## 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 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 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 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.

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.

## Reading the Comics, October 30, 2018: I Spot An Error Edition

The edition title says it all. Comic Strip Master Command sent me enough strips the past week for two editions and I made an unhappy discovery about one of the comics in today’s.

Dave Coverly’s Speed Bump for the 28th is your anthropomorphic-numerals joke for the week. We get to know the lowest common denominator from fractions. It’s easier to compute anything with a fraction in it if you can put everything under a common denominator. But it’s also — usually — easier to work with smaller denominators than larger ones. It’s always okay to multiply a number by 1. It may not help, but it can always be done. This has the result of multiplying both the numerator and denominator by the same number. So suppose you have something that’s written in terms of sixths, and something else written in terms of eighths. You can multiply the first thing by four-fourths, and the second thing by three-thirds. Then both fractions are in terms of 24ths and your calculation is, probably, easier.

So this strip is the rare one where I have to say the joke doesn’t work on mathematical grounds. Coverly was mislead by the association between “lowest” and “smallest”. 2 is going to be the lowest common denominator very rarely. Everything in the problem needs to be in terms of even denominators to start with, and even that won’t guarantee it. I hate to do that, since the point of a comic strip is humor and getting any mathematics right is a bonus. But in this case, knowing the terminology shatters the joke. Coverly would have a mathematically valid joke were 9 offering the consolation “you’re not always the greatest common divisor”, the largest number that goes into a set of numbers. But nobody thinks being called the “greatest” anything ever needs consolation, so the joke would fail all but mathematics class.

Randy Glasbergen’s Glasbergen Cartoons for the 29th is a joke of the why-learn-mathematics model. “Because we always have done this” is not a reason compelling by the rules of deductive logic. It can have great practical value. Experience can encode things which are hard to state explicitly, or to untangle from one another. And an experienced system will have workarounds for the most obvious problems, ones that a new system will not have. And any attempt at educational reform, however well-planned or meant, must answer parents’ reasonable question of why their child should be your test case.

I do sometimes see algebra attacked as being too little-useful for the class time given. I could see good cases made for spending the time on other fields of mathematics. (Probability and statistics always stands out as potentially useful; the subjects were born from things people urgently needed to know.) I’m not competent to judge those arguments and so shall not.

Carl Skanberg’s That New Carl Smell for the 29th is a riff on jokes about giving more than 100%. Interpreting this giving-more-than-everything as running a deficit is a reasonable one. I’ve given my usual talk about “100% of what?” enough times now; I don’t need to repeat it until I think of something fresh to say.

Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney for the 30th uses mathematics — story problems, specifically — as icons of intelligence. I can’t speak to the Mensa experience, but intellectual types trying to out-do each other? Yes, that’s a thing that happens. I mostly dodge attempts to put me to a fun mathematics puzzle. I’m embarrassed by how long it can take me to actually do one of these, when put on the spot. (I have a similar reaction to people testing my knowledge of trivia in the stuff I actually do know a ridiculous amount about.) Mostly I hope Dave Coverly doesn’t think I’m being this kid.

All of the Reading the Comics posts should be at this link. Essays that include Speed Bump are at this link. I don’t usually have a problem with it. Essays discussing Glasbergen Cartoons should be at this link. They won’t include Glasbergen’s longrunning The Better Half comic, which as far as I can find only ever appeared here the one time anyway. It’s a new tag anyway. Essays with a mention of That New Carl Smell are at this link. It’s a new tag, though, so give it some time if you want to read anything else. Essays with a mention of Mustard and Boloney are at this link. And my Fall 2018 Mathematics A-To-Z should continue for the rest of this calendar year. And it is open for requests for more of the alphabet. Thanks for reading.

## Reading the Comics, June 13, 2018: Wild Squirrel Edition

I have another Reading the Comics post with a title that’s got nothing to do with the post. It has got something to do with how I spent my weekend. Not sure if I’ll ever get around to explaining that since there’s not much mathematical content to that weekend. I’m not sure whether the nonsense titles are any better than trying to find a theme in what Comic Strip Master Command has sent the past week. It takes time to pick something when anything would do, after all.

Scott Hilburn’s The Argyle Sweater for the 10th is the anthropomorphic numerals strip for the week. Also arithmetic symbols. The &div; sign is known as “the division symbol”, although now and then people try to promote it as the “obelus”. They’re not wrong to call it that, although they are being the kind of person who tries to call the # sign the “octothorp”. Sometimes social media pass around the false discovery that the &div; sign is a representation of a fraction, $\frac{a}{b}$, with the numbers replaced by dots. It’s a good mnemonic for linking fractions and division. But it’s wrong to say that’s what the symbol means. &div; started being used for division in Western Europe in the mid-17th century, in competition with many symbols, including / (still in common use), : (used in talking about ratios or odds), – (not used in this context anymore, and just confusing if you do try to use it so). And &div; was used in northern Europe to mean “subtraction” for several centuries after this.

Tom Toles’s Randolph Itch, 2am for the 11th is a repeat; the too-short-lived strip has run through several cycles since I started doing these summaries. But it is also one of the great pie chart jokes ever and I have no intention of not telling people to enjoy it.

Pie charts, and the also-mentioned bar charts, come to us originally from the economist William Playfair, who in the late 1700s and early 1800s devised nearly all the good ways to visualize data. But we know them thanks to Florence Nightingale. Among her other works, she recognized in these charts good ways to represent her studies about Crimean War medicine and about sanitation in India. Nightingale was in 1859 named the first woman in the Royal Statistical Society, and was named an honorary member of the American Statistical Association in 1874.

Olivia Jaimes’s Nancy for the 12th uses arithmetic as iconic for classwork nobody wants to do. Algebra, too; I understand the reluctance to start. Simultaneous solutions; the challenge is to find sets of values ‘x’ and ‘y’ that make both equations true together. That second equation is a good break, though. $16 x^2 - 4y^2 = 0$ makes it easy to write what ‘y’ has to be in terms of ‘x’. Then you can replace the ‘y’ in the first equation with its expression in terms of ‘x’. In a slightly tedious moment, it’s going to turn out there’s multiple sets of answers. Four sets, if I haven’t missed something. But they’ll be clearly related to each other. Even attractively arranged.

$x^2 + y^2 = 3$ is an equation that’s true if the numbers ‘x’ and ‘y’ are coordinates of the points on a circle. This is if the coordinates are using the Cartesian coordinate system for the plane, which is such a common thing to do that mathematicians can forget they’re doing that. The circle has radius $\sqrt{3}$. So you can look at the first equation and draw a circle and write down a note that its radius is $\sqrt{3}$ and you’ve got it. $16x^2 - 4y^2 = 0$ looks like an equation that’s true if the numbers ‘x’ and ‘y’ are coordinates of the points on a hyperbola. Again in the Cartesian coordinate system. But I have to feel a little uncomfortable saying this. If the equation were (say) $16x^2 - 4y^2 = 1$ then it’d certainly be a hyperbola, which mostly looks like a mirror-symmetric pair of arcs. But equalling zero? That’s called a “degenerate hyperbola”, which makes it sound like the hyperbola is doing something wrong. Unfortunate word, but one we’re stuck with.

The description just reflects that the hyperbola is boring in some way. In this case, it’s boring because the ‘x’ and ‘y’ that make the equation true are just the points on a pair of straight lines that go through the origin, the point with coordinates (0, 0). And they’re going to be mirror-images of each other around the x- and the y-axis. So it seems like a waste to use the form of a hyperbola when we could do just as well using the forms of straight lines to describe the same points. This hyperbola will look like an X, although it might be a pretty squat ‘x’ or a pretty narrow one or something. Depends on the exact equation.

So. The solutions for ‘x’ and ‘y’ are going to be on the points that are on both a circle centered around the origin and on an X centered around the origin. This is a way to see why I would expect four solutions. Also they they would look about the same. There’d be an answer with positive ‘x’ and positive ‘y’, and then three more answers. One answer has ‘x’ with the same size but a minus sign. One answer has ‘y’ with the same size but a minus sign. One has both ‘x’ and ‘y’ with the same values but minus signs.

Sorry I wasn’t there to partner with.

Dave Coverly’s Speed Bump for the 12th is a Rubik’s Cube joke. Here it merges the idea with the struggles of scheduling anything anymore. I’m not sure that the group-theory operations of lining up a Rubik’s cube can be reinterpreted as the optimization problems of scheduling stuff. But there are all sorts of astounding and surprising links between mathematical problems. So I wouldn’t rule it out.

John Allen’s Nest Heads for the 13th is a lotteries joke. I’m less dogmatic than are many mathematicians about the logic of participating in a lottery. At least in the ones as run by states and regional authorities the chance of a major payout are, yes, millions to one against. There can be jackpots large enough that the expectation value of playing becomes positive. In this case the reward for that unlikely outcome is so vast that it covers the hundreds of millions of times you play and lose. But even then, you have the question of whether doing something that just won’t pay out is worth it. My taste is to say that I shall do much more foolish things with my disposable income than buying a couple tickets each year. And while I would like to win the half-billion-dollar jackpot that would resolve all my financial woes and allow me to crush those who had me imprisoned in the Château d’If, I’d also be coming out ahead if I won, like, one of the petty $10,000 prizes. ## Reading the Comics, October 21, 2017: Education Week Edition Comic Strip Master Command had a slow week for everyone. This is odd since I’d expect six to eight weeks ago, when the comics were (probably) on deadline, most (United States) school districts were just getting back to work. So education-related mathematics topics should’ve seemed fresh. I think I can make that fit. No way can I split this pile of comics over two days. Hector D Cantu and Carlos Castellanos’s Baldo for the 17th has Gracie quizzed about percentages of small prices, apparently as a test of her arithmetic. Her aunt has other ideas in mind. It’s hard to dispute that this is mathematics people use in real life. The commenters on GoComics got into an argument about whether Gracie gave the right answers, though. That is, not that 20 percent of$5.95 is anything about $1.19. But did Tia Carmen want to know what 20 percent of$5.95, or did she want to know what $5.95 minus 20 percent of that price was? Should Gracie have answered$4.76 instead? It took me a bit to understand what the ambiguity was, but now that I see it, I’m glad I didn’t write a multiple-choice test with both $1.19 and$4.76 as answers. I’m not sure how to word the questions to avoid ambiguity yet still sound like something one of the hew-mons might say.

Dan Thompson’s Brevity for the 19th uses the blackboard and symbols on it as how a mathematician would prove something. In this case, love. Arithmetic’s a good visual way of communicating the mathematician at work here. I don’t think a mathematician would try arguing this in arithmetic, though. I mean if we take the premise at face value. I’d expect an argument in statistics, so, a mathematician showing various measures of … feelings or something. And tests to see whether it’s plausible this cluster of readings could come out by some reason other than love. If that weren’t used, I’d expect an argument in propositional logic. And that would have long strings of symbols at work, but they wouldn’t look like arithmetic. They look more like Ancient High Martian. Just saying.

Reza Farazmand’s Poorly Drawn Lines for the 20th you maybe already saw going around your social media. It’s well-designed for that. Also for grad students’ office doors.

Dave Coverly’s Speed Bump for the 20th is designed with crossover appeal in mind and I wonder if whoever does Reading the Comics for English Teacher Jokes is running this same strip in their collection for the week.

Darrin Bell’s Candorville for the 21st sees Lemont worry that he’s forgotten how to do long division. And, fair enough: any skill you don’t use in long enough becomes stale, whether it’s division or not. You have to keep in practice and, in time, have to decide what you want to keep in practice about. (That said, I have a minor phobia about forgetting how to prove the Contraction Mapping Theorem, as several professors in grad school stressed how it must always be possible to give a coherent proof of that, even if you’re startled awake in the middle of the night by your professor.) Me, I would begin by estimating what 4,858.8 divided by 297.492 should be. 297.492 is very near 300. And 4,858.8 is a little over 4800. And that’s suggestive because it’s obvious that 48 divided by 3 is 16. Well, it’s obvious to me. So I would expect the answer to be “a little more than 16” and, indeed, it’s about 16.3.

(Don’t read the comments on GoComics. There’s some slide-rule-snobbishness, and some snark about the uselessness of the skill or the dumbness of Facebook readers, and one comment about too many people knowing how to multiply by someone who’s reading bad population-bomb science fiction of the 70s.)

## Reading the Comics, June 24, 2017: Saturday Morning Breakfast Cereal Edition

Somehow this is not the title of every Reading The Comics review! But it is for this post and we’ll explore why below.

Dave Coverly’s Speed Bump for the 18th is not exactly an anthropomorphic-numerals joke. It is about making symbols manifest in the real world, at least. The greater-than and less-than signs as we know them were created by the English mathematician Thomas Harriot, and introduced to the world in his posthumous Artis Analyticae Praxis (1631). He also had an idea of putting a . between the numerals of an expression and the letters multiplied by them, for example, “4.x” to mean four times x. We mostly do without that now, taking multiplication as assumed if two meaningful quantities are put next to one another. But we will use, now, a vertically-centered dot to separate terms multiplied together when that helps our organization. The equals sign we trace to the 16th century mathematician Robert Recorde, whose 1557 Whetsone of Witte uses long but recognizable equals signs. The = sign went into hibernation after that, though, until the 17th century and it took some time to quite get well-used. So it often is with symbols.

Ted Shearer’s Quincy for the 25th of April, 1978 and rerun the 19th of June, starts from the history of zero. It’s worth noting there are a couple of threads woven together in the concept of zero. One is the idea of “nothing”, which we’ve had just forever. I mean, the idea that there isn’t something to work with. Another is the idea of the … well, the additive identity, there being some number that’s one less than one and two less than two. That you can add to anything without changing the thing. And then there’s symbols. There’s the placeholder for “there are no examples of this quantity here”. There’s the denotation of … well, the additive identity. All these things are zeroes, and if you listen closely, they are not quite the same thing. Which is not weird. Most words mean a collection of several concepts. We’re lucky the concepts we mean by “zero” are so compatible in meaning. Think of the poor person trying to understand the word “bear”, or “cleave”.

John Deering’s Strange Brew for the 19th is a “New Math” joke, fittingly done with cavemen. Well, numerals were new things once. Amusing to me is that — while I’m not an expert — in quite a few cultures the symbol for “one” was pretty much the same thing, a single slash mark. It’s hard not to suppose that numbers started out with simple tallies, and the first thing to tally might get dressed up a bit with serifs or such but is, at heart, the same thing you’d get jabbing a sharp thing into a soft rock.

Guy Gilchrist’s Today’s Dogg for the 19th I’m sure is a rerun and I think I’ve featured it here before. So be it. It’s silly symbol-play and dog arithmetic. It’s a comic strip about how dogs are cute; embrace it or skip it.

Zach Weinersmith’s Saturday Morning Breakfast Cereal is properly speaking reruns when it appears on GoComics.com. For whatever reason Weinersmith ran a patch of mathematics strips there this past week. So let me bundle all that up. On the 19th he did a joke mathematicians get a lot, about how the only small talk anyone has about mathematics is how they hated mathematics. I’m not sure mathematicians have it any better than any other teachers, though. Have you ever known someone to say, “My high school gym class gave me a greater appreciation of the world”? Or talk about how grade school history opened their eyes to the wonders of the subject? It’s a sad thing. But there are a lot of things keeping teachers from making students feel joy in their subjects.

For the 21st Weinersmith makes a statisticians joke. I can wrangle some actual mathematics out of an otherwise correctly-formed joke. How do we ever know that something is true? Well, we gather evidence. But how do we know the evidence is relevant? Even if the evidence is relevant, how do we know we’ve interpreted it correctly? Even if we have interpreted it correctly, how do we know that it shows what we want to know? Statisticians become very familiar with hypothesis testing, which amounts to the question, “does this evidence indicate that some condition is implausibly unlikely”? And they can do great work with that. But “implausibly unlikely” is not the same thing as “false”. A person knowledgeable enough and honest turns out to have few things that can be said for certain.

The June 23rd strip I’ve seen go around Mathematics Twitter several times, as see above tweet, about the ways in which mathematical literacy would destroy modern society. It’s a cute and flattering portrait of mathematics’ power, probably why mathematicians like passing it back and forth. But … well, how would “logic” keep people from being fooled by scams? What makes a scam work is that the premise seems logical. And real-world problems — as opposed to logic-class problems — are rarely completely resolvable by deductive logic. There have to be the assumptions, the logical gaps, and the room for humbuggery that allow hoaxes and scams to slip through. And does anyone need a logic class to not “buy products that do nothing”? And what is “nothing”? I have more keychains than I have keys to chain, even if we allow for emergencies and reasonable unexpected extra needs. This doesn’t stop my buying keychains as souvenirs. Does a Penn Central-logo keychain “do nothing” merely because it sits on the windowsill rather than hold any sort of key? If so, was my love foolish to buy it as a present? Granted that buying a lottery ticket is a foolish use of money; is my life any worse for buying that than, say, a peanut butter cup that I won’t remember having eaten a week afterwards? As for credit cards — It’s not clear to me that people max out their credit cards because they don’t understand they will have to pay it back with interest. My experience has been people max out their credit cards because they have things they must pay for and no alternative but going further into debt. That people need more money is a problem of society, yes, but it’s not clear to me that a failure to understand differential equations is at the heart of it. (Also, really, differential equations are overkill to understand credit card debt. A calculator with a repeat-the-last-operation feature and ten minutes to play is enough.)

## Late April, Early May Math Comics

I’ve got enough new mathematics-themed comic strips to assemble them into a fresh post. It’s a challenge to time these rightly; I don’t want to waste everyone’s time with a set weekly post, particularly since the syndicated comics might just not have anything. On the other hand, waiting until a set number of strips have passed before my eyes seems likely to just encourage me to wonder how marginally a strip can touch mathematics before I include it. Dave Coverly’s Speed Bump, from the 6th of May, is a fine marginal case: there’s a mathematics problem in it, but it’s not at all a mathematics strip. It’s just very easy to put a math problem on the chalkboard and have it be understood the scenario is “student with no idea how to answer”.