Reading the Comics, July 20, 2019: What Are The Chances Edition

The temperature’s cooled. So let me get to the comics that, Saturday, I thought were substantial enough to get specific discussion. It’s possible I was overestimating how much there was to say about some of these. These are the risks I take.

Paige Braddock’s Jane’s World for the 15th sees Jane’s niece talk about enjoying mathematics. I’m glad to see. You sometimes see comic strip characters who are preposterously good at mathematics. Here I mean Jason and Marcus over in Bill Amend’s FoxTrot. But even they don’t often talk about why mathematics is appealing. There is no one answer for all people. I suspect even for a single person the biggest appeal changes over time. That mathematics seems to offer certainty, though, appeals to many. Deductive logic promises truths that can be known independent of any human failings. (The catch is actually doing a full proof, because that takes way too many boring steps. Mathematicians more often do enough of a prove to convince anyone that the full proof could be produced if needed.)

Paige Braddock’s Jane’s World for the 15th of July, 2019. The comic originally ran, if I’m reading the dates right, the 28th of October, 2002. Essays mentioning Jane’s World should appear at this link. I think that so far the only mention would be Sunday’s post, when I pointed out the existence of this storyline.

Alexa also enjoys math for there always being a right answer. Given her age there probably always is. There are mathematical questions for which there is no known right answer. Some of these are questions for which we just don’t know the answer, like, “is there an odd perfect number?” Some of these are more like value judgements, though. Is Euclidean geometry or non-Euclidean geometry more correct? The answer depends on what you want to do. There’s no more a right answer to that question than there is a right answer to “what shall I eat for dinner”.

Jane is disturbed by the idea of there being a right answer that she doesn’t know. She would not be happy to learn about “existence proofs”. This is a kind of proof in which the goal is not to find an answer. It’s just to show that there is an answer. This might seem pointless. But there are problems for which there can’t be an answer. If an answer’s been hard to find, it’s worth checking whether there are answers to find.

Art Sansom and Chip Sansom’s The Born Loser for the 16th of July, 2019. There are a couple of essays mentioning The Born Loser, gathered at this link.

Art Sansom and Chip Sansom’s The Born Loser for the 16th builds on comparing the probability of winning the lottery to that of being hit by lightning. This comparison’s turned up a couple of times, including in Mister Boffo and The Wandering Melon, when I learned that Peter McCathie had both won the lottery and been hit by lightning.

Pab Sungenis’s New Adventures of Queen Victoria for the 17th of July, 2019. I thought I mentioned this strip more than it seems I have. Well, the essays inspired by something in New Adventures of Queen Victoria should be at this link.

Pab Sungenis’s New Adventures of Queen Victoria for the 17th is maybe too marginal for full discussion. It’s just reeling off a physics-major joke. The comedy is from it being a pun: Planck’s Constant is a number important in many quantum mechanics problems. It’s named for Max Planck, one of the pioneers of the field. The constant is represented in symbols as either or as . The constant is equal to and might be used even more often. It turns out appears all over the place in quantum mechanics, so it’s convenient to write it with fewer symbols. is maybe properly called the reduced Planck’s constant, although in my physics classes I never encountered anyone calling it “reduced”. We just accepted there were these two Planck’s Constants and trusted context to make clear which one we wanted. It was . Planck’s Constant made some news among mensuration fans recently. The International Bureau of Weights and Measures chose to fix the value of this constant. This, through various physics truths, thus fixes the mass of the kilogram in terms of physical constants. This is regarded as better than the old method, where we just had a lump of metal that we used as reference.

Jonathan Lemon’s Rabbits Against Magic for the 17th of July, 2019. This comic is trying to become the next Andertoons. Essays mentioninng Rabbits Against Magic are at this link.

Jonathan Lemon’s Rabbits Against Magic for the 17th is another probability joke. If a dropped piece of toast is equally likely to land butter-side-up or butter-side-down, then it’s quite unlikely to have it turn up the same way twenty times in a row. There’s about one chance in 524,288 of doing it in a string of twenty toast-flips. (That is, of twenty butter-side-up or butter-side-down in a row. If all you want is twenty butter-side-up, then there’s one chance in 1,048,576.) It’s understandable that Eight-Ball would take Lettuce to be quite lucky just now.

But there’s problems with the reasoning. First is the supposition that toast is as likely to fall butter-side-up as butter-side-down. I have a dim recollection of a mid-2000s pop physics book explaining why, given how tall a table usually is, a piece of toast is more likely to make half a turn — to land butter-side-down — before falling. Lettuce isn’t shown anywhere near a table, though. She might be dropping toast from a height that makes butter-side-up more likely. And there’s no reason to suppose that luck in toast-dropping connects to any formal game of chance. Or that her luck would continue to hold: even if she can drop the toast consistently twenty times there’s not much reason to think she could do it twenty-five times, or even twenty-one.

And then there’s this, a trivia that’s flawed but striking. Suppose that all seven billion people in the world have, at some point, tossed a coin at least twenty times. Then there should be seven thousand of them who had the coin turn up tails every single one of the first twenty times they’ve tossed a coin. And, yes, not everyone in the world has touched a coin, much less tossed it twenty times. But there could reasonably be quite a few people who grew up just thinking that every time you toss a coin it comes up tails. That doesn’t mean they’re going to have any luck gambling.

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Thanks for waiting for me. The weather looks like I should have my next Reading the Comics post at this link, and on time. I’ll let you know if circumstances change.

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Reading the Comics, July 12, 2019: Ricci Tensor Edition

So a couple days ago I was chatting with a mathematician friend. He mentioned how he was struggling with the Ricci Tensor. Not the definition, not exactly, but its point. What the Ricci Tensor was for, and why it was a useful thing. He wished he knew of a pop mathematics essay about the thing. And this brought, slowly at first, to my mind that I knew of one. I wrote such a pop-mathematics essay about the Ricci Tensor, as part of my 2017 A To Z sequence. In it, I spend several paragraphs admitting that I’m not sure I understand what the Ricci tensor is for, and why it’s a useful thing.

Daniel Beyer’s Long Story Short for the 11th of July, 2019. Essays inspired by something mentioned in Long Story Short should be at this link.

Daniel Beyer’s Long Story Short for the 11th mentions some physics hypotheses. These are ideas about how the universe might be constructed. Like many such cosmological thoughts they blend into geometry. The no-boundary proposal, also known as the Hartle-Hawking state (for James Hartle and Stephen Hawking), is a hypothesis about the … I want to write “the start of time”. But I am not confident that this doesn’t beg the question. Well, we think we know what we mean by “the start of the universe”. A natural question in mathematical physics is, what was the starting condition? At the first moment that there was anything, what did it look like? And this becomes difficult to answer, difficult to even discuss, because part of the creation of the universe was the creation of spacetime. In this no-boundary proposal, the shape of spacetime at the creation of the universe is such that there just isn’t a “time” dimension at the “moment” of the Big Bang. The metaphor I see reprinted often about this is how there’s not a direction south of the south pole, even though south is otherwise a quite understandable concept on the rest of the Earth. (I agree with this proposal, but I feel like analogy isn’t quite tight enough.)

Still, there are mathematical concepts which seem akin to this. What is the start of the positive numbers, for example? Any positive number you might name has some smaller number we could have picked instead, until we fall out of the positive numbers altogether and into zero. For a mathematical physics concept there’s absolute zero, the coldest temperature there is. But there is no achieving absolute zero. The thermodynamical reasons behind this are hard to argue. (I’m not sure I could put them in a two-thousand-word essay, not the way I write.) It might be that the “moment of the Big Bang” is similarly inaccessible but, at least for the correct observer, incredibly close by.

The Weyl Curvature is a creation of differential geometry. So it is important in relativity, in describing the curve of spacetime. It describes several things that we can think we understand. One is the tidal forces on something moving along a geodesic. Moving along a geodesic is the general-relativity equivalent of moving in a straight line at a constant speed. Tidal forces are those things we remember reading about. They come from the Moon, sometimes the Sun, sometimes from a black hole a theoretical starship is falling into. Another way we are supposed to understand it is that it describes how gravitational waves move through empty space, space which has no other mass in it. I am not sure that this is that understandable, but it feels accessible.

The Weyl tensor describes how the shapes of things change under tidal forces, but it tracks no information about how the volume changes. The Ricci tensor, in contrast, tracks how the volume of a shape changes, but not the shape. Between the Ricci and the Weyl tensors we have all the information about how the shape of spacetime affects the things within it.

Ted Baum, writing to John Baez, offers a great piece of advice in understanding what the Weyl Tensor offers. Baum compares the subject to electricity and magnetism. If one knew all the electric charges and current distributions in space, one would … not quite know what the electromagnetic fields were. This is because there are electromagnetic waves, which exist independently of electric charges and currents. We need to account for those to have a full understanding of electromagnetic fields. So, similarly, the Weyl curvature gives us this for gravity. How is a gravitational field affected by waves, which exist and move independently of some source?

I am not sure that the Weyl Curvature is truly, as the comic strip proposes, a physics hypothesis “still on the table”. It’s certainly something still researched, but that’s because it offers answers to interesting questions. But that’s also surely close enough for the comic strip’s needs.

Dave Coverly’s Speed Bump for the 11th of July, 2019. Essays which discuss something that appeared in Speed Bump should be at this link.

Dave Coverly’s Speed Bump for the 11th is a wordplay joke, and I have to admit its marginality. I can’t say it’s false for people who (presumably) don’t work much with coefficients to remember them after a long while. I don’t do much with French verb tenses, so I don’t remember anything about the pluperfect except that it existed. (I have a hazy impression that I liked it, but not an idea why. I think it was something in the auxiliary verb.) Still, this mention of coefficients nearly forms a comic strip synchronicity with Mike Thompson’s Grand Avenue for the 11th, in which a Math Joke allegedly has a mistaken coefficient as its punch line.

Mike Thompson’s Grand Avenue for the 12th of July, 2019. The fair number of essays in which I complain about Grand Avenue I gather at this link.

Mike Thompson’s Grand Avenue for the 12th is the one I’m taking as representative for the week, though. The premise has been that Gabby and Michael were sent to Math Camp. They do not want to go to Math Camp. They find mathematics to be a bewildering set of arbitrary and petty rules to accomplish things of no interest to them. From their experience, it’s hard to argue. The comic has, since I started paying attention to it, consistently had mathematics be a chore dropped on them. And not merely from teachers who want them to solve boring story problems. Their grandmother dumps workbooks on them, even in the middle of summer vacation, presenting it as a chore they must do. Most comic strips present mathematics as a thing students would rather not do, and that’s both true enough and a good starting point for jokes. But I don’t remember any that make mathematics look so tedious. Anyway, I highlight this one because of the Math Camp jokes it, and the coefficients mention above, are the most direct mention of some mathematical thing. The rest are along the lines of the strip from the 9th, asserting that the “Math Camp Activity Board” spelled the last word wrong. The joke’s correct but it’s not mathematical.

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So I had to put this essay to bed before I could read Saturday’s comics. Were any of them mathematically themed? I may know soon! And were there comic strips with some mention of mathematics, but too slight for me to make a paragraph about? What could be even slighter than the mathematical content of the Speed Bump and the Grand Avenue I did choose to highlight? Please check the Reading the Comics essay I intend to publish Tuesday. I’m curious myself.

Particle Physics Made Hard

A friend was playing with that cute little particle-physics simulator idea I mentioned last week. And encountered a problem. With a little bit of thought, I was able to not solve the problem. But I was able to explain why it was a subtler and more difficult problem than they had realized. These are the moments that make me feel justified calling myself a mathematician.

The proposed simulation was simple enough: imagine a bunch of particles that interact by rules that aren’t necessarily symmetric. Like, the attraction particle A exerts on particle B isn’t the same as what B exerts on A. Or there are multiple species of particles. So (say) red particles are attracted to blue but repelled by green. But green is attracted to red and repelled by blue twice as strongly as red is attracted to blue. Your choice.

Give a mathematician a perfectly good model of something. She’ll have the impulse to try tinkering with it. One reliable way to tinker with it is to change the domain on which it works. If your simulation supposes you have particles moving on the plane, then, what if they were in space instead? Or on the surface of a sphere? Or what if something was strange about the plane? My friend had this idea: what if the particles were moving on the surface of a cube?

And the problem was how to find the shortest distance between two particles on the surface of a cube. The distance matters since most any attraction rule depends on the distance. This may be as simple as “particles more than this distance apart don’t interact in any way”. The obvious approach, or if you prefer the naive approach, is to pretend the cube is a sphere and find distances that way. This doesn’t get it right, not if the two points are on different faces of the cube. If they’re on adjacent faces, ones which share an edge — think the floor and the wall of a room — it seems straightforward enough. My friend got into trouble with points on opposite faces. Think the floor and the ceiling.

This problem was posed (to the public) in January 1905 by Henry Ernest Dudeney. Dudeney was a newspaper columnist with an exhaustive list of mathematical puzzles. A couple of the books collecting them are on Project Gutenberg. The puzzles show their age in spots. Some in language; some in problems that ask to calculate money in pounds-shillings-and-pence. Many of them are chess problems. But many are also still obviously interesting, and worth thinking about. This one, I was able to find, was a variation of The Spider and the Fly, problem 75 in The Canterbury Puzzles:

Inside a rectangular room, measuring 30 feet in length and 12 feet in width and height, a spider is at a point on the middle of one of the end walls, 1 foot from the ceiling, as at A; and a fly is on the opposite wall, 1 foot from the floor in the centre, as shown at B. What is the shortest distance that the spider must crawl in order to reach the fly, which remains stationary? Of course the spider never drops or uses its web, but crawls fairly.

(Also I admire Dudeney’s efficient closing off of the snarky, problem-breaking answer someone was sure to give. It suggests experienced thought about how to pose problems.)

What makes this a puzzle, even a paradox, is that the obvious answer is wrong. At least, what seems like the obvious answer is to start at point A, move to one of the surfaces connecting the spider’s and the fly’s starting points, and from that move to the fly’s surface. But, no: you get a shorter answer by using more surfaces. Going on a path that seems like it wanders more gets you a shorter distance. The solution’s presented here, along with some follow-up problems. In this case, the spider’s shortest path uses five of the six surfaces of the room.

The approach to finding this is an ingenious one. Imagine the room as a box, and unfold it into something flat. Then find the shortest distance on that flat surface. Then fold the box back up. It’s a good trick. It turns out to be useful in many problems. Mathematical physicists often have reason to ponder paths of things on flattenable surfaces like this. Sometimes they’re boxes. Sometimes they’re toruses, the shape of a doughnut. This kind of unfolding often makes questions like “what’s the shortest distance between points” easier to solve.

There are wrinkles to the unfolding. Of course there are. How interesting would it be if there weren’t? The wrinkles amount to this. Imagine you start at the corner of the room, and walk up a wall at a 45 degree angle to the horizon. You’ll get to the far corner eventually, if the room has proportions that allow it. All right. But suppose you walked up at an angle of 30 degrees to the horizon? At an angle of 75 degrees? You’ll wind your way around the walls (and maybe floor and ceiling) some number of times, each path you start with. Probably different numbers of times. Some path will be shortest, and that’s fine. But … like, think about the path that goes along the walls and ceiling and floor three times over. The room, unfolded into a flat panel, has only one floor and one ceiling and each wall once. The straight line you might be walking goes right off the page.

And this is the wrinkle. You might need to tile the room. In a column of blocks (like in Dudeney’s solution) every fourth block might be the floor, with, between any two of them, a ceiling. This is fine, and what’s needed. It can be a bit dizzying to imagine such a state of affairs. But if you’ve ever zoomed a map of the globe out far enough that you see Australia six times over then you’ve understood how this works.

I cannot attest that this has helped my friend in the slightest. I am glad that my friend wanted to think about the surface of the cube. The surface of a dodecahedron would be far, far past my ability to help with.

A Neat Fake Particle Physics Simulator

A friend sent me this video, after realizing that I had missed an earlier mention of it and thought it weird I never commented on it. And I wanted to pass it on, partly because it’s neat and partly because I haven’t done enough writing about topics besides the comics recently.

Particle Life: A Game Of Life Made Of Particles is, at least in video form, a fascinating little puzzle. The Game of Life referenced is one that anybody reading a pop mathematics blog is likely to know. But here goes. The Game of Life is this iterative process. We look at a grid of points, with each point having one of a small set of possible states. Traditionally, just two. At each iteration we go through every grid location. We might change that state. Whether we do depends on some simple rules. In the original Game of Life it’s (depending on your point of view) two or either three rules. A common variation is to include “mutations”, where a location’s state changes despite what the other rules would dictate. And the fascinating thing is that these very simple rules can yield incredibly complicated and beautiful patterns. It’s a neat mathematical refutation of the idea that life is so complicated that it must take a supernatural force to generate. It turns out that many things following simple rules can produce complicated patterns. We will often call them “unpredictable”, although (unless we do have mutations) they are literally perfectly predictable. They’re just chaotic, with tiny changes in the starting conditions often resulting in huge changes in behavior quickly.

This Particle Life problem is built on similar principles. The model is different. Instead of grid locations there are a cloud of particles. The rules are a handful of laws of attraction-or-repulsion. That is, that each particle exerts a force on all the other particles in the system. This is very like the real physics, of clouds of asteroids or of masses of electrically charged gasses or the like. But, like, a cloud of asteroids has everything following the same rule, everything attracts everything else with an intensity that depends on their distance apart. Masses of charged particles follow two rules, particles attracting or repelling each other with an intensity that depends on their distance apart.

This simulation gets more playful. There can be many kinds of particles. They can follow different and non-physically-realistic rules. Like, a red particle can be attracted to a blue, while a blue particle is repelled by a red. A green particle can be attracted to a red with twice the intensity that a red particle’s attracted to a green. Whatever; set different rules and you create different mock physics.

The result is, as the video shows, particles moving in “unpredictable” ways. Again, here, it’s “unpredictable” in the same way that I couldn’t predict when my birthday will next fall on a Tuesday. That is to say, it’s absolutely predictable; it’s just not obvious before you do the calculations. Still, it’s wonderful watching and tinkering with, if you have time to create some physics simulators. There’s source code for one in C++ that you might use. If you’re looking for little toy projects to write on your own, I suspect this would be a good little project to practice your Lua/LOVE coding, too.

Reading the Comics, May 25, 2019: Slighter Comics Edition.

It turned out to be Thursday. These things happen. The comics for the second half of last week were more marginal

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th is a joke about holographic cosmology, proving that there are such things as jokes about holographic cosmology. Cosmology is about the big picture stuff, like, why there is a universe and why it looks like that. It’s a rather mathematical field, owing to the difficulty of doing controlled experiments. Holograms are that same technology used back in the 80s to put shoddy three-dimensional-ish pictures of eagles on credit cards. (In the United States. I imagine they were other animals in other countries.) Holograms, at least when they’re well-made, encode the information needed to make a three-dimensional image in a two-dimensional surface. (Please pretend that anything made of matter is two-dimensional like that.)

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th of May, 2019. Always glad to discuss Saturday Morning Breakfast Cereal, as you can see from these essays.

Holographic cosmology is a mathematical model for the universe. It represents the things in a space with a description of information on the boundary of this space. This seems bizarre and it won’t surprise you that key inspiration was in the strange physics of black holes. Properties of everything which falls into a black hole manifest in the event horizon, the boundary between normal space and whatever’s going on inside the black hole. The black hole is this three-dimensional volume, but in some way everything there is to say about it is the two-dimensional edge.

Dr Leonard Susskind did much to give this precise mathematical form. You didn’t think the character name was just a bit of whimsy, did you? Susskind’s work showed how the information of a particle falling into a black hole — information here meaning stuff like its position and momentum — turn into oscillations in the event horizon. The holographic principle argues this can be extended to ordinary space, the whole of the regular universe. Is this so? It’s hard to say. It’s a corner of string theory. It’s difficult to run experiments that prove very much. And we are stuck with an epistemological problem. If all the things in the universe and their interactions are equally well described as a three-dimensional volume or as a two-dimensional surface, which is “real”? It may seem intuitively obvious that we experience a three-dimensional space. But that intuition is a way we organize our understanding of our experiences. That’s not the same thing as truth.

Gene Weingarten, Dan Weingarten, and David Clark’s Barney and Clyde for the 22nd of May, 2019. Essays which mention some aspect of Barney and Clyde should appear at this link.

Gene Weingarten, Dan Weingarten, and David Clark’s Barney and Clyde for the 22nd is a joke about power, and how it can coerce someone out of truth. Arithmetic serves as an example of indisputable truth. It could be any deductive logic statement, or for that matter a definition. Arithmetic is great for the comic purpose needed here, though. Anyone can understand, at least the simpler statements, and work out their truth or falsity. And need very little word balloon space for it.

Bill Griffith’s Zippy the Pinhead for the 25th of May, 2019. My attempts to form a quite rational and faintly linear discussion out of Zippy the Pinhead should be gathered here.

Bill Griffith’s Zippy the Pinhead for the 25th also features a quick mention of algebra as the height of rationality. Also as something difficult to understand. Most fields are hard to understand, when you truly try. But algebra works well for this writing purpose. Anyone who’d read Zippy the Pinhead has an idea of what understanding algebra would be like, the way they might not have an idea of holographic cosmology.

Teresa Logan’s Laughing Redhead Comics for the 25th of May, 2019. This one is a new tag. So there’s just the one Laughing Redhead Comics essay at this link. But that might change any day now!

Teresa Logan’s Laughing Redhead Comics for the 25th is the Venn diagram joke for the week, this one with a celebrity theme. Your choice whether the logic of the joke makes sense. Ryan Reynolds and John Krasinski are among those celebrities that I keep thinking I don’t know, but that it turns out I do know. Ryan Gosling I’m still not sure about.

And then there are a couple strips too slight even to appear in this collection. Dean Young and John Marshall’s Blondie on the 22nd did a lottery joke, with discussion of probability along the way. (And I hadn’t had a tag for ‘Blondie’ before, so that’s an addition which someday will baffle me.) Bob Shannon’s Tough Town for the 23rd mentions mathematics teaching. It’s in service of a pun.

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And now I’ve had the past week covered. The next Reading the Comics post should be at this link come Sunday.

Reading the Comics, March 26, 2019: March 26, 2019 Edition

And we had another of those peculiar days where a lot of strips are on-topic enough for me to talk about.

Eric the Circle, this one by Kyle, for the 26th has a bit of mathematical physics in it. This is the kind of diagram you’ll see all the time, at least if you do the mathematics that tells you where things will be and when. The particular example is an easy problem, a thing rolling down an inclined plane. But the work done for it applies to more complicated problems. The question it’s for is, “what happens when this thing slides down the plane?” And that depends on the forces at work. There’s gravity, certainly . If there were something else it’d be labelled. Gravity’s represented with that arrow pointing straight down. That gives us the direction. The label (Eric)(g) gives us how strong this force is.

Eric the Circle, by Kyle, for the 26th of March, 2019. Essays inspired at all by Eric the Circle are at this link.

Where the diagram gets interesting, and useful, are those dashed lines ending in arrows. One of those lines is, or at least means to be, parallel to the incline. The other is perpendicular to it. These both reflect gravity. We can represent the force of gravity as a vector. That means, we can represent the force of gravity as the sum of vectors. This is like how we can can write “8” or we can write “3 + 5”, depending on what’s more useful for what we’re doing. (For example, if you wanted to work out “67 + 8”, you might be better off doing “67 + 3 + 5”.) The vector parallel to the plane and the one perpendicular to the plane add up to the original gravity vector.

The force that’s parallel to the plane is the only force that’ll actually accelerate Eric. The force perpendicular to the plane just … keeps it snug against the plane. (Well, it can produce friction. We try not to deal with that in introductory physics because it is so hard. At most we might look at whether there’s enough friction to keep Eric from starting to slide downhill.) The magnitude of the force parallel to the plane, and perpendicular to the plane, are easy enough to work out. These two forces and the original gravity can be put together into a little right triangle. It’s the same shape but different size to the right triangle made by the inclined plane plus a horizontal and a vertical axis. So that’s how the diagram knows the parallel force is the original gravity times the sine of x. And that the perpendicular force is the original gravity times the cosine of x.

The perpendicular force is often called the “normal” force. This because mathematical physicists noticed we had only 2,038 other, unrelated, things called “normal”.

Rick Detorie’s One Big Happy for the 26th sees Ruthie demand to know who this Venn person was. Fair question. Mathematics often gets presented as these things that just are. That someone first thought about these things gets forgotten.

Rick Detorie’s One Big Happy for the 26th of March, 2019. This is a rerun from … 2007, I want to say? There are two separate feeds, one of current and one of several-years-old, strips on the web. Essays including One Big Happy, current or years-old reruns, should be at this link.

John Venn, who lived from 1834 to 1923 — he died the 4th of April, it happens — was an English mathematician and philosopher and logician and (Anglican) priest. This is not a rare combination of professions. From 1862 he was a lecturer in Moral Science at Cambridge. This included work in logic, yes. But he also worked on probability questions. Wikipedia credits his 1866 Logic Of Chance with advancing the frequentist interpretation of probability. This is one of the major schools of thought about what the “probability of an event” is. It’s the one where you list all the things that could possibly happen, and consider how many of those are the thing you’re interested in. So, when you do a problem like “what’s the probability of rolling two six-sided dice and getting a total of four”? You’re doing a frequentist probability problem.

Venn Diagrams he presented to the world around 1880. These show the relationships between different sets. And the relationships of mathematical logic problems they represent. Venn, if my sources aren’t fibbing, didn’t take these diagrams to be a new invention of his own. He wrote of them as “Euler diagrams”. Venn diagrams, properly, need to show all the possible intersections of all the sets in play. You just mark in some way the intersections that happen to have nothing in them. Euler diagrams don’t require this overlapping. The name “Venn diagram” got attached to these pictures in the early 20th century. Euler here is Leonhard Euler, who created every symbol and notation mathematicians use for everything, and who has a different “Euler’s Theorem” that’s foundational to every field of mathematics, including the ones we don’t yet know exist. I exaggerate by 0.04 percent here.

Although we always start Venn diagrams off with circles, they don’t have to be. Circles are good shapes if you have two or three sets. It gets hard to represent all the possible intersections with four circles, though. This is when you start seeing weirder shapes. Wikipedia offers some pictures of Venn diagrams for four, five, and six sets. Meanwhile Mathworld has illustrations for seven- and eleven-set Venn diagrams. At this point, the diagrams are more for aesthetic value than to clarify anything, though. You could draw them with squares. Some people already do. Euler diagrams, particularly, are often squares, sometimes with rounded corners.

Venn had his other projects, too. His biography at St Andrews writes of his composing The Biographical History of Gonville and Caius College (Cambridge). And then he had another history of the whole Cambridge University. It also mentions his skills in building machines, though only cites one, a device for bowling cricket balls. The St Andrews biography says that in 1909 “Venn’s machine clean bowled one of [the Australian Cricket Team’s] top stars four times”. I do not know precisely what it means but I infer it to be a pretty good showing for the machine. His Wikipedia biography calls him a “passionate gardener”. Apparently the Cambridgeshire Horticultural Society awarded him prizes for his roses in July 1885 and for white carrots in September that year. And that he was a supporter of votes for women.

Ashleigh Brilliant’s Pot-Shots for the 26th of March, 2019. The strip originally appeared sometime in 1979. Essays discussing anything from Pot-Shots should appear at this link.

Ashleigh Brilliant’s Pot-Shots for the 26th makes a cute and true claim about percentiles. That a person will usually be in the upper 99% of whatever’s being measured? Hard to dispute. But, measure enough things and eventually you’ll fall out of at least one of them. How many things? This is easy to calculate if we look at different things that are independent of each other. In that case we could look at 69 things before there we’d expect a 50% chance of at least one not being in the upper 99%.

It’s getting that independence that’s hard. There’s often links between things. For example, a person’s height does not tell us much about their weight. But it does tell us something. A person six foot, ten inches tall is almost certainly not also 35 pounds, even though a person could be that size or could be that weight. A person’s scores on a reading comprehension test and their income? But test-taking results and wealth are certainly tied together. Age and income? Most of us have a bigger income at 46 than at 6. This is part of what makes studying populations so hard.

T Shepherd’s Snow Sez for the 26th of March, 2019. Essays including an appearance of Essays inspired at all by Snow Sez should be gathered at this link. They will be, anyway; this is a new tag.

T Shepherd’s Snow Sez for the 26th is finally a strip I can talk about briefly, for a change. Snow does a bit of arithmetic wordplay, toying with what an expression like “1 + 1” might represent.

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There were a lot of mathematically-themed comic strips last week. There’ll be another essay soon, and it should appear at this link. And then there’s always Sunday, as long as I stay ahead of deadline. I am never ahead of deadline.

Six Or Arguably Four Things For Pi Day

I hope you’ll pardon me for being busy. I haven’t had the chance to read all the Pi Day comic strips yet today. But I’d be a fool to let the day pass without something around here. I confess I’m still not sure that Pi Day does anything lasting to encourage people to think more warmly of mathematics. But there is probably some benefit if people temporarily think more fondly of the subject. Certainly I’ll do more foolish things than to point at things and say, “pi, cool, huh?” this week alone.

I’ve got a couple of essays that discuss π some. The first noteworthy one is Calculating Pi Terribly, discussing a way to calculate the value of π using nothing but a needle, a tile floor, and a hilariously excessive amount of time. Or you can use an HTML5-and-JavaScript applet and slightly less time, and maybe even experimentally calculate the digits of π to two decimal places, if you get lucky.

Tom Toles’s Randolph Itch, 2am for the 11th of June, 2018. I’m not sure when it did first run, past that it was in 2000, but I’ve featured it at least two times before, both of those in 2015, peculiarly. So in short I have no idea how GoComics picks its reruns for this strip.

In Calculating Pi Less Terribly I showed a way to calculate π that’s … well, you see where that sentence was going. This is a method that uses an alternating series. To get π exactly correct you have to do an infinite amount of work. But if you just want π to a certain precision, all right. This will even tell you how much work you have to do. There are other formulas that will get you digits of π with less work, though, and maybe I’ll write up one of those sometime.

Dave Whamond’s Reality Check for the 27th of October, 2018. Does the weight count if the jack-o-lantern is wearing sneakers?

And the last of the relevant essays I’ve already written is an A To Z essay about normal numbers. I don’t know whether π is a normal number. No human, to the best of my knowledge, does. Well, anyone with an opinion on the matter would likely say, of course it’s normal. There’s fantastic reasons to think it is. But none of those amount to a proof it is.

Scott Hilburn’s The Argyle Sweater for the 14th of March, 2018. Also a free probability question, if you’re going to assume that every second of the year is equally likely to be the time of birth.

That’s my three items. After that I’d like to share … I don’t know whether to classify this as one or three pieces. They’re YouTube videos which a couple months ago everybody in the world was asking me if I’d seen. Now it’s your turn. I apologize if you too got this, a couple months ago, but don’t worry. You can tell people you watched and not actually do it. I’ll alibi you.

Mark Parisi’s Off The Mark for the 14th of March, 2018. Really the book seems a little short for that.

It’s a string of videos posted on youTube by 3Blue1Brown. The first lays out the matter with a neat physics problem. Imagine you have an impenetrable wall, a frictionless floor, and two blocks. One starts at rest. The other is sliding towards the first block and the wall. How many times will one thing collide with another? That is, will one block collide with another block, or will one block collide with a wall?

Michael Cavna’s Warped for the 14th of March, 2018. Yes, but have you seen Pythagoras and his golden thigh?

The answer seems like it should depend on many things. What it actually depends on is the ratio of the masses of the two blocks. If they’re the same mass, then there are three collisions. You can probably work that sequence out in your head and convince yourself it’s right. If the outer block has ten times the mass of the inner block? There’ll be 31 collisions before all the hits are done. You might work that out by hand. I did not. You will not work out what happens if the outer block has 100 times the mass of the inner block. That’ll be 314 collisions. If the outer block has 1,000 times the mass of the inner block? 3,141 collisions. You see where this is going.

John Zakour and Scott Roberts’s Working Daze for the 15th of March, 2018. No, you should never read the comments, but here, really, don’t read the comments.

The second video in the sequence explains why the digits of π turn up in this. And shows how to calculate this. You could, in principle, do this all using Newtonian mechanics. You will not live long enough to finish that, though.

Daniel Beyer’s Long Story Short for the 14th of March, 2015.

The video shows a way that saves an incredible load of work. But you save on that tedious labor by having to think harder. Part of it is making use of conservation laws, that energy and linear momentum are conserved in collisions. But part is by recasting the problem. Recast it into “phase space”. This uses points in an abstract space to represent different configurations of a system. Like, how fast blocks are moving, and in what direction. The recasting of the problem turns something that’s impossibly tedious into something that’s merely … well, it’s still a bit tedious. But it’s much less hard work. And it’s a good chance to show off you remember the Inscribed Angle Theorem. You do remember the Inscribed Angle Theorem, don’t you? The video will catch you up. It’s a good show of how phase spaces can make physics problems so much more manageable.

John Hambrock’s The Brilliant Mind of Edison Lee for the 14th of March, 2016. The strip is like this a lot.

The third video recasts the problem yet again. In this form, it’s about rays of light reflecting between mirrors. And this is a great recasting. That blocks bouncing off each other and walls should have anything to do with light hitting mirrors seems ridiculous. But set out your phase space, and look hard at what collisions and reflections are like, and you see the resemblance. The sort of trick used to make counting reflections easy turns up often in phase spaces. It also turns up in physics problems on toruses, doughnut shapes. You might ask when do we ever do anything on a doughnut shape. Well, real physical doughnuts, not so much. But problems where there are two independent quantities, and both quantities are periodic? There’s a torus lurking in there. There might be a phase space using that shape, and making your life easier by doing so.

Scott Hilburn’s The Argyle Sweater for the 14th of March, 2017. And while the strip is true, arguably, 2 goes on forever also; it’s just not very interesting how it does.

That’s my promised four or maybe six items. Pardon, please, now, as I do need to get back to reading the comics.

Reading the Comics, December 22, 2018: Christmas Break Edition

There were just enough mathematically-themed comic strips last week for me to make two posts out of it. This current week? Is looking much slower, at least as of Wednesday night. But that’s a problem for me to worry about on Sunday.

Eric the Circle for the 20th, this one by Griffinetsabine, mentions a couple of shapes. That’s enough for me, at least on a slow comics week. There is a fictional tradition of X marking the spot. It can be particularly credited to Robert Louis Stevenson’s Treasure Island. Any symbol could be used to note a special place on maps, certainly. Many maps are loaded with a host of different symbols to convey different information. Circles and crosses have the advantage of being easy to draw and difficult to confuse for one another. Squares, triangles, and stars are good too.

Eric the Circle for the 20th of December, 2018, this one by Griffinetsabine. This and other essays featuring Eric the Circle are at this link.

Bill Whitehead’s Free Range for the 22nd spoofs Wheel of Fortune with “theoretical mathematics”. Making a game out of filling in parts of a mathematical expression isn’t ridiculous, although it is rather niche. I don’t see how the revealed string of mathematical expressions build to a coherent piece, but perhaps a few further pieces would help.

Bill Whitehead’s Free Range for the 22nd of December, 2018. Appearances by Free Range should be at this link.

The parts shown are all legitimate enough expressions. Well, like is only true for some specific numbers ‘a’ and ‘b’, but you can find solutions. is just an expression, not picking out any particular values of ‘b’ or ‘x’ or ‘y’ as interesting. But in conjunction with or other expressions there might be something useful. On the second row is a graph, highlighting a region underneath a curve (and above the x-axis) between two vertical lines. This is often the sort of thing looked at in calculus. It also turns up in probability, as the area under a curve like this can show the chance that an experiment will turn up something in a range of values. And is a straightforward differential equation. Its solution is a family of similar-looking polynomials.

Mark Pett’s Lucky Cow for the 22nd of December, 2018. This and other discussions inspired by Lucky Cow should be at this link.

Mark Pett’s Lucky Cow for the 22nd has run before. I’ve even made it the title strip for a Reading the Comics post back in 2014. So it’s probably time to drop this from my regular Reading the Comics reporting. The physicists comes running in with the left half of the time-dependent Schrödinger Equation. This is all over quantum mechanics. In this form, quantum mechanics contains information about how a system behaves by putting it into a function named . Its value depends on space (‘x’). It can also depend on time (‘t’). The physicists pretends to not be able to complete this. Neil arranges to give the answer.

Schrödinger’s Equation looks very much like a diffusion problem. Normal diffusion problems don’t have that which appears in the part of Neil’s answer. But this form of equation turns up a lot. If you have something that acts like a fluid — and heat counts — then a diffusion problem is likely important in understanding it.

And, yes, the setup reminds me of a mathematical joke that I only encounter in lists of mathematics jokes. That one I told the last time this strip came up in the rotation. You might chuckle, or at least be convinced that it is a correctly formed joke.

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Each of the Reading the Comics posts should all be at this link. And I have finished the alphabet in my Fall 2018 Mathematics A To Z glossary. There should be a few postscript thoughts to come this week, though.

Reading the Comics, August 4, 2018: August 4, 2018 Edition

And finally, at last, there’s a couple of comics left over from last week and that all ran the same day. If I hadn’t gone on forever about negative Kelvin temperatures I might have included them in the previous essay. That’s all right. These are strips I expect to need relatively short discussions to explore. Watch now as I put out 2,400 words explaining Wavehead misinterpreting the teacher’s question.

Dave Whamond’s Reality Check for the 4th is proof that my time spent writing about which is better, large numbers or small last week wasn’t wasted. There I wrote about four versus five for Beetle Bailey. Here it’s the same joke, but with compound words. Well, that’s easy to take care of.

Dave Whamond’s Reality Check for the 4th of August, 2018. I’m sure it’s a coincidence that the tall-headed person shares a name with the cartoonist.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 4th is driving me slightly crazy. The equation on the board looks like an electrostatics problem to me. The ‘E’ is a common enough symbol for the strength of an electric field. And the funny-looking K’s look to me like the Greek kappa. This often represents the dielectric constant. That measures how well an electric field can move through a material. The upside-down triangles, known in the trade as Delta, describe — well, that’s getting complicated. By themselves, they describe measuring “how much the thing right after this changes in different directions”. When there’s a x symbol between the Delta and the thing, it measures something called the “curl”. This roughly measures how much the field inspires things caught up in it to turn. (Don’t try passing this off to your thesis defense committee.) The Delta x Delta x E describes the curl of the curl of E. Oh, I don’t like visualizing that. I don’t blame you if you don’t want to either.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 4th of August, 2018. Really not clear what the cry for help would be about. Just treat the rod as a limiting case of an enormous number of small spheres placed end to end and you’re done.

Anyway. So all this looks like it’s some problem about a rod inside an electric field. Fine enough. What I don’t know and can’t work out is what the problem is studying exactly. So I can’t tell you whether the equation, so far as we see it, is legitimately something to see in class. Envisioning a rod that’s infinitely thin is a common enough mathematical trick, though. Three-dimensional objects are hard to deal with. They have edges. These are fussy to deal with. Making sure the interior, the boundary, and the exterior match up in a logically consistent way is tedious. But a wire? A plane? A single point? That’s easy. They don’t have an interior. You don’t have to match up the complicated stuff.

For real world problems, yeah, you have to deal with the interior. Or you have to work out reasons why the interiors aren’t important in your problem. And it can be that your object is so small compared to the space it has to work in that the fact it’s not infinitely thin or flat or smooth just doesn’t matter. Mathematical models, such as give us equations, are a blend of describing what really is there and what we can work with.

Mike Shiell’s The Wandering Melon for the 4th of August, 2018. Still, impressive watchband that it’s stood up to all that trouble.

Mike Shiell’s The Wandering Melon for the 4th is a probability joke, about two events that nobody’s likely to experience. The chance any individual will win a lottery is tiny, but enough people play them that someone wins just about any given week. The chance any individual will get struck by lightning is tiny too. But it happens to people. The combination? Well, that’s obviously impossible.

In July of 2015, Peter McCathie had this happen. He survived a lightning strike first. And then won the Atlantic Lotto 6/49. This was years apart, but the chance of both happening the same day, or same week? … Well, the world is vast and complicated. Unlikely things will happen.

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And that’s all that I have for the past week. Come Sunday I should have my next Reading the Comics post, and you can find it and other essays at this link. Other essays that mention Reality Check are at this link. The many other essays which talk about Saturday Morning Breakfast Cereal are at this link. And other essays about The Wandering Melon are at this link. Thanks.

Without Tipping My Hand To My Plans For The Next Couple Weeks

I wanted to get this out of the way before I did it:

• Laying Some Groundwork
• Why Stuff Can’t Orbit
• It Turns Out Spinning Matters
• On The L
• Why Physics Doesn’t Work And What To Do About It
• Circles and Where To Find Them
• ALL the Circles
• Introducing Stability
• How The Spring In The Cosmos Behaves
• Where Time Comes From And How It Changes Things
• In Search Of Closure
• How Fast Is An Orbit?
• To Close A Loop

And the supplemental reading:

• How Differential Calculus Works
• How Mathematical Physics Works
• Everything Interesting There Is To Say About Springs
• What Second Derivatives Are And What They Can Do For You

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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.

Reading the Comics, November 4, 2017: Slow, Small Week Edition

It was a slow week for mathematically-themed comic strips. What I have are meager examples. Small topics to discuss. The end of the week didn’t have anything even under loose standards of being on-topic. Which is fine, since I lost an afternoon of prep time to thunderstorms that rolled through town and knocked out power for hours. Who saw that coming? … If I had, I’d have written more the day before.

Mac King and Bill King’s Magic in a Minute for the 29th of October looks like a word problem. Well, it is a word problem. It looks like a problem about extrapolating a thing (price) from another thing (quantity). Well, it is an extrapolation problem. The fun is in figuring out what quantities are relevant. Now I’ve spoiled the puzzle by explaining it all so.

Olivia Walch’s Imogen Quest for the 30th doesn’t say it’s about a mathematics textbook. But it’s got to be. What other kind of textbook will have at least 28 questions in a section and only give answers to the odd-numbered problems in back? You never see that in your social studies text.

Eric the Circle for the 30th, this one by Dennill, tests how slow a week this was. I guess there’s a geometry joke in Jane Austen? I’ll trust my literate readers to tell me. My doing the world’s most casual search suggests there’s no mention of triangles in Pride and Prejudice. The previous might be the most ridiculously mathematics-nerdy thing I have written in a long while.

Tony Murphy’s It’s All About You for the 31st does some advanced-mathematics name-dropping. In so doing, it’s earned a spot taped to the door of two people in any mathematics department with more than 24 professors across the country. Or will, when they hear there was a gap unification theory joke in the comics. I’m not sure whether Murphy was thinking of anything particular in naming the subject “gap unification theory”. It sounds like a field of mathematical study. But as far as I can tell there’s just one (1) paper written that even says “gap unification theory”. It’s in partition theory. Partition theory is a rich and developed field, which seems surprising considering it’s about breaking up sets of the counting numbers into smaller sets. It seems like a time-waster game. But the game sneaks into everything, so the field turns out to be important. Gap unification, in the paper I can find, is about studying the gaps between these smaller sets.

There’s also a “band-gap unification” problem. I could accept this name being shortened to “gap unification” by people who have to say its name a lot. It’s about the physics of semiconductors, or the chemistry of semiconductors, as you like. The physics or chemistry of them is governed by the energies that electrons can have. Some of these energies are precise levels. Some of these energies are bands, continuums of possible values. When will bands converge? When will they not? Ask a materials science person. Going to say that’s not mathematics? Don’t go looking at the papers.

Whether partition theory or materials since it seems like a weird topic. Maybe Murphy just put together words that sounded mathematical. Maybe he has a friend in the field.

Bill Amend’s FoxTrot Classics for the 1st of November is aiming to be taped up to the high school teacher’s door. It’s easy to show how the square root of two is irrational. Takes a bit longer to show the square root of three is. Turns out all the counting numbers are either perfect squares — 1, 4, 9, 16, and so on — or else have irrational square roots. There’s no whole number with a square root of, like, something-and-three-quarters or something-and-85-117ths. You can show that, easily if tediously, for any particular whole number. What’s it look like to show for all the whole numbers that aren’t perfect squares already? (This strip originally ran the 8th of November, 2006.)

Guy Gilchrist’s Nancy for the 1st does an alphabet soup joke, so like I said, it’s been a slow week around here.

John Zakour and Scott Roberts’s Maria’s Day for the 2nd is really just mathematics being declared hated, so like I said, it’s been a slow week around here.

Reading the Comics, October 14, 2017: Physics Equations Edition

So that busy Saturday I promised for the mathematically-themed comic strips? Here it is, along with a Friday that reached the lowest non-zero levels of activity.

Stephan Pastis’s Pearls Before Swine for the 13th is one of those equations-of-everything jokes. Naturally it features a panel full of symbols that, to my eye, don’t parse. There are what look like syntax errors, for example, with the one that anyone could see the { mark that isn’t balanced by a }. But when someone works rough they will, often, write stuff that doesn’t quite parse. Think of it as an artist’s rough sketch of a complicated scene: the lines and anatomy may be gibberish, but if the major lines of the composition are right then all is well.

Most attempts to write an equation for everything are really about writing a description of the fundamental forces of nature. We trust that it’s possible to go from a description of how gravity and electromagnetism and the nuclear forces go to, ultimately, a description of why chemistry should work and why ecologies should form and there should be societies. There are, as you might imagine, a number of assumed steps along the way. I would accept the idea that we’ll have a unification of the fundamental forces of physics this century. I’m not sure I would believe having all the steps between the fundamental forces and, say, how nerve cells develop worked out in that time.

Mark Anderson’s Andertoons makes it overdue appearance for the week on the 14th, with a chalkboard word-problem joke. Amusing enough. And estimating an answer, getting it wrong, and refining it is good mathematics. It’s not just numerical mathematics that will look for an approximate solution and then refine it. As a first approximation, 15 minus 7 isn’t far off 10. And for mental arithmetic approximating 15 minus 7 as 10 is quite justifiable. It could be made more precise if a more exact answer were needed.

Maria Scrivan’s Half Full for the 14th I’m going to call the anthropomorphic geometry joke for the week. If it’s not then it’s just wordplay and I’d have no business including it here.

Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 14th tosses in the formula describing how strong the force of gravity between two objects is. In Newtonian gravity, which is why it’s the Newton Police. It’s close enough for most purposes. I’m not sure how this supports the cause of world peace.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 14th names Riemann’s Quaternary Conjecture. I was taken in by the panel, trying to work out what the proposed conjecture could even mean. The reason it works is that Bernhard Riemann wrote like 150,000 major works in every field of mathematics, and about 149,000 of them are big, important foundational works. The most important Riemann conjecture would be the one about zeroes of the Riemann Zeta function. This is typically called the Riemann Hypothesis. But someone could probably write a book just listing the stuff named for Riemann, and that’s got to include a bunch of very specific conjectures.

Reading the Comics, October 4, 2017: Time-Honored Traditions Edition

It was another busy week in mathematically-themed comic strips last week. Busy enough I’m comfortable rating some as too minor to include. So it’s another week where I post two of these Reading the Comics roundups, which is fine, as I’m still recuperating from the Summer 2017 A To Z project. This first half of the week includes a lot of rerun comics, and you’ll see why my choice of title makes sense.

Lincoln Pierce’s Big Nate: First Class for the 1st of October reprints the strip from the 2nd of October, 1993. It’s got a well-formed story problem that, in the time-honored tradition of this setup, is subverted. I admit I kind of miss the days when exams would have problems typed out in monospace like this.

Ashleigh Brilliant’s Pot-Shots for the 1st is a rerun from sometime in 1975. And it’s an example of the time-honored tradition of specifying how many statistics are made up. Here it comes in at 43 percent of statistics being “totally worthless” and I’m curious how the number attached to this form of joke changes over time.

The Joey Alison Sayers Comic for the 2nd uses a blackboard with mathematics — a bit of algebra and a drawing of a sphere — as the designation for genius. That’s all I have to say about this. I remember being set straight about the difference between ponies and horses and it wasn’t by my sister, who’s got a professional interest in the subject.

Mark Pett’s Lucky Cow rerun for the 2nd is a joke about cashiers trying to work out change. As one of the GoComics.com commenters mentions, the probably best way to do this is to count up from the purchase to the amount you have to give change for. That is, work out $12.43 to $12.50 is seven cents, then from $12.50 to $13.00 is fifty more cents (57 cents total), then from $13.00 to $20.00 is seven dollars ($7.57 total) and then from $20 to $50 is thirty dollars ($37.57 total).

It does make me wonder, though: what did Neil enter as the amount tendered, if it wasn’t $50? Maybe he hit “exact change” or whatever the equivalent was. It’s been a long, long time since I worked a cash register job and while I would occasionally type in the wrong amount of money, the kinds of errors I would make would be easy to correct for. (Entering $30 instead of $20 for the tendered amount, that sort of thing.) But the cash register works however Mark Pett decides it works, so who am I to argue?

Keith Robinson’s Making It rerun for the 2nd includes a fair bit of talk about ratios and percentages, and how to inflate percentages. Also about the underpaying of employees by employers.

Mark Anderson’s Andertoons for the 3rd continues the streak of being Mark Anderson Andertoons for this sort of thing. It has the traditional form of the student explaining why the teacher’s wrong to say the answer was wrong.

Brian Fies’s The Last Mechanical Monster for the 4th includes a bit of legitimate physics in the mad scientist’s captioning. Ballistic arcs are about a thing given an initial speed in a particular direction, moving under constant gravity, without any of the complicating problems of the world involved. No air resistance, no curvature of the Earth, level surfaces to land on, and so on. So, if you start from a given height (‘y₀‘) and a given speed (‘v’) at a given angle (‘θ’) when the gravity is a given strength (‘g’), how far will you travel? That’s ‘d’. How long will you travel? That’s ‘t’, as worked out here.

(I should maybe explain the story. The mad scientist here is the one from the first, Fleischer Studios, Superman cartoon. In it the mad scientist sends mechanical monsters out to loot the city’s treasures and whatnot. As the cartoon has passed into the public domain, Brian Fies is telling a story of that mad scientist, finally out of jail, salvaging the one remaining usable robot. Here, training the robot to push aside bank tellers has gone awry. Also, the ground in his lair is not level.)

Tom Toles’s Randolph Itch, 2 am rerun for the 4th uses the time-honored tradition of Albert Einstein needing a bit of help for his work.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 4th uses the time-honored tradition of little bits of physics equations as designation of many deep thoughts. And then it gets into a bit more pure mathematics along the way. It also reflects the time-honored tradition of people who like mathematics and physics supposing that those are the deepest and most important kinds of thoughts to have. But I suppose we all figure the things we do best are the things it’s important to do best. It’s traditional.

And by the way, if you’d like more of these Reading the Comics posts, I put them all in the category ‘Comic Strips’ and I just now learned the theme I use doesn’t show categories for some reason? This is unsettling and unpleasant. Hm.

Reading the Comics, September 24, 2017: September 24, 2017 Edition

Comic Strip Master Command sent a nice little flood of comics this week, probably to make sure that I transitioned from the A To Z project to normal activity without feeling too lost. I’m going to cut the strips not quite in half because I’m always delighted when I can make a post that’s just a single day’s mathematically-themed comics. Last Sunday, the 24th of September, was such a busy day. I’m cheating a little on what counts as noteworthy enough to talk about here. But people like comic strips, and good on them for liking them.

Norm Feuti’s Gil for the 24th sees Gil discover and try to apply some higher mathematics. There’s probably a good discussion about what we mean by division to explain why Gil’s experiment didn’t pan out. I would pin it down to eliding the difference between “dividing in half” and “dividing by a half”, which is a hard one. Terms that seem almost alike but mean such different things are probably the hardest part of mathematics.

Norm Feuti’s Gil for the 24th of September, 2017, didn’t appear on Gocomics.com or Comics Kingdom, my usual haunts for these comics. But I started reading the strip when it was on Comics Kingdom, and keep reading its reruns. Feuti has continued the comic strip on his own web site, and posts it on Twitter. So it’s quite easy to pick the strip back up, if you have a Twitter account or can read RSS from it. I assume you can read RSS from it.

Russell Myers’s Broom Hilda looks like my padding. But the last panel of the middle row gets my eye. The squirrels talk about how on the equinox night and day “can never be of identical length, due to the angular size of the sun and atmospheric refraction”. This is true enough for the equinox. While any spot on the Earth might see twelve hours facing the sun and twelve hours facing away, the fact the sun isn’t a point, and that the atmosphere carries light around to the “dark” side of the planet, means daylight lasts a little longer than night.

Ah, but. This gets my mathematical modelling interest going. Because it is true that, at least away from the equator, there’s times of year that day is way shorter than night. And there’s times of year that day is way longer than night. Shouldn’t there be some time in the middle when day is exactly equal to night?

The easy argument for is built on the Intermediate Value Theorem. Let me define a function, with domain each of the days of the year. The range is real numbers. It’s defined to be the length of day minus the length of night. Let me say it’s in minutes, but it doesn’t change things if you argue that it’s seconds, or milliseconds, or hours, if you keep parts of hours in also. So, like, 12.015 hours or something. At the height of winter, this function is definitely negative; night is longer than day. At the height of summer, this function is definitely positive; night is shorter than day. So therefore there must be some time, between the height of winter and the height of summer, when the function is zero. And therefore there must be some day, even if it isn’t the equinox, when night and day are the same length

There’s a flaw here and I leave that to classroom discussions to work out. I’m also surprised to learn that my onetime colleague Dr Helmer Aslaksen’s grand page of mathematical astronomy and calendar essays doesn’t seem to have anything about length of day calculations. But go read that anyway; you’re sure to find something fascinating.

Mike Baldwin’s Cornered features an old-fashioned adding machine being used to drown an audience in calculations. Which makes for a curious pairing with …

Bill Amend’s FoxTrot, and its representation of “math hipsters”. I hate to encourage Jason or Marcus in being deliberately difficult. But there are arguments to make for avoiding digital calculators in favor of old-fashioned — let’s call them analog — calculators. One is that people understand tactile operations better, or at least sooner, than they do digital ones. The slide rule changes multiplication and division into combining or removing lengths of things, and we probably have an instinctive understanding of lengths. So this should train people into anticipating what a result is likely to be. This encourages sanity checks, verifying that an answer could plausibly be right. And since a calculation takes effort, it encourages people to think out how to arrange the calculation to require less work. This should make it less vulnerable to accidents.

I suspect that many of these benefits are what you get in the ideal case, though. Slide rules, and abacuses, are no less vulnerable to accidents than anything else is. And if you are skilled enough with the abacus you have no trouble multiplying 18 by 7, you probably would not find multiplying 17 by 8 any harder, and wouldn’t notice if you mistook one for the other.

Jef Mallett’s Frazz asserts that numbers are cool but the real insight is comparisons. And we can argue that comparisons are more basic than numbers. We can talk about one thing being bigger than another even if we don’t have a precise idea of numbers, or how to measure them. See every mathematics blog introducing the idea of different sizes of infinity.

Bill Whitehead’s Free Range features Albert Einstein, universal symbol for really deep thinking about mathematics and physics and stuff. And even a blackboard full of equations for the title panel. I’m not sure whether the joke is a simple absent-minded-professor joke, or whether it’s a relabelled joke about Werner Heisenberg. Absent-minded-professor jokes are not mathematical enough for me, so let me point once again to American Cornball. They’re the first subject in Christopher Miller’s encyclopedia of comic topics. So I’ll carry on as if the Werner Heisenberg joke were the one meant.

Heisenberg is famous, outside World War II history, for the Uncertainty Principle. This is one of the core parts of quantum mechanics, under which there’s a limit to how precisely one can know both the position and momentum of a thing. To identify, with absolutely zero error, where something is requires losing all information about what its momentum might be, and vice-versa. You see the application of this to a traffic cop’s question about knowing how fast someone was going. This makes some neat mathematics because all the information about something is bundled up in a quantity called the Psi function. To make a measurement is to modify the Psi function by having an “operator” work on it. An operator is what we call a function that has domains and ranges of other functions. To measure both position and momentum is equivalent to working on Psi with one operator and then another. But these operators don’t commute. You get different results in measuring momentum and then position than you do measuring position and then momentum. And so we can’t know both of these with infinite precision.

There are pairs of operators that do commute. They’re not necessarily ones we care about, though. Like, the total energy commutes with the square of the angular momentum. So, you know, if you need to measure with infinite precision the energy and the angular momentum of something you can do it. If you had measuring tools that were perfect. You don’t, but you could imagine having them, and in that case, good. Underlying physics wouldn’t spoil your work.

Probably the panel was an absent-minded professor joke.

The Summer 2017 Mathematics A To Z: Young Tableau

I never heard of today’s entry topic three months ago. Indeed, three weeks ago I was still making guesses about just what Gaurish, author of For the love of Mathematics,, was asking about. It turns out to be maybe the grand union of everything that’s ever been in one of my A To Z sequences. I overstate, but barely.

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.

Young Tableau.

The specific thing that a Young Tableau is is beautiful in its simplicity. It could almost be a recreational mathematics puzzle, except that it isn’t challenging enough.

Start with a couple of boxes laid in a row. As many or as few as you like.

Now set another row of boxes. You can have as many as the first row did, or fewer. You just can’t have more. Set the second row of boxes — well, your choice. Either below the first row, or else above. I’m going to assume you’re going below the first row, and will write my directions accordingly. If you do things the other way you’re following a common enough convention. I’m leaving it on you to figure out what the directions should be, though.

Now add in a third row of boxes, if you like. Again, as many or as few boxes as you like. There can’t be more than there are in the second row. Set it below the second row.

And a fourth row, if you want four rows. Again, no more boxes in it than the third row had. Keep this up until you’ve got tired of adding rows of boxes.

How many boxes do you have? I don’t know. But take the numbers 1, 2, 3, 4, 5, and so on, up to whatever the count of your boxes is. Can you fill in one number for each box? So that the numbers are always increasing as you go left to right in a single row? And as you go top to bottom in a single column? Yes, of course. Go in order: ‘1’ for the first box you laid down, then ‘2’, then ‘3’, and so on, increasing up to the last box in the last row.

Can you do it in another way? Any other order?

Except for the simplest of arrangements, like a single row of four boxes or three rows of one box atop another, the answer is yes. There can be many of them, turns out. Seven boxes, arranged three in the first row, two in the second, one in the third, and one in the fourth, have 35 possible arrangements. It doesn’t take a very big diagram to get an enormous number of possibilities. Could be fun drawing an arbitrary stack of boxes and working out how many arrangements there are, if you have some time in a dull meeting to pass.

Let me step away from filling boxes. In one of its later, disappointing, seasons Futurama finally did a body-swap episode. The gimmick: two bodies could only swap the brains within them one time. So would it be possible to put Bender’s brain back in his original body, if he and Amy (or whoever) had already swapped once? The episode drew minor amusement in mathematics circles, and a lot of amazement in pop-culture circles. The writer, a mathematics major, found a proof that showed it was indeed always possible, even after many pairs of people had swapped bodies. The idea that a theorem was created for a TV show impressed many people who think theorems are rarer and harder to create than they necessarily are.

It was a legitimate theorem, and in a well-developed field of mathematics. It’s about permutation groups. These are the study of the ways you can swap pairs of things. I grant this doesn’t sound like much of a field. There is a surprising lot of interesting things to learn just from studying how stuff can be swapped, though. It’s even of real-world relevance. Most subatomic particles of a kind — electrons, top quarks, gluons, whatever — are identical to every other particle of the same kind. Physics wouldn’t work if they weren’t. What would happen if we swap the electron on the left for the electron on the right, and vice-versa? How would that change our physics?

A chunk of quantum mechanics studies what kinds of swaps of particles would produce an observable change, and what kind of swaps wouldn’t. When the swap doesn’t make a change we can describe this as a symmetric operation. When the swap does make a change, that’s an antisymmetric operation. And — the Young Tableau that’s a single row of two boxes? That matches up well with this symmetric operation. The Young Tableau that’s two rows of a single box each? That matches up with the antisymmetric operation.

How many ways could you set up three boxes, according to the rules of the game? A single row of three boxes, sure. One row of two boxes and a row of one box. Three rows of one box each. How many ways are there to assign the numbers 1, 2, and 3 to those boxes, and satisfy the rules? One way to do the single row of three boxes. Also one way to do the three rows of a single box. There’s two ways to do the one-row-of-two-boxes, one-row-of-one-box case.

What if we have three particles? How could they interact? Well, all three could be symmetric with each other. This matches the first case, the single row of three boxes. All three could be antisymmetric with each other. This matches the three rows of one box. Or you could have two particles that are symmetric with each other and antisymmetric with the third particle. Or two particles that are antisymmetric with each other but symmetric with the third particle. Two ways to do that. Two ways to fill in the one-row-of-two-boxes, one-row-of-one-box case.

This isn’t merely a neat, aesthetically interesting coincidence. I wouldn’t spend so much time on it if it were. There’s a matching here that’s built on something meaningful. The different ways to arrange numbers in a set of boxes like this pair up with a select, interesting set of matrices whose elements are complex-valued numbers. You might wonder who introduced complex-valued numbers, let alone matrices of them, into evidence. Well, who cares? We’ve got them. They do a lot of work for us. So much work they have a common name, the “symmetric group over the complex numbers”. As my leading example suggests, they’re all over the place in quantum mechanics. They’re good to have around in regular physics too, at least in the right neighborhoods.

These Young Tableaus turn up over and over in group theory. They match up with polynomials, because yeah, everything is polynomials. But they turn out to describe polynomial representations of some of the superstar groups out there. Groups with names like the General Linear Group (square matrices), or the Special Linear Group (square matrices with determinant equal to 1), or the Special Unitary Group (that thing where quantum mechanics says there have to be particles whose names are obscure Greek letters with superscripts of up to five + marks). If you’d care for more, here’s a chapter by Dr Frank Porter describing, in part, how you get from Young Tableaus to the obscure baryons.

Porter’s chapter also lets me tie this back to tensors. Tensors have varied ranks, the number of different indicies you can have on the things. What happens when you swap pairs of indices in a tensor? How many ways can you swap them, and what does that do to what the tensor describes? Please tell me you already suspect this is going to match something in Young Tableaus. They do this by way of the symmetries and permutations mentioned above. But they are there.

As I say, three months ago I had no idea these things existed. If I ever ran across them it was from seeing the name at MathWorld’s list of terms that start with ‘Y’. The article shows some nice examples (with each rows a atop the previous one) but doesn’t make clear how much stuff this subject runs through. I can’t fit everything in to a reasonable essay. (For example: the number of ways to arrange, say, 20 boxes into rows meeting these rules is itself a partition problem. Partition problems are probability and statistical mechanics. Statistical mechanics is the flow of heat, and the movement of the stars in a galaxy, and the chemistry of life.) I am delighted by what does fit.

The Summer 2017 Mathematics A To Z: Volume Forms

I’ve been reading Elke Stangl’s Elkemental Force blog for years now. Sometimes I even feel social-media-caught-up enough to comment, or at least to like posts. This is relevant today as I discuss one of the Stangl’s suggestions for my letter-V topic.

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.

Volume Forms.

So sometime in pre-algebra, or early in (high school) algebra, you start drawing equations. It’s a simple trick. Lay down a coordinate system, some set of axes for ‘x’ and ‘y’ and maybe ‘z’ or whatever letters are important. Look to the equation, made up of x’s and y’s and maybe z’s and so. Highlight all the points with coordinates whose values make the equation true. This is the logical basis for saying (eg) that the straight line “is” .

A short while later, you learn about polar coordinates. Instead of using ‘x’ and ‘y’, you have ‘r’ and ‘θ’. ‘r’ is the distance from the center of the universe. ‘θ’ is the angle made with respect to some reference axis. It’s as legitimate a way of describing points in space. Some classrooms even have a part of the blackboard (whiteboard, whatever) with a polar-coordinates “grid” on it. This looks like the lines of a dartboard. And you learn that some shapes are easy to describe in polar coordinates. A circle, centered on the origin, is ‘r = 2’ or something like that. A line through the origin is ‘θ = 1’ or whatever. The line that we’d called before? … That’s … some mess. And now … that’s not even a line. That’s some kind of spiral. Two spirals, really. Kind of wild.

And something to bother you a while. is an equation that looks the same as . You’ve changed the names of the variables, but not how they relate to each other. But one is a straight line and the other a spiral thing. How can that be?

The answer, ultimately, is that the letters in the equations aren’t these content-neutral labels. They carry meaning. ‘x’ and ‘y’ imply looking at space a particular way. ‘r’ and ‘θ’ imply looking at space a different way. A shape has different representations in different coordinate systems. Fair enough. That seems to settle the question.

But if you get to calculus the question comes back. You can integrate over a region of space that’s defined by Cartesian coordinates, x’s and y’s. Or you can integrate over a region that’s defined by polar coordinates, r’s and θ’s. The first time you try this, you find … well, that any region easy to describe in Cartesian coordinates is painful in polar coordinates. And vice-versa. Way too hard. But if you struggle through all that symbol manipulation, you get … different answers. Eventually the calculus teacher has mercy and explains. If you’re integrating in Cartesian coordinates you need to use “dx dy”. If you’re integrating in polar coordinates you need to use “r dr dθ”. If you’ve never taken calculus, never mind what this means. What is important is that “r dr dθ” looks like three things multiplied together, while “dx dy” is two.

We get this explained as a “change of variables”. If we want to go from one set of coordinates to a different one, we have to do something fiddly. The extra ‘r’ in “r dr dθ” is what we get going from Cartesian to polar coordinates. And we get formulas to describe what we should do if we need other kinds of coordinates. It’s some work that introduces us to the Jacobian, which looks like the most tedious possible calculation ever at that time. (In Intro to Differential Equations we learn we were wrong, and the Wronskian is the most tedious possible calculation ever. This is also wrong, but it might as well be true.) We typically move on after this and count ourselves lucky it got no worse than that.

None of this is wrong, even from the perspective of more advanced mathematics. It’s not even misleading, which is a refreshing change. But we can look a little deeper, and get something good from doing so.

The deeper perspective looks at “differential forms”. These are about how to encode information about how your coordinate system represents space. They’re tensors. I don’t blame you for wondering if they would be. A differential form uses interactions between some of the directions in a space. A volume form is a differential form that uses all the directions in a space. And satisfies some other rules too. I’m skipping those because some of the symbols involved I don’t even know how to look up, much less make WordPress present.

What’s important is the volume form carries information compactly. As symbols it tells us that this represents a chunk of space that’s constant no matter what the coordinates look like. This makes it possible to do analysis on how functions work. It also tells us what we would need to do to calculate specific kinds of problem. This makes it possible to describe, for example, how something moving in space would change.

The volume form, and the tools to do anything useful with it, demand a lot of supporting work. You can dodge having to explicitly work with tensors. But you’ll need a lot of tensor-related materials, like wedge products and exterior derivatives and stuff like that. If you’ve never taken freshman calculus don’t worry: the people who have taken freshman calculus never heard of those things either. So what makes this worthwhile?

Yes, person who called out “polynomials”. Good instinct. Polynomials are usually a reason for any mathematics thing. This is one of maybe four exceptions. I have to appeal to my other standard answer: “group theory”. These volume forms match up naturally with groups. There’s not only information about how coordinates describe a space to consider. There’s ways to set up coordinates that tell us things.

That isn’t all. These volume forms can give us new invariants. Invariants are what mathematicians say instead of “conservation laws”. They’re properties whose value for a given problem is constant. This can make it easier to work out how one variable depends on another, or to work out specific values of variables.

For example, classical physics problems like how a bunch of planets orbit a sun often have a “symplectic manifold” that matches the problem. This is a description of how the positions and momentums of all the things in the problem relate. The symplectic manifold has a volume form. That volume is going to be constant as time progresses. That is, there’s this way of representing the positions and speeds of all the planets that does not change, no matter what. It’s much like the conservation of energy or the conservation of angular momentum. And this has practical value. It’s the subject that brought my and Elke Stangl’s blogs into contact, years ago. It also has broader applicability.

There’s no way to provide an exact answer for the movement of, like, the sun and nine-ish planets and a couple major moons and all that. So there’s no known way to answer the question of whether the Earth’s orbit is stable. All the planets are always tugging one another, changing their orbits a little. Could this converge in a weird way suddenly, on geologic timescales? Might the planet might go flying off out of the solar system? It doesn’t seem like the solar system could be all that unstable, or it would have already. But we can’t rule out that some freaky alignment of Jupiter, Saturn, and Halley’s Comet might not tweak the Earth’s orbit just far enough for catastrophe to unfold. Granted there’s nothing we could do about the Earth flying out of the solar system, but it would be nice to know if we face it, we tell ourselves.

But we can answer this numerically. We can set a computer to simulate the movement of the solar system. But there will always be numerical errors. For example, we can’t use the exact value of π in a numerical computation. 3.141592 (and more digits) might be good enough for projecting stuff out a day, a week, a thousand years. But if we’re looking at millions of years? The difference can add up. We can imagine compensating for not having the value of π exactly right. But what about compensating for something we don’t know precisely, like, where Jupiter will be in 16 million years and two months?

Symplectic forms can help us. The volume form represented by this space has to be conserved. So we can rewrite our simulation so that these forms are conserved, by design. This does not mean we avoid making errors. But it means we avoid making certain kinds of errors. We’re more likely to make what we call “phase” errors. We predict Jupiter’s location in 16 million years and two months. Our simulation puts it thirty degrees farther in its circular orbit than it actually would be. This is a less serious mistake to make than putting Jupiter, say, eight-tenths as far from the Sun as it would really be.

Volume forms seem, at first, a lot of mechanism for a small problem. And, unfortunately for students, they are. They’re more trouble than they’re worth for changing Cartesian to polar coordinates, or similar problems. You know, ones that the student already has some feel for. They pay off on more abstract problems. Tracking the movement of a dozen interacting things, say, or describing a space that’s very strangely shaped. Those make the effort to learn about forms worthwhile.

The Summer 2017 Mathematics A To Z: Ricci Tensor

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

Art courtesy of Thomas K Dye, creator of the web comic Newshounds. He has a Patreon for those able to support his work. He’s also open for commissions, starting from US$10.

Ricci Tensor.

Imagine if nothing existed.

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

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

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

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

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

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

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

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

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

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

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

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

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

The Summer 2017 Mathematics A To Z: Morse Theory

Today’s A To Z entry is a change of pace. It dives deeper into analysis than this round has been. The term comes from Mr Wu, of the Singapore Maths Tuition blog, whom I thank for the request.

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.

Morse Theory.

An old joke, as most of my academia-related ones are. The young scholar says to his teacher how amazing it was in the old days, when people were foolish, and thought the Sun and the Stars moved around the Earth. How fortunate we are to know better. The elder says, ah yes, but what would it look like if it were the other way around?

There are many things to ponder packed into that joke. For one, the elder scholar’s awareness that our ancestors were no less smart or perceptive or clever than we are. For another, the awareness that there is a problem. We want to know about the universe. But we can only know what we perceive now, where we are at this moment. Even a note we’ve written in the past, or a message from a trusted friend, we can’t take uncritically. What we know is that we perceive this information in this way, now. When we pay attention to our friends in the philosophy department we learn that knowledge is even harder than we imagine. But I’ll stop there. The problem is hard enough already.

We can put it in a mathematical form, one that seems immune to many of the worst problems of knowledge. In this form it looks something like this: if what can we know about the universe, if all we really know is what things in that universe are doing near us? The things that we look at are functions. The universe we’re hoping to understand is the domain of the functions. One filter we use to see the universe is Morse Theory.

We don’t look at every possible function. Functions are too varied and weird for that. We look at functions whose range is the real numbers. And they must be smooth. This is a term of art. It means the function has derivatives. It has to be continuous. It can’t have sharp corners. And it has to have lots of derivatives. The first derivative of a smooth function has to also be continuous, and has to also lack corners. And the derivative of that first derivative has to be continuous, and to lack corners. And the derivative of that derivative has to be the same. A smooth function can can differentiate over and over again, infinitely many times. None of those derivatives can have corners or jumps or missing patches or anything. This is what makes it smooth.

Most functions are not smooth, in much the same way most shapes are not circles. That’s all right. There are many smooth functions anyway, and they describe things we find interesting. Or we think they’re interesting, anyway. Smooth functions are easy for us to work with, and to know things about. There’s plenty of smooth functions. If you’re interested in something else there’s probably a smooth function that’s close enough for practical use.

Morse Theory builds on the “critical points” of these smooth functions. A critical point, in this context, is one where the derivative is zero. Derivatives being zero usually signal something interesting going on. Often they show where the function changes behavior. In freshman calculus they signal where a function changes from increasing to decreasing, so the critical point is a maximum. In physics they show where a moving body no longer has an acceleration, so the critical point is an equilibrium. Or where a system changes from one kind of behavior to another. And here — well, many things can happen.

So take a smooth function. And take a critical point that it’s got. (And, erg. Technical point. The derivative of your smooth function, at that critical point, shouldn’t be having its own critical point going on at the same spot. That makes stuff more complicated.) It’s possible to approximate your smooth function near that critical point with, of course, a polynomial. It’s always polynomials. The shape of these polynomials gives you an index for these points. And that can tell you something about the shape of the domain you’re on.

At least, it tells you something about what the shape is where you are. The universal model for this — based on skimming texts and papers and popularizations of this — is of a torus standing vertically. Like a doughnut that hasn’t tipped over, or like a tire on a car that’s working as normal. I suspect this is the best shape to use for teaching, as anyone can understand it while it still shows the different behaviors. I won’t resist.

Imagine slicing this tire horizontally. Slice it close to the bottom, below the central hole, and the part that drops down is a disc. At least, it could be flattened out tolerably well to a disc.

Slice it somewhere that intersects the hole, though, and you have a different shape. You can’t squash that down to a disc. You have a noodle shape. A cylinder at least. That’s different from what you got the first slice.

Slice the tire somewhere higher. Somewhere above the central hole, and you have … well, it’s still a tire. It’s got a hole in it, but you could imagine patching it and driving on. There’s another different shape that we’ve gotten from this.

Imagine we were confined to the surface of the tire, but did not know what surface it was. That we start at the lowest point on the tire and ascend it. From the way the smooth functions around us change we can tell how the surface we’re on has changed. We can see its change from “basically a disc” to “basically a noodle” to “basically a doughnut”. We could work out what the surface we’re on has to be, thanks to how these smooth functions around us change behavior.

Occasionally we mathematical-physics types want to act as though we’re not afraid of our friends in the philosophy department. So we deploy the second thing we know about Immanuel Kant. He observed that knowing the force of gravity falls off as the square of the distance between two things implies that the things should exist in a three-dimensional space. (Source: I dunno, I never read his paper or book or whatever and dunno I ever heard anyone say they did.) It’s a good observation. Geometry tells us what physics can happen, but what physics does happen tells us what geometry they happen in. And it tells the philosophy department that we’ve heard of Immanuel Kant. This impresses them greatly, we tell ourselves.

Morse Theory is a manifestation of how observable physics teaches us the geometry they happen on. And in an urgent way, too. Some of Edward Witten’s pioneering work in superstring theory was in bringing Morse Theory to quantum field theory. He showed a set of problems called the Morse Inequalities gave us insight into supersymmetric quantum mechanics. The link between physics and doughnut-shapes may seem vague. This is because you’re not remembering that mathematical physics sees “stuff happening” as curves drawn on shapes which represent the kind of problem you’re interested in. Learning what the shapes representing the problem look like is solving the problem.

If you’re interested in the substance of this, the universally-agreed reference is J Milnor’s 1963 text Morse Theory. I confess it’s hard going to read, because it’s a symbols-heavy textbook written before the existence of LaTeX. Each page reminds one why typesetters used to get hazard pay, and not enough of it.

Reading the Comics, August 9, 2017: Pets Doing Mathematics Edition

I had just enough comic strips to split this week’s mathematics comics review into two pieces. I like that. It feels so much to me like I have better readership when I have many days in a row with posting something, however slight. The A to Z is good for three days a week, and if comic strips can fill two of those other days then I get to enjoy a lot of regular publication days. … Though last week I accidentally set the Sunday comics post to appear on Monday, just before the A To Z post. I’m curious how that affected my readers. That nobody said anything is ominous.

Niklas Eriksson’s Carpe Diem for the 7th of August, 2017. I have to agree the border collies haven’t worked out the point of fetch. I also question whether they’ve worked out the simple ballistics of the tossed stick. If the variables mean what they suggest they mean, then dimensional analysis suggests they’ve got at least three fiascos going on here. Maybe they have an idiosyncratic use for variables like ‘v’.

Niklas Eriksson’s Carpe Diem for the 7th of August uses mathematics as the signifier for intelligence. I’m intrigued by how the joke goes a little different: while the border collies can work out the mechanics of a tossed stick, they haven’t figured out what the point of fetch is. But working out people’s motivations gets into realms of psychology and sociology and economics. There the mathematics might not be harder, but knowing that one is calculating a relevant thing is. (Eriksson’s making a running theme of the intelligence of border collies.)

Nicole Hollander’s Sylvia rerun for the 7th tosses off a mention that “we’re the first generation of girls who do math”. And that therefore there will be a cornucopia of new opportunities and good things to come to them. There’s a bunch of social commentary in there. One is the assumption that mathematics skill is a liberating thing. Perhaps it is the gloom of the times but I doubt that an oppressed group developing skills causes them to be esteemed. It seems more likely to me to make the skills become devalued. Social justice isn’t a matter of good exam grades.

Then, too, it’s not as though women haven’t done mathematics since forever. Every mathematics department on a college campus has some faded posters about Emmy Noether and Sofia Kovalevskaya and maybe Sophie Germaine. Probably high school mathematics rooms too. Again perhaps it’s the gloom of the times. But I keep coming back to the goddess’s cynical dismissal of all this young hope.

Mort Walker and Dik Browne’s Hi and Lois for the 10th of February, 1960 and rerun the 8th portrays arithmetic as a grand-strategic imperative. Well, it means education as a strategic imperative. But arithmetic is the thing Dot uses. I imagine because it is so easy to teach as a series of trivia and quiz about. And it fits in a single panel with room to spare.

Mort Walker and Dik Browne’s Hi and Lois for the 10th of February, 1960 and rerun the 8th of August, 2017. Remember: you’re only young once, but you can be geopolitically naive forever!

Paul Trap’s Thatababy for the 8th is not quite the anthropomorphic-numerals joke of the week. It circles around that territory, though, giving a couple of odd numbers some personality.

Brian Anderson’s Dog Eat Doug for the 9th finally justifies my title for this essay, as cats ponder mathematics. Well, they ponder quantum mechanics. But it’s nearly impossible to have a serious thought about that without pondering its mathematics. This doesn’t mean calculation, mind you. It does mean understanding what kinds of functions have physical importance. And what kinds of things one can do to functions. Understand them and you can discuss quantum mechanics without being mathematically stupid. And there’s enough ways to be stupid about quantum mechanics that any you can cut down is progress.

Reading the Comics, July 30, 2017: Not Really Mathematics edition

It’s been a busy enough week at Comic Strip Master Command that I’ll need to split the results across two essays. Any other week I’d be glad for this, since, hey, free content. But this week it hits a busy time and shouldn’t I have expected that? The odd thing is that the mathematics mentions have been numerous but not exactly deep. So let’s watch as I make something big out of that.

Mark Tatulli’s Heart of the City closed out its “Math Camp” storyline this week. It didn’t end up having much to do with mathematics and was instead about trust and personal responsibility issues. You know, like stories about kids who aren’t learning to believe in themselves and follow their dreams usually are. Since we never saw any real Math Camp activities we don’t get any idea what they were trying to do to interest kids in mathematics, which is a bit of a shame. My guess would be they’d play a lot of the logic-driven puzzles that are fun but that they never get to do in class. The story established that what I thought was an amusement park was instead a fair, so, that might be anywhere Pennsylvania or a couple of other nearby states.

Rick Kirkman and Jerry Scott’s Baby Blues for the 25th sees Hammie have “another” mathematics worksheet accident. Could be any subject, really, but I suppose it would naturally be the one that hey wait a minute, why is he doing mathematics worksheets in late July? How early does their school district come back from summer vacation, anyway?

Rick Kirkman and Jerry Scott’s Baby Blues for the 25th of July, 2017 Almost as alarming: Hammie is clearly way behind on his “faking plausible excuses” homework. If he doesn’t develop the skills to make a credible reason why he didn’t do something how is he ever going to dodge texts from people too important not to reply to?

Olivia Walch’s Imogen Quest for the 26th uses a spot of mathematics as the emblem for teaching. In this case it’s a bit of physics. And an important bit of physics, too: it’s the time-dependent Schrödinger Equation. This is the one that describes how, if you know the total energy of the system, and the rules that set its potential and kinetic energies, you can work out the function Ψ that describes it. Ψ is a function, and it’s a powerful one. It contains probability distributions: how likely whatever it is you’re modeling is to have a particle in this region, or in that region. How likely it is to have a particle with this much momentum, versus that much momentum. And so on. Each of these we find by applying a function to the function Ψ. It’s heady stuff, and amazing stuff to me. Ψ somehow contains everything we’d like to know. And different functions work like filters that make clear one aspect of that.

Dan Thompson’s Brevity for the 26th is a joke about Sesame Street‘s Count von Count. Also about how we can take people’s natural aptitudes and delights and turn them into sad, droning unpleasantness in the service of corporate overlords. It’s fun.

Steve Sicula’s Home and Away rerun for the 26th is a misplaced Pi Day joke. It originally ran the 22nd of April, but in 2010, before Pi Day was nearly so much a thing.

Doug Savage’s Savage Chickens for the 26th proves something “scientific” by putting numbers into it. Particularly, by putting statistics into it. Understandable impulse. One of the great trends of the past century has been taking the idea that we only understand things when they are measured. And this implies statistics. Everything is unique. Only statistical measurement lets us understand what groups of similar things are like. Does something work better than the alternative? We have to run tests, and see how the something and the alternative work. Are they so similar that the differences between them could plausibly be chance alone? Are they so different that it strains belief that they’re equally effective? It’s one of science’s tools. It’s not everything which makes for science. But it is stuff easy to communicate in one panel.

Neil Kohney’s The Other End for the 26th is really a finance joke. It’s about the ways the finance industry can turn one thing into a dazzling series of trades and derivative trades. But this is a field that mathematics colonized, or that colonized mathematics, over the past generation. Mathematical finance has done a lot to shape ideas of how we might study risk, and probability, and how we might form strategies to use that risk. It’s also done a lot to shape finance. Pretty much any major financial crisis you’ve encountered since about 1990 has been driven by a brilliant new mathematical concept meant to govern risk crashing up against the fact that humans don’t behave the way some model said they should. Nor could they; models are simplified, abstracted concepts that let hard problems be approximated. Every model has its points of failure. Hopefully we’ll learn enough about them that major financial crises can become as rare as, for example, major bridge collapses or major airplane disasters.