From my First A-to-Z: Tensor

Of course I can’t just take a break for the sake of having a break. I feel like I have to do something of interest. So why not make better use of my several thousand past entries and repost one? I’d just reblog it except WordPress’s system for that is kind of rubbish. So here’s what I wrote, when I was first doing A-to-Z’s, back in summer of 2015. Somehow I was able to post three of these a week. I don’t know how.

I had remembered this essay as mostly describing the boring part of tensors, that we usually represent them as grids of numbers and then symbols with subscripts and superscripts. I’m glad to rediscover that I got at why we do such things to numbers and subscripts and superscripts.

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Tensor.

The true but unenlightening answer first: a tensor is a regular, rectangular grid of numbers. The most common kind is a two-dimensional grid, so that it looks like a matrix, or like the times tables. It might be square, with as many rows as columns, or it might be rectangular.

It can also be one-dimensional, looking like a row or a column of numbers. Or it could be three-dimensional, rows and columns and whole levels of numbers. We don’t try to visualize that. It can be what we call zero-dimensional, in which case it just looks like a solitary number. It might be four- or more-dimensional, although I confess I’ve never heard of anyone who actually writes out such a thing. It’s just so hard to visualize.

You can add and subtract tensors if they’re of compatible sizes. You can also do something like multiplication. And this does mean that tensors of compatible sizes will form a ring. Of course, that doesn’t say why they’re interesting.

Tensors are useful because they can describe spatial relationships efficiently. The word comes from the same Latin root as “tension”, a hint about how we can imagine it. A common use of tensors is in describing the stress in an object. Applying stress in different directions to an object often produces different effects. The classic example there is a newspaper. Rip it in one direction and you get a smooth, clean tear. Rip it perpendicularly and you get a raggedy mess. The stress tensor represents this: it gives some idea of how a force put on the paper will create a tear.

Tensors show up a lot in physics, and so in mathematical physics. Technically they show up everywhere, since vectors and even plain old numbers (scalars, in the lingo) are kinds of tensors, but that’s not what I mean. Tensors can describe efficiently things whose magnitude and direction changes based on where something is and where it’s looking. So they are a great tool to use if one wants to represent stress, or how well magnetic fields pass through objects, or how electrical fields are distorted by the objects they move in. And they describe space, as well: general relativity is built on tensors. The mathematics of a tensor allow one to describe how space is shaped, based on how to measure the distance between two points in space.

My own mathematical education happened to be pretty tensor-light. I never happened to have courses that forced me to get good with them, and I confess to feeling intimidated when a mathematical argument gets deep into tensor mathematics. Joseph C Kolecki, with NASA’s Glenn (Lewis) Research Center, published in 2002 a nice little booklet “An Introduction to Tensors for Students of Physics and Engineering”. This I think nicely bridges some of the gap between mathematical structures like vectors and matrices, that mathematics and physics majors know well, and the kinds of tensors that get called tensors and that can be intimidating.

Reading the Comics, September 19, 2017: Visualization Edition

Comic Strip Master Command apparently doesn’t want me talking about the chances of Friday’s Showcase Showdown. They sent me enough of a flood of mathematically-themed strips that I don’t know when I’ll have the time to talk about the probability of that episode. (The three contestants spinning the wheel all tied, each spinning $1.00. And then in the spin-off, two of the three contestants also spun $1.00. And this after what was already a perfect show, in which the contestants won all six of the pricing games.) Well, I’ll do what comic strips I can this time, and carry on the last week of the Summer 2017 A To Z project, and we’ll see if I can say anything timely for Thursday or Saturday or so.

Jim Scancarelli’s Gasoline Alley for the 17th is a joke about the student embarrassing the teacher. It uses mathematics vocabulary for the specifics. And it does depict one of those moments that never stops, as you learn mathematics. There’s always more vocabulary. There’s good reasons to have so much vocabulary. Having names for things seems to make them easier to work with. We can bundle together ideas about what a thing is like, and what it may do, under a name. I suppose the trouble is that we’ve accepted a convention that we should define terms before we use them. It’s nice, like having the dramatis personae listed at the start of the play. But having that list isn’t the same as saying why anyone should care. I don’t know how to balance the need to make clear up front what one means and the need to not bury someone under a heap of similar-sounding names.

Mac King and Bill King’s Magic in a Minute for the 17th is another puzzle drawn from arithmetic. Look at it now if you want to have the fun of working it out, as I can’t think of anything to say about it that doesn’t spoil how the trick is done. The top commenter does have a suggestion about how to do the problem by breaking one of the unstated assumptions in the problem. This is the kind of puzzle created for people who want to motivate talking about parity or equivalence classes. It’s neat when you can say something of substance about a problem using simple information, though.

Terri Libenson’s Pajama Diaries for the 18th of September, 2017. When I first read this I assumed that of course the base of the triangle had length 4 and the second leg, at a 45-degree angle to that, had length 2, and I wondered if those numbers could be consistent for a triangle to exist. Of course they could, though. There is a bit of fun to be had working out whether a particular triangle could exist from knowing its side lengths, though.

Terri Libenson’s Pajama Diaries for the 18th uses trigonometry as the marker for deep thinking. It comes complete with a coherent equation, too. It gives the area of a triangle with two legs that meet at a 45 degree angle. I admit I am uncomfortable with promoting the idea that people who are autistic have some super-reasoning powers. (Also with the pop-culture idea that someone who spots things others don’t is probably at least a bit autistic.) I understand wanting to think someone’s troubles have some compensation. But people are who they are; it’s not like they need to observe some “balance”.

Lee Falk and Wilson McCoy’s The Phantom for the 10th of August, 1950 was rerun Monday. It’s a side bit of joking about between stories. And it uses knowledge of mathematics — and an interest in relativity — as signifier of civilization. I can only hope King Hano does better learning tensors on his own than I do.

Lee Falk and Wilson McCoy’s The Phantom for the 10th of August, 1950 and rerun the 18th of September, 2017. For my money, just reading a mathematics book doesn’t take. I need to take notes, as if it were in class. I don’t quite copy the book, but it comes close.

Mike Thompson’s Grand Avenue for the 18th goes back to classrooms and stuff for clever answers that subvert the teacher. And I notice, per the title given this edition, that the teacher’s trying to make the abstractness of three minus two tangible, by giving it an example. Which pairs it with …

Will Henry’s Wallace the Brace for the 18th, wherein Wallace asserts that arithmetic is easier if you visualize real things. I agree it seems to help with stuff like basic arithmetic. I wouldn’t want to try taking the cosine of an apple, though. Separating the quantity of a thing from the kind of thing measured is one of those subtle breakthroughs. It’s one of the ways that, for example, modern calculations differ from those of the Ancient Greeks. But it does mean thinking of numbers in, we’d say, a more abstract way than they did, and in a way that seems to tax us more.

Wallace the Brave recently had a book collection published, by the way. I mention because this is one of a handful of comics with a character who likes pinball, and more, who really really loves the Williams game FunHouse. This is an utterly correct choice for favorite pinball game. It’s one of the games that made me a pinball enthusiast.

Ryan North’s Dinosaur Comics rerun for the 19th I mention on loose grounds. In it T-Rex suggests trying out an alternate model for how gravity works. The idea, of what seems to be gravity “really” being the shade cast by massive objects in a particle storm, was explored in the late 17th and early 18th century. It avoids the problem of not being able to quite say what propagates gravitational attraction. But it also doesn’t work, analytically. We would see the planets orbit differently if this were how gravity worked. And there’s the problem about mass and energy absorption, as pointed out in the comic. But it can often be interesting or productive to play with models that don’t work. You might learn something about models that do, or that could.

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.

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 End 2016 Mathematics A To Z: General Covariance

Today’s term is another request, and another of those that tests my ability to make something understandable. I’ll try anyway. The request comes from Elke Stangl, whose “Research Notes on Energy, Software, Life, the Universe, and Everything” blog I first ran across years ago, when she was explaining some dynamical systems work.

General Covariance

So, tensors. They’re the things mathematicians get into when they figure vectors just aren’t hard enough. Physics majors learn about them too. Electrical engineers really get into them. Some material science types too.

You maybe notice something about those last three groups. They’re interested in subjects that are about space. Like, just, regions of the universe. Material scientists wonder how pressure exerted on something will get transmitted. The structure of what’s in the space matters here. Electrical engineers wonder how electric and magnetic fields send energy in different directions. And physicists — well, everybody who’s ever read a pop science treatment of general relativity knows. There’s something about the shape of space something something gravity something equivalent acceleration.

So this gets us to tensors. Tensors are this mathematical structure. They’re about how stuff that starts in one direction gets transmitted into other directions. You can see how that’s got to have something to do with transmitting pressure through objects. It’s probably not too much work to figure how that’s relevant to energy moving through space. That it has something to do with space as just volume is harder to imagine. But physics types have talked about it quite casually for over a century now. Science fiction writers have been enthusiastic about it almost that long. So it’s kind of like the Roman Empire. It’s an idea we hear about early and often enough we’re never really introduced to it. It’s never a big new idea we’re presented, the way, like, you get specifically told there was (say) a War of 1812. We just soak up a couple bits we overhear about the idea and carry on as best our lives allow.

But to think of space. Start from somewhere. Imagine moving a little bit in one direction. How far have you moved? If you started out in this one direction, did you somehow end up in a different one? Now imagine moving in a different direction. Now how far are you from where you started? How far is your direction from where you might have imagined you’d be? Our intuition is built around a Euclidean space, or one close enough to Euclidean. These directions and distances and combined movements work as they would on a sheet of paper, or in our living room. But there is a difference. Walk a kilometer due east and then one due north and you will not be in exactly the same spot as if you had walked a kilometer due north and then one due east. Tensors are efficient ways to describe those little differences. And they tell us something of the shape of the Earth from knowing these differences. And they do it using much of the form that matrices and vectors do, so they’re not so hard to learn as they might be.

That’s all prelude. Here’s the next piece. We go looking at transformations. We take a perfectly good coordinate system and a point in it. Now let the light of the full Moon shine upon it, so that it shifts to being a coordinate werewolf. Look around you. There’s a tensor that describes how your coordinates look here. What is it?

You might wonder why we care about transformations. What was wrong with the coordinates we started with? But that’s because mathematicians have lumped a lot of stuff into the same name of “transformation”. A transformation might be something as dull as “sliding things over a little bit”. Or “turning things a bit”. It might be “letting a second of time pass”. Or “following the flow of whatever’s moving”. Stuff we’d like to know for physics work.

“General covariance” is a term that comes up when thinking about transformations. Suppose we have a description of some physics problem. By this mostly we mean “something moving in space” or “a bit of light moving in space”. That’s because they’re good building blocks. A lot of what we might want to know can be understood as some mix of those two problems.

Put your description through the same transformation your coordinate system had. This will (most of the time) change the details of how your problem’s represented. But does it change the overall description? Is our old description no longer even meaningful?

I trust at this point you’ve nodded and thought something like “well, that makes sense”. Give it another thought. How could we not have a “generally covariant” description of something? Coordinate systems are our impositions on a problem. We create them to make our lives easier. They’re real things in exactly the same way that lines of longitude and latitude are real. If we increased the number describing the longitude of every point in the world by 14, we wouldn’t change anything real about where stuff was or how to navigate to it. We couldn’t.

Here I admit I’m stumped. I can’t think of a good example of a system that would look good but not be generally covariant. I’m forced to resort to metaphors and analogies that make this essay particularly unsuitable to use for your thesis defense.

So here’s the thing. Longitude is a completely arbitrary thing. Measuring where you are east or west of some prime meridian might be universal, or easy for anyone to tumble onto. But the prime meridian is a cultural choice. It’s changed before. It may change again. Indeed, Geographic Information Services people still work with many different prime meridians. Most of them are for specialized purposes. Stuff like mapping New Jersey in feet north and east of some reference, for which Greenwich would make the numbers too ugly. But if our planet is mapped in an alien’s records, that map has at its center some line almost surely not Greenwich.

But latitude? Latitude is, at least, less arbitrary. That we measure it from zero to ninety degrees, north or south, is a cultural choice. (Or from -90 to 90 degrees. Same thing.) But that there’s a north pole and a south pole? That’s true as long as the planet is rotating. And that’s forced on us. If we tried to describe the Earth as rotating on an axis between Paris and Mexico City, we would … be fighting an uphill struggle, at least. It’s hard to see any problem that might make easier, apart from getting between Paris and Mexico City.

In models of the laws of physics we don’t really care about the north or south pole. A planet might have them or might not. But it has got some privileged stuff that just has to be so. We can’t have stuff that makes the speed of light in a vacuum change. And we have to make sense of a block of space that hasn’t got anything in it, no matter, no light, no energy, no gravity. I think those are the important pieces actually. But I’ll defer, growling angrily, to an expert in general relativity or non-Euclidean coordinates if I’ve misunderstood.

It’s often put that “general covariance” is one of the requirements for a scheme to describe General Relativity. I shall risk sounding like I’m making a joke and say that depends on your perspective. One can use different philosophical bases for describing General Relativity. In some of them you can see general covariance as a result rather than use it as a basic assumption. Here’s a 1993 paper by Dr John D Norton that describes some of the different ways to understand the point of general covariance.

By the way the term “general covariance” comes from two pieces. The “covariance” is because it describes how changes in one coordinate system are reflected in another. It’s “general” because we talk about coordinate transformations without knowing much about them. That is, we’re talking about transformations in general, instead of some specific case that’s easy to work with. This is why the mathematics of this can be frightfully tricky; we don’t know much about the transformations we’re working with. For a parallel, it’s easy to tell someone how to divide 14 into 112. It’s harder to tell them how to divide absolutely any number into absolutely any other number.

Quite a bit of mathematical physics plays into geometry. Gravity physicists mostly see as a problem of geometry. People who like reading up on science take that as given too. But many problems can be understood as a point or a blob of points in some kind of space, and how that point moves or that blob evolves in time. We don’t see “general covariance” in these other fields exactly. But we do see things that resemble it. It’s an idea with considerable reach.

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I’m not sure how I feel about this. For most of my essays I’ve kept away from equations, even for the Why Stuff Can Orbit sequence. But this is one of those subjects it’s hard to be exact about without equations. I might revisit this in a special all-symbols, calculus-included, edition. Depends what my schedule looks like.

Reading the Comics, June 26, 2015: June 23, 2016 Plus Golden Lizards Edition

And now for the huge pile of comic strips that had some mathematics-related content on the 23rd of June. I admit some of them are just using mathematics as a stand-in for “something really smart people do”. But first, another moment with the Magic Realism Bot:

A watchmaker realises that a golden lizard is controlling mathematics.

— Magic Realism Bot (@MagicRealismBot) June 25, 2016

So, you know, watch the lizards and all.

Tom Batiuk’s Funky Winkerbean name-drops E = mc² as the sort of thing people respect. If the strip seems a little baffling then you should know that Mason’s last name is Jarr. He was originally introduced as a minor player in a storyline that wasn’t about him, so the name just had to exist. But since then Tom Batiuk’s decided he likes the fellow and promoted him to major-player status. And maybe Batiuk regrets having a major character with a self-consciously Funny Name, which is an odd thing considering he named his long-running comic strip for original lead character Funky Winkerbean.

Tom Batiuk’s Funky Winkerbean for the 23rd of June, 2016. They’re in the middle of filming one or possibly two movies about the silver-age comic book hero Starbuck Jones. This is all the comic strip is about anymore, so if you go looking for its old standbys — people dying — or its older standbys — band practice being rained on — sorry, you’ll have to look somewhere else. That somewhere else would be the yellowed strips taped to the walls in the teachers lounge.

Charlie Podrebarac’s CowTown depicts the harsh realities of Math Camp. I assume they’re the realities. I never went to one myself. And while I was on the Physics Team in high school I didn’t make it over to the competitive mathematics squad. Yes, I noticed that the not-a-numbers-person Jim Smith can’t come up with anything other than the null symbol, representing nothing, not even zero. I like that touch.

Ryan North’s Dinosaur Comics rerun is about Richard Feynman, the great physicist whose classic memoir What Do You Care What Other People Think? is hundreds of pages of stories about how awesome he was. Anyway, the story goes that Feynman noticed one of the sequences of digits in π and thought of the joke which T-Rex shares here.

π is believed but not proved to be a “normal” number. This means several things. One is that any finite sequence of digits you like should appear in its representation, somewhere. Feynman and T-Rex look for the sequence ‘999999’, which sure enough happens less than eight hundred digits past the decimal point. Lucky stroke there. There’s no reason to suppose the sequence should be anywhere near the decimal point. There’s no reason to suppose the sequence has to be anywhere in the finite number of digits of π that humanity will ever know. (This is why Carl Sagan’s novel Contact, which has as a plot point the discovery of a message apparently encoded in the digits of π, is not building on a stupid idea. That any finite message exists somewhere is kind-of certain. That it’s findable is not.)

e, mentioned in the last panel, is similarly thought to be a normal number. It’s also not proved to be. We are able to say that nearly all numbers are normal. It’s in much the way we can say nearly all numbers are irrational. But it is hard to prove that any numbers are. I believe that the only numbers humans have proved to be normal are a handful of freaks created to show normal numbers exist. I don’t know of any number that’s interesting in its own right that’s also been shown to be normal. We just know that almost all numbers are.

But it is imaginable that π or e aren’t. They look like they’re normal, based on how their digits are arranged. It’s an open question and someone might make a name for herself by answering the question. It’s not an easy question, though.

Missy Meyer’s Holiday Doodles breaks the news to me the 23rd was SAT Math Day. I had no idea and I’m not sure what that even means. The doodle does use the classic “two trains leave Chicago” introduction, the “it was a dark and stormy night” of Boring High School Algebra word problems.

Stephan Pastis’s Pearls Before Swine is about everyone who does science and mathematics popularization, and what we worry someone’s going to reveal about us. Um. Except me, of course. I don’t do this at all.

Ashleigh Brilliant’s Pot-Shots rerun is a nice little averages joke. It does highlight something which looks paradoxical, though. Typically if you look at the distributions of values of something that can be measured you get a bell cure, like Brilliant drew here. The value most likely to turn up — the mode, mathematicians say — is also the arithmetic mean. “The average”, is what everybody except mathematicians say. And even they say that most of the time. But almost nobody is at the average.

Looking at a drawing, Brilliant’s included, explains why. The exact average is a tiny slice of all the data, the “population”. Look at the area in Brilliant’s drawing underneath the curve that’s just the blocks underneath the upside-down fellow. Most of the area underneath the curve is away from that.

There’s a lot of results that are close to but not exactly at the arithmetic mean. Most of the results are going to be close to the arithmetic mean. Look at how many area there is under the curve and within four vertical lines of the upside-down fellow. That’s nearly everything. So we have this apparent contradiction: the most likely result is the average. But almost nothing is average. And yet almost everything is nearly average. This is why statisticians have their own departments, or get to make the mathematics department brand itself the Department of Mathematics and Statistics.

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: tensor

Tensor.

The true but unenlightening answer first: a tensor is a regular, rectangular grid of numbers. The most common kind is a two-dimensional grid, so that it looks like a matrix, or like the times tables. It might be square, with as many rows as columns, or it might be rectangular.

It can also be one-dimensional, looking like a row or a column of numbers. Or it could be three-dimensional, rows and columns and whole levels of numbers. We don’t try to visualize that. It can be what we call zero-dimensional, in which case it just looks like a solitary number. It might be four- or more-dimensional, although I confess I’ve never heard of anyone who actually writes out such a thing. It’s just so hard to visualize.

You can add and subtract tensors if they’re of compatible sizes. You can also do something like multiplication. And this does mean that tensors of compatible sizes will form a ring. Of course, that doesn’t say why they’re interesting.

Tensors are useful because they can describe spatial relationships efficiently. The word comes from the same Latin root as “tension”, a hint about how we can imagine it. A common use of tensors is in describing the stress in an object. Applying stress in different directions to an object often produces different effects. The classic example there is a newspaper. Rip it in one direction and you get a smooth, clean tear. Rip it perpendicularly and you get a raggedy mess. The stress tensor represents this: it gives some idea of how a force put on the paper will create a tear.

Tensors show up a lot in physics, and so in mathematical physics. Technically they show up everywhere, since vectors and even plain old numbers (scalars, in the lingo) are kinds of tensors, but that’s not what I mean. Tensors can describe efficiently things whose magnitude and direction changes based on where something is and where it’s looking. So they are a great tool to use if one wants to represent stress, or how well magnetic fields pass through objects, or how electrical fields are distorted by the objects they move in. And they describe space, as well: general relativity is built on tensors. The mathematics of a tensor allow one to describe how space is shaped, based on how to measure the distance between two points in space.

My own mathematical education happened to be pretty tensor-light. I never happened to have courses that forced me to get good with them, and I confess to feeling intimidated when a mathematical argument gets deep into tensor mathematics. Joseph C Kolecki, with NASA’s Glenn (Lewis) Research Center, published in 2002 a nice little booklet “An Introduction to Tensors for Students of Physics and Engineering”. This I think nicely bridges some of the gap between mathematical structures like vectors and matrices, that mathematics and physics majors know well, and the kinds of tensors that get called tensors and that can be intimidating.

Reading the Comics, May 4, 2015: Hatless Aliens Edition

I have to make two confessions for this round of mathematics comic strips. First is that I was busy for like two days and missed about a jillion comic strips. So this is the first part of some catching-up to do. The second is that I don’t have a favorite of this bunch. The most interesting, I suppose, is the Mr Boffo, because it lets me get into a little trivia about Albert Einstein. But there’s not any in this bunch that made me smile much or that gave me a juicy topic to discuss. Maybe tomorrow.

Steve Breen and Mike Thompson’s Grand Avenue ran a week of snarky-answers-to-word-problems strips. April 28th, April 30th, and May 2nd featured mathematics questions. This must reflect how easy it is to undermine the logic of a mathematics question. The April 27th strip is about using Roman numerals, which I suppose is arithmetic. I’m not sure there’s much point to learning Roman numerals. We don’t do any calculations using the Roman numeral scheme except to show why Arabic numerals are better. All you get from Roman numerals is an ability to read building cornerstones and movie copyright dates. At least learning cursive handwriting provides the learner with a way to make illegible notes.

Reading The Comics, November 9, 2014: Finally, A Picture Edition

I knew if I kept going long enough some cartoonist not on Gocomics.com would have to mention mathematics. That finally happened with one from Comics Kingdom, and then one from the slightly freak case of Rick Detorie’s One Big Happy. Detorie’s strip is on Gocomics.com, but a rerun from several years ago. He has a different one that runs on the normal daily pages. This is for sound economic reasons: actual newspapers pay much better than the online groupings of them (considering how cheap Comics Kingdom and Gocomics are for subscribers I’m not surprised) so he doesn’t want his current strips run on Gocomics.com. As for why his current strips do appear on, for example, the fairly good online comics page of AZcentral.com, that’s a good question, and one that deserves a full answer.

Vic Lee’s Pardon My Planet for the 6th of November, 2014.

Vic Lee’s Pardon My Planet (November 9), which broke the streak of Comics Kingdom not making it into these pages, builds around a quote from Einstein I never heard of before but which sounds like the sort of vaguely inspirational message that naturally attaches to famous names. The patient talks about the difficulty of finding something in “the middle of four-dimensional curved space-time”, although properly speaking it could be tricky finding anything within a bounded space, whether it’s curved or not. The generic mathematics problem you’d build from this would be to have some function whose maximum in a region you want to find (if you want the minimum, just multiply your function by minus one and then find the maximum of that), and there’s multiple ways to do that. One obvious way is the mathematical equivalent of getting to the top of a hill by starting from wherever you are and walking the steepest way uphill. Another way is to just amble around, picking your next direction at random, always taking directions that get you higher and usually but not always refusing directions that bring you lower. You can probably see some of the obvious problems with either approach, and this is why finding the spot you want can be harder than it sounds, even if it’s easy to get started looking.

Reuben Bolling’s Super Fun-Pak Comix (November 6), which is technically a rerun since the Super Fun-Pak Comix have been a longrunning feature in his Tom The Dancing Bug pages, is primarily a joke about the Heisenberg Uncertainty Principle, that there is a limit to what information one can know about the universe. This limit can be understood mathematically, though. The wave formulation of quantum mechanics describes everything there is to know about a system in terms of a function, called the state function and normally designated Ψ, the value of which can vary with location and time. Determining the location or the momentum or anything about the system is done by a process called “applying an operator to the state function”. An operator is a function that turns one function into another, which sounds like pretty sophisticated stuff until you learn that, like, “multiply this function by minus one” counts.

In quantum mechanics anything that can be observed has its own operator, normally a bit tricker than just “multiply this function by minus one” (although some are not very much harder!), and applying that operator to the state function is the mathematical representation of making that observation. If you want to observe two distinct things, such as location and momentum, that’s a matter of applying the operator for the first thing to your state function, and then taking the result of that and applying the operator for the second thing to it. And here’s where it gets really interesting: it doesn’t have to, but it can depend what order you do this in, so that you get different results applying the first operator and then the second from what you get applying the second operator and then the first. The operators for location and momentum are such a pair, and the result is that we can’t know to arbitrary precision both at once. But there are pairs of operators for which it doesn’t make a difference. You could, for example, know both the momentum and the electrical charge of Scott Baio simultaneously to as great a precision as your Scott-Baio-momentum-and-electrical-charge-determination needs are, and the mathematics will back you up on that.

Ruben Bolling’s Tom The Dancing Bug (November 6), meanwhile, was a rerun from a few years back when it looked like the Large Hadron Collider might never get to working and the glitches started seeming absurd, as if an enormous project involving thousands of people and millions of parts could ever suffer annoying setbacks because not everything was perfectly right the first time around. There was an amusing notion going around, illustrated by Bolling nicely enough, that perhaps the results of the Large Hadron Collider would be so disastrous somehow that the universe would in a fit of teleological outrage prevent its successful completion. It’s still a funny idea, and a good one for science fiction stories: Isaac Asimov used the idea in a short story dubbed “Thiotimoline and the Space Age”, published 1959, which resulted in the attempts to manipulate a compound which dissolves before it adds water might have accidentally sent hurricanes Carol, Edna, and Diane into New England in 1954 and 1955.

Chip Sansom’s The Born Loser (November 7) gives me a bit of a writing break by just being a pun strip that you can save for next March 14.

Dan Thompson’s Brevity (November 7), out of reruns, is another pun strip, though with giant monsters.

Francesco Marciuliano’s Medium Large (November 7) is about two of the fads of the early 80s, those of turning everything into a breakfast cereal somehow and that of playing with Rubik’s Cubes. Rubik’s Cubes have long been loved by a certain streak of mathematicians because they are a nice tangible representation of group theory — the study of things that can do things that look like addition without necessarily being numbers — that’s more interesting than just picking up a square and rotating it one, two, three, or four quarter-turns. I still think it’s easier to just peel the stickers off (and yet, the die-hard Rubik’s Cube Popularizer can point out there’s good questions about polarity you can represent by working out the rules of how to peel off only some stickers and put them back on without being detected).

Rick Detorie’s One Big Happy for the 9th of November, 2014.

Rick Detorie’s One Big Happy (November 9), and I’m sorry, readers about a month in the future from now, because that link’s almost certainly expired, is another entry in the subject of word problems resisted because the thing used to make the problem seem less abstract has connotations that the student doesn’t like.

Fred Wagner’s Animal Crackers (November 9) is your rare comic that could be used to teach positional notation, although when you actually pay attention you realize it doesn’t actually require that.

Mac and Bill King’s Magic In A Minute (November 9) shows off a mathematically-based slight-of-hand trick, describing a way to make it look like you’re reading your partner-monkey’s mind. This is probably a nice prealgebra problem to work out just why it works. You could also consider this a toe-step into the problem of encoding messages, finding a way to send information about something in a way that the original information can be recovered, although obviously this particular method isn’t terribly secure for more than a quick bit of stage magic.

Reading the Comics, August 29, 2014: Recurring Jokes Edition

Well, I did say we were getting to the end of summer. It’s taken only a couple days to get a fresh batch of enough mathematics-themed comics to include here, although the majority of them are about mathematics in ways that we’ve seen before, sometimes many times. I suppose that’s fair; it’s hard to keep thinking of wholly original mathematics jokes, after all. When you’ve had one killer gag about “537”, it’s tough to move on to “539” and have it still feel fresh.

Tom Toles’s Randolph Itch, 2 am (August 27, rerun) presents Randolph suffering the nightmare of contracting a case of entropy. Entropy might be the 19th-century mathematical concept that’s most achieved popular recognition: everyone knows it as some kind of measure of how disorganized things are, and that it’s going to ever increase, and if pressed there’s maybe something about milk being stirred into coffee that’s linked with it. The mathematical definition of entropy is tied to the probability one will find whatever one is looking at in a given state. Work out the probability of finding a system in a particular state — having particles in these positions, with these speeds, maybe these bits of magnetism, whatever — and multiply that by the logarithm of that probability. Work out that product for all the possible ways the system could possibly be configured, however likely or however improbable, just so long as they’re not impossible states. Then add together all those products over all possible states. (This is when you become grateful for learning calculus, since that makes it imaginable to do all these multiplications and additions.) That’s the entropy of the system. And it applies to things with stunning universality: it can be meaningfully measured for the stirring of milk into coffee, to heat flowing through an engine, to a body falling apart, to messages sent over the Internet, all the way to the outcomes of sports brackets. It isn’t just body parts falling off.

Randy Glasbergen’s _The Better Half_ For the 28th of August, 2014.

Randy Glasbergen’s The Better Half (August 28) does the old joke about not giving up on algebra someday being useful. Do teachers in other subjects get this? “Don’t worry, someday your knowledge of the Panic of 1819 will be useful to you!” “Never fear, someday they’ll all look up to you for being able to diagram a sentence!” “Keep the faith: you will eventually need to tell someone who only speaks French that the notebook of your uncle is on the table of your aunt!”

Eric the Circle (August 28, by “Gilly” this time) sneaks into my pages again by bringing a famous mathematical symbol into things. I’d like to make a mention of the links between mathematics and music which go back at minimum as far as the Ancient Greeks and the observation that a lyre string twice as long produced the same note one octave lower, but lyres and strings don’t fit the reference Gilly was going for here. Too bad.

Zach Weinersmith’s Saturday Morning Breakfast Cereal (August 28) is another strip to use a “blackboard full of mathematical symbols” as visual shorthand for “is incredibly smart stuff going on”. The symbols look to me like they at least started out as being meaningful — they’re the kinds of symbols I expect in describing the curvature of space, and which you can find by opening up a book about general relativity — though I’m not sure they actually stay sensible. (It’s not the kind of mathematics I’ve really studied.) However, work in progress tends to be sloppy, the rough sketch of an idea which can hopefully be made sound.

Anthony Blades’s Bewley (August 29) has the characters stare into space pondering the notion that in the vastness of infinity there could be another of them out there. This is basically the same existentially troublesome question of the recurrence of the universe in enough time, something not actually prohibited by the second law of thermodynamics and the way entropy tends to increase with the passing of time, but we have already talked about that.

Reading the Comics, November 13, 2013

For this week’s round of comic strips there’s almost a subtler theme than “they mention math in some way”: several have got links to statistical mechanics and the problem of recurrence. I’m not sure what’s gotten into Comic Strip Master Command that they sent out instructions to do comics that I can tie to the astounding interactions of infinities and improbable events, but it makes me wonder if I need to write a few essays about it.

Gene Weingarten, Dan Weingarten, and David Clark’s Barney and Clyde (October 30) summons the classic “infinite monkeys” problem of probability for its punch line. The concept — that if you had something producing strings of letters at random (taken to be monkeys because, I suppose, it’s assumed they would hit every key without knowing what sensibly comes next), it would, given enough time, produce any given result. The idea goes back a long way, and it’s blessed with a compelling mental image even though typewriters are a touch old-fashioned these days.

It seems to have gotten its canonical formulation in Émile Borel’s 1913 article “Statistical Mechanics and Irreversibility”, as you might expect since statistical mechanics brings up the curious problem of entropy. In short: every physical interaction, say, when two gases — let’s say clear air and some pink smoke as 1960s TV shows used to knock characters out — mix, is time-reversible. Look at the interaction of one clear-gas molecule and one pink-gas molecule and you can’t tell whether it’s playing forward or backward. But look at the entire room and it’s obvious whether they’re mixing or unmixing. How can something be time-reversible at every step of every interaction but not in whole?

The idea got a second compelling metaphor with Jorge Luis Borges’s Library of Babel, with a bit more literary class and in many printings fewer monkeys.