The Summer 2017 Mathematics A To Z: Topology


Today’s glossary entry comes from Elke Stangl, author of the Elkemental Force blog. I’ll do my best, although it would have made my essay a bit easier if I’d had the chance to do another topic first. We’ll get there.

Summer 2017 Mathematics A to Z, featuring a coati (it's kind of the Latin American raccoon) looking over alphabet blocks, with a lot of equations in the background.
Art courtesy of Thomas K Dye, creator of the web comic Newshounds. He has a Patreon for those able to support his work. He’s also open for commissions, starting from US$10.

Topology.

Start with a universe. Nice thing to have around. Call it ‘M’. I’ll get to why that name.

I’ve talked a fair bit about weird mathematical objects that need some bundle of traits to be interesting. So this will change the pace some. Here, I request only that the universe have a concept of “sets”. OK, that carries a little baggage along with it. We have to have intersections and unions. Those come about from having pairs of sets. The intersection of two sets is all the things that are in both sets simultaneously. The union of two sets is all the things that are in one set, or the other, or both simultaneously. But it’s hard to think of something that could have sets that couldn’t have intersections and unions.

So from your universe ‘M’ create a new collection of things. Call it ‘T’. I’ll get to why that name. But if you’ve formed a guess about why, then you know. So I suppose I don’t need to say why, now. ‘T’ is a collection of subsets of ‘M’. Now let’s suppose these four things are true.

First. ‘M’ is one of the sets in ‘T’.

Second. The empty set ∅ (which has nothing at all in it) is one of the sets in ‘T’.

Third. Whenever two sets are in ‘T’, their intersection is also in ‘T’.

Fourth. Whenever two (or more) sets are in ‘T’, their union is also in ‘T’.

Got all that? I imagine a lot of shrugging and head-nodding out there. So let’s take that. Your universe ‘M’ and your collection of sets ‘T’ are a topology. And that’s that.

Yeah, that’s never that. Let me put in some more text. Suppose we have a universe that consists of two symbols, say, ‘a’ and ‘b’. There’s four distinct topologies you can make of that. Take the universe plus the collection of sets {∅}, {a}, {b}, and {a, b}. That’s a topology. Try it out. That’s the first collection you would probably think of.

Here’s another collection. Take this two-thing universe and the collection of sets {∅}, {a}, and {a, b}. That’s another topology and you might want to double-check that. Or there’s this one: the universe and the collection of sets {∅}, {b}, and {a, b}. Last one: the universe and the collection of sets {∅} and {a, b} and nothing else. That one barely looks legitimate, but it is. Not a topology: the universe and the collection of sets {∅}, {a}, and {b}.

The number of toplogies grows surprisingly with the number of things in the universe. Like, if we had three symbols, ‘a’, ‘b’, and ‘c’, there would be 29 possible topologies. The universe of the three symbols and the collection of sets {∅}, {a}, {b, c}, and {a, b, c}, for example, would be a topology. But the universe and the collection of sets {∅}, {a}, {b}, {c}, and {a, b, c} would not. It’s a good thing to ponder if you need something to occupy your mind while awake in bed.

With four symbols, there’s 355 possibilities. Good luck working those all out before you fall asleep. Five symbols have 6,942 possibilities. You realize this doesn’t look like any expected sequence. After ‘4’ the count of topologies isn’t anything obvious like “two to the number of symbols” or “the number of symbols factorial” or something.

Are you getting ready to call me on being inconsistent? In the past I’ve talked about topology as studying what we can know about geometry without involving the idea of distance. How’s that got anything to do with this fiddling about with sets and intersections and stuff?

So now we come to that name ‘M’, and what it’s finally mnemonic for. I have to touch on something Elke Stangl hoped I’d write about, but a letter someone else bid on first. That would be a manifold. I come from an applied-mathematics background so I’m not sure I ever got a proper introduction to manifolds. They appeared one day in the background of some talk about physics problems. I think they were introduced as “it’s a space that works like normal space”, and that was it. We were supposed to pretend we had always known about them. (I’m translating. What we were actually told would be that it “works like R3”. That’s how mathematicians say “like normal space”.) That was all we needed.

Properly, a manifold is … eh. It’s something that works kind of like normal space. That is, it’s a set, something that can be a universe. And it has to be something we can define “open sets” on. The open sets for the manifold follow the rules I gave for a topology above. You can make a collection of these open sets. And the empty set has to be in that collection. So does the whole universe. The intersection of two open sets in that collection is itself in that collection. The union of open sets in that collection is in that collection. If all that’s true, then we have a manifold.

And now the piece that makes every pop mathematics article about topology talk about doughnuts and coffee cups. It’s possible that two topologies might be homeomorphic to each other. “Homeomorphic” is a term of art. But you understand it if you remember that “morph” means shape, and suspect that “homeo” is probably close to “homogenous”. Two things being homeomorphic means you can match their parts up. In the matching there’s nothing left over in the first thing or the second. And the relations between the parts of the first thing are the same as the relations between the parts of the second thing.

So. Imagine the snippet of the number line for the numbers larger than -π and smaller than π. Think of all the open sets you can use to cover that. It will have a set like “the numbers bigger than 0 and less than 1”. A set like “the numbers bigger than -π and smaller than 2.1”. A set like “the numbers bigger than 0.01 and smaller than 0.011”. And so on.

Now imagine the points that exist on a circle, if you’ve omitted one point. Let’s say it’s the unit circle, centered on the origin, and that what we’re leaving out is the point that’s exactly to the left of the origin. The open sets for this are the arcs that cover some part of this punctured circle. There’s the arc that corresponds to the angles from 0 to 1 radian measure. There’s the arc that corresponds to the angles from -π to 2.1 radians. There’s the arc that corresponds to the angles from 0.01 to 0.011 radians. You see where this is going. You see why I say we can match those sets on the number line to the arcs of this punctured circle. There’s some details to fill in here. But you probably believe me this could be done if I had to.

There’s two (or three) great branches of topology. One is called “algebraic topology”. It’s the one that makes for fun pop mathematics articles about imaginary rubber sheets. It’s called “algebraic” because this field makes it natural to study the holes in a sheet. And those holes tend to form groups and rings, basic pieces of Not That Algebra. The field (I’m told) can be interpreted as looking at functors on groups and rings. This makes for some neat tying-together of subjects this A To Z round.

The other branch is called “differential topology”, which is a great field to study because it sounds like what Mister Spock is thinking about. It inspires awestruck looks where saying you study, like, Bayesian probability gets blank stares. Differential topology is about differentiable functions on manifolds. This gets deep into mathematical physics.

As you study mathematical physics, you stop worrying about ever solving specific physics problems. Specific problems are petty stuff. What you like is solving whole classes of problems. A steady trick for this is to try to find some properties that are true about the problem regardless of what exactly it’s doing at the time. This amounts to finding a manifold that relates to the problem. Consider a central-force problem, for example, with planets orbiting a sun. A planet can’t move just anywhere. It can only be in places and moving in directions that give the system the same total energy that it had to start. And the same linear momentum. And the same angular momentum. We can match these constraints to manifolds. Whatever the planet does, it does it without ever leaving these manifolds. To know the shapes of these manifolds — how they are connected — and what kinds of functions are defined on them tells us something of how the planets move.

The maybe-third branch is “low-dimensional topology”. This is what differential topology is for two- or three- or four-dimensional spaces. You know, shapes we can imagine with ease in the real world. Maybe imagine with some effort, for four dimensions. This kind of branches out of differential topology because having so few dimensions to work in makes a lot of problems harder. We need specialized theoretical tools that only work for these cases. Is that enough to count as a separate branch? It depends what topologists you want to pick a fight with. (I don’t want a fight with any of them. I’m over here in numerical mathematics when I’m not merely blogging. I’m happy to provide space for anyone wishing to defend her branch of topology.)

But each grows out of this quite general, quite abstract idea, also known as “point-set topology”, that’s all about sets and collections of sets. There is much that we can learn from thinking about how to collect the things that are possible.

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The Summer 2017 Mathematics A To Z: Ricci Tensor


Today’s is technically a request from Elke Stangl, author of the Elkemental Force blog. I think it’s also me setting out my own petard for self-hoisting, as my recollection is that I tossed off a mention of “defining the Ricci Tensor” as the sort of thing that’s got a deep beauty that’s hard to share with people. And that set off the search for where I had written about the Ricci Tensor. I hadn’t, and now look what trouble I’m in. Well, here goes.

Summer 2017 Mathematics A to Z, featuring a coati (it's kind of the Latin American raccoon) looking over alphabet blocks, with a lot of equations in the background.
Art courtesy of Thomas K Dye, creator of the web comic Newshounds. He has a Patreon for those able to support his work. He’s also open for commissions, starting from US$10.

Ricci Tensor.

Imagine if nothing existed.

You’re not doing that right, by the way. I expect what you’re thinking of is a universe that’s a big block of space that doesn’t happen to have any things clogging it up. Maybe you have a natural sense of volume in it, so that you know something is there. Maybe you even imagine something with grid lines or reticules or some reference points. What I imagine after a command like that is a sort of great rectangular expanse, dark and faintly purple-tinged, with small dots to mark its expanse. That’s fine. This is what I really want. But it’s not really imagining nothing existing. There’s space. There’s some sense of where things would be, if they happened to be in there. We’d have to get rid of the space to have “nothing” exist. And even then we have logical problems that sound like word games. (How can nothing have a property like “existing”? Or a property like “not existing”?) This is dangerous territory. Let’s not step there.

So take the empty space that’s what mathematics and physics people mean by “nothing”. What do we know about it? Unless we’re being difficult, it’s got some extent. There are points in it. There’s some idea of distance between these points. There’s probably more than one dimension of space. There’s probably some sense of time, too. At least we’re used to the expectation that things would change if we watched. It’s a tricky sense to have, though. It’s hard to say exactly what time is. We usually fall back on the idea that we know time has passed if we see something change. But if there isn’t anything to see change? How do we know there’s still time passing?

You maybe already answered. We know time is passing because we can see space changing. One of the legs of Modern Physics is geometry, how space is shaped and how its shape changes. This tells us how gravity works, and how electricity and magnetism propagate. If there were no matter, no energy, no things in the universe there would still be some kind of physics. And interesting physics, since the mathematics describing this stuff is even subtler and more challenging to the intuition than even normal Euclidean space. If you’re going to read a pop mathematics blog like this, you’re very used to this idea.

Probably haven’t looked very hard at the idea, though. How do you tell whether space is changing if there’s nothing in it? It’s all right to imagine a coordinate system put on empty space. Coordinates are our concept. They don’t affect the space any more than the names we give the squirrels in the yard affect their behavior. But how to make the coordinates move with the space? It seems question-begging at least.

We have a mathematical gimmick to resolve this. Of course we do. We call it a name like a “test mass” or a “test charge” or maybe just “test particle”. Imagine that we drop into space a thing. But it’s only barely a thing. It’s tiny in extent. It’s tiny in mass. It’s tiny in charge. It’s tiny in energy. It’s so slight in every possible trait that it can’t sully our nothingness. All it does is let us detect it. It’s a good question how. We have good eyes. But now, we could see the particle moving as the space it’s in moves.

But again we can ask how. Just one point doesn’t seem to tell us much. We need a bunch of test particles, a whole cloud of them. They don’t interact. They don’t carry energy or mass or anything. They just carry the sense of place. This is how we would perceive space changing in time. We can ask questions meaningfully.

Here’s an obvious question: how much volume does our cloud take up? If we’re going to be difficult about this, none at all, since it’s a finite number of particles that all have no extent. But you know what we mean. Draw a ball, or at least an ellipsoid, around the test particles. How big is that? Wait a while. Draw another ball around the now-moved test particles. How big is that now?

Here’s another question: has the cloud rotated any? The test particles, by definition, don’t have mass or anything. So they don’t have angular momentum. They aren’t pulling one another to the side any. If they rotate it’s because space has rotated, and that’s interesting to consider. And another question: might they swap positions? Could a pair of particles that go left-to-right swap so they go right-to-left? That I ask admits that I want to allow the possibility.

These are questions about coordinates. They’re about how one direction shifts to other directions. How it stretches or shrinks. That is to say, these are questions of tensors. Tensors are tools for many things, most of them about how things transmit through different directions. In this context, time is another direction.

All our questions about how space moves we can describe as curvature. How do directions fall away from being perpendicular to one another? From being parallel to themselves? How do their directions change in time? If we have three dimensions in space and one in time — a four-dimensional “manifold” — then there’s 20 different “directions” each with maybe their own curvature to consider. This may seem a lot. Every point on this manifold has this set of twenty numbers describing the curvature of space around it. There’s not much to do but accept that, though. If we could do with fewer numbers we would, but trying cheats us out of physics.

Ten of the numbers in that set are themselves a tensor. It’s known as the Weyl Tensor. It describes gravity’s equivalent to light waves. It’s about how the shape of our cloud will change as it moves. The other ten numbers form another tensor. That is, a thousand words into the essay, the Ricci Tensor. The Ricci Tensor describes how the volume of our cloud will change as the test particles move along. It may seem odd to need ten numbers for this, but that’s what we need. For three-dimensional space and one-dimensional time, anyway. We need fewer for two-dimensional space; more, for more dimensions of space.

The Ricci Tensor is a geometric construct. Most of us come to it, if we do, by way of physics. It’s a useful piece of general relativity. It has uses outside this, though. It appears in the study of Ricci Flows. Here space moves in ways akin to how heat flows. And the Ricci Tensor appears in projective geometry, in the study of what properties of shapes don’t depend on how we present them.

It’s still tricky stuff to get a feeling for. I’m not sure I have a good feel for it myself. There’s a long trail of mathematical symbols leading up to these tensors. The geometry of them becomes more compelling in four or more dimensions, which taxes the imagination. Yann Ollivier here has a paper that attempts to provide visual explanations for many of the curvatures and tensors that are part of the field. It might help.

The Summer 2017 Mathematics A To Z: Open Set


Today’s glossary entry is another request from Elke Stangl, author of the Elkemental Force blog. I’m hoping this also turns out to be a well-received entry. Half of that is up to you, the kind reader. At least I hope you’re a reader. It’s already gone wrong, as it was supposed to be Friday’s entry. I discovered I hadn’t actually scheduled it while I was too far from my laptop to do anything about that mistake. This spoils the nice Monday-Wednesday-Friday routine of these glossary entries that dates back to the first one I ever posted and just means I have to quit forever and not show my face ever again. Sorry, Ulam Spiral. Someone else will have to think of you.

Summer 2017 Mathematics A to Z, featuring a coati (it's kind of the Latin American raccoon) looking over alphabet blocks, with a lot of equations in the background.
Art courtesy of Thomas K Dye, creator of the web comic Newshounds. He has a Patreon for those able to support his work. He’s also open for commissions, starting from US$10.

Open Set.

Mathematics likes to present itself as being universal truths. And it is. At least if we allow that the rules of logic by which mathematics works are universal. Suppose them to be true and the rest follows. But we start out with intuition, with things we observe in the real world. We’re happy when we can remove the stuff that’s clearly based on idiosyncratic experience. We find something that’s got to be universal.

Sets are pretty abstract things, as mathematicians use the term. They get to be hard to talk about; we run out of simpler words that we can use. A set is … a bunch of things. The things are … stuff that could be in a set, or else that we’d rule out of a set. We can end up better understanding things by drawing a picture. We draw the universe, which is a rectangular block, sometimes with dashed lines as the edges. The set is some blotch drawn on the inside of it. Some shade it in to emphasize which stuff we want in the set. If we need to pick out a couple things in the universe we drop in dots or numerals. If we’re rigorous about the drawing we could create a Venn Diagram.

When we do this, we’re giving up on the pure mathematical abstraction of the set. We’re replacing it with a territory on a map. Several territories, if we have several sets. The territories can overlap or be completely separate. We’re subtly letting our sense of geography, our sense of the spaces in which we move, infiltrate our understanding of sets. That’s all right. It can give us useful ideas. Later on, we’ll try to separate out the ideas that are too bound to geography.

A set is open if whenever you’re in it, you can’t be on its boundary. We never quite have this in the real world, with territories. The border between, say, New Jersey and New York becomes this infinitesimally slender thing, as wide in space as midnight is in time. But we can, with some effort, imagine the state. Imagine being as tiny in every direction as the border between two states. Then we can imagine the difference between being on the border and being away from it.

And not being on the border matters. If we are not on the border we can imagine the problem of getting to the border. Pick any direction; we can move some distance while staying inside the set. It might be a lot of distance, it might be a tiny bit. But we stay inside however we might move. If we are on the border, then there’s some direction in which any movement, however small, drops us out of the set. That’s a difference in kind between a set that’s open and a set that isn’t.

I say “a set that’s open and a set that isn’t”. There are such things as closed sets. A set doesn’t have to be either open or closed. It can be neither, a set that includes some of its borders but not other parts of it. It can even be both open and closed simultaneously. The whole universe, for example, is both an open and a closed set. The empty set, with nothing in it, is both open and closed. (This looks like a semantic trick. OK, if you’re in the empty set you’re not on its boundary. But you can’t be in the empty set. So what’s going on? … The usual. It makes other work easier if we call the empty set ‘open’. And the extra work we’d have to do to rule out the empty set doesn’t seem to get us anything interesting. So we accept what might be a trick.) The definitions of ‘open’ and ‘closed’ don’t exclude one another.

I’m not sure how this confusing state of affairs developed. My hunch is that the words ‘open’ and ‘closed’ evolved independent of each other. Why do I think this? An open set has its openness from, well, not containing its boundaries; from the inside there’s always a little more to it. A closed set has its closedness from sequences. That is, you can consider a string of points inside a set. Are these points leading somewhere? Is that point inside your set? If a string of points always leads to somewhere, and that somewhere is inside the set, then you have closure. You have a closed set. I’m not sure that the terms were derived with that much thought. But it does explain, at least in terms a mathematician might respect, why a set that isn’t open isn’t necessarily closed.

Back to open sets. What does it mean to not be on the boundary of the set? How do we know if we’re on it? We can define sets by all sorts of complicated rules: complex-valued numbers of size less than five, say. Rational numbers whose denominator (in lowest form) is no more than ten. Points in space from which a satellite dropped would crash into the moon rather than into the Earth or Sun. If we have an idea of distance we could measure how far it is from a point to the nearest part of the boundary. Do we need distance, though?

No, it turns out. We can get the idea of open sets without using distance. Introduce a neighborhood of a point. A neighborhood of a point is an open set that contains that point. It doesn’t have to be small, but that’s the connotation. And we get to thinking of little N-balls, circle or sphere-like constructs centered on the target point. It doesn’t have to be N-balls. But we think of them so much that we might as well say it’s necessary. If every point in a set has a neighborhood around it that’s also inside the set, then the set’s open.

You’re going to accuse me of begging the question. Fair enough. I was using open sets to define open sets. This use is all right for an intuitive idea of what makes a set open, but it’s not rigorous. We can give in and say we have to have distance. Then we have N-balls and we can build open sets out of balls that don’t contain the edges. Or we can try to drive distance out of our idea of open sets.

We can do it this way. Start off by saying the whole universe is an open set. Also that the union of any number of open sets is also an open set. And that the intersection of any finite number of open sets is also an open set. Does this sound weak? So it sounds weak. It’s enough. We get the open sets we were thinking of all along from this.

This works for the sets that look like territories on a map. It also works for sets for which we have some idea of distance, however strange it is to our everyday distances. It even works if we don’t have any idea of distance. This lets us talk about topological spaces, and study what geometry looks like if we can’t tell how far apart two points are. We can, for example, at least tell that two points are different. Can we find a neighborhood of one that doesn’t contain the other? Then we know they’re some distance apart, even without knowing what distance is.

That we reached so abstract an idea of what an open set is without losing the idea’s usefulness suggests we’re doing well. So we are. It also shows why Nicholas Bourbaki, the famous nonexistent mathematician, thought set theory and its related ideas were the core of mathematics. Today category theory is a more popular candidate for the core of mathematics. But set theory is still close to the core, and much of analysis is about what we can know from the fact of sets being open. Open sets let us explain a lot.

A Leap Day 2016 Mathematics A To Z: Riemann Sphere


To my surprise nobody requested any terms beginning with `R’ for this A To Z. So I take this free day to pick on a concept I’d imagine nobody saw coming.

Riemann Sphere.

We need to start with the complex plane. This is just, well, a plane. All the points on the plane correspond to a complex-valued number. That’s a real number plus a real number times i. And i is one of those numbers which, squared, equals -1. It’s like the real number line, only in two directions at once.

Take that plane. Now put a sphere on it. The sphere has radius one-half. And it sits on top of the plane. Its lowest point, the south pole, sits on the origin. That’s whatever point corresponds to the number 0 + 0i, or as humans know it, “zero”.

We’re going to do something amazing with this. We’re going to make a projection, something that maps every point on the sphere to every point on the plane, and vice-versa. In other words, we can match every complex-valued number to one point on the sphere. And every point on the sphere to one complex-valued number. Here’s how.

Imagine sitting at the north pole. And imagine that you can see through the sphere. Pick any point on the plane. Look directly at it. Shine a laser beam, if that helps you pick the point out. The laser beam is going to go into the sphere — you’re squatting down to better look through the sphere — and come out somewhere on the sphere, before going on to the point in the plane. The point where the laser beam emerges? That’s the mapping of the point on the plane to the sphere.

There’s one point with an obvious match. The south pole is going to match zero. They touch, after all. Other points … it’s less obvious. But some are easy enough to work out. The equator of the sphere, for instance, is going to match all the points a distance of 1 from the origin. So it’ll have the point matching the number 1 on it. It’ll also have the point matching the number -1, and the point matching i, and the point matching -i. And some other numbers.

All the numbers that are less than 1 from the origin, in fact, will have matches somewhere in the southern hemisphere. If you don’t see why that is, draw some sketches and think about it. You’ll convince yourself. If you write down what convinced you and sprinkle the word “continuity” in here and there, you’ll convince a mathematician. (WARNING! Don’t actually try getting through your Intro to Complex Analysis class doing this. But this is what you’ll be doing.)

What about the numbers more than 1 from the origin? … Well, they all match to points on the northern hemisphere. And tell me that doesn’t stagger you. It’s one thing to match the southern hemisphere to all the points in a circle of radius 1 away from the origin. But we can match everything outside that little circle to the northern hemisphere. And it all fits in!

Not amazed enough? How about this: draw a circle on the plane. Then look at the points on the Riemann sphere that match it. That set of points? It’s also a circle. A line on the plane? That’s also a line on the sphere. (Well, it’s a geodesic. It’s the thing that looks like a line, on spheres.)

How about this? Take a pair of intersecting lines or circles in the plane. Look at what they map to. That mapping, squashed as it might be to the northern hemisphere of the sphere? The projection of the lines or circles will intersect at the same angles as the original. As much as space gets stretched out (near the south pole) or squashed down (near the north pole), angles stay intact.

OK, but besides being stunning, what good is all this?

Well, one is that it’s a good thing to learn on. Geometry gets interested in things that look, at least in places, like planes, but aren’t necessarily. These spheres are, and the way a sphere matches a plane is obvious. We can learn the tools for geometry on the Möbius strip or the Klein bottle or other exotic creations by the tools we prove out on this.

And then physics comes in, being all weird. Much of quantum mechanics makes sense if you imagine it as things on the sphere. (I admit I don’t know exactly how. I went to grad school in mathematics, not in physics, and I didn’t get to the physics side of mathematics much at that time.) The strange ways distance can get mushed up or stretched out have echoes in relativity. They’ll continue having these echoes in other efforts to explain physics as geometry, the way that string theory will.

Also important is that the sphere has a top, the north pole. That point matches … well, what? It’s got to be something infinitely far away from the origin. And this make sense. We can use this projection to make a logically coherent, sensible description of things “approaching infinity”, the way we want to when we first learn about infinitely big things. Wrapping all the complex-valued numbers to this ball makes the vast manageable.

It’s also good numerical practice. Computer simulations have problems with infinitely large things, for the obvious reason. We have a couple of tools to handle this. One is to model a really big but not infinitely large space and hope we aren’t breaking anything. One is to create a “tiling”, making the space we are able to simulate repeat itself in a perfect grid forever and ever. But recasting the problem from the infinitely large plane onto the sphere can also work. This requires some ingenuity, to be sure we do the recasting correctly, but that’s all right. If we need to run a simulation over all of space, we can often get away with doing a simulation on a sphere. And isn’t that also grand?

The Riemann named here is Bernhard Riemann, yet another of those absurdly prolific 19th century mathematicians, especially considering how young he was when he died. His name is all over the fundamentals of analysis and geometry. When you take Introduction to Calculus you get introduced pretty quickly to the Riemann Sum, which is how we first learn how to calculate integrals. It’s that guy. General relativity, and much of modern physics, is based on advanced geometries that again fall back on principles Riemann noticed or set out or described so well that we still think of them as he discovered.

A Summer 2015 Mathematics A To Z: y-axis


y-axis.

It’s easy to tell where you are on a line. At least it is if you have a couple tools. One is a reference point. Another is the ability to say how far away things are. Then if you say something is a specific distance from the reference point you can pin down its location to one of at most two points. If we add to the distance some idea of direction we can pin that down to at most one point. Real numbers give us a good sense of distance. Positive and negative numbers fit the idea of orientation pretty well.

To tell where you are on a plane, though, that gets tricky. A reference point and a sense of how far things are help. Knowing something is a set distance from the reference point tells you something about its position. But there’s still an infinite number of possible places the thing could be, unless it’s at the reference point.

The classic way to solve this is to divide space into a couple directions. René Descartes made his name for himself — well, with many things. But one of them, in mathematics, was to describe the positions of things by components. One component describes how far something is in one direction from the reference point. The next component describes how far the thing is in another direction.

This sort of scheme we see as laying down axes. One, conventionally taken to be the horizontal or left-right axis, we call the x-axis. The other direction — one perpendicular, or orthogonal, to the x-axis — we call the y-axis. Usually this gets drawn as the vertical axis, the one running up and down the sheet of paper. That’s not required; it’s just convention.

We surely call it the x-axis in echo of the use of x as the name for a number whose value we don’t know right away. (That, too, is a convention Descartes gave us.) x carries with it connotations of the unknown, the sought-after, the mysterious thing to be understood. The next axis we name y because … well, that’s a letter near x and we don’t much need it for anything else, I suppose. If we need another direction yet, if we want something in space rather than a plane, then the third axis we dub the z-axis. It’s perpendicular to the x- and the y-axis directions.

These aren’t the only names for these directions, though. It’s common and often convenient to describe positions of things using vector notation. A vector describes the relative distance and orientation of things. It’s compact symbolically. It lets one think of the position of things as a single variable, a single concept. Then we can talk about a position being a certain distance in the direction of the x-axis plus a certain distance in the direction of the y-axis. And, if need be, plus some distance in the direction of the z-axis.

The direction of the x-axis is often written as \hat{i} , and the direction of the y-axis as \hat{j} . The direction of the z-axis if needed gets written \hat{k} . The circumflex there indicates two things. First is that the thing underneath it is a vector. Second is that it’s a vector one unit long. A vector might have any length, including zero. It’s convenient to make some mention when it’s a nice one unit long.

Another popular notation is to write the direction of the x-axis as the vector \hat{e}_1 , and the y-axis as the vector \hat{e}_2 , and so on. This method offers several advantages. One is that we can talk about the vector \hat{e}_j , that is, some particular direction without pinning down just which one. That’s the equivalent of writing “x” or “y” for a number we don’t want to commit ourselves to just yet. Another is that we can talk about axes going off in two, or three, or four, or more directions without having to pin down how many there are. And then we don’t have to think of what to call them. x- and y- and z-axes make sense. w-axis sounds a little odd but some might accept it. v-axis? u-axis? Nobody wants that, trust me.

Sometimes people start the numbering from \hat{e}_0 so that the y-axis is the direction \hat{e}_1 . Usually it’s either clear from context or else it doesn’t matter.

A Summer 2015 Mathematics A To Z: n-tuple


N-tuple.

We use numbers to represent things we want to think about. Sometimes the numbers represent real-world things: the area of our backyard, the number of pets we have, the time until we have to go back to work. Sometimes the numbers mean something more abstract: an index of all the stuff we’re tracking, or how its importance compares to other things we worry about.

Often we’ll want to group together several numbers. Each of these numbers may measure a different kind of thing, but we want to keep straight what kind of thing it is. For example, we might want to keep track of how many people are in each house on the block. The houses have an obvious index number — the street number — and the number of people in each house is just what it says. So instead of just keeping track of, say, “32” and “34” and “36”, and “3” and “2” and “3”, we would keep track of pairs: “32, 3”, and “34, 2”, and “36, 3”. These are called ordered pairs.

They’re not called ordered because the numbers are in order. They’re called ordered because the order in which the numbers are recorded contains information about what the numbers mean. In this case, the first number is the street address, and the second number is the count of people in the house, and woe to our data set if we get that mixed up.

And there’s no reason the ordering has to stop at pairs of numbers. You can have ordered triplets of numbers — (32, 3, 2), say, giving the house number, the number of people in the house, and the number of bathrooms. Or you can have ordered quadruplets — (32, 3, 2, 6), say, house number, number of people, bathroom count, room count. And so on.

An n-tuple is an ordered set of some collection of numbers. How many? We don’t care, or we don’t care to say right now. There are two popular ways to pronounce it. One is to say it the way you say “multiple” only with the first syllable changed to “enn”. Others say it about the same, but with a long u vowel, so, “enn-too-pull”. I believe everyone worries that everyone else says it the other way and that they sound like they’re the weird ones.

You might care to specify what your n is for your n-tuple. In that case you can plug in a value for that n right in the symbol: a 3-tuple is an ordered triplet. A 4-tuple is that ordered quadruplet. A 26-tuple seems like rather a lot but I’ll trust that you know what you’re trying to study. A 1-tuple is just a number. We might use that if we’re trying to make our notation consistent with something else in the discussion.

If you’re familiar with vectors you might ask: so, an n-tuple is just a vector? It’s not quite. A vector is an n-tuple, but in the same way a square is a rectangle. It has to meet some extra requirements. To be a vector we have to be able to add corresponding numbers together and get something meaningful out of it. The ordered pair (32, 3) representing “32 blocks north and 3 blocks east” can be a vector. (32, 3) plus (34, 2) can give us us (66, 5). This makes sense because we can say, “32 blocks north, 3 blocks east, 34 more blocks north, 2 more blocks east gives us 66 blocks north, 5 blocks east.” At least it makes sense if we don’t run out of city. But to add together (32, 3) plus (34, 2) meaning “house number 32 with 3 people plus house number 34 with 2 people gives us house number 66 with 5 people”? That’s not good, whatever town you’re in.

I think the commonest use of n-tuples is to talk about vectors, though. Vectors are such useful things.

A Summer 2015 Mathematics A To Z: measure


Measure.

Before painting a room you should spackle the walls. This fills up small holes and cracks. My father is notorious for using enough spackle to appreciably diminish the room’s volume. (So says my mother. My father disagrees.) I put spackle on as if I were paying for it myself, using so little my father has sometimes asked when I’m going to put any on. I’ll get to mathematics in the next paragraph.

One of the natural things to wonder about a set — a collection of things — is how big it is. The “measure” of a set is how we describe how big a set is. If we’re looking at a set that’s a line segment within a longer line, the measure pretty much matches our idea of length. If we’re looking at a shape on the plane, the measure matches our idea of area. A solid in space we expect has a measure that’s like the volume.

We might say the cracks and holes in a wall are as big as the amount of spackle it takes to fill them. Specifically, we mean it’s the least bit of spackle needed to fill them. And similarly we describe the measure of a set in terms of how much it takes to cover it. We even call this “covering”.

We use the tool of “cover sets”. These are sets with a measure — a length, a volume, a hypervolume, whatever — that we know. If we look at regular old normal space, these cover sets are typically circles or spheres or similar nice, round sets. They’re familiar. They’re easy to work with. We don’t have to worry about how to orient them, the way we might if we had square or triangular covering sets. These covering sets can be as small or as large as you need. And we suppose that we have some standard reference. This is a covering set with measure 1, this with measure 1/2, this with measure 24, this with measure 1/72.04, and so on. (If you want to know what units these measures are in, they’re “units of measure”. What we’re interested in is unchanged whether we measure in “inches” or “square kilometers” or “cubic parsecs” or something else. It’s just longer to say.)

You can imagine this as a game. I give you a set; you try to cover it. You can cover it with circles (or spheres, or whatever fits the space we’re in) that are big, or small, or whatever size you like. You can use as many as you like. You can cover more than just the things in the set I gave you. The only absolute rule is you must not miss anything, even one point, in the set I give you. Find the smallest total area of the covering circles you use. That smallest total area that covers the whole set is the measure of that set.

Generally, measure matches pretty well the intuitive feel we might have for length or area or volume. And the idea extends to things that don’t really have areas. For example, we can study the probability of events by thinking of the space of all possible outcomes of an experiment, like all the ways twenty coins might come up. We find the measure of the set of outcomes we’re interested in, like all the sets that have ten tails. The probability of the outcome we’re interested in is the measure of the set we’re interested in divided by the measure of the set of all possible outcomes. (There’s more work to do to make this quite true. In an advanced probability course we do this work. Please trust me that we could do it if we had to. Also you see why we stride briskly past the discussion of units. What unit would make sense for measuring “the space of all possible outcomes of an experiment” anyway?)

But there are surprises. For example, there’s the Cantor set. The easiest way to make the Cantor set is to start with a line of length 1 — of measure 1 — and take out the middle third. This produces two line segments of length, measure, 1/3 each. Take out the middle third of each of those segments. This leaves four segments each of length 1/9. Take out the middle third of each of those four segments, producing eight segments, and so on. If you do this infinitely many times you’ll create a set that has no measure; it fills no volume, it has no length. And yet you can prove there are just as many points in this set as there are in a real normal space. Somehow merely having a lot of points doesn’t mean they fill space.

Measure is useful not just because it can give us paradoxes like that. We often want to say how big sets, or subsets, of whatever we’re interested in are. And using measure lets us adapt things like calculus to become more powerful. We’re able to say what the integral is for functions that are much more discontinuous, more chopped up, than ones that high school or freshman calculus can treat, for example. The idea of measure takes length and area and such and makes it more abstract, giving it great power and applicability.

A Summer 2015 Mathematics A To Z: bijection


Bijection.

To explain this second term in my mathematical A to Z challenge I have to describe yet another term. That’s function. A non-mathematician’s idea a function is something like “a line with a bunch of x’s in it, and maybe also a cosine or something”. That’s fair enough, although it’s a bit like defining chemistry as “mixing together colored, bubbling liquids until something explodes”.

By a function a mathematician means a rule describing how to pair up things found in one set, called the domain, with the things found in another set, called the range. The domain and the range can be collections of anything. They can be counting numbers, real numbers, letters, shoes, even collections of numbers or sets of shoes. They can be the same kinds of thing. They can be different kinds of thing.

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Reading the Comics, September 28, 2014: Punning On A Sunday Edition


I honestly don’t intend this blog to become nothing but talk about the comic strips, but then something like this Sunday happens where Comic Strip Master Command decided to send out math joke priority orders and what am I to do? And here I had a wonderful bit about the natural logarithm of 2 that I meant to start writing sometime soon. Anyway, for whatever reason, there’s a lot of punning going on this time around; I don’t pretend to explain that.

Jason Poland’s Robbie and Bobby (September 25) puns off of a “meth lab explosion” in a joke that I’ve seen passed around Twitter and the like but not in a comic strip, possibly because I don’t tend to read web comics until they get absorbed into the Gocomics.com collective.

Brian Boychuk and Ron Boychuk’s The Chuckle Brothers (September 26) shows how an infinity pool offers the chance to finally, finally, do a one-point perspective drawing just like the art instruction book says.

Bill Watterson’s Calvin and Hobbes (September 27, rerun) wrapped up the latest round of Calvin not learning arithmetic with a gag about needing to know the difference between the numbers of things and the values of things. It also surely helps the confusion that the (United States) dime is a tiny coin, much smaller in size than the penny or nickel that it far out-values. I’m glad I don’t have to teach coin values to kids.

Zach Weinersmith’s Saturday Morning Breakfast Cereal (September 27) mentions Lagrange points. These are mathematically (and physically) very interesting because they come about from what might be the first interesting physics problem. If you have two objects in the universe, attracting one another gravitationally, then you can describe their behavior perfectly and using just freshman or even high school calculus. For that matter, describing their behavior is practically what Isaac Newton invented his calculus to do.

Add in a third body, though, and you’ve suddenly created a problem that just can’t be done by freshman calculus, or really, done perfectly by anything but really exotic methods. You’re left with approximations, analytic or numerical. (Karl Fritiof Sundman proved in 1912 that one could create an infinite series solution, but it’s not a usable solution. To get a desired accuracy requires so many terms and so much calculation that you’re better off not using it. This almost sounds like the classical joke about mathematicians, coming up with solutions that are perfect but unusable. It is the most extreme case of a possible-but-not-practical solution I’m aware of, if stories I’ve heard about its convergence rate are accurate. I haven’t tried to follow the technique myself.)

But just because you can’t solve every problem of a type doesn’t mean you can’t solve some of them, and the ones you do solve might be useful anyway. Joseph-Louis Lagrange did that, studying the problem of one large body — like a sun, or a planet — and one middle-sized body — a planet, or a moon — and one tiny body — like an asteroid, or a satellite. If the middle-sized body is orbiting the large body in a nice circular orbit, then, there are five special points, dubbed the Lagrange points. A satellite that’s at one of those points (with the right speed) will keep on orbiting at the same rotational speed that the middle body takes around the large body; that is, the system will turn as if the large, middle, and tiny bodies were fixed in place, relative to each other.

Two of these spots, dubbed numbers 4 and 5, are stable: if your tiny body is not quite in the right location that’s all right, because it’ll stay nearby, much in the same way that if you roll a ball into a pit it’ll stay in the pit. But three of these spots, numbers 1, 2, and 3, are unstable: if your tiny body is not quite on those spots, it’ll fall away, in much the same way if you set a ball on the peak of the roof it’ll roll off one way or another.

When Lagrange noticed these points there wasn’t any particular reason to think of them as anything but a neat mathematical construct. But the points do exist, and they can be stable even if the medium body doesn’t have a perfectly circular orbit, or even if there are other planets in the universe, which throws off the nice simple calculations yet. Something like 1700 asteroids are known to exist in the number 4 and 5 Lagrange points for the Sun and Jupiter, and there are a handful known for Saturn and Neptune, and apparently at least five known for Mars. For Earth apparently there’s just the one known to exist, catchily named 2010 TK7, discovered in October 2010, although I’d be surprised if that were the only one. They’re just small.

Professor Peter Peddle has the crazy idea of studying boxing scientifically and preparing strategy accordingly.
Elliot Caplin and John Cullen Murphy’s Big Ben Bolt, from the 23rd of August, 1953 (rerun the 28th of September, 2014).

Elliot Caplin and John Cullen Murphy’s Big Ben Bolt (September 28, originally run August 23, 1953) has been on the Sunday strips now running a tale about a mathematics professor, Peter Peddle, who’s threatening to revolutionize Big Ben Bolt’s boxing world by reducing it to mathematical abstraction; past Sunday strips have even shown the rather stereotypically meek-looking professor overwhelming much larger boxers. The mathematics described here is nonsense, of course, but it’d be asking a bit of the comic strip writers to have a plausible mathematical description of the perfect boxer, after all.

But it’s hard for me anyway to not notice that the professor’s approach is really hard to gainsay. The past generation of baseball, particularly, has been revolutionized by a very mathematical, very rigorous bit of study, looking at questions like how many pitches can a pitcher actually throw before he loses control, and where a batter is likely to hit based on past performance (of this batter and of batters in general), and how likely is this player to have a better or a worse season if he’s signed on for another year, and how likely is it he’ll have a better enough season than some cheaper or more promising player? Baseball is extremely well structured to ask these kinds of questions, with football almost as good for it — else there wouldn’t be fantasy football leagues — and while I am ignorant of modern boxing, I would be surprised if a lot of modern boxing strategy weren’t being studied in Professor Peddle’s spirit.

Eric the Circle (September 28), this one by Griffinetsabine, goes to the Shapes Singles Bar for a geometry pun.

Bill Amend’s FoxTrot (September 28) (and not a rerun; the strip is new runs on Sundays) jumps on the Internet Instructional Video bandwagon that I’m sure exists somewhere, with child prodigy Jason Fox having the idea that he could make mathematics instruction popular enough to earn millions of dollars. His instincts are probably right, too: instructional videos that feature someone who looks cheerful and to be having fun and maybe a little crazy — well, let’s say eccentric — are probably the ones that will be most watched, at least. It’s fun to see people who are enjoying themselves, and the odder they act the better up to a point. I kind of hate to point out, though, that Jason Fox in the comic strip is supposed to be ten years old, implying that (this year, anyway) he was born nine years after Bob Ross died. I know that nothing ever really goes away anymore, but, would this be a pop culture reference that makes sense to Jason?

Tom Thaves’s Frank and Ernest (September 28) sets up the idea of Euclid as a playwright, offering a string of geometry puns.

Jef Mallet’s Frazz (September 28) wonders about why trains show up so often in story problems. I’m not sure that they do, actually — haven’t planes and cars taken their place here, too? — although the reasons aren’t that obscure. Questions about the distance between things changing over time let you test a good bit of arithmetic and algebra while being naturally about stuff it’s reasonable to imagine wanting to know. What more does the homework-assigner want?

Zach Weinersmith’s Saturday Morning Breakfast Cereal (September 28) pops back up again with the prospect of blowing one’s mind, and it is legitimately one of those amazing things, that e^{i \pi} = -1 . It is a remarkable relationship between a string of numbers each of which are mind-blowing in their ways — negative 1, and pi, and the base of the natural logarithms e, and dear old i (which, multiplied by itself, is equal to negative 1) — and here they are all bundled together in one, quite true, relationship. I do have to wonder, though, whether anyone who would in a social situation like this understand being told “e raised to the i times pi power equals negative one”, without the framing of “we’re talking now about exponentials raised to imaginary powers”, wouldn’t have already encountered this and had some of the mind-blowing potential worn off.

Reblog: The Math That Saved Apollo 13


GCDXY here presents images from the Apollo 13 flight checklist. This is itself a re-representing of images that Gizmodo posted when Apollo 13 Commander James Lovell sold the checklist last year, but I’m just coming across this now. And it nicely combines the mathematics and the space history interests I so enjoy.

The particular calculations done here were shown in one of many, many, outstanding scenes in the movie Apollo 13. However, the movie presents the calculations as being done on slide rule, when the computations needed are mostly addition and subtraction. It is possible to use slide rules to do addition and subtraction, but that’s really the hard way to do it; slide rules are for multiplication, division, and raising numbers to powers.

But considerable calculation for Apollo (and Gemini, and Mercury) was done without electronic computers, and the movie would have missed out on presenting an important point if it didn’t have the scene. So the movie achieved that strange state of conveying something true about what happened by showing it in a way it all but certainly did not.

gcdxy

Two hours after a service module’s oxygen tank explosion on Apollo 13, Commander James Lovell did calculations that helped put the ship back on course so that they could return back to Earth. They needed to establish the right course to use the Moon’s gravity to get back home. Check out the article on Gizmodo from November 2011.

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Reblog: Kant & Leibniz on Space and Implications in Geometry


Mathematicians and philosophers are fairly content to share credit for Rene Descartes, possibly because he was able to provide catchy, easy-to-popularize cornerstones for both fields.

Immanuel Kant, these days at least, is almost exclusively known as a philosopher, and that he was also a mathematician and astronomer is buried in the footnotes. If you stick to math and science popularizations you’ll probably pick up (as I did) that Kant was one of the co-founders of the nebular hypothesis, the basic idea behind our present understanding of how solar systems form, and maybe, if the book has room, that Kant had the insight that knowing gravitation falls off by an inverse-square rule implies that we live in a three-dimensional space.

Frank DeVita here writes some about Kant (and Wilhelm Leibniz)’s model of how we understand space and geometry. It’s not technical in the mathematics sense, although I do appreciate the background in Kant’s philosophy which my Dearly Beloved has given me. In the event I’d like to offer it as a way for mathematically-minded people to understand more of an important thinker they may not have realized was in their field.

Frank DeVita

        

Kant’s account of space in the Prolegomena serves as a cornerstone for his thought and comes about in a discussion of the transcendental principles of mathematics that precedes remarks on the possibility of natural science and metaphysics. Kant begins his inquiry concerning the possibility of ‘pure’ mathematics with an appeal to the nature of mathematical knowledge, asserting that it rests upon no empirical basis, and thus is a purely synthetic product of pure reason (§6). He also argues that mathematical knowledge (pure mathematics) has the unique feature of first exhibiting its concepts in a priori intuition which in turn makes judgments in mathematics ‘intuitive’ (§7.281). For Kant, intuition is prior to our sensibility and the activity of reason since the former does not grasp ‘things in themselves,’ but rather only the things that can be perceived by the senses. Thus, what we can perceive is based…

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Wednesday, June 6, 1962 – Food Contract, Boilerplate Purchase


The Manned Spacecraft Center has awarded to the Whirlpool Corporation Research Laboratories of Saint Joseph, Michigan, a contract to provide the food and waste management systems for Project Gemini. Whirlpool is to provide the water dispenser, food storage, and waste storage devices. The food and the zero-gravity feeding devices, however, are to be provided by the United States Army Quartermaster Corps Food and Container Institute, of Chicago. The Life Systems Division of the Manned Spacecraft Center is responsible for directing the program.
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Friday, June 1, 1962 – Operations Coordination Meeting, Astronaut Applications Close


Today’s was the first spacecraft operations coordination meeting. Presented at it was a list of all the aerospace ground equipment required for Gemini spacecraft handling and checkout before flight.

June 1 was also the nominal closing date for applications to be a new astronaut. Applications were opened April 18. The plan is to select between five and ten new astronauts to augment the Mercury 7.

Tuesday, May 29, 1962 – Ejection Seat Plans


The development testing plans for the Gemini spacecraft ejection seat were settled in a meeting between representatives of McDonnell, Weber Aircraft, the Gemini Procurement Office, Life Systems Division, Gemini Project Office, and the US Naval Ordnance Test Station at China Lake, California.
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Thursday, May 24, 1962 – Parachute testing starts


At the Naval Parachute Facility in El Centro, California, North American completed a successful drop test of the emergency parachute recovery system, using a half-scale test vehicle.
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Wednesday, May 23, 1962 – Paraglider Wing Wind Tunnel Test


The Ames Research Center has begun the first wind tunnel test of the inflatable paraglider wing, using a half-scale model of the wing intended to bring Gemini flights (after the first one) to a touchdown on the ground. This is the first large-scale paraglider wing in the full-scale test facility. The objective of the test program, to run over two months, are to understand the basic aerodynamic and loads data for the wing and spacecraft system, and to identify potential aerodynamic and design problems.
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Monday, May 21, 1962 – Launch Pad 19 For Gemini; Pulse Code Modulation for Telemetry


The pulse code modulation method is to be used for transmitting Project Gemini telemetry. McDonnell has awarded an $8 million subcontract to Electro-Mechanical Research, Inc, of Sarasota, Florida, for this digital transmission system. The system will use a pulse code modulation subsystem, an onboard tape recorder, and a pair of VHF transmitters, and be capable of transmitting data in real or delayed time.
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Friday, May 18, 1962 – Parachute Landing System


McDonnell has subcontracted the parachute landing system to be used for the first Gemini flight to Northrop Ventura. The cost is estimated at $1,829,272. The design for this flight is to use a single parachute system, a ring-sail parachute with diameter 84.2 feet. Later flights are to use the paraglider system under development. Earlier meetings have worked out a provisional schedule of events for the parachute landing.

Thursday, May 17, 1962 – Retrorocket and Parachute Decisions


The meeting about the retrograde rocket motors has concluded the design should be changed to provide about three times the thrust level. This will allow retrorocket aborts at altitudes as low as between 72,000 and 75,000 feet. The meeting was between representatives of McDonnell and the Gemini Project Office.
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Wednesday, May 16, 1962 – Retrorockets, Parachutes, and Interface Group


Representatives of the Gemini Project Office and of McDonnell are meeting to discuss retrograde rockets for the Gemini spacecraft. These rockets are currently to be provided by Thiokol.
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