If we have to restrict ourselves to a finite picture, what happens here is that it goes around in a loop and reconnects to itself twice as high, so it looks discontinuous. But it's not. It turns out to be, in fact, a cylinder and it is in fact orientable.

Taking a cue from a problem in Spivak's excellent 5-volume series, A Comprehensive Introduction to Differential Geometry, we can imagine that instead of starting out with just a line for the vertical dimension, we could have a whole space ℝⁿ, and identify the initial and final spaces by an invertible linear transformation T : ℝⁿ → ℝⁿ, namely, identifying (0,v) to (1,Tv). If T has positive determinant, then we have not made anything topologically new: it's S¹ × ℝⁿ. But to actually (constructively) prove this is a more difficult task. But one way it can be done is by choosing a global frame, n continuous vector fields that are a basis at every point. And one way to do that is to somehow connect the transformation T to the identity. This can be done any number of ways, since, in fancy-schmancy speak, the orientation-preserving elements of the general linear group form a path-connected component containing the identity. But one quickly implementable realization is to use the SVD! Namely, factor the matrix T as UΣV* with U and V orthogonal and Σ the singular values, which must all be positive if the matrix has positive determinant. Since it is a diagonal matrix of positive numbers, Σ can be raised to any real power t. And U, V are respectively exponentials of some antisymmetric matrices ξ and η (which are not unique). So given some basis vectors v_i at 0, we consider the section

t ⟼ (t, exp(tξ)Σ^texp(–tη)v_i).

At time t = 0 it is just (t, v_i). At time t = 1, it is (1, UΣV*v_i) = (1, Tv_i), which is identified with (0,v_i). (Warning: Although we could make things more notationally uniform by writing Σ^t = exp(tσ) where σ is the diagonal matrix consisting of the logarithms of the original singular values, we must resist the temptation to compute the result as exp(t(ξ+σ–η)), despite it being called an "exponential" map, as that only works when all the matrices commute!!) The path is continuous and so we have proved that there is indeed a continuous, globally defined frame, and thus our bundle is trivial. Beautiful? Well, where are the pictures? The picture above is given by considering the case n = 2, that of a plane, and the transformation by scaling by 2. If we take circular set in the initial plane, we can see how this transformation acts (here the U and V are the identity and Σ is just twice the identiy) and make it go 'round and 'round (without identifying), we get something that is reminiscent of nested tori (but not really, as it is just one torus spiraling into itself!). We're not done with the awesomeness of this example quite yet though—I've got some other math to get back to! But there are more interesting visualizations from this example, so stay tuned!

Some Unnested, Sprinkled Donuts

Great talk on Tuesday by Professor Anil Hirani on harmonic differential forms and generally bringing back things like geometry, topology, and algebra back into numerical analysis. Here's a sampling of what was talked about.

Harmonic differential forms are the basic topological obstructions to finding of solutions of Poisson's equation (∆u=f). Solutions to Poisson's equation represent steady-state phenomena (the scalar version representing heat, and the vector or differential form versions representing flows). On a closed surface, if we have a source term f, it must balance out (sum to zero over the whole surface), so that as much material is being created as it is being destroyed. Otherwise, of course, we could not have a steady state (more and more stuff just keeps getting added!). However, if the domain has some kind of topological nontriviality, such as holes (which could be represented as obstacles of some sort), different kinds of steady-state solutions can exist without sources. This is because stuff can flow "around" the holes, our "out of a cavity" (even in the idealized case that it's a single point and infintesimally small). But if the holes are not there, then it is often true that zero is the only field that satisfies the (reasonable) demand that the flow be continuous at that point. On other things besides donuts, we have planar domains with holes (lotus root slices), whose harmonic fields do vanish somewhere.

And here is an example of flow out of a cavity: 

We call this example the Koosh [Thanks to fellow math homie Adam for locating the slide with this... I didn't get a pic of it during the talk. He'll be one of our guest bloggers from time to time!]

The topological object that describes these fields more precisely is the de Rham cohomology of the space, equal to the "locally constant fields modulo the trivially constant ones" as described by Bott and Tu in their nice book Differential Forms in Algebraic Topology (Prof. Hirani recommends that if there's any class that one should take in graduate school, it's Algebraic Topology. I have my reservations about that, but he also said that we might not know what he is talking about when he says that!). In any case, there is a notion of closed form, which is a form which when integrated over a surface of the appropriate dimension, it is unchanged under small perturbations of the surface, holding its boundary fixed. Too large of a perturbation may cause the surface to cross over one of these topological nontrivialities, which will in fact cause a jump in the value of the integral. On the other hand, if the form is exact, it has what is known as a potential, which and the integral is unchanged regardless of the surface, so long as it has the same boundary, by virtue of Stokes's Theorem.

As abstract as this all seems, it has some concrete interpretations, for example, rankings of sports teams (or the Netflix Problem), where everybody ranks things by comparing two things, and we want to figure out some sort of global, overall ranking. Cohomology tells us the level of inconsistency one may achieve---one can get into situations where A is better than B is better than C is better than A. This also has applications in economics. In order to take advantage of all of this for applications, we obviously have to have a method of computing these things. Geometry allows us to use a very common, bread-and-butter tool of scientists everywhere: least squares. Finding a harmonic form is equivalent to finding a form within the same cohomology class of minimal norm. It is quite beautiful how all these techniques work well together!

Welcome to Nested Tori!

The goal of this blog is will be to collect some AWESOME visualizations in mathematics and other scientific disciplines, and show how it is possible to get them. Even if you aren't the biggest math fan, stick around anyway and enjoy the view!

Clifford tori, a foliation of $S^3$ and stereographically projected to $\mathbb{R}^3$

Another view, rendered in Mathematica, this time sliced with a vertical plane, rather than restricting parameter values directly

More details here!