As mentioned, ‘X’ is a difficult letter for a glossary project. There aren’t many mathematical terms that start with the letter, as much as it is the default variable name. Making things better is that many of the terms that do are important ones. Xor, from my 2015 A-to-Z, is an example of this. It’s one of the major pieces of propositional logic, and anyone working in logic gets familiar with it really fast.
The letter ‘X’ is a problem for this sort of glossary project. At least around the fourth time you do one, as you exhaust the good terms that start with the letter X. In 2018, I went to the Extreme Value Theorem, using the 1990s Rule that x- and ex- were pretty much the same thing. The Extreme Value Theorem is one of those little utility theorems. On a quick look it seems too obvious to tell us anything useful. It serves a role in proofs that do tell us interesting, surprising things.
Today (the 26th of November) is the Thanksgiving holiday in the United States. The holiday’s set, by law since 1941, to the fourth Thursday in November. (Before then it was customarily the last Thursday in November, but set by Presidential declaration. After Franklin Delano Roosevelt set the holiday to the third Thursday in November, to extend the 1939 and 1940 Christmas-shopping seasons — a decision Republican Alf Landon characterized as Hitlerian — the fourth Thursday was encoded in law.)
Any know-it-all will tell you, though, how the 13th of the month is very slightly more likely to be a Friday than any other day of the week. This is because the Gregorian calendar has that peculiar century-year leap day rule. It throws off the regular progression of the dates through the week. It takes 400 years for the calendar to start repeating itself. How does this affect the fourth Thursday of November? (A month which, this year, did have a Friday the 13th.)
It turns out, it changes things in subtle ways. Thanksgiving, by the current rule, can be any date between the 22nd and 28th; it’s most likely to be any of the 22nd, 24th, or 26th. (This implies that the 13th of November is equally likely to be a Friday, Wednesday, or Monday, a result that surprises me too.) So here’s how often which date is Thanksgiving. This if we pretend the current United States definition of Thanksgiving will be in force for 400 years unchanged:
|November||Will Be Thanksgiving|
|times in 400 years|
I hope this helps your planning, somehow.
Today’s is another topic suggested by Mr Wu, author of the Singapore Maths Tuition blog. The Wronskian is named for Józef Maria Hoëne-Wroński, a Polish mathematician, born in 1778. He served in General Tadeusz Kosciuszko’s army in the 1794 Kosciuszko Uprising. After being captured and forced to serve in the Russian army, he moved to France. He kicked around Western Europe and its mathematical and scientific circles. I’d like to say this was all creative and insightful, but, well. Wikipedia describes him trying to build a perpetual motion machine. Trying to square the circle (also impossible). Building a machine to predict the future. The St Andrews mathematical biography notes his writing a summary of “the general solution of the fifth degree [polynomial] equation”. This doesn’t exist.
Both sources, though, admit that for all that he got wrong, there were flashes of insight and brilliance in his work. The St Andrews biography particularly notes that Wronski’s tables of logarithms were well-designed. This is a hard thing to feel impressed by. But it’s hard to balance information so that it’s compact yet useful. He wrote about the Wronskian in 1812; it wouldn’t be named for him until 1882. This was 29 years after his death, but it does seem likely he’d have enjoyed having a familiar thing named for him. I suspect he wouldn’t enjoy my next paragraph, but would enjoy the fight with me about it.
The Wronskian is a thing put into Introduction to Ordinary Differential Equations courses because students must suffer in atonement for their sins. Those who fail to reform enough must go on to the Hessian, in Partial Differential Equations.
To be more precise, the Wronskian is the determinant of a matrix. The determinant you find by adding and subtracting products of the elements in a matrix together. It’s not hard, but it is tedious, and gets more tedious pretty fast as the matrix gets bigger. (In Big-O notation, it’s the order of the cube of the matrix size. This is rough, for things humans do, although not bad as algorithms go.) The matrix here is made up of a bunch of functions and their derivatives. The functions need to be ones of a single variable. The derivatives, you need first, second, third, and so on, up to one less than the number of functions you have.
If you have two functions, and , you need their first derivatives, and . If you have three functions, , , and , you need first derivatives, , , and , as well as second derivatives, , , and . If you have functions and here I’ll call them , you need derivatives, and so on through . You see right away this is a fun and exciting thing to calculate. Also why in intro to differential equations you only work this out with two or three functions. Maybe four functions if the class has been really naughty.
Go through your functions and your derivatives and make a big square matrix. And then you go through calculating the derivative. This involves a lot of multiplying strings of these derivatives together. It’s a lot of work. But at least doing all this work gets you older.
So one will ask why do all this? Why fit it into every Intro to Ordinary Differential Equations textbook and why slip it in to classes that have enough stuff going on?
One answer is that if the Wronskian is not zero for some values of the independent variable, then the functions that went into it are linearly independent. Mathematicians learn to like sets of linearly independent functions. We can treat functions like directions in space. Linear independence assures us none of these functions are redundant, pointing a way we already can describe. (Real people see nothing wrong in having north, east, and northeast as directions. But mathematicians would like as few directions in our set as possible.) The Wronskian being zero for every value of the independent variable seems like it should tell us the functions are linearly dependent. It doesn’t, not without some more constraints on the functions.
This is fine, but who cares? And, unfortunately, in Intro it’s hard to reach a strong reason to care. To this major, the emphasis on linearly independent functions felt misplaced. It’s the sort of thing we care about in linear algebra. Or some course where we talk about vector spaces. Differential equations do lead us into vector spaces. It’s hard to find a corner of analysis that doesn’t.
Every ordinary differential equation has a secret picture. This is a vector field. One axis in the field is the independent variable of the function. The other axes are the value of the function. And maybe its derivatives, depending on how many derivatives are used in the ordinary differential equation. To solve one particular differential equation is to find one path in this field. People who just use differential equations will want to find one path.
Mathematicians tend to be fine with finding one path. But they want to find what kinds of paths there can be. Are there paths which the differential equation picks out, by making paths near it stay near? Or by making paths that run away from it? And here is the value of the Wronskian. The Wronskian tells us about the divergence of this vector field. This gives us insight to how these paths behave. It’s in the same way that knowing where high- and low-pressure systems are describes how the weather will change. The Wronskian, by way of a thing called Liouville’s Theorem that I haven’t the strength to describe today, ties in to the Hamiltonian. And the Hamiltonian we see in almost every mechanics problem of note.
You can see where the mathematics PhD, or the physicist, would find this interesting. But what about the student, who would look at the symbols evoked by those paragraphs above with reasonable horror?
And here’s the second answer for what the Wronskian is good for. It helps us solve ordinary differential equations. Like, particular ones. An ordinary differential equation will (normally) have several linearly independent solutions. If you know all but one of those solutions, it’s possible to calculate the Wronskian and, from that, the last of the independent solutions. Since a big chunk of mathematics — particularly for science or engineering — is solving differential equations you see why this is something valuable. Allow that it’s tedious. Tedious work we can automate, or give to research assistant to do.
One then asks what kind of differential equation would have all-but-one answer findable, and yield that last one only by long efforts of hard work. So let me show you an example ordinary differential equation:
Here , , and are some functions that depend only on the independent variable, . Don’t know what they are; don’t care. The differential equation is a lot easier of and are constants, but we don’t insist on that.
This equation has a close cousin, and one that’s easier to solve than the original. Is cousin is called a homogeneous equation:
The left-hand-side, the parts with the function that we want to find, is the same. It’s the right-hand-side that’s different, that’s a constant zero. This is what makes the new equation homogenous. This homogenous equation is easier and we can expect to find two functions, and , that solve it. If and are constant this is even easy. Even if they’re not, if you can find one solution, the Wronskian lets you generate the second.
That’s nice for the homogenous equation. But if we care about the original, inhomogenous one? The Wronskian serves us there too. Imagine that the inhomogenous solution has any solution, which we’ll call . (The ‘p’ stands for ‘particular’, as in “the solution for this particular ”.) But also has to solve that inhomogenous differential equation. It seems startling but if you work it out, it’s so. (The key is the derivative of the sum of functions is the same as the sum of the derivative of functions.) also has to solve that inhomogenous differential equation. In fact, for any constants and , it has to be that is a solution.
I’ll skip the derivation; you have Wikipedia for that. The key is that knowing these homogenous solutions, and the Wronskian, and the original , will let you find the that you really want.
My reading is that this is more useful in proving things true about differential equations, rather than particularly solving them. It takes a lot of paper and I don’t blame anyone not wanting to do it. But it’s a wonder that it works, and so well.
Don’t make your instructor so mad you have to do the Wronskian for four functions.
This and all the others in My 2020 A-to-Z essays should be at this link. All the essays from every A-to-Z series should be at this link. Thank you for reading.
And let me tease other W-words I won’t be repeating for my essay this week with the Well-Ordering Principle, discussed in the summer of 2017. This is one of those little properties that some sets of numbers, like whole numbers, have and that others, like the rationals, don’t. It doesn’t seem like anything much, which is often a warning that the concept sneaks into a lot of interesting work. On re-reading my own work, I got surprised, which I hope speaks better of the essay than it does of me.
No reason not to keep showing off old posts while I prepare new ones. A Summer 2015 Mathematics A To Z: well-posed problem shows off one of the set of things mathematicians describe as “well”. Well-posedness is one of those things mathematicians learn to look for in problems, and to recast problems so that they have it. The essay also shows off how much I haven’t been able to settle on rules about how to capitalize subject lines.
I’m happy to be back with long-form pieces. This week’s is another topic suggested by Mr Wu, of the Singapore Maths Tuition blog.
This is easy. The velocity is the first derivative of the position. First derivative with respect to time, if you must know. That hardly needed an extra week to write.
Yes, there’s more. There is always more. Velocity is important by itself. It’s also important for guiding us into new ideas. There are many. One idea is that it’s often the first good example of vectors. Many things can be vectors, as mathematicians see them. But the ones we think of most often are “some magnitude, in some direction”.
The position of things, in space, we describe with vectors. But somehow velocity, the changes of positions, seems more significant. I suspect we often find static things below our interest. I remember as a physics major that my Intro to Mechanics instructor skipped Statics altogether. There are many important things, like bridges and roofs and roller coaster supports, that we find interesting because they don’t move. But the real Intro to Mechanics is stuff in motion. Balls rolling down inclined planes. Pendulums. Blocks on springs. Also planets. (And bridges and roofs and roller coaster supports wouldn’t work if they didn’t move a bit. It’s not much though.)
So velocity shows us vectors. Anything could, in principle, be moving in any direction, with any speed. We can imagine a thing in motion inside a room that’s in motion, its net velocity being the sum of two vectors.
And they show us derivatives. A compelling answer to “what does differentiation mean?” is “it’s the rate at which something changes”. Properly, we can take the derivative of any quantity with respect to any variable. But there are some that make sense to do, and position with respect to time is one. Anyone who’s tried to catch a ball understands the interest in knowing.
We take derivatives with respect to time so often we have shorthands for it, by putting a ‘ mark after, or a dot above, the variable. So if x is the position (and it often is), then is the velocity. If we want to emphasize we think of vectors, is the position and the velocity.
Velocity has another common shorthand. This is , or if we want to emphasize its vector nature, . Why a name besides the good enough ? It helps us avoid misplacing a ‘ mark in our work, for one. And giving velocity a separate symbol encourages us to think of the velocity as independent from the position. It’s not — not exactly — independent. But knowing that a thing is in the lawn outside tells us nothing about how it’s moving. Velocity affects position, in a process so familiar we rarely consider how there’s parts we don’t understand about it. But velocity is also somehow also free of the position at an instant.
Velocity also guides us into a first understanding of how to take derivatives. Thinking of the change in position over smaller and smaller time intervals gets us to the “instantaneous” velocity by doing only things we can imagine doing with a ruler and a stopwatch.
Velocity has a velocity. , also known as . Or, if we’re sure we won’t lose a ‘ mark, . Once we are comfortable thinking of how position changes in time we can think of other changes. Velocity’s change in time we call acceleration. This is also a vector, more abstract than position or velocity. Multiply the acceleration by the mass of the thing accelerating and we have a vector called the “force”. That, we at least feel we understand, and can work with.
Acceleration has a velocity too, a rate of change in time. It’s called the “jerk” by people telling you the change in acceleration in time is called the “jerk”. (I don’t see the term used in the wild, but admit my experience is limited.) And so on. We could, in principle, keep taking derivatives of the position and keep finding new changes. But most physics problems we find interesting use just a couple of derivatives of the position. We can label them, if we need, , where n is some big enough number like 4.
We can bundle them in interesting ways, though. Come back to that mention of treating position and velocity of something as though they were independent coordinates. It’s a useful perspective. Imagine the rules about how particles interacting with one another and with their environment. These usually have explicit roles for position and velocity. (Granting this may reflect a selection bias. But these do cover enough interesting problems to fill a career.)
So we create a new vector. It’s made of the positition and the velocity. We’d write it out as . The superscript-T there, “transposition”, lets us use the tools of matrix algebra. This vector describes a point in phase space. Phase space is the collection of all the physically possible positions and velocities for the system.
What’s the derivative, in time, of this point in phase space? Glad to say we can do this piece by piece. The derivative of a vector is the derivative of each component of a vector. So the derivative of is , or, . This acceleration itself depends on, normally, the positions and velocities. So we can describe this as for some function . You are surely impressed with this symbol-shuffling. You are less sure why this bother.
The bother is a trick of ordinary differential equations. All differential equations are about how a function-to-be-determined and its derivatives relate to one another. In ordinary differential equations, the function-to-be-determined depends on a single variable. Usually it’s called x or t. There may be many derivatives of f. This symbol-shuffling rewriting takes away those higher-order derivatives. We rewrite the equation as a vector equation of just one order. There’s some point in phase space, and we know what its velocity is. That we do because in this form many problems can be written as a matrix problem: . Or approximate our problem as a matrix problem. This lets us bring in linear algebra tools, and that’s worthwhile.
It also lets us bring in numerical tools. Numerical mathematics has developed many methods to solve the ordinary differential equation . Most of them extend to . The result is a classic mathematician’s trick. We can recast a problem as one we have better tools to solve.
It calls on a more abstract idea of what a “velocity” might be. We can explain what the thing that’s “moving” and what it’s moving through are, given time. But the instincts we develop from watching ordinary things move help us in these new territories. This is also a classic mathematician’s trick. It may seem like all mathematicians do is develop tricks to extend what they already do. I can’t say this is wrong.
Thank you all for reading and for putting up with my gap week. This and all of my 2020 A-to-Z essays should be at this link. All the essays from every A-to-Z series should be at this link.
I decided to let the V essay slide to Wednesday. This will make the end of the 2020 A-to-Z run a week later than I originally imagined, but that’s all right. It’ll all end in 2020 unless there’s another unexpected delay.
I have gotten several good suggestions for the letters W and X, but I’m still open to more, preferably for X. And I would like any thoughts anyone would like to share for the last letters of the alphabet. If you have an idea for a mathematical term starting with either letter, please let me know in comments. Also please let me know about any blogs or other projects you have, so that I can give them my modest boost with the essay. I’m open to revisiting topics I’ve already discussed, if I can think of something new to say or if I’ve forgotten I wrote them about them already.
Topics I’ve already covered, starting with the letter ‘Y’, are:
- Y-Axis (2015)
- Yukawa Potential (Leap Day 2016)
- Yang Hui’s Triangle (End 2016)
- Young Tableau (2017)
- Yamada Polynomial (2018)
- The Game of ‘Y’ (2019)
Topics I’ve already covered, starting with the letter ‘Z’, are:
I have accepted that this week, at least, I do not have it in me to write an A-to-Z essay. I’ll be back to it next week, I think. I don’t know whether I’ll publish my usual I-meant-this-to-be-800-words-and-it’s-three-times-that piece on Monday or on Wednesday, but it’ll be sometime next week. And, events personal and public allowing, I’ll continue weekly from there. Should still finish the essay series before 2020 finishes. I say this assuming that 2020 will in fact finish.
But now let me look back on a time when I could produce essays with an almost machine-like reliability, except for when I forgot to post them. My 2019 Mathematics A To Z: Versine is such an essay. The versine is a function that had a respectably long life in a niche of computational computing. Cheap electronic computers wiped out that niche. The reasons that niche ever existed, though, still apply, just to different problems. Knowing of past experiences can help us handle future problems.
I am not writing another duplicate essay. I intend to have an A-to-Z essay for the week. I just haven’t had the time or energy to write anything so complicated as an A-to-Z since the month began. Things are looking up, though, and I hope to have something presentable for Friday.
So let me just swap my publication slots around, and share an older essay, as I would have on Friday. My 2018 Mathematics A To Z: Volume was suggested by Ray Kassinger, of the popular web comic Housepets!, albeit as a Mystery Science Theater 3000 reference. It’s a great topic, though. It’s one of those things everyone instinctively understands. But making that instinct precise demands we accept some things that seem absurd. It’s a great example of what mathematics can do, given a chance.
And, then, many of the U- entries in an A-to-Z are negations. Unbounded, from the summer 2015 sequence, is a good example of that. It’s also a concept worth knowing, since a lot of properties of analysis depend on whether you have an unbounded set or not. Or an unbounded function.
In looking over past A-to-Z’s I notice a lot of my U- entries are the negation of something. Unknots, for example. Or unbounded. English makes this construction hard to avoid. Any interesting property is also interesting when it’s absent. But there are also mathematical terms that start with a U on their own terms. The Summer 2017 Mathematics A To Z: Ulam’s Spiral shows off one of them. Stanislaw Ulam’s spiral is one of those things we find as a curious graphical adjunct to prime numbers. The essay also features one of my many pieces in praise of boredom.
I assume that last week I disappointed Mr Wu, of the Singapore Maths Tuition blog, last week when I passed on a topic he suggested to unintentionally rewrite a good enough essay. I hope to make it up this week with a piece of linear algebra.
A Unitary Matrix — note the article; there is not a singular the Unitary Matrix — starts with a matrix. This is an ordered collection of scalars. The scalars we call elements. I can’t think of a time I ever saw a matrix represented except as a rectangular grid of elements, or as a capital letter for the name of a matrix. Or a block inside a matrix. In principle the elements can be anything. In practice, they’re almost always either real numbers or complex numbers. To speak of Unitary Matrixes invokes complex-valued numbers. If a matrix that would be Unitary has only real-valued elements, we call that an Orthogonal Matrix. It’s not wrong to call an Orthogonal matrix “Unitary”. It’s like pointing to a known square, though, and calling it a parallelogram. Your audience will grant that’s true. But it wonder what you’re getting at, unless you’re talking about a bunch of parallelograms and some of them happen to be squares.
As with polygons, though, there are many names for particular kinds of matrices. The flurry of them settles down on the Intro to Linear Algebra student and it takes three or four courses before most of them feel like familiar names. I will try to keep the flurry clear. First, we’re talking about square matrices, ones with the same number of rows as columns.
Start with any old square matrix. Give it the name U because you see where this is going. There are a couple of new matrices we can derive from it. One of them is the complex conjugate. This is the matrix you get by taking the complex conjugate of every term. So, if one element is , in the complex conjugate, that element would be . Reverse the plus or minus sign of the imaginary component. The shorthand for “the complex conjugate to matrix U” is . Also we’ll often just say “the conjugate”, taking the “complex” part as implied.
Start back with any old square matrix, again called U. Another thing you can do with it is take the transposition. This matrix, U-transpose, you get by keeping the order of elements but changing rows and columns. That is, the elements in the first row become the elements in the first column. The elements in the second row become the elements in the second column. Third row becomes the third column, and so on. The diagonal — first row, first column; second row, second column; third row, third column; and so on — stays where it was. The shorthand for “the transposition of U” is .
You can chain these together. If you start with U and take both its complex-conjugate and its transposition, you get the adjoint. We write that with a little dagger: . For a wonder, as matrices go, it doesn’t matter whether you take the transpose or the conjugate first. It’s the same . You may ask how people writing this out by hand never mistake for . This is a good question and I hope to have an answer someday. (I would write it as in my notes.)
And the last thing you can maybe do with a square matrix is take its inverse. This is like taking the reciprocal of a number. When you multiply a matrix by its inverse, you get the Identity Matrix. Not every matrix has an inverse, though. It’s worse than real numbers, where only zero doesn’t have a reciprocal. You can have a matrix that isn’t all zeroes and that doesn’t have an inverse. This is part of why linear algebra mathematicians command the big money. But if a matrix U has an inverse, we write that inverse as .
The Identity Matrix is one of a family of square matrices. Every element in an identity matrix is zero, except on the diagonal. That is, the element at row one, column one, is the number 1. The element at row two, column two is also the number 1. Same with row three, column three: another one. And so on. This is the “identity” matrix because it works like the multiplicative identity. Pick any matrix you like, and multiply it by the identity matrix; you get the original matrix right back. We use the name for an identity matrix. If we have to be clear how many rows and columns the matrix has, we write that as a subscript: or or or so on.
So this, finally, lets me say what a Unitary Matrix is. It’s any square matrix U where the adjoint, is the same matrix as the inverse, . It’s wonderful to learn you have a Unitary Matrix. Not just because, most of the time, finding the inverse of a matrix is a long and tedious procedure. Here? You have to write the elements in a different order and change the plus-or-minus sign on the imaginary numbers. The only way it would be easier if you had only real numbers, and didn’t have to take the conjugates.
That’s all a nice heap of terms. What makes any of them important, other than so Intro to Linear Algebra professors can test their students?
Well, you know mathematicians. If we like something like this, it’s usually because it holds out the prospect of turning a hard problems into easier ones. So it is. Start out with any old matrix. Call it A. Then there exist some unitary matrixes, call them U and V. And their product does something wonderful: is a “diagonal” matrix. A diagonal matrix has zeroes for every element except the diagonal ones. That is, row one, column one; row two, column two; row three, column three; and so on. The elements that trace a path from the upper-left to the lower-right corner of the matrix. (The diagonal from the upper-right to the lower-left we have nothing to do with.) Everything we might do with matrices is easier on a diagonal matrix. So we process our matrix A into this diagonal matrix D. Process it by whatever the heck we’re doing. If we then multiply this by the inverses of U and V? If we calculate ? We get whatever our process would have given us had we done it to A. And, since U and V are unitary matrices, it’s easy to find these inverses. Wonderful!
Also this sounds like I just said Unitary Matrixes are great because they solve a problem you never heard of before.
The 20th Century’s first great use for Unitary Matrixes, and I imagine the impulse for Mr Wu’s suggestion, was quantum mechanics. (A later use would be data compression.) Unitary Matrixes help us calculate how quantum systems evolve. This should be a little easier to understand if I use a simple physics problem as demonstration.
So imagine three blocks, all the same mass. They’re connected in a row, left to right. There’s two springs, one between the left and the center mass, one between the center and the right mass. The springs have the same strength. The blocks can only move left-to-right. But, within those bounds, you can do anything you like with the blocks. Move them wherever you like and let go. Let them go with a kick moving to the left or the right. The only restraint is they can’t pass through one another; you can’t slide the center block to the right of the right block.
This is not quantum mechanics, by the way. But it’s not far, either. You can turn this into a fine toy of a molecule. For now, though, think of it as a toy. What can you do with it?
A bunch of things, but there’s two really distinct ways these blocks can move. These are the ways the blocks would move if you just hit it with some energy and let the system do what felt natural. One is to have the center block stay right where it is, and the left and right blocks swinging out and in. We know they’ll swing symmetrically, the left block going as far to the left as the right block goes to the right. But all these symmetric oscillations look about the same. They’re one mode.
The other is … not quite antisymmetric. In this mode, the center block moves in one direction and the outer blocks move in the other, just enough to keep momentum conserved. Eventually the center block switches direction and swings the other way. But the outer blocks switch direction and swing the other way too. If you’re having trouble imagining this, imagine looking at it from the outer blocks’ point of view. To them, it’s just the center block wobbling back and forth. That’s the other mode.
And it turns out? It doesn’t matter how you started these blocks moving. The movement looks like a combination of the symmetric and the not-quite-antisymmetric modes. So if you know how the symmetric mode evolves, and how the not-quite-antisymmetric mode evolves? Then you know how every possible arrangement of this system evolves.
So here’s where we get to quantum mechanics. Suppose we know the quantum mechanics description of a system at some time. This we can do as a vector. And we know the Hamiltonian, the description of all the potential and kinetic energy, for how the system evolves. The evolution in time of our quantum mechanics description we can see as a unitary matrix multiplied by this vector.
The Hamiltonian, by itself, won’t (normally) be a Unitary Matrix. It gets the boring name H. It’ll be some complicated messy thing. But perhaps we can find a Unitary Matrix U, so that is a diagonal matrix. And then that’s great. The original H is hard to work with. The diagonalized version? That one we can almost always work with. And then we can go from solutions on the diagonalized version back to solutions on the original. (If the function describes the evolution of , then describes the evolution of .) The work that U (and ) does to H is basically what we did with that three-block, two-spring model. It’s picking out the modes, and letting us figure out their behavior. Then put that together to work out the behavior of what we’re interested in.
There are other uses, besides time-evolution. For instance, an important part of quantum mechanics and thermodynamics is that we can swap particles of the same type. Like, there’s no telling an electron that’s on your nose from an electron that’s in one of the reflective mirrors the Apollo astronauts left on the Moon. If they swapped positions, somehow, we wouldn’t know. It’s important for calculating things like entropy that we consider this possibility. Two particles swapping positions is a permutation. We can describe that as multiplying the vector that describes what every electron on the Earth and Moon is doing by a Unitary Matrix. Here it’s a matrix that does nothing but swap the descriptions of these two electrons. I concede this doesn’t sound thrilling. But anything that goes into calculating entropy is first-rank important.
As with time-evolution and with permutation, though, any symmetry matches a Unitary Matrix. This includes obvious things like reflecting across a plane. But it also covers, like, being displaced a set distance. And some outright obscure symmetries too, such as the phase of the state function . I don’t have a good way to describe what this is, physically; we can’t observe it directly. This symmetry, though, manifests as the conservation of electric charge, a thing we rather like.
This, then, is the sort of problem that draws Unitary Matrixes to our attention.
Thank you for reading. This and all of my 2020 A-to-Z essays should be at this link. All the essays from every A-to-Z series should be at this link. Next week, I hope to have something to say for the letter V.
I’m still only doing short reviews of my readership figures. These are nice easy posts to make, and strangely popular, but they do take time and I’m never sure why people find them interesting. I think it’s all from other bloggers, happy to know how much better their blogs are doing.
Granted that: I had, for me, a really well-read month. According to WordPress, there were 3,043 pages viewed here in October 2020. This is way above the twelve-month running average of 2,381.5 views per month. Also this is the second-largest number of page views I’ve gotten since October 2019. That month, too, was part of an A-to-Z sequence. I wrote something that got referenced on some actually popular web site, though, last year. This year, all I can figure is spillover of people on my other blog wanting to know what’s going on with Mark Trail.
(If you read any web site that regularly talks about Mark Trail, poke around the comments. There’s people upset about the new artist. It’s not my intention to mock them; anything you like changing out from under you is upsetting. But it is soothing to see people worrying about, ultimately, a guy who punches smugglers while giant squirrels talk. On my other blog I plan to have a full plot recap of that in about two weeks.)
There were more unique visitors in October 2020 than any other month besides October 2019, also. WordPress recorded 2,161 unique visitors, well above the twelve-month running average of 1,644.2. It’s much the same for interactions as well: 79 things were liked, compared to the running average of 59.8, and 18 comments, above the 17.1 running average.
October was another month of 18 posts, and I have a running average of 17.6 posts per month now. I’m surprised by that too. I feel like any month that isn’t an A-to-Z sequence I have twelve posts, but there we go. This all means the per-post October averages were above the per-post running averages.
What were the most popular recent posts? Here recent means “from September or October”? That I’m glad to share:
- Playful Math Education Blog Carnival 141 which was September’s only entry in the most popular posts. And, well, I’d hope this would rate high; it’s an accessible and high-profile piece.
- My All 2020 Mathematics A to Z: Renormalization which is about something that’s all over a part of quantum mechanics, but doesn’t start with that.
- My All 2020 Mathematics A to Z: Statistics which I’m also glad for because writing this one felt urgent.
- My All 2020 Mathematics A to Z: Quadratic Form in which I finally learned what a “form” is.
- I’m looking for S, T, and U topics for the All 2020 A-to-Z which … I’m not, anymore, but I am still up for W and X, if you have ideas.
All told, in October I published 12,937 words, down a bit from September. This was an average of 718.7 words per posting in October, which still brings my year-to-date average post length up to 697 words. It had been 694 at the start of October.
As of the start of November I’ve published 1,554 posts here. They’ve gathered 116,811 page views. I like how nearly but not quite palindromic that number is. It even almost but not quite stays the same under a 180 degree rotation. These pages overall have drawn 66,030 logged unique visitors.
I still plan to have an A-to-Z essay this and every Wednesday until we run out of alphabet. You can see that, and all this year’s essays, at this link. This year’s and all past A-to-Z essays should be at this link.
You can be a regular reader here adding my essays’ feed to whatever your RSS reader is. If you don’t have an RSS reader, get a free account at Dreamwidth or Livejournal. You can add any RSS feed to your friends page from https://www.dreamwidth.org/feeds/ or https://www.livejournal.com/syn as you like. Or if you have a WordPress account you can add this to your Reading page by clicking the “Follow Nebusresearch” button on the page. Or
My essays are announcedon Twitter as @nebusj. Don’t try to talk with me there. I haven’t had the energy to work out why Safari only sometimes will let Twitter load. If you actually want to social-media talk with me look to the mathematics-themed Mathstodon and my account @firstname.lastname@example.org. If you really need me, leave a comment. Thank you all for reading.
I know, it’s strange for me to not post another piece about tiling. But My 2019 Mathematics A To Z: Taylor Series is going to be a good utility essay, useful for a long while to come. Taylor Series represent one of the standard mathematician tricks. This is to rewrite a thing we want to do as a sum of things it’s easy to do. This can make our problem into a long series of little problems. But the advantage is we know what to do with all those little problems. It’s often a worthwhile trade.
Well, this is just embarrassing.
I’ve always held out the option that I would revisit a topic sometime. I thought it would most likely be taking some essay from one of my earliest A-to-Z’s where, with a half-decade’s more experience in pop mathematics writing, I could do much better. And at the request of someone who felt that, like, my piece on duals was foggy. It is, but nobody’s ever cared enough about duals to say anything.
So I went looking at what previous T topics I’d written about here. Usually I pick them the Sunday or Monday of a week, since that’s easy to do. This week, I didn’t have the time until Thursday when I looked and found I wrote up “Tiling” for the 2018 A-to-Z. In about November of that year, too. And after casting aside a suggestion from Mr Wu of the Singapore Maths Tuition blog, although that time at least I was responding to a specific topic suggestion. 2020, you know?
Well, now that the deed is done, I can see what I learned from it anyway. First is picking out the archive pieces before I write the week’s essay. Second is how my approach differed in the 2020 essay. The broad picture is similar enough. The most interesting differences are that in the 2020 essay I look at more specifics. Like, just when Robert Berger found his aperiodic tiling of the plane. And what the Wang Tiles are that he found them with. Or, a very brief sketch of how to show Penrose (rhomboid) tiling is aperiodic. This changes the shape of the essay. Also it makes the essay longer, but that might also might reflect that in 2018 I was publishing two essays a week. This year I’m doing one, and somehow still putting out as many words per week.
I like the greater focus on specifics, although that might just reflect that I’m usually happiest with something I just wrote. As I get distance from it, I come to feel the whole thing’s so bad as to be humiliating. When it’s far enough in the past, usually, I come around again and feel it’s pretty good, and maybe that I don’t know how to write like that anymore. The 2018 essay is, to me, only embarrassing in stuff that I glossed over that in 2020 I made specific. Not to worry, though. I still get foggy and elliptical about important topics anyway.
Mr Wu, author of the Singapore Maths Tuition blog, had an interesting suggestion for the letter T: Talent. As in mathematical talent. It’s a fine topic but, in the end, too far beyond my skills. I could share some of the legends about mathematical talent I’ve received. But what that says about the culture of mathematicians is a deeper and more important question.
So I picked my own topic for the week. I do have topics for next week — U — and the week after — V — chosen. But the letters W and X? I’m still open to suggestions. I’m open to creative or wild-card interpretations of the letters. Especially for X and (soon) Z. Thanks for sharing any thoughts you care to.
Think of a floor. Imagine you are bored. What do you notice?
What I hope you notice is that it is covered. Perhaps by carpet, or concrete, or something homogeneous like that. Let’s ignore that. My floor is covered in small pieces, repeated. My dining room floor is slats of wood, about three and a half feet long and two inches wide. The slats are offset from the neighbors so there’s a pleasant strong line in one direction and stippled lines in the other. The kitchen is squares, one foot on each side. This is a grid we could plot high school algebra functions on. The bathroom is more elaborate. It has white rectangles about two inches long, tan rectangles about two inches long, and black squares. Each rectangle is perpendicular to ones of the other color, and arranged to bisect those. The black squares fill the gaps where no rectangle would fit.
Move from my house to pure mathematics. It’s easy to turn the floor of a room into abstract mathematics. We start with something to tile. Usually this is the infinite, two-dimensional plane. The thing you get if you have a house and forget the walls. Sometimes we look to tile the hyperbolic plane, a different geometry that we of course represent with a finite circle. (Setting particular rules about how to measure distance makes this equivalent to a funny-shaped plane.) Or the surface of a sphere, or of a torus, or something like that. But if we don’t say otherwise, it’s the plane.
What to cover it with? … Smaller shapes. We have a mathematical tiling if we have a collection of not-overlapping open sets. And if those open sets, plus their boundaries, cover the whole plane. “Cover” here means what “cover” means in English, only using more technical words. These sets — these tiles — can be any shape. We can have as many or as few of them as we like. We can even add markings to the tiles, give them colors or patterns or such, to add variety to the puzzles.
(And if we want, we can do this in other dimensions. There are good “tiling” questions to ask about how to fill a three-dimensional space, or a four-dimensional one, or more.)
Having an unlimited collection of tiles is nice. But mathematicians learn to look for how little we need to do something. Here, we look for the smallest number of distinct shapes. As with tiling an actual floor, we can get all the tiles we need. We can rotate them, too, to any angle. We can flip them over and put the “top” side “down”, something kitchen tiles won’t let us do. Can we reflect them? Use the shape we’d get looking at the mirror image of one? That’s up to whoever’s writing this paper.
What shapes will work? Well, squares, for one. We can prove that by looking at a sheet of graph paper. Rectangles would work too. We can see that by drawing boxes around the squares on our graph paper. Two-by-one blocks, three-by-two blocks, 40-by-1 blocks, these all still cover the paper and we can imagine covering the plane. If we like, we can draw two-by-two squares. Squares made up of smaller squares. Or repeat this: draw two-by-one rectangles, and then group two of these rectangles together to make two-by-two squares.
We can take it on faith that, oh, rectangles π long by e wide would cover the plane too. These can all line up in rows and columns, the way our squares would. Or we can stagger them, like bricks or my dining room’s wood slats are.
How about parallelograms? Those, it turns out, tile exactly as well as rectangles or squares do. Grids or staggered, too. Ah, but how about trapezoids? Surely they won’t tile anything. Not generally, anyway. The slanted sides will, most of the time, only fit in weird winding circle-like paths.
Unless … take two of these trapezoid tiles. We’ll set them down so the parallel sides run horizontally in front of you. Rotate one of them, though, 180 degrees. And try setting them — let’s say so the longer slanted line of both trapezoids meet, edge to edge. These two trapezoids come together. They make a parallelogram, although one with a slash through it. And we can tile parallelograms, whether or not they have a slash.
OK, but if you draw some weird quadrilateral shape, and it’s not anything that has a more specific name than “quadrilateral”? That won’t tile the plane, will it?
It will! In one of those turns that surprises and impresses me every time I run across it again, any quadrilateral can tile the plane. It opens up so many home decorating options, if you get in good with a tile maker.
That’s some good news for quadrilateral tiles. How about other shapes? Triangles, for example? Well, that’s good news too. Take two of any identical triangle you like. Turn one of them around and match sides of the same length. The two triangles, bundled together like that, are a quadrilateral. And we can use any quadrilateral to tile the plane, so we’re done.
How about pentagons? … With pentagons, the easy times stop. It turns out not every pentagon will tile the plane. The pentagon has to be of the right kind to make it fit. If the pentagon is in one of these kinds, it can tile the plane. If not, not. There are fifteen families of tiling known. The most recent family was discovered in 2015. It’s thought that there are no other convex pentagon tilings. I don’t know whether the proof of that is generally accepted in tiling circles. And we can do more tilings if the pentagon doesn’t need to be convex. For example, we can cut any parallelogram into two identical pentagons. So we can make as many pentagons as we want to cover the plane. But we can’t assume any pentagon we like will do it.
Hexagons look promising. First, a regular hexagon tiles the plane, as strategy games know. There are also at least three families of irregular hexagons that we know can tile the plane.
And there the good times end. There are no convex heptagons or octagons or any other shape with more sides that tile the plane.
Not by themselves, anyway. If we have more than one tile shape we can start doing fine things again. Octagons assisted by squares, for example, will tile the plane. I’ve lived places with that tiling. Or something that looks like it. It’s easier to install if you have square tiles with an octagon pattern making up the center, and triangle corners a different color. These squares come together to look like octagons and squares.
And this leads to a fun avenue of tiling. Hao Wang, in the early 60s, proposed a sort of domino-like tiling. You may have seen these in mathematics puzzles, or in toys. Each of these Wang Tiles, or Wang Dominoes, is a square. But the square is cut along the diagonals, into four quadrants. Each quadrant is a right triangle. Each quadrant, each triangle, is one of a finite set of colors. Adjacent triangles can have the same color. You can place down tiles, subject only to the rule that the tile edge has to have the same color on both sides. So a tile with a blue right-quadrant has to have on its right a tile with a blue left-quadrant. A tile with a white upper-quadrant on its top has, above it, a tile with a white lower-quadrant.
In 1961 Wang conjectured that if a finite set of these tiles will tile the plane, then there must be a periodic tiling. That is, if you picked up the plane and slid it a set horizontal and vertical distance, it would all look the same again. This sort of translation is common. All my floors do that. If we ignore things like the bounds of their rooms, or the flaws in their manufacture or installation or where a tile broke in some mishap.
This is not to say you couldn’t arrange them aperiodically. You don’t even need Wang Tiles for that. Get two colors of square tile, a white and a black, and lay them down based on whether the next decimal digit of π is odd or even. No; Wang’s conjecture was that if you had tiles that you could lay down aperiodically, then you could also arrange them to set down periodically. With the black and white squares, lay down alternate colors. That’s easy.
In 1964, Robert Berger proved Wang’s conjecture was false. He found a collection of Wang Tiles that could only tile the plane aperiodically. In 1966 he published this in the Memoirs of the American Mathematical Society. The 1964 proof was for his thesis. 1966 was its general publication. I mention this because while doing research I got irritated at how different sources dated this to 1964, 1966, or sometimes 1961. I want to have this straightened out. It appears Berger had the proof in 1964 and the publication in 1966.
I would like to share details of Berger’s proof, but haven’t got access to the paper. What fascinates me about this is that Berger’s proof used a set of 20,426 different tiles. I assume he did not work this all out with shards of construction paper, but then, how to get 20,426 of anything? With computer time as expensive as it was in 1964? The mystery of how he got all these tiles is worth an essay of its own and regret I can’t write it.
Berger conjectured that a smaller set might do. Quite so. He himself reduced the set to 104 tiles. Donald Knuth in 1968 modified the set down to 92 tiles. In 2015 Emmanuel Jeandel and Michael Rao published a set of 11 tiles, using four colors. And showed by computer search that a smaller set of tiles, or fewer colors, would not force some aperiodic tiling to exist. I do not know whether there might be other sets of 11, four-colored, tiles that work. You can see the set at the top of Wikipedia’s page on Wang Tiles.
These Wang Tiles, all squares, inspired variant questions. Could there be other shapes that only aperiodically tile the plane? What if they don’t have to be squares? Raphael Robinson, in 1971, came up with a tiling using six shapes. The shapes have patterns on them too, usually represented as colored lines. Tiles can be put down only in ways that fit and that make the lines match up.
Among my readers are people who have been waiting, for 1800 words now, for Roger Penrose. It’s now that time. In 1974 Penrose published an aperiodic tiling, one based on pentagons and using a set of six tiles. You’ve never heard of that either, because soon after he found a different set, based on a quadrilateral cut into two shapes. The shapes, as with Wang Tiles or Robinson’s tiling, have rules about what edges may be put against each other. Penrose — and independently Robert Ammann — also developed another set, this based on a pair of rhombuses. These have rules about what edges may tough one another, and have patterns on them which must line up.
The Penrose tiling became, and stayed famous. (Ammann, an amateur, never had much to do with the mathematics community. He died in 1994.) Martin Gardner publicized it, and it leapt out of mathematicians’ hands into the popular culture. At least a bit. That it could give you nice-looking floors must have helped.
To show that the rhombus-based Penrose tiling is aperiodic takes some arguing. But it uses tools already used in this essay. Remember drawing rectangles around several squares? And then drawing squares around several of these rectangles? We can do that with these Penrose-Ammann rhombuses. From the rhombus tiling we can draw bigger rhombuses. Ones which, it turns out, follow the same edge rules that the originals do. So that we can go again, grouping these bigger rhombuses into even-bigger rhombuses. And into even-even-bigger rhombuses. And so on.
What this gets us is this: suppose the rhombus tiling is periodic. Then there’s some finite-distance horizontal-and-vertical move that leaves the pattern unchanged. So, the same finite-distance move has to leave the bigger-rhombus pattern unchanged. And this same finite-distance move has to leave the even-bigger-rhombus pattern unchanged. Also the even-even-bigger pattern unchanged.
Keep bundling rhombuses together. You get eventually-big-enough-rhombuses. Now, think of how far you have to move the tiles to get a repeat pattern. Especially, think how many eventually-big-enough-rhombuses it is. This distance, the move you have to make, is less than one eventually-big-enough rhombus. (If it’s not you aren’t eventually-big-enough yet. Bundle them together again.) And that doesn’t work. Moving one tile over without changing the pattern makes sense. Moving one-half a tile over? That doesn’t. So the eventually-big-enough pattern can’t be periodic, and so, the original pattern can’t be either. This is explained in graphic detail a nice Powerpoint slide set from Professor Alexander F Ritter, A Tour Of Tilings In Thirty Minutes.
It’s possible to do better. In 2010 Joshua E S Socolar and Joan M Taylor published a single tile that can force an aperiodic tiling. As with the Wang Tiles, and Robinson shapes, and the Penrose-Ammann rhombuses, markings are part of it. They have to line up so that the markings — in two colors, in the renditions I’ve seen — make sense. With the Penrose tilings, you can get away from the pattern rules for the edges by replacing them with little notches. The Socolar-Taylor shape can make a similar trade. Here the rules are complex enough that it would need to be a three-dimensional shape, one that looks like the dilithium housing of the warp core. You can see the tile — in colored, marked form, and also in three-dimensional tile shape — at the PDF here. It’s likely not coming to the flooring store soon.
It’s all wonderful, but is it useful? I could go on a few hundred words about, particularly, crystals and quasicrystals. These are important for materials science. Especially these days as we have harnessed slightly-imperfect crystals to be our computers. I don’t care. These are lovely to look at. If you see nothing appealing in a great heap of colors and polygons spread over the floor there are things we cannot communicate about. Tiling is a delight; what more do you need?
Thanks for your attention. This and all of my 2020 A-to-Z essays should be at this link. All the essays from every A-to-Z series should be at this link. See you next week, I hope.
It’s the final week of the month and so the Playful Math Education Blog Carnival has published again. The host this time is Denise Gaskins, who also runs the project. You can read the 142nd Blog Carnival essay here, and find something interesting or delightful or useful.
If you’d like to host the carnival some month, you can sign up here. Most of 2021 is open, as I write this. I’ve had the chance to host several times. It’s a novel challenge, and something that pushes me to look beyond my usual familiar mathematics reading. You might find it similarly energizing.
It feels to me like I did a lot of functional analysis terms in the Leap Day 2016 series. Its essay for the letter ‘S’, Surjective Map, is one of them. We have many ways of dividing up the kinds of functions we have. One of them is in how they use their range. A function has a set called the domain, and a set called the range, and they might be the same set, yes. The function pairs things in the domain with things in the range. Everything in the domain has to pair with something in the range. But we allow having things in the range that aren’t paired to anything in the domain. So we have jargon that tells us, quickly, whether there are unmatched pieces in the range.
Sometimes I write an essay and know it’s something I’m going to refer back to a lot. Sometimes I know it’s just going to sink without trace. Often these deserve it; the subject is something particular and not well-connected to other topics. Sometimes, one sinks without a trace and for not much good reason. Smooth is one of those curiously sunk pieces. It’s about a concept important to analysis. And also a piece that shows my obsession with pointing out cultural factors in mathematics: we care about ‘smooth’ because we’ve found it a useful thing to highlight. And yet it’s gotten no comments, only an average number of likes, and I don’t seem to have linked back to it in any essays where it might be useful. I may have forgotten I wrote the thing. So here’s a referral that maybe will help me remember I have it on hand, ready for future use.