My All 2020 Mathematics A to Z: Jacobi Polynomials


Mr Wu, author of the Singapore Maths Tuition blog, gave me a good nomination for this week’s topic: the j-function of number theory. Unfortunately I concluded I didn’t understand the function well enough to write about it. So I went to a topic of my own choosing instead.

The Jacobi Polynomials discussed here are named for Carl Gustav Jacob Jacobi. Jacobi lived in Prussia in the first half of the 19th century. Though his career was short, it was influential. I’ve already discussed the Jacobian, which describes how changes of variables change volume. He has a host of other things named for him, most of them in matrices or mathematical physics. He was also a pioneer in those elliptic curves you hear so much about these days.

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Jacobi Polynomials.

Jacobi Polynomials are a family of functions. Polynomials, it happens; this is a happy case where the name makes sense. “Family” is the name mathematicians give to a bunch of functions that have some similarity. This often means there’s a parameter, and each possible value of the parameter describes a different function in the family. For example, we talk about the family of sine functions, S_n(z) . For every integer n we have the function S_n(z) = \sin(n z) where z is a real number between -π and π.

We like a family because every function in it gives us some nice property. Often, the functions play nice together, too. This is often something like mutual orthogonality. This means two different representatives of the family are orthogonal to one another. “Orthogonal” means “perpendicular”. We can talk about functions being perpendicular to one another through a neat mechanism. It comes from vectors. It’s easy to use vectors to represent how to get from one point in space to another. From vectors we define a dot product, a way of multiplying them together. A dot product has to meet a couple rules that are pretty easy to do. And if you don’t do anything weird? Then the dot product between two vectors is the cosine of the angle made by the end of the first vector, the origin, and the end of the second vector.

Functions, it turns out, meet all the rules for a vector space. (There are not many rules to make a vector space.) And we can define something that works like a dot product for two functions. Take the integral, over the whole domain, of the first function times the second. This meets all the rules for a dot product. (There are not many rules to make a dot product.) Did you notice me palm that card? When I did not say “the dot product is take the integral …”? That card will come back. That’s for later. For now: we have a vector space, we have a dot product, we can take arc-cosines, so why not define the angle between functions?

Mostly we don’t because we don’t care. Where we do care? We do like functions that are at right angles to one another. As with most things mathematicians do, it’s because it makes life easier. We’ll often want to describe properties of a function we don’t yet know. We can describe the function we don’t yet know as the sum of coefficients — some fixed real number — times basis functions that we do know. And then our problem of finding the function changes to one of finding the coefficients. If we picked a set of basis functions that are all orthogonal to one another, the finding of these coefficients gets easier. Analytically and numerically: we can often turn each coefficient into its own separate problem. Let a different computer, or at least computer process, work on each coefficient and get the full answer much faster.

The Jacobi Polynomials have three coefficients. I see them most often labelled α, β, and n. Likely you imagine this means it’s a huge family. It is huger than that. A zoologist would call this a superfamily, at least. Probably an order, possibly a class.

It turns out different relationships of these coefficients give you families of functions. Many of these families are noteworthy enough to have their own names. For example, if α and β are both zero, then the Jacobi functions are a family also known as the Legendre Polynomials. This is a great set of orthogonal polynomials. And the roots of the Legendre Polynomials give you information needed for Gaussian quadrature. Gaussian quadrature is a neat trick for numerically integrating a function. Take a weighted sum of the function you’re integrating evaluated at a set of points. This can get a very good — maybe even perfect — numerical estimate of the integral. The points to use, and the weights to use, come from a Legendre polynomial.

If α and β are both -\frac{1}{2} then the Jacobi Polynomials are the Chebyshev Polynomials of the first kind. (There’s also a second kind.) These are handy in approximation theory, describing ways to better interpolate a polynomial from a set of data. They also have a neat, peculiar relationship to the multiple-cosine formulas. Like, \cos(2\theta) = 2\cos^2(\theta) - 1 . And the second Chebyshev polynomial is T_2(x) = 2x^2 - 1 . Imagine sliding between x and cos(\theta) and you see the relationship. cos(3\theta) = 4 \cos^3(\theta) - 3\cos(\theta) and T_3(x) = 4x^3 - 3x . And so on.

Chebyshev Polynomials have some superpowers. One that’s most amazing is accelerating convergence. Often a numerical process, such as finding the solution of an equation, is an iterative process. You can’t find the answer all at once. You instead find an approximation and do something that improves it. Each time you do the process, you get a little closer to the true answer. This can be fine. But, if the problem you’re working on allows it, you can use the first couple iterations of the solution to figure out where this is going. The result is that you can get very good answers using the same amount of computer time you needed to just get decent answers. The trade, of course, is that you need to understand Chebyshev Polynomials and accelerated convergence. We always have to make trades like that.

Back to the Jacobi Polynomials family. If α and β are the same number, then the Jacobi functions are a family called the Gegenbauer Polynomials. These are great in mathematical physics, in potential theory. You can turn the gravitational or electrical potential function — that one-over-the-distance-squared force — into a sum of better-behaved functions. And they also describe zonal spherical harmonics. These let you represent functions on the surface of a sphere as the sum of coefficients times basis functions. They work in much the way the terms of a Fourier series do.

If β is zero and there’s a particular relationship between α and n that I don’t want to get into? The Jacobi Polynomials become the Zernike Polynomials, which I never heard of before this paragraph either. I read they are the tools you need to understand optics, and particularly how lenses will alter the light passing through.

Since the Jacobi Polynomials have a greater variety of form than even poison ivy has, you’ll forgive me not trying to list them. Or even listing a representative sample. You might also ask how they’re related at all.

Well, they all solve the same differential equation, for one. Not literally a single differential equation. A family of differential equations, where α and β and n turn up in the coefficients. The formula using these coefficients is the same in all these differential equations. That’s a good reason to see a relationship. Or we can write the Jacobi Polynomials as a series, a function made up of the sum of terms. The coefficients for each of the terms depends on α and β and n, always in the same way. I’ll give you that formula. You won’t like it and won’t ever use it. The Jacobi Polynomial for a particular α, β, and n is the polynomial

P_n^{(\alpha, \beta)}(z) = (n+\alpha)!(n + \beta)!\sum_{s=0}^n \frac{1}{s!(n + \alpha - s)!(\beta + s)!(n - s)!}\left(\frac{z-1}{2}\right)^{n-s}\left(\frac{z + 1}{2}\right)^s

Its domain, by the way, is the real numbers from -1 to 1. We need something for the domain. It turns out there’s nothing you can do on the real numbers that you can’t fit into the domain from -1 to 1 anyway. (If you have to do something on, say, the interval from 10 to 54? Do a change of variable, scaling things down and moving them, and use -1 to 1. Then undo that change when you’re done.) The range is the real numbers, as you’d expect.

(You maybe noticed I used ‘z’ for the independent variable there, rather than ‘x’. Usually using ‘z’ means we expect this to be a complex number. But ‘z’ here is definitely a real number. This is because we can also get to the Jacobi Polynomials through the hypergeometric series, a function I don’t want to get into. But for the hypergeometric series we are open to the variable being a complex number. So many references carry that ‘z’ back into Jacobi Polynomials.)

Another thing which links these many functions is recurrence. If you know the Jacobi Polynomial for one set of parameters — and you do; P_0^{(\alpha, \beta)}(z) = 1 — you can find others. You do this in a way rather like how you find new terms in the Fibonacci series by adding together terms you already know. These formulas can be long. Still, if you know P_{n-1}^{(\alpha, \beta)} and P_{n-2}^{(\alpha, \beta)} for the same α and β? Then you can calculate P_n^{(\alpha, \beta)} with nothing more than pen, paper, and determination. If it helps,

P_1^{(\alpha, \beta)}(z) = (\alpha + 1) + (\alpha + \beta + 2)\frac{z - 1}{2}

and this is true for any α and β. You’ll never do anything with that. This is fine.

There is another way that all these many polynomials are related. It goes back to their being orthogonal. We measured orthogonality by a dot product. Back when I palmed that card I told you was the integral of the two functions multiplied together. This is indeed a dot product. We can define others. We make those others by taking a weighted integral of the product of these two functions. That is, integrate the two functions times a third, a weight function. Of course there’s reasons to do this; they amount to deciding that some parts of the domain are more important than others. The weight function can be anything that meets a few rules. If you want to get the Jacobi Polynomials out of them, you start with the function P_0^{(\alpha, \beta)}(z) = 1 and the weight function

w_n(z) = (1 - z)^{\alpha} (1 + z)^{\beta}

As I say, though, you’ll never use that. If you’re eager and ready to leap into this work you can use this to build a couple Legendre Polynomials. Or Chebyshev Polynomials. For the full Jacobi Polynomials, though? Use, like, the command JacobiP[n, a, b, z] in Mathematica, or jacobiP(n, a, b, z) in Matlab. Other people have programmed this for you. Enjoy their labor.

In my work I have not used the full set of Jacobi Polynomials much. There’s more of them than I need. I do rely on the Legendre Polynomials, and the Chebyshev Polynomials. Other mathematicians use other slices regularly. It is stunning to sometimes look and realize that these many functions, different as they look, are reflections of one another, though. Mathematicians like to generalize, and find one case that covers as many things as possible. It’s rare that we are this successful.


I thank you for reading this. All of this year’s A-to-Z essays should be available at this link. The essays from every A-to-Z sequence going back to 2015 should be at this link. And I’m already looking ahead to the M, N, and O essays that I’ll be writing the day before publication instead of the week before like I want! I appreciate any nominations you have, even ones I can’t cover fairly.

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.

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.
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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.

Dice and Compass Games


By the way, I wasn’t the only one to write about that dice problem the other day. Jim Doherty, with the MrDardy blog, also spoke about it. He’s actively teaching, and hopes to report what his classes made of it. He writes regularly about the teaching experience and the experiments to try to make it better.

This did get me into a fun bit of Twitter chatter about the odds of bloggers writing about the same question like this. I can’t imagine the question having a real answer, though. We both wrote about it because we saw the same initial question on Twitter. But we saw it because we both try following stuff in the mathematics blogosphere. Among other things, that seeks out and connects fun problems like this. And it’s a problem easy to write up.

In a bit more of mathematical puttering-about news, here’s a pleasant little tool for making geometric constructions. It’s got compass-and-straightedge, as well as protractor-and-ruler, features. I admit I’m not sure I have a practical use for it, but it’s pretty and fun.

And you can do amazing things with compass-and-straightedge constructions. For my money, the most amazing thing is quadrature. That’s starting from some other shape and constructing a square with the same area. There are shapes it’s easy to do this for: rectangles, triangles, polygons of all sorts. There are shapes it’s impossible to do this for: circles, most famously. And then there are shapes you’d think would be impossible but aren’t, such as certain lunes. These are crescent-moon shapes. If circles are impossible (and they are), wouldn’t you think a shape with edges are the arcs of two different circles would be impossible too? And yet, they’re possible, for at least the right lunes.

Here’s one. Draw a half-circle. Let’s say, for convenience, that you’ve drawn the upper half of one. Now draw the vertical line from the center of the circle to its top point. Then draw the line connecting the leftmost corner to the top corner. This will be the hypotenuse of a right triangle with two 45-degree angles.

Next, draw the half-circle that fits on that hypotenuse, and that points outward, past the edge of the original half-circle. The lune of interest is the one between the original half-circle and the new one. And you can, using only compass and straightedge, produce a square with exactly the same area as that curved shape. If that’s not remarkable enough, it’s the same area as that triangle we had to start out. But we can not, using compass and straightedge, make a square that’s the same area as that little wedge between lune and triangle.

The quadrature of the triangle isn’t too hard to work out, if you start from scratch. (If you don’t know how to start, try starting with the area of a rectangle instead.) The lune, I’ll admit, I didn’t figure out by myself, but it’s not absurd. That the remaining wedge is impossible you won’t prove on your own. I’m not sure how I would explain it, not in only a few essays.

And with that hook, I’d like to toss in one last appeal for any requests for the Winter 2016 Mathematics A To Z. Before you pull out calendars on me and work out how long three-a-week essays might last, remember that I live in a state that typically gets a long winter. Letters are filling up, but many are still open. And last time around I had to really dig to find a good y- or z- term. If you want a sure in, those are good letters to think up.