Archive for May, 2010|Monthly archive page

Interlude about geometry

Before I try to explain what’s up with three-dimensional manifolds, I need to talk a bit about geometry, specifically spherical vs. Euclidean vs. hyperbolic geometry.

Euclidean geometry is “high-school” geometry, the geometry of the plane put into axiomatic form by Euclid of Alexandria. By which I mean, Euclid replaced the study of actual physical points and lines with the study of formal mathematical objects called “points” and “lines”. Euclid’s “points” and “lines” were inspired by real-world points and lines, and have properties based on the properties of real-world points and lines. For example, there’s a single, unique “line” through any two distinct “points”. But Euclid’s “points” and “lines” are just a formalism. They have the same relationship to real-world points and lines that poems have to roses.

And like poems, Euclid’s formalism permits analogies.

What if, instead of a plane, we consider the surface of the sphere? And instead of straight lines in the plane, what if we define a “line” to be a great circle on the surface of the sphere, such as the equator? Then we get spherical geometry instead of Euclidean geometry. The “lines” in spherical geometry don’t look like the “lines” in Euclidean geometry, but they act like lines in lots of important ways: they intersect in points, they subtend angles with one another, their segments have well-defined lengths, and so on. The formalisms of Euclid’s geometry are just a language, and by applying that same language to spherical geometry we can highlight the similarities between the two.

At the same time, we discover the differences. Triangles, for example: in Euclidean geometry, the angles of a triangle always add up to 180 degrees, but in spherical geometry they always add up to more than 180 degrees. Spherical geometry is finite: no shape can have an area greater than the area of the entire sphere. There’s no such thing as similar triangles in spherical geometry. Also, most famously (if that’s the word) there’s no such thing as parallel lines.

Hyperbolic geometry, like spherical geometry, uses the language of “points” and “lines” to describe a completely different space from the Euclidean geometry. But hyperbolic geometry is much stranger and weirder than spherical geometry. It’s hard to visualize: it doesn’t live on any surface in Euclidean space the way that spherical geometry lives on the sphere. And it turns the properties of spherical geometry on their heads. The angles of a triangle in hyperbolic geometry always add up to less than 180 degrees. Hyperbolic geometry is infinite…but bizarrely, no triangle can have area greater than pi. (Circles, on the other hand, grow as large as they want, exponentially with respect to their radius.) There are no similar triangles in hyperbolic geometry, but there area infinitely many lines “parallel” to any given line. And so on.

It’s just bizarre. It’s beautiful and it’s frustrating and it’s bizarre, and I’ve been studying it for so long that I’m more comfortable with it than I am with Euclidean geometry most of the time. I live in a Euclidean world, but I tend to think in a hyperbolic one.

Next time I’ll try to explain what this has to do with manifolds.

Topology and manifolds, part two

Continuing my detailed answer to the question, “So what do you do?”…

When a mathematician sees the definition of a mathematical object, they want to know what all the possible examples of that object are. It’s a pretty hard-wired reflex. So what are all the possible examples of manifolds?

Well, that’s a rather huge question. One-dimensional manifolds are easy: the circle and the real line. That’s it. Two-dimensional manifolds, i.e. surfaces, are harder, but the question has largely been answered. If we restrict ourselves to compact two-dimensional manifolds, then the question was answered at the beginning of the twentieth century, although the i’s weren’t dotted and the t’s weren’t crossed until the 1920’s.

I haven’t explained what compact means yet. Compact is, loosely speaking, a geometrical notion of finiteness. Of the two one-dimensional manifolds, the circle is compact, but the real line is not, because of the way it extends to infinity. Compact manifolds are an important subclass of manifolds, if only because they’re easier to study.

The list of compact two-dimensional manifolds starts with the sphere, e.g. the surface of the earth. Next is the torus, e.g. the surface of a doughnut. After that, picture the surface of a doughnut with two holes. Mathematicians usually just call this a “two-holed torus” or a “genus-two surface” if they’re getting technical. (I once misread an old topology book which used the term “quoit” and thought the word referred to something with the shape of a genus-two surface, but it turns out a “quoit” is just a ring. Pity.) Then there’s a three-holed torus, a four-holed torus, and so on.

That’s not the complete list. Have you ever seen a Klein bottle? Wonderful stuff. A Klein bottle is an example of a non-orientable surface, a class of two-dimensional manifolds that can’t actually be constructed in three-dimensional space. (The examples at the link cheat a bit by letting the surface pass through itself.) There’s a list of non-orientable two-dimensional manifolds that parallels the list of many-holed tori, of which the Klein bottle is the second on the list. The first is the projective plane, which is a Moebius band glued to a disk in a wonderfully mind-bending way which I won’t go into here.

So: sphere, torus, two-holed torus, and so on, plus projective plane, Klein bottle, and so on. Two simple, if infinite, lists containing every possible compact two-dimensional manifold. What’s more, if you present a mathematician with some arbitrary compact two-dimensional manifold, it takes only a few very simple calculations before the mathematician can tell you where in the two lists the manifold has to appear. It’s the best possible outcome to this kind of mathematical problem.

What about three-dimensional manifolds?

Oy. You have no idea.

Topology and manifolds, part one

One additional caveat: this is not going to be as detailed as it could be. If you’re looking for a detailed technical online resource I figure Wikipedia, Wolfram MathWorld, and half a dozen other websites already have that covered. This is just me, writing my detailed answer to “So what do you do?” Onward.

What is a manifold? It’s something on which we can do a generalized form of geometry1.

“Geometry” is a good starting point. It’s something most people can visualize neatly: triangles, circles, and angles are familiar concepts. For many people it’s also their first and only exposure to proofs, the specialized, stilted form of language that is mathematicians’ bread and butter. Euclid of Alexandria is so famous for these kinds of formal arguments that the high school geometry of triangles, circles, and angles is properly known as Euclidean geometry.

The problem is, Euclidean geometry is a bit too abstract. It takes place on an infinite, flat plane, an ideal object bearing no resemblance to that thing we build skyscrapers, tennis courts, and particle accelerators on: the surface of the Earth, which is neither infinite nor flat. The surface of the Earth looks infinite…okay, really big…and it looks kinda-sorta flat depending on where you are, but that’s just because we’re really tiny and close to the ground. In reality, it’s a surface not at all like the ideal Euclidean plane. For that and other reasons, it’s necessary to generalize the mathematics of the Euclidean plane to other, more interesting shapes.

That’s where manifolds come in. A manifold is something which, like the surface of the Earth, looks like the Euclidean plane if you’re very small and standing right up next to it.

The technical definition of a manifold involves charts and atlases: a manifold is something which has an atlas made of charts, where each chart looks likes a portion of the Euclidean plane. The charts necessarily overlap, and together they map the entire manifold. On any one chart we can attempt to apply the mathematics of the Euclidean plane, but we can’t necessarily do that to the entire manifold, much like how we can lay flat any chart in an atlas of the Earth but can’t lay flat an entire globe.

The technical definition also has a very great deal to say about how those charts overlap. The short version is, how smoothly the charts overlap determines how much and what kind of mathematics you can do on the manifold. But let’s not go there.

I also should point out that what I’ve talking about so far are two-dimensional manifolds. We can define, say, three-dimensional manifolds by using “charts” which look like portions of Euclidean three-dimensional space. How such charts can come together to form an entire atlas is already much harder to visualize than in the two-dimensional case. It’s also a lot more interesting, in many ways. More on that later.

1And calculus. In fact calculus on manifolds is probably more important than geometry on manifolds. But the word “calculus” tends to frighten people.

Topology and manifolds, introduction

Be a pure mathematician for any length of time and you will come to dread the question “So what do you do?” Okay, perhaps “dread” is too strong a word. You’ll at least learn to limit your answer, because the questioner is almost certainly not asking “Would you tell me in detail about your job?” More likely, what the questioner is really asking is “Would you please summarize what you do in the hope that it will provide a hook for further small talk?” (It won’t.)

The correct answer is “I’m a mathematician.” Occasionally people will ask for more detail, at which point you trot out a brief summary of your field of study that you’ve refined over and over again with family members. Very, very rarely, you’ll be asked to explain in detail. Cherish these moments.

Because those moments come up far too infrequently, I thought I might write a detailed explanation here. More to come in future posts.

(Yeah, this post is just to shame me into writing the other posts later.)

Late to the party, as always

Hi ho!

What will this space be filled with in the future? Wit? Profundity? Pathos? Horror? Cat macros?

Probably not cat macros.

Possibly topology. Possibly computer code related to topology. Probably various banal things as strikes my fancy.