My 2019 Mathematics A To Z: Quadrature


I got a good nomination for a Q topic, thanks again to goldenoj. It was for Qualitative/Quantitative. Either would be a good topic, but they make a natural pairing. They describe the things mathematicians look for when modeling things. But ultimately I couldn’t find an angle that I liked. So rather than carry on with an essay that wasn’t working I went for a topic of my own. Might come back around to it, though, especially if nothing good presents itself for the letter X, which will probably need to be a wild card topic anyway.

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Quadrature.

We like comparing sizes. I talked about that some with norms. We do the same with shapes, though. We’d like to know which one is bigger than another, and by how much. We rely on squares to do this for us. It could be any shape, but we in the western tradition chose squares. I don’t know why.

My guess, unburdened by knowledge, is the ancient Greek tradition of looking at the shapes one can make with straightedge and compass. The easiest shape these tools make is, of course, circles. But it’s hard to find a circle with the same area as, say, any old triangle. Squares are probably a next-best thing. I don’t know why not equilateral triangles or hexagons. Again I would guess that the ancient Greeks had more rectangular or square rooms than the did triangles or hexagons, and went with what they knew.

So that’s what lurks behind that word “quadrature”. It may be hard for us to judge whether this pentagon is bigger than that octagon. But if we find squares that are the same size as the pentagon and the octagon, great. We can spot which of the squares is bigger, and by how much.

Straightedge-and-compass lets you find the quadrature for many shapes. Like, take a rectangle. Let me call that ABCD. Let’s say that AB is one of the long sides and BC one of the short sides. OK. Extend AB, outwards, to another point that I’ll call E. Pick E so that the length of BE is the same as the length of BC.

Next, bisect the line segment AE. Call that point F. F is going to be the center of a new semicircle, one with radius FE. Draw that in, on the side of AE that’s opposite the point C. Because we are almost there.

Extend the line segment CB upwards, until it touches this semicircle. Call the point where it touches G. The line segment BG is the side of a square with the same area as the original rectangle ABCD. If you know enough straightedge-and-compass geometry to do that bisection, you know enough to turn BG into a square. If you’re not sure why that’s the correct length, you can get there quickly. Use a little algebra and the Pythagorean theorem.

Neat, yeah, I agree. Also neat is that you can use the same trick to find the area of a parallelogram. A parallelogram has the same area as a square with the same bases and height between them, you remember. So take your parallelogram, draw in some perpendiculars to share that off into a rectangle, and find the quadrature of that rectangle. you’ve got the quadrature of your parallelogram.

Having the quadrature of a parallelogram lets you find the quadrature of any triangle. Pick one of the sides of the triangle as the base. You have a third point not on that base. Draw in the parallel to that base that goes through that third point. Then choose one of the other two sides. Draw the parallel to that side which goes through the other point. Look at that: you’ve got a parallelogram with twice the area of your original triangle. Bisect either the base or the height of this parallelogram, as you like. Then follow the rules for the quadrature of a parallelogram, and you have the quadrature of your triangle. Yes, you’re doing a lot of steps in-between the triangle you started with and the square you ended with. Those steps don’t count, not by this measure. Getting the results right matters.

And here’s some more beauty. You can find the quadrature for any polygon. Remember how you can divide any polygon into triangles? Go ahead and do that. Find the quadrature for every one of those triangles then. And you can create a square that has an area as large as all those squares put together. I’ll refrain from saying quite how, because realizing how is such a delight, one of those moments that at least made me laugh at how of course that’s how. It’s through one of those things that even people who don’t know mathematics know about.

With that background you understand why people thought the quadrature of the circle ought to be possible. Moreso when you know that the lune, a particular crescent-moon-like shape, can be squared. It looks so close to a half-circle that it’s obvious the rest should be possible. It’s not, and it took two thousand years and a completely different idea of geometry to prove it. But it sure looks like it should be possible.

Along the way to modernity quadrature picked up a new role. This is as part of calculus. One of the legs of calculus is integration. There is an interpretation of what the (definite) integral of a function means so common that we sometimes forget it doesn’t have to be that. This is to say that the integral of a function is the area “underneath” the curve. That is, it’s the area bounded by the limits of integration, by the horizontal axis, and by the curve represented by the function. If the function is sometimes less than zero, within the limits of integration, we’ll say that the integral represents the “net area”. Then we allow that the net area might be less than zero. Then we ignore the scolding looks of the ancient Greek mathematicians.

No matter. We love being able to find “the” integral of a function. This is a new function, and evaluating it tells us what this net area bounded by the limits of integration is. Finding this is “integration by quadrature”. At least in books published back when they wrote words like “to-day” or “coördinate”. My experience is that the term’s passed out of the vernacular, at least in North American Mathematician’s English.

Anyway the real flaw is that there are, like, six functions we can find the integral for. For the rest, we have to make do with approximations. This gives us “numerical quadrature”, a phrase which still has some currency.

And with my prologue about compass-and-straightedge quadrature you can see why it’s called that. Numerical integration schemes often rely on finding a polynomial with a part that looks like a graph of the function you’re interested in. The other edges look like the limits of the integration. Then the area of that polygon should be close to the area “underneath” this function. So it should be close to the integral of the function you want. And we’re old hands at how the quadrature of polygons, since we talked that out like five hundred words ago.

Now, no person ever has or ever will do numerical quadrature by compass-and-straightedge on some function. So why call it “numerical quadrature” instead of just “numerical integration”? Style, for one. “Quadrature” as a word has a nice tone, clearly jargon but not threateningly alien. Also “numerical integration” often connotes the solving differential equations numerically. So it can clarify whether you’re evaluating integrals or solving differential equations. If you think that’s a distinction worth making. Evaluating integrals and solving differential equations are similar together anyway.

And there is another adjective that often attaches to quadrature. This is Gaussian Quadrature. Gaussian Quadrature is, in principle, a fantastic way to do numerical integration perfectly. For some problems. For some cases. The insight which justifies it to me is one of those boring little theorems you run across in the chapter introducing How To Integrate. It runs something like this. Suppose ‘f’ is a continuous function, with domain the real numbers and range the real numbers. Suppose a and b are the limits of integration. Then there’s at least one point c, between a and b, for which:

\int_a^b f(x) dx = f(c) \cdot (b - a)

So if you could pick the right c, any integration would be so easy. Evaluate the function for one point and multiply it by whatever b minus a is. The catch is, you don’t know what c is.

Except there’s some cases where you kinda do. Like, if f is a line, rising or falling with a constant slope from a to b? Then have c be the midpoint of a and b.

That won’t always work. Like, if f is a parabola on the region from a to b, then c is not going to be the midpoint. If f is a cubic, then the midpoint is probably not c. And so on. And if you don’t know what kind of function f is? There’s no guessing where c will be.

But. If you decide you’re only trying to certain kinds of functions? Then you can do all right. If you decide you only want to integrate polynomials, for example, then … well, you’re not going to find a single point c for this. But what you can find is a set of points between a and b. Evaluate the function for those points. And then find a weighted average by rules I’m not getting into here. And that weighted average will be exactly that integral.

Of course there’s limits. The Gaussian Quadrature of a function is only possible if you can evaluate the function at arbitrary points. If you’re trying to integrate, like, a set of sample data it’s inapplicable. The points you pick, and the weighting to use, depend on what kind of function you want to integrate. The results will be worse the less your function is like what you supposed. It’s tedious to find what these points are for a particular assumption of function. But you only have to do that once, or look it up, if you know (say) you’re going to use polynomials of degree up to six or something like that.

And there are variations on this. They have names like the Chevyshev-Gauss Quadrature, or the Hermite-Gauss Quadrature, or the Jacobi-Gauss Quadrature. There are even some that don’t have Gauss’s name in them at all.

Despite that, you can get through a lot of mathematics not talking about quadrature. The idea implicit in the name, that we’re looking to compare areas of different things by looking at squares, is obsolete. It made sense when we worked with numbers that depended on units. One would write about a shape’s area being four times another shape’s, or the length of its side some multiple of a reference length.

We’ve grown comfortable thinking of raw numbers. It makes implicit the step where we divide the polygon’s area by the area of some standard reference unit square. This has advantages. We don’t need different vocabulary to think about integrating functions of one or two or ten independent variables. We don’t need wordy descriptions like “the area of this square is to the area of that as the second power of this square’s side is to the second power of that square’s side”. But it does mean we don’t see squares as intermediaries to understanding different shapes anymore.


Thank you again for reading. This essay and all the others written for the Fall 2019 A to Z should be at this link. This should include, later this week, something for the letter R. And all of the A to Z essays ought to be at this link.

My 2018 Mathematics A To Z: Volume


Ray Kassinger, of the popular web comic Housepets!, had a silly suggestion when I went looking for topics. In one episode of Mystery Science Theater 3000, Crow T Robot gets the idea that you could describe the size of a space by the number of turkeys which fill it. (It’s based on like two minor mentions of “turkeys” in the show they were watching.)

I liked that episode. I’ve got happy memories of the time when I first saw it. I thought the sketch in which Crow T Robot got so volume-obsessed was goofy and dumb in the fun-nerd way.

I accept Mr Kassinger’s challenge only I’m going to take it seriously.

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Art by Thomas K Dye, creator of the web comics Newshounds, Something Happens, and Infinity Refugees. His current project is Projection Edge. And you can get Projection Edge six months ahead of public publication by subscribing to his Patreon. And he’s on Twitter as @Newshoundscomic.

Volume.

How big is a thing?

There is a legend about Thomas Edison. He was unimpressed with a new hire. So he hazed the college-trained engineer who deeply knew calculus. He demanded the engineer tell him the volume within a light bulb. The engineer went to work, making measurements of the shape of the bulb’s outside. And then started the calculations. This involves a calculus technique called “volumes of rotation”. This can tell the volume within a rotationally symmetric shape. It’s tedious, especially if the outer edge isn’t some special nice shape. Edison, fed up, took the bulb, filled it with water, poured that out into a graduated cylinder and said that was the answer.

I’m skeptical of legends. I’m skeptical of stories about the foolish intellectual upstaged by the practical man-of-action. And I’m skeptical of Edison because, jeez, I’ve read biographies of the man. Even the fawning ones make him out to be yeesh.

But the legend’s Edison had a point. If the volume of a shape is not how much stuff fits inside the shape, what is it? And maybe some object has too complicated a shape to find its volume. Can we think of a way to produce something with the same volume, but that is easier? Sometimes we can. When we do this with straightedge and compass, the way the Ancient Greeks found so classy, we call this “quadrature”. It’s called quadrature from its application in two dimensions. It finds, for a shape, a square with the same area. For a three-dimensional object, we find a cube with the same volume. Cubes are easy to understand.

Straightedge and compass can’t do everything. Indeed, there’s so much they can’t do. Some of it is stuff you’d think it should be able to, like, find a cube with the same volume as a sphere. Integration gives us a mathematical tool for describing how much stuff is inside a shape. It’s even got a beautiful shorthand expression. Suppose that D is the shape. Then its volume V is:

V = \int\int\int_D dV

Here “dV” is the “volume form”, a description of how the coordinates we describe a space in relate to the volume. The \int\int\int is jargon, meaning, “integrate over the whole volume”. The subscript “D” modifies that phrase by adding “of D” to it. Writing “D” is shorthand for “these are all the points inside this shape, in whatever coordinate system you use”. If we didn’t do that we’d have to say, on each \int sign, what points are inside the shape, coordinate by coordinate. At this level the equation doesn’t offer much help. It says the volume is the sum of infinitely many, infinitely tiny pieces of volume. True, but that doesn’t give much guidance about whether it’s more or less than two cups of water. We need to get more specific formulas, usually. We need to pick coordinates, for example, and say what coordinates are inside the shape. A lot of the resulting formulas can’t be integrated exactly. Like, an ellipsoid? Maybe you can integrate that. Don’t try without getting hazard pay.

We can approximate this integral. Pick a tiny shape whose volume is easy to know. Fill your shape with duplicates of it. Count the duplicates. Multiply that count by the volume of this tiny shape. Done. This is numerical integration, sometimes called “numerical quadrature”. If we’re being generous, we can say the legendary Edison did this, using water molecules as the tiny shape. And working so that he didn’t need to know the exact count or the volume of individual molecules. Good computational technique.

It’s hard not to feel we’re begging the question, though. We want the volume of something. So we need the volume of something else. Where does that volume come from?

Well, where does an inch come from? Or a centimeter? Whatever unit you use? You pick something to use as reference. Any old thing will do. Which is why you get fascinating stories about choosing what to use. And bitter arguments about which of several alternatives to use. And we express the length of something as some multiple of this reference length.

Volume works the same way. Pick a reference volume, something that can be one unit-of-volume. Other volumes are some multiple of that unit-of-volume. Possibly a fraction of that unit-of-volume.

Usually we use a reference volume that’s based on the reference length. Typically, we imagine a cube that’s one unit of length on each side. The volume of this cube with sides of length 1 unit-of-length is then 1 unit-of-volume. This seems all nice and orderly and it’s surely not because mathematicians have paid off by six-sided-dice manufacturers.

Does it have to be?

That we need some reference volume seems inevitable. We can’t very well say the area of something is ten times nothing-in-particular. Does that reference volume have to be a cube? Or even a rectangle or something else? It seems obvious that we need some reference shape that tiles, that can fill up space by itself … right?

What if we don’t?

I’m going to drop out of three dimensions a moment. Not because it changes the fundamentals, but because it makes something easier. Specifically, it makes it easier if you decide you want to get some construction paper, cut out shapes, and try this on your own. What this will tell us about area is just as true for volume. Area, for a two-dimensional sapce, and volume, for a three-dimensional, describe the same thing. If you’ll let me continue, then, I will.

So draw a figure on a clean sheet of paper. What’s its area? Now imagine you have a whole bunch of shapes with reference areas. A bunch that have an area of 1. That’s by definition. That’s our reference area. A bunch of smaller shapes with an area of one-half. By definition, too. A bunch of smaller shapes still with an area of one-third. Or one-fourth. Whatever. Shapes with areas you know because they’re marked on them.

Here’s one way to find the area. Drop your reference shapes, the ones with area 1, on your figure. How many do you need to completely cover the figure? It’s all right to cover more than the figure. It’s all right to have some of the reference shapes overlap. All you need is to cover the figure completely. … Well, you know how many pieces you needed for that. You can count them up. You can add up the areas of all these pieces needed to cover the figure. So the figure’s area can’t be any bigger than that sum.

Can’t be exact, though, right? Because you might get a different number if you covered the figure differently. If you used smaller pieces. If you arranged them better. This is true. But imagine all the possible reference shapes you had, and all the possible ways to arrange them. There’s some smallest area of those reference shapes that would cover your figure. Is there a more sensible idea for what the area of this figure would be?

And put this into three dimensions. If we start from some reference shapes of volume 1 and maybe 1/2 and 1/3 and whatever other useful fractions there are? Doesn’t this covering make sense as a way to describe the volume? Cubes or rectangles are easy to imagine. Tetrahedrons too. But why not any old thing? Why not, as the Mystery Science Theater 3000 episode had it, turkeys?

This is a nice, flexible, convenient way to define area. So now let’s see where it goes all bizarre. We know this thanks to Giuseppe Peano. He’s among the late-19th/early-20th century mathematicians who shaped modern mathematics. They did this by showing how much of our mathematics broke intuition. Peano was (here) exploring what we now call fractals. And noted a family of shapes that curl back on themselves, over and over. They’re beautiful.

And they fill area. Fill volume, if done in three dimensions. It seems impossible. If we use this covering scheme, and try to find the volume of a straight line, we get zero. Well, we find that any positive number is too big, and from that conclude that it has to be zero. Since a straight line has length, but not volume, this seems fine. But a Peano curve won’t go along with this. A Peano curve winds back on itself so much that there is some minimum volume to cover it.

This unsettles. But this idea of volume (or area) by covering works so well. To throw it away seems to hobble us. So it seems worth the trade. We allow ourselves to imagine a line so long and so curled up that it has a volume. Amazing.


And now I get to relax and unwind and enjoy a long weekend before coming to the letter ‘W’. That’ll be about some topic I figure I can whip out a nice tight 500 words about, and instead, produce some 1541-word monstrosity while I wonder why I’ve had no free time at all since August. Tuesday, give or take, it’ll be available at this link, as are the rest of these glossary posts. Thanks for reading.

The Summer 2017 Mathematics A To Z: Integration


One more mathematics term suggested by Gaurish for the A-To-Z today, and then I’ll move on to a couple of others. Today’s is a good one.

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.

Integration.

Stand on the edge of a plot of land. Walk along its boundary. As you walk the edge pay attention. Note how far you walk before changing direction, even in the slightest. When you return to where you started consult your notes. Contained within them is the area you circumnavigated.

If that doesn’t startle you perhaps you haven’t thought about how odd that is. You don’t ever touch the interior of the region. You never do anything like see how many standard-size tiles would fit inside. You walk a path that is as close to one-dimensional as your feet allow. And encoded in there somewhere is an area. Stare at that incongruity and you realize why integrals baffle the student so. They have a deep strangeness embedded in them.

We who do mathematics have always liked integration. They grow, in the western tradition, out of geometry. Given a shape, what is a square that has the same area? There are shapes it’s easy to find the area for, given only straightedge and compass: a rectangle? Easy. A triangle? Just as straightforward. A polygon? If you know triangles then you know polygons. A lune, the crescent-moon shape formed by taking a circular cut out of a circle? We can do that. (If the cut is the right size.) A circle? … All right, we can’t do that, but we spent two thousand years trying before we found that out for sure. And we can do some excellent approximations.

That bit of finding-a-square-with-the-same-area was called “quadrature”. The name survives, mostly in the phrase “numerical quadrature”. We use that to mean that we computed an integral’s approximate value, instead of finding a formula that would get it exactly. The otherwise obvious choice of “numerical integration” we use already. It describes computing the solution of a differential equation. We’re not trying to be difficult about this. Solving a differential equation is a kind of integration, and we need to do that a lot. We could recast a solving-a-differential-equation problem as a find-the-area problem, and vice-versa. But that’s bother, if we don’t need to, and so we talk about numerical quadrature and numerical integration.

Integrals are built on two infinities. This is part of why it took so long to work out their logic. One is the infinity of number; we find an integral’s value, in principle, by adding together infinitely many things. The other is an infinity of smallness. The things we add together are infinitesimally small. That we need to take things, each smaller than any number yet somehow not zero, and in such quantity that they add up to something, seems paradoxical. Their geometric origins had to be merged into that of arithmetic, of algebra, and it is not easy. Bishop George Berkeley made a steady name for himself in calculus textbooks by pointing this out. We have worked out several logically consistent schemes for evaluating integrals. They work, mostly, by showing that we can make the error caused by approximating the integral smaller than any margin we like. This is a standard trick, or at least it is, now that we know it.

That “in principle” above is important. We don’t actually work out an integral by finding the sum of infinitely many, infinitely tiny, things. It’s too hard. I remember in grad school the analysis professor working out by the proper definitions the integral of 1. This is as easy an integral as you can do without just integrating zero. He escaped with his life, but it was a close scrape. He offered the integral of x as a way to test our endurance, without actually doing it. I’ve never made it through that.

But we do integrals anyway. We have tools on our side. We can show, for example, that if a function obeys some common rules then we can use simpler formulas. Ones that don’t demand so many symbols in such tight formation. Ones that we can use in high school. Also, ones we can adapt to numerical computing, so that we can let machines give us answers which are near enough right. We get to choose how near is “near enough”. But then the machines decide how long we’ll have to wait to get that answer.

The greatest tool we have on our side is the Fundamental Theorem of Calculus. Even the name promises it’s the greatest tool we might have. This rule tells us how to connect integrating a function to differentiating another function. If we can find a function whose derivative is the thing we want to integrate, then we have a formula for the integral. It’s that function we found. What a fantastic result.

The trouble is it’s so hard to find functions whose derivatives are the thing we wanted to integrate. There are a lot of functions we can find, mind you. If we want to integrate a polynomial it’s easy. Sine and cosine and even tangent? Yeah. Logarithms? A little tedious but all right. A constant number raised to the power x? Also tedious but doable. A constant number raised to the power x2? Hold on there, that’s madness. No, we can’t do that.

There is a weird grab-bag of functions we can find these integrals for. They’re mostly ones we can find some integration trick for. An integration trick is some way to turn the integral we’re interested in into a couple of integrals we can do and then mix back together. A lot of a Freshman Calculus course is a heap of tricks we’ve learned. They have names like “u-substitution” and “integration by parts” and “trigonometric substitution”. Some of them are really exotic, such as turning a single integral into a double integral because that leads us to something we can do. And there’s something called “differentiation under the integral sign” that I don’t know of anyone actually using. People know of it because Richard Feynman, in his fun memoir What Do You Care What Other People Think: 250 Pages Of How Awesome I Was In Every Situation Ever, mentions how awesome it made him in so many situations. Mathematics, physics, and engineering nerds are required to read this at an impressionable age, so we fall in love with a technique no textbook ever mentions. Sorry.

I’ve written about all this as if we were interested just in areas. We’re not. We like calculating lengths and volumes and, if we dare venture into more dimensions, hypervolumes and the like. That’s all right. If we understand how to calculate areas, we have the tools we need. We can adapt them to as many or as few dimensions as we need. By weighting integrals we can do calculations that tell us about centers of mass and moments of inertial, about the most and least probable values of something, about all quantum mechanics.

As often happens, this powerful tool starts with something anyone might ponder: what size square has the same area as this other shape? And then think seriously about it.