## Reading the Comics, December 19, 2018: Andertoons Is Back Edition

I had not wanted to mention, for fear of setting off a panic. But Mark Anderson’s Andertoons, which I think of as being in every Reading the Comics post, hasn’t been around lately. If I’m not missing something, it hasn’t made an appearance in three months now. I don’t know why, and I’ve been trying not to look too worried by it. Mostly I’ve been forgetting to mention the strange absence. This even though I would think any given Tuesday or Friday that I should talk about the strip not having anything for me to write about. Fretting about it would make a great running theme. But I have never spotted a running theme before it’s finished. In any event the good news is that the long drought has ended, and Andertoons reappears this week. Yes, I’m hoping that it won’t be going to long between appearances this time.

Jef Mallett’s Frazz for the 16th talks about probabilities. This in the context of assessing risks. People are really bad at estimating probabilities. We’re notoriously worse at assessing risks, especially when it’s a matter of balancing a present cost like “fifteen minutes waiting while the pharmacy figures out whether insurance will pay for the flu shot” versus a nebulous benefit like “lessened chance of getting influenza, or at least having a less severe influenza”. And it’s asymmetric, too. We view improbable but potentially enormous losses differently from the way we view improbable but potentially enormous gains. And it’s hard to make the rationally-correct choice reliably, not when there are so many choices of this kind every day.

Tak Bui’s PC and Pixel for the 16th features a wall full of mathematical symbols, used to represent deep thought about a topic. The symbols are gibberish, yes. I’m not sure that an actual “escape probability” could be done in a legible way, though. Or even what precisely Professor Phillip might be calculating. I imagine it would be an estimate of the various ways he might try to escape, and what things might affect that. This might be for the purpose of figuring out what he might do to maximize his chances of a successful escape. Although I wouldn’t put it past the professor to just be quite curious what the odds are. There’s a thrill in having a problem solved, even if you don’t use the answer for anything.

Ruben Bolling’s Super-Fun-Pak Comix for the 18th has a trivia-panel-spoof dubbed Amazing Yet Tautological. One could make an argument that most mathematics trivia fits into this category. At least anything about something that’s been proven. Anyway, whether this is a tautological strip depends on what the strip means by “average” in the phrase “average serving”. There’s about four jillion things dubbed “average” and each of them has a context in which they make sense. The thing intended here, and the thing meant if nobody says anything otherwise, is the “arithmetic mean”. That’s what you get from adding up everything in a sample (here, the amount of egg salad each person in America eats per year) and dividing it by the size of the sample (the number of people in America that year). Another “average” which would make sense, but would break this strip, would be the median. That would be the amount of egg salad that half of all Americans eat more than, and half eat less than. But whether every American could have that big a serving really depends on what that median is. The “mode”, the most common serving, would also be a reasonable “average” to expect someone to talk about.

Mark Anderson’s Andertoons for the 19th is that strip’s much-awaited return to my column here. It features solid geometry, which is both an important part of geometry and also a part that doesn’t get nearly as much attention as plane geometry. It’s reductive to suppose the problem is that it’s harder to draw solids than planar figures. I suspect that’s a fair part of the problem, though. Mathematicians don’t get much art training, not anymore. And while geometry is supposed to be able to rely on pure reasoning, a good picture still helps. And a bad picture will lead us into trouble.

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, December 3, 2016: Cute Little Jokes Edition

Comic Strip Master Command apparently wanted me to have a bunch of easy little pieces that don’t inspire rambling essays. Message received!

Mark Litzler’s Joe Vanilla for the 27th is a wordplay joke in which any mathematical content is incidental. It could be anything put in a positive light; numbers are just easy things to arrange so. From the prominent appearance of ‘3’ and ‘4’ I supposed Litzler was using the digits of π, but if he is, it’s from some part of π that I don’t recognize. (That would be any part after the seventeenth digit. I’m not obsessive about π digits.)

Samson’s Dark Side Of The Horse is whatever the equivalent of punning is for Roman Numerals. I like Horace blushing.

John Deering’s Strange Brew for the 28th is a paint-by-numbers joke, and one I don’t see done often. And there is beauty in the appearance of mathematics. It’s not appreciated enough. I think looking at the tables of integral formulas on the inside back cover of a calculus book should prove the point, though. All those rows of integral signs and sprawls of symbols after show this abstract beauty. I’ve surely mentioned the time when the creative-arts editor for my undergraduate leftist weekly paper asked for a page of mathematics or physics work to include as an picture, too. I used the problem that inspired my “Why Stuff Can Orbit” sequence over on my mathematics blog. The editor loved the look of it all, even if he didn’t know what most of it meant.

Niklas Eriksson’s Carpe Diem for the 29th is a joke about life, I suppose. It uses a sprawled blackboard full of symbols to play the part of the proof. It’s gibberish, of course, although I notice how many mathematics cliches get smooshed into it. There’s a 3.1451 — I assume that’s a garbed digits of π — under a square root sign. There’s an “E = mc”, I suppose a garbled bit of Einstein’s Famous Equation in there. There’s a “cos 360”. 360 evokes the number of degrees in a circle, but mathematicians don’t tend to use degrees. There’s analytic reasons why we find it nicer to use radians, for which the equivalent would be “cos 2π”. If we wrote that at all, since the cosine of 2π is one of the few cosines everyone knows. Every mathematician knows. It’s 1. Well, maybe the work just got to that point and it hasn’t been cleaned up.

Eriksson’s Carpe Diem reappears the 30th, with a few blackboards with less mathematics to suggest someone having a creative block. It does happen to us all. My experience is mathematicians don’t tend to say “Eureka” when we do get a good idea, though. It’s more often some vague mutterings and “well what if” while we form the idea. And then giggling or even laughing once we’re sure we’ve got something. This may be just me and my friends. But it is a real rush when we have it.

Dan Collins’s Looks Good On Paper for the 29t tells the Möbius strip joke. It’s a well-rendered one, though; I like that there is a readable strip in there and that it’s distorted to fit the geometry.

Henry Scarpelli and Craig Boldman’s Archie rerun for the 2nd of December tosses off the old gag about not needing mathematics now that we have calculators. It’s not a strip about that, and that’s fine.

Mark Anderson’s Andertoons finally appeared the 2nd. It’s a resistant-student joke. And a bit of wordplay.

Ruben Bolling’s Super-Fun-Pak Comix from the 2nd featured an installment of Tautological But True. One might point out they’re using “average” here to mean “arithmetic mean”. There probably isn’t enough egg salad consumed to let everyone have a median-sized serving. And I wouldn’t make any guesses about the geometric mean serving. But the default meaning of “average” is the arithmetic mean. Anyone using one of the other averages would say so ahead of time or else is trying to pull something.

## The End 2016 Mathematics A To Z: Ergodic

This essay follows up on distributions, mentioned back on Wednesday. This is only one of the ideas which distributions serve. Do you have a word you’d like to request? I figure to close ‘F’ on Saturday afternoon, and ‘G’ is already taken. But give me a request for a free letter soon and I may be able to work it in.

## Ergodic.

There comes a time a physics major, or a mathematics major paying attention to one of the field’s best non-finance customers, first works on a statistical mechanics problem. Instead of keeping track of the positions and momentums of one or two or four particles she’s given the task of tracking millions of particles. It’s listed as a distribution of all the possible values they can have. But she still knows what it really is. And she looks at how to describe the way this distribution changes in time. If she’s the slightest bit like me, or anyone I knew, she freezes up this. Calculate the development of millions of particles? Impossible! She tries working out what happens to just one, instead, and hopes that gives some useful results.

And then it does.

It’s a bit much to call this luck. But it is because the student starts off with some simple problems. Particles of gas in a strong box, typically. They don’t interact chemically. Maybe they bounce off each other, but she’s never asked about that. She’s asked about how they bounce off the walls. She can find the relationship between the volume of the box and the internal gas pressure on the interior and the temperature of the gas. And it comes out right.

She goes on to some other problems and it suddenly fails. Eventually she re-reads the descriptions of how to do this sort of problem. And she does them again and again and it doesn’t feel useful. With luck there’s a moment, possibly while showering, that the universe suddenly changes. And the next time the problem works out. She’s working on distributions instead of toy little single-particle problems.

But the problem remains: why did it ever work, even for that toy little problem?

It’s because some systems of things are ergodic. It’s a property that some physics (or mathematics) problems have. Not all. It’s a bit hard to describe clearly. Part of what motivated me to take this topic is that I want to see if I can explain it clearly.

Every part of some system has a set of possible values it might have. A particle of gas can be in any spot inside the box holding it. A person could be in any of the buildings of her city. A pool ball could be travelling in any direction on the pool table. Sometimes that will change. Gas particles move. People go to the store. Pool balls bounce off the edges of the table.

These values will have some kind of distribution. Look at where the gas particle is now. And a second from now. And a second after that. And so on, to the limits of human knowledge. Or to when the box breaks open. Maybe the particle will be more often in some areas than in others. Maybe it won’t. Doesn’t matter. It has some distribution. Over time we can say how often we expect to find the gas particle in each of its possible places.

The same with whatever our system is. People in buildings. Balls on pool tables. Whatever.

Now instead of looking at one particle (person, ball, whatever) we have a lot of them. Millions of particle in the box. Tens of thousands of people in the city. A pool table that somehow supports ten thousand balls. Imagine they’re all settled to wherever they happen to be.

So where are they? The gas particle one is easy to imagine. At least for a mathematics major. If you’re stuck on it I’m sorry. I didn’t know. I’ve thought about boxes full of gas particles for decades now and it’s hard to remember that isn’t normal. Let me know if you’re stuck, and where you are. I’d like to know where the conceptual traps are.

But back to the gas particles in a box. Some fraction of them are in each possible place in the box. There’s a distribution here of how likely you are to find a particle in each spot.

How does that distribution, the one you get from lots of particles at once, compare to the first, the one you got from one particle given plenty of time? If they agree the system is ergodic. And that’s why my hypothetical physics major got the right answers from the wrong work. (If you are about to write me to complain I’m leaving out important qualifiers let me say I know. Please pretend those qualifiers are in place. If you don’t see what someone might complain about thank you, but it wouldn’t hurt to think of something I might be leaving out here. Try taking a shower.)

The person in a building is almost certainly not an ergodic system. There’s buildings any one person will never ever go into, however possible it might be. But nearly all buildings have some people who will go into them. The one-person-with-time distribution won’t be the same as the many-people-at-once distribution. Maybe there’s a way to qualify things so that it becomes ergodic. I doubt it.

The pool table, now, that’s trickier to say. For a real pool table no, of course not. An actual ball on an actual table rolls to a stop pretty soon, either from the table felt’s friction or because it drops into a pocket. Tens of thousands of balls would form an immobile heap on the table that would be pretty funny to see, now that I think of it. Well, maybe those are the same. But they’re a pretty boring same.

Anyway when we talk about “pool tables” in this context we don’t mean anything so sordid as something a person could play pool on. We mean something where the table surface hasn’t any friction. That makes the physics easier to model. It also makes the game unplayable, which leaves the mathematical physicist strangely unmoved. In this context anyway. We also mean a pool table that hasn’t got any pockets. This makes the game even more unplayable, but the physics even easier. (It makes it, really, like a gas particle in a box. Only without that difficult third dimension to deal with.)

And that makes it clear. The one ball on a frictionless, pocketless table bouncing around forever maybe we can imagine. A huge number of balls on that frictionless, pocketless table? Possibly trouble. As long as we’re doing imaginary impossible unplayable pool we could pretend the balls don’t collide with each other. Then the distributions of what ways the balls are moving could be equal. If they do bounce off each other, or if they get so numerous they can’t squeeze past one another, well, that’s different.

An ergodic system lets you do this neat, useful trick. You can look at a single example for a long time. Or you can look at a lot of examples at one time. And they’ll agree in their typical behavior. If one is easier to study than the other, good! Use the one that you can work with. Mathematicians like to do this sort of swapping between equivalent problems a lot.

The problem is it’s hard to find ergodic systems. We may have a lot of things that look ergodic, that feel like they should be ergodic. But proved ergodic, with a logic that we can’t shake? That’s harder to do. Often in practice we will include a note up top that we are assuming the system to be ergodic. With that “ergodic hypothesis” in mind we carry on with our work. It gives us a handle on a lot of problems that otherwise would be beyond us.

## Reading the Comics, January 15, 2015: Electric Brains and Klein Bottles Edition

I admit I don’t always find a theme running through Comic Strip Master Command’s latest set of mathematically-themed comics. The edition names are mostly so that I can tell them apart when I see a couple listed in the Popular Posts roundup anyway.

Jimmy Hatlo’s Little Iodine is a vintage comic strip from the 1950s. It strikes me as an unlicensed adaptation of Baby Schnooks, but that’s not something for me to worry about. The particular strip, originally from the 7th of November, 1954 (and just run the 12th of January this year) interests me for its ancient views of computers. It’s from the days they were called “electric brains”. I’m also impressed that the machine on display early on is able to work out the “square root of 7921 x2 y2”. The square root of 7921 is no great feat. Being able to work with the symbols of x and y without knowing what they stand for, though, does impress me. I’m not sure there were computers which could handle that sort of symbolic manipulation in 1954. That sort of ability to work with a quantity by name rather than value is what we would buy Mathematica for, if we could afford it. It’s also at least a bit impressive that someone knows the square of 89 offhand. All told, I think this is my favorite of this essay’s set of strips. But it’s a weak field considering none of them are “students giving a snarky reply to a homework/exam/blackboard question”.

Joe Martin’s Willy and Ethel for the 13th of January is a percentages joke. Some might fault it for talking about people giving 110 percent, but of course, what is “100 percent”? If it’s the standard amount of work being done then it does seem like ten people giving 110 percent gets the job done as quickly as eleven people doing 100 percent. If work worked like that.

Steve Sicula’s Home and Away for the 13th (a rerun from the 8th of October, 2004) gives a wrongheaded application of a decent principle. The principle is that of taking several data points and averaging their value. The problem with data is that it’s often got errors in it. Something weird happened and it doesn’t represent what it’s supposed to. Or it doesn’t represent it well. By averaging several data points together we can minimize the influence of a fluke reading. Or if we’re measuring something that changes in time, we might use a running average of the last several sampled values. In this way a short-term spike or a meaningless flutter will be minimized. We can avoid wasting time reacting to something that doesn’t matter. (The cost of this, though, is that if a trend is developing we will notice it later than we otherwise would.) Still, sometimes a data point is obviously wrong.

Zach Weinersmith’s Saturday Morning Breakfast Cereal wanted my attention, and so on the 13th it did a joke about Zeno’s Paradox. There are actually four classic Zeno’s Paradoxes, although the one riffed on here I think is the most popular. This one — the idea that you can’t finish something (leaving a room is the most common form) because you have to get halfway done, and have to get halfway to being halfway done, and halfway to halfway to halfway to being done — is often resolved by people saying that Zeno just didn’t understand that an infinite series could converge. That is, that you can add together infinitely many numbers and get a finite number. I’m inclined to think Zeno did not, somehow, think it was impossible to leave rooms. What the paradoxes as a whole get to are questions about space and time: they’re either infinitely divisible or they’re not. And either way produces effects that don’t seem to quite match our intuitions.

The next day Saturday Morning Breakfast Cereal does a joke about Klein bottles. These are famous topological constructs. At least they’re famous in the kinds of places people talk about topological constructs. It’s much like the Möbius strip, a ribbon given a twist and joined back to its edge. The Klein bottle similarly you can imagine as a cylinder stretched out into the fourth dimension, given a twist, then joined back to itself. We can’t really do this, what with it being difficult to craft four-dimensional objects. But we can imagine this, and it creates an object that doesn’t have a boundary, and has only one side. There’s not an inside or an outside. There’s no making this in the real world, but we can make nice-looking approximations, usually as bottles.

Ruben Bolling’s Super-Fun-Pak Comix for the 13th of January is an extreme installment of Chaos Butterfly. The trouble with touching Chaos Butterfly to cause disasters is that you don’t know — you can’t know — what would have happened had you not touched the butterfly. You change your luck, but there’s no way to tell whether for the better or worse. One of the commenters at Gocomics.com alludes to this problem.

Jon Rosenberg’s Scenes From A Multiverse for the 13th of January makes quite literal quantum mechanics talk about probability waves and quantum foam and the like. The wave formulation of quantum mechanics, the most popular and accessible one, describes what’s going on in equations that look much like the equations for things diffusing into space. And quantum mechanical problems are often solved by supposing that the probability distribution we’re interested in can be broken up into a series of sinusoidal waves. Representing a complex function as a set of waves is a common trick, not just in quantum mechanics, because it works so well so often. Sinusoidal waves behave in nice, predictable ways for most differential equations. So converting a hard differential equation problem into a long string of relatively easy differential equation problems is usually a good trade.

Tom Thaves’s Frank and Ernest for the 14th of January ties together the baffling worlds of grammar and negative numbers. It puts Frank and Ernest on panel with Euclid, who’s a fair enough choice to represent the foundation of (western) mathematics. He’s famous for the geometry we now call Euclidean. That’s the common everyday kind of blackboards and tabletops and solid cubes and spheres. But among his writings are compilations of arithmetic, as understood at the time. So if we know anyone in Ancient Greece to have credentials to talk about negative numbers it’s him. But the choice of Euclid traps the panel into an anachronism: the Ancient Greeks just didn’t think of negative numbers. They could work through “a lack of things” or “a shortage of something”, but a negative? That’s a later innovation. But it’s hard to think of a good rewriting of the joke. You might have Isaac Newton be consulted, but Newton makes normal people think of gravity and physics, confounding the mathematics joke. There’s a similar problem with Albert Einstein. Leibniz or Gauss should be good, but I suspect they’re not the household names that even Euclid is. And if we have to go “less famous mathematician than Gauss” we’re in real trouble. (No, not Andrew Wiles. Normal people know him as “the guy that proved Fermat’s thing”, and that’s too many words to fit on panel.) Perhaps the joke can’t be made to read cleanly and make good historic sense.

## Reading the Comics, December 11, 2015: So, That Didn’t Work Edition

I’d hoped that running a slightly-too-soon edition of Reading the Comics would let me have a better-sized edition for later in this week. Then everybody did comics for the 11th of December. I can have a series of awkward-sized essays or just run what I have. I wonder which I’ll do.

Aaron McGruder’s The Boondocks for the 6th of December is a student-resisting-the-problem joke. It ran originally the 24th of September, 2000, if the copyright information is right. The original problem — “what is 24 divided by 4 minus 2” — is a reasonable one for at least some level of elementary school. (I’m vague on just what grade Caesar is supposed to be in. It’s a problem for any strip with wise-beyond-their-years children. Peanuts plays with this by having the kids give book reports on Peter Rabbit and Tess of the d’Urbervilles.) What makes it a challenge is that you know to know the order of operations. Should you divide 24 by 4 first, and subtract 2 from that, or should you take 4 minus 2 and then divide 24 by whatever that number is?

Absent any confounding information, you should always do multiplication and division before you do addition and subtraction. So this suggests 24 divided by 4, giving us 6, and then subtract 2, giving us 4. The only relevant confounding information, though, would be the direction to do something else first. That’s indicated by putting something in parentheses. (Or brackets, if you have so many parentheses the symbols are getting confusing.) A thing in parentheses has higher priority and should be calculated first. But there’s no way to tell parentheses in dialogue. The best the teacher could do is say something like “24 divided by the quantity four minus two”, or even, “24 divided by parenthesis four minus two close parenthesis”. That’s awkward but it is what we resort to even in the mathematics department.

Eric the Circle for the 6th of December, this one by “Scooterpiggy”, is the anthropomorphic-numerals joke this essay. You might fuss that there’s a difference between a circle and zero. The earliest examples of zero seem to have been simple dots. But the circle, or at least elliptical, shape of zero grew pretty fast. Maybe in a couple of centuries. Maybe there’s something in the empty loop that suggests what it stands for.

Tom Thaves’s Frank and Ernest for the 6th of December tosses in a statistics pun for the final panel. The statistics use of “median” is the number that half the data is less than and half the data is greater than. It’s one of several quantities that get called an “average”. In this case it’s average because if you picked a data point at random you’d be as likely to be above as below the median. In data sets that aren’t too weird, that will usually be pretty close to the arithmetic mean. The arithmetic mean is the thing normal people mean by “average”. It’ll also typically be near the most common value. That most common value mathematicians and statisticians call the “mode”.

I don’t know if the use of “median” for the middle strip of a divided road shares an etymology with the statistics use of the word. It might be one use might have inspired the other, perhaps as metaphor. But the similarity between “being in the middle of the data” and “being in the middle of the street” is straightforward for English.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 6th of December pinpoints a common failure mode of experts. (The strip almost surely ran before, sometime. The only method I have to find out when, though, is to post an incorrect date and make someone correct me. So let me say it originally ran on Singapore National Day, 2009.) Mathematics is especially prone to it. It’s so seductive to teach something the way an expert sees it. This is usually in a rigorously thought-out, open-ended, flexible method. After all, why would you ever teach something that wasn’t exactly right, with “right” being “the ways experts see things”? A teacher knows the answer: the expert understanding of a thing is hard to get to. That’s why having it takes expertise. The comic strip’s explanation of fractions is correct and reasonable. But it brings up why Bertrand Russell and Alfred North Whitehead needed over four hundred pages to establish 1 + 1 equals 2. That’s a lot of intellectual scaffolding for the quality of paint job required. Sometimes it’s easier to start with a quick and dirty explanation, and then go back later and rebuild the understanding if a student needs it.

Rick Stromoski’s Soup to Nutz for the 7th of December puts forth a kind of Zeno’s paradox problem in the guise of compound interest. If doing something increases life expectancy by a certain percentage, then, how much of the extra time one gets do you need to be immortal? I’m amused by this although I can’t imagine modest alcohol consumption increasing lifespan by 20 percent. (I assume 20 percent of the average expected lifespan.) If the effect were anything near that big the actuaries would have noticed and ordered people to drink long ago.

On looking at all this, I think I’ll save the December 11th strips for later. This is enough text for this early in the morning.

## Reading the Comics, January 24, 2015: Many, But Not Complicated Edition

I’m sorry to have fallen behind on my mathematics-comics posts, but I’ve been very busy wielding a cudgel at Microsoft IIS all week in the service of my day job. And since I telecommute it’s quite hard to convincingly threaten the server, however much it deserves it. Sorry. Comic Strip Master Command decided to send me three hundred billion gazillion strips, too, so this is going to be a bit of a long post.

Jenny Campbell’s Flo and Friends (January 19) is almost a perfect example of the use of calculus as a signifier of “something really intelligent people think of”. Which is flattening to mathematicians, certainly, although I worry that attitude does make people freeze up in panic when they hear that they have to take calculus.

The Amazing Yet Tautological feature of Ruben Bolling’s Super-Fun-Pak Comix (January 19) lives up to its title, at least provided we are all in agreement about what “average” means. From context this seems to be the arithmetic mean — that’s usually what people, mathematicians included mean by “average” if they don’t specify otherwise — although you can produce logical mischief by slipping in an alternate average, such as the “median” — the amount that half the results are less than and half are greater than — or the “mode” — the most common result. There are other averages too, but they’re not so often useful. On the 21st Super-Fun-Pak Comix returned with another installation of Chaos Butterfly, by the way.

## What Do I Need If The Final Is Worth 40 Percent?

I suspected that my little pair of articles about what scores you need on the final to pass a class (or get an A, or such) would prove useful, which is almost as good as being informative. I noticed in the search queries bringing people to my pages a question about what a person needed for a course where the final was 40 percent of the class score. I hadn’t put that in my original set of tables, and if the searcher followed my first article — about how to work out what you need for any weighting of the final exam — then she or he got what was needed. I just didn’t think of finals being quite that heavily weighted. But, what the heck, if people want to see the tables worth 40 percent, it’s easy enough to generate them, and I added that to the collection of scores-needed tables. Good luck next term.

## What Do I Need To Get An A In This Class?

After writing my bit about how to figure out what grade you need to pass your class, I thought some more and realized that while everything in it is true, it’s not necessarily helpful, because people get panicky at formulas. So I thought to make up some tables showing what you need, if you go in with a certain grade, to pass, or get a B, or an A, or what not, for different weightings of the final exam. That’s easy enough to do especially once I set up a Matlab (well, an Octave) script to build the tables for me.