Greg Evans’s Luann Againn for the 12th features some poor tutoring on Gunther’s part. Usually a person isn’t stuck for what the answer to a problem is; they’re stuck on how to do it correctly. Maybe on how to do it efficiently. But tutoring is itself a skill, and it’s a hard one to learn. We don’t get enough instruction in how to do it.
The problem Luann’s doing is one of simplifying an expression. I remember doing a lot of this, in middle school algebra like that. Simplifying expressions does not change their value; we don’t create new ideas by writing them. So why simplify?
Any grammatically correct expression for a concept may be as good as any other grammatically correct expression. This is as true in writing as it is in mathematics. So what is good writing? There are a thousand right answers. One trait that I think most good writing has is that it makes concepts feel newly accessible. It frames something in a way which makes ideas easier to see. So it is with simplifying algebraic expressions. Finding a version of a formula that makes clearer what you would like to do makes the formula more useful.
Simplifying like this, putting an expression into the fewest number of terms, is common. It typically makes it easier to calculate with a formula. We calculate with formulas all the time. It often makes it easier to compare one formula to another. We compare formulas some of the time. So we practice simplifying like this a lot. Occasionally we’ll have a problem where this simplification is counter-productive and we’d do better to write out something as, to make up an example, instead. Someone who’s gotten good at simplifications, to the point it doesn’t take very much work, is likely to spot cases where one wants to keep part of the expression un-simplified.
Chen Weng’s Messycow Comics for the 13th starts off with some tut-tutting of lottery players. Objectively, yes, money put on a lottery ticket is wasted; even, for example, pick-three or pick-four daily games are so unlikely to pay any award as to be worth it. But I cannot make myself believe that this is necessarily a more foolish thing to do with a couple dollars than, say, buying a candy bar or downloading a song you won’t put on any playlists.
And as the Cow points out, the chance of financial success in art — in any creative field — is similarly ridiculously slight. Even skilled people need a stroke of luck to make it, and even then, making it is a marginal matter. (There is a reason I haven’t quit my job to support myself by blog-writing.) People are terrible at estimating probabilities, especially in situations that are even slightly complicated.
Finally we get to last week’s comics. This past one wasn’t nearly so busy a week for mathematically-themed comic strips. But there’s still just enough that I can split them across two days. This fits my schedule well, too.
Rick Detorie’s One Big Happy for the 9th is trying to be the anthropomorphized numerals joke of the week. It’s not quite there, but it also uses some wordplay. … And I’ll admit being impressed any of the kids could do much with turning any of the numerals into funny pictures. I remember once having a similar assignment, except that we were supposed to use the shape of our state, New Jersey, as the basis for the picture. I grant I am a dreary and literal-minded person. But there’s not much that the shape of New Jersey resembles besides itself, “the shape of Middlesex County, New Jersey”, and maybe a discarded sock. I’m not still upset about this.
The choice of ‘z’ to mean a snore is an arbitrary choice, no more inherent to the symbol than that ‘2’ should mean two. Christopher Miller’s American Cornball, which tracks a lot of (American) comedic conventions of the 20th century, notes a 1911 comic postcard representing snoring as “Z-Z-Z-Z-R-R-R-R-Z-Z-Z-Z-R-R-R-R”, which captures how the snore is more than a single prolonged sound.
Dave Blazek’s Loose Parts for the 11th has the traditional blackboard full of symbols. And two mathematics-types agreeing that they could make up some more symbols. Well, mathematics is full of symbols. Each was created by someone. Each had a point, which was to express some concept better. Usually the goal is to be more economical: it’s fewer strokes of the pen to write = instead of “equals”, and = is quicker even than “eq”. Or we want to talk a lot about a complicated concept, which is how we get, say, for “a representative of the set of angles with sine equal to x”.
I suspect every mathematician has made up a couple symbols in their notes. In the excitement of working out a problem there’ll be something they want to refer to a lot. That gets reduced to an acronym or a repeated scribble soon enough. Sometimes it’s done by accident: for a while when I needed a dummy variable I would call on “ksee”, a Greek letter so obscure that it does not even exist. It looks like a cross between zeta and xi. The catch is, always, getting anyone else to use the symbol. Most of these private symbols stay private, because they don’t do work that can’t be better done by a string of symbols we already have (letters included). Or at least they don’t to well enough to be worth the typesetting trouble. I’d be surprised if any of the students I used “ksee” in front of reused the letter, even if they did find a need for a dummy variable. Founding a field, or writing a definitive text in a field, helps your chances.
I am curious how the modern era of digital typesetting will affect symbol creation. It’s relatively easy to put in a new symbol — or to summon one in the Unicode universe not currently used for mathematics — in a document and have it copied. Certainly it’s easy compared to what it was like in typewriter and Linotype days, when you might need to rely on a friend who knows a guy at the type foundry. On the other hand, it’s hard enough to get the raw file in LaTeX — a long-established standard mathematics typesetting computer language — from another person and have it actually work, even without adding in new symbols. I don’t see that changing just because several people have found that a bubble tea emoji quite helps their paper on sedimentation rates.
Pedro Martin’s Mexikid Stories for the 11th recounts childhood memories and anxieties of being matched, boys versus girls, in various activities. This includes mathematics quizzes. Here, the mathematics is done as a class game, which is a neat coincidence as I’d been thinking of similar public mathematics quiz-games that I’d done. I liked them, but then, I was almost always at top or second in the class rankings, and — after the initial couple rounds — never fell below third. My recent thoughts were for how much less fun this must have been for the kids in 26th place, especially if they’re ones who can do the work just fine, given time and space. We do value speed, in working, and that comes from practicing a task so often that we do it in the slightest time possible. And we value ability to perform under pressure, so we put people into anxiety-producing states until they can do a particular task anyway.
With this essay, I finally finish the comic strips from the first full week of February. You know how these things happen. I’ll get to the comics from last week soon enough, at an essay gathered under this link. For now, some pictures with words:
Art Sansom and Chip Sansom’s The Born Loser for the 7th builds on one of the probability questions people often use. That is the probability of an event, in the weather forecast. Predictions for what the weather will do are so common that it takes work to realize there’s something difficult about the concept. The weather is a very complicated fluid-dynamics problem. It’s almost certainly chaotic. A chaotic system is deterministic, but unpredictable, because to get a meaningful prediction requires precision that’s impossible to ever have in the real world. The slight difference between the number π and the number 3.1415926535897932 throws calculations off too quickly. Nevertheless, it implies that the “chance” of snow on the weekend means about the same thing as the “chance” that Valentinte’s Day was on the weekend this year. The way the system is set up implies it will be one or the other. This is a probability distribution, yes, but it’s a weird one.
What we talk about when we say the “chance” of snow or Valentine’s on a weekend day is one of ignorance. It’s about our estimate that the true value of something is one of the properties we find interesting. Here, past knowledge can guide us. If we know that the past hundred times the weather was like this on Friday, snow came on the weekend less than ten times, we have evidence that suggests these conditions don’t often lead to snow. This is backed up, these days, by numerical simulations which are not perfect models of the weather. But they are ones that represent something very like the weather, and that stay reasonably good for several days or a week or so.
And we have the question of whether the forecast is right. Observing this fact is used as the joke here. Still, there must be some measure of confidence in a forecast. Around here, the weather forecast is for a cold but not abnormally cold week ahead. This seems likely. A forecast that it was to jump into the 80s and stay there for the rest of February would be so implausible that we’d ignore it altogether. A forecast that it would be ten degrees (Fahrenheit) below normal, or above, though? We could accept that pretty easily.
Proving a forecast is wrong takes work, though. Mostly it takes evidence. If we look at a hundred times the forecast was for a 10% chance of snow, and it actually snowed 11% of the time, is it implausible that the forecast was right? Not really, not any more than a coin coming up tails 52 times out of 100 would be suspicious. If it actually snowed 20% of the time? That might suggest that the forecast was wrong. If it snowed 80% of the time? That suggests something’s very wrong with the forecasting methods. It’s hard to say one forecast is wrong, but we can have a sense of what forecasters are more often right than others are.
Doug Savage’s Savage Chickens for the 7th is a cute little bit about counting. Counting things out is an interesting process; for some people, hearing numbers said aloud will disrupt their progress. For others, it won’t, but seeing numbers may disrupt it instead.
Niklas Eriksson’s Carpe Diem for the 8th is a bit of silliness about the mathematical sense of animals. Studying how animals understand number is a real science, and it turns up interesting results. It shouldn’t be surprising that animals can do a fair bit of counting and some geometric reasoning, although it’s rougher than even our untrained childhood expertise. We get a good bit of our basic mathematical ability from somewhere, because we’re evolved to notice some things. It’s silly to suppose that dogs would be able to state the Pythagorean Theorem, at least in a form that we recognize. But it is probably someone’s good research problem to work out whether we can test whether dogs understand the implications of the theorem, and whether it helps them go about dog work any.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th speaks of the “Cinnamon Roll Delta Function”. The point is clear enough on its own. So let me spoil a good enough bit of fluff by explaining that it’s a reference to something. There is, lurking in mathematical physics, a concept called the “Dirac delta function”, named for that innovative and imaginative fellow Paul Dirac. It has some weird properties. Its domain is … well, it has many domains. The real numbers. The set of ordered pairs of real numbers, R2. The set of ordered triples of real numbers, R3. Basically any space you like, there’s a Dirac delta function for it. The Dirac delta function is equal to zero everywhere in this domain, except at one point, the “origin”. At that one function, though? There it’s equal to …
Here we step back a moment. We really, really, really want to say that it’s infinitely large at that point, which is what Weinersmith’s graph shows. If we’re being careful, we don’t say that though. Because if we did say that, then we would lose the thing that we use the Dirac delta function for. The Dirac delta function, represented with δ, is a function with the property that for any set D, in the domain, that you choose to integrate over
whenever the origin is inside the interval of integration D. It’s equal to 0 if the origin is not inside the interval of integration. This, whatever the set is. If we use the ordinary definitions for what it means to integrate a function, and say that the delta function is “infinitely big” at the origin, then this won’t happen; the integral will be zero everywhere.
This is one of those cases where physicists worked out new mathematical concepts, and the mathematicians had to come up with a rationalization by which this made sense. This because the function is quite useful. It allows us, mathematically, to turn descriptions of point particles into descriptions of continuous fields. And vice-versa: we can turn continuous fields into point particles. It turns out we like to do this a lot. So if we’re being careful we don’t say just what the Dirac delta function “is” at the origin, only some properties about what it does. And if we’re being further careful we’ll speak of it as a “distribution” rather than a function.
But colloquially, we think of the Dirac delta function as one that’s zero everywhere, except for the one point where it’s somehow “a really big infinity” and we try to not look directly at it.
The sharp-eyed observer may notice that Weinersmith’s graph does not put the great delta spike at the origin, that is, where the x-axis represents zero. This is true. We can create a delta-like function with a singular spot anywhere we like by the process called “translation”. That is, if we would like the function to be zero everywhere except at the point , then we define a function and are done. Translation is a simple step, but it turns out to be useful all the time.
Besides kids doing homework there were a good ten or so comic strips with enough mathematical content for me to discuss. So let me split that over a couple of days; I don’t have the time to do them all in one big essay.
Sandra Bell-Lundy’s Between Friends for the 2nd is declared to be a Venn Diagram joke. As longtime readers of these columns know, it’s actually an Euler Diagram: a Venn Diagram requires some area of overlap between all combinations of the various sets. Two circles that never touch, or as these two do touch at a point, don’t count. They do qualify as Euler Diagrams, which have looser construction requirements. But everything’s named for Euler, so that’s a less clear identifier.
John Kovaleski’s Daddy Daze for the 2nd talks about probability. Particularly about the probability of guessing someone’s birthday. This is going to be about one chance in 365, or 366 in leap years. Birthdays are not perfectly uniformly distributed through the year. The 13th is less likely than other days in the month for someone to be born; this surely reflects a reluctance to induce birth on an unlucky day. Births are marginally more likely in September than in other months of the year; this surely reflects something having people in a merry-making mood in December. These are tiny effects, though, and to guess any day has about one chance in 365 of being someone’s birthday will be close enough.
If the child does this long enough there’s almost sure to be a match of person and birthday. It’s not guaranteed in the first 365 cards given out, or even the first 730, or more. But, if the birthdays of passers-by are independent — one pedestrian’s birthday has nothing to do with the next’s — then, overall, about one-365th of all cards will go to someone whose birthday it is. (This also supposes that we won’t see things like the person picked saying that while it’s not their birthday, it is their friend’s, here.) This, the Law of Large Numbers, one of the cornerstones of probability, guarantees us.
Mark Anderson’s Andertoons for the 2nd is the Mark Anderson’s Andertoons for the week. And it’s a Venn Diagram joke, at least if the two circles are “really” there. Diplopia is what most of us would call double vision, seeing multiple offset copies of a thing. So the Venn diagram might be an optical illusion on the part of the businessman and the reader.
Dave Blazek’s Loose Parts for the 3rd is an anthropomorphic mathematical symbols joke. I suppose it’s algebraic symbols. We usually get to see the ‘x’ and ‘y’ axes in (high school) algebra, used to differentiate two orthogonal axes. The axes can be named anything. If ‘x’ and ‘y’ won’t do, we might move to using and . In linear algebra, when we might want to think about Euclidean spaces with possibly enormously many dimensions, we may change the names to and . (We could use subscripts of 0 and 1, although I do not remember ever seeing someone do that.)
Morrie Turner’s Wee Pals for the 3rd is a repeat, of course. Turner died several years ago and no one continued the strip. But it is also a repeat that I have discussed in these essays before, which likely makes this a good reason to drop Wee Pals from my regular reading here. There are 42 distinct ways to add (positive) whole numbers up to make ten, when you remember that you can add three or four or even six numbers together to do it. The study of how many different ways to make the same sum is a problem of partitioning. This might not seem very interesting, but if you try to guess how many ways there are to add up to 9 or 11 or 15, you’ll notice it’s a harder problem than it appears.
And for all that, there’s still some more comic strips to review. I will probably slot those in to Sunday, and start taking care of this current week’s comic strips on … probably Tuesday. Please check in at this link Sunday, and Tuesday, and we’ll see what I do.
There were a bunch of comic strips mentioning some kind of mathematical theme last week. I need to clear some out. So I’ll start with some of the marginal mentions. Many of these involve having to deal with exams or quizzes.
There are different ways to find square roots. (I can guarantee that Skip wasn’t expected to use this one.) The term ‘root’ derives from an idea that the root of a number is the thing that generates it: 3 is a square root of 9 because multiplying 3’s together gives you 9. ‘Square’ is I have always only assumed because multiplying a number by itself will give you the area of a square with sides of length that number. This is such an obvious word origin, though, that I am reflexively suspicious. Word histories are usually subtle and capricious things.
The strip for the 8th closing the storyline has a nice example of using “billion” as a number so big as to be magical, capable of anything. Big numbers can do strange and contrary-to-intuition things. But they can be reasoned out.
Tony Cochran’s Agnes for the 4th sees the title character figuring she could sell her “personal smartness”. Her best friend Trout wonders if that’s tutoring math or something. (Incidentally, Agnes is one of the small handful of strips to capture what made Calvin and Hobbes great; I recommend giving it a try.)
Charles Schulz’s Peanuts Begins for the 5th sees Charlie Brown working problems on the board. He’s stuck for what to do until he recasts the problem as scoring in football and golf. We may giggle at this, but I support his method. It’s convinced him the questions are worth solving, the most important thing to doing them at all. And it’s gotten him to the correct answers. Casting these questions as sports problems is the building of falsework: it helps one do the task, and then is taken away (or hidden) from the final product. Everyone who does mathematics builds some falsework like this. If we do a particular problem, or kind of problem, often enough we get comfortable enough with the main work that we don’t need the falsework anymore. So it is likely to be for Charlie Brown.
There’s some comic strips that get mentioned here all the time. Then there’s comic strips that I have been reading basically my whole life, and that never give me a thread to talk about. Although I’ve been reading comic strips for their mathematics content for a long while now, somehow, I am still surprised when these kinds of comic strip are not the same thing. So here’s the end of last week’s comics, almost in time for next week to start:
Kevin Fagan’s Drabble for the 28th has Penny doing “math” on colors. Traditionally I use an opening like this to mention group theory. In that we study things that can be added together, in ways like addition works on the integers. Colors won’t quite work like this, unfortunately. A group needs an element that’s an additive identity. This works like zero: it can be added to anything without changing its value. There isn’t a color that you can mix with other colors that leaves the other color unchanged, though. Even white or clear will dilute the original color.
If you’ve thought of the clever workaround, that each color can be the additive identity to itself, you get credit for ingenuity. Unfortunately, to be a group there has to be a lone additive identity. Having more than one makes a structure that’s so unlike the integers that mathematicians won’t stand for it. I also don’t know of any interesting structures that have more than one additive identity. This suggests that nobody has found a problem that they represent well. But the strip suggests maybe it could tell us something useful for colors. I don’t know.
Tom Armstrong’s Marvin for the 28th is a strip which follows from the discovery that “fake news” is a thing that people say. Here the strip uses a bit of arithmetic as the sort of incontrovertibly true thing that Marvin is dumb to question. Well, that 1 + 1 equals 2 is uncontrovertibly true, unless we are looking at some funny definitions of ‘1’ or ‘plus’ or something. I remember, as a kid, being quite angry with a book that mentioned “one cup of popcorn plus one cup of water does not give us two cups of soggy popcorn”, although I didn’t know how to argue the point.
Hilary Price and Rina Piccolo’s Rhymes with Orange for the 30th is … well, I’m in this picture and I don’t like it. I come from a long line of people who cover every surface with stuff. But as for what surface area is? … Well, there’s a couple of possible definitions. One that I feel is compelling is to think of covering sets. Take a shape that’s set, by definition, to have an area of 1 unit of area. What is the smallest number of those unit shapes which will cover the original shape? Cover is a technical term here. But also, here, the ordinary English word describes what we need it for. How many copies of the unit shape do you need to exactly cover up the whole original shape? That’s your area. And this fits to the mother’s use of surfaces in the comic strip neatly enough.
Bud Fisher’s Mutt and Jeff for the 31st is a rerun of vintage unknown to me. I’m not sure whether it’s among the digitally relettered strips. The lettering’s suspiciously neat, but, for example, there’s at least three different G’s in there. Anyway, it’s an old joke about adding together enough gas-saving contraptions that it uses less than zero gas. So far as it’s tenable at all, it comes from treating percentage savings from different schemes as additive, instead of multiplying together. Also, I suppose, that the savings are independent, that (in this case) Jeff’s new gas saving ten percent still applies even with the special spark plugs or the new carburettor [sic]. The premise is also probably good for a word problem, testing out understanding of percentages and multiplication, which is just a side observation here.
This wraps up last week’s mathematically-themed comic strips. This week I can tell you already was a bonanza week. When I start getting to its comics I should have an essay at this link. Thanks for reading.
I apologize for missing Sunday. I wasn’t able to make the time to write about last week’s mathematically-themed comic strips. But I’m back in the swing of things. Here are some of the comic strips that got my attention.
Still, we do keep discovering things we didn’t know were numbers before. The earliest number notations, in the western tradition, for example, used letters to represent numbers. This did well for counting numbers, up to a large enough total. But it required idiosyncratic treatment if you wanted to handle large numbers. Hindu-Arabic numerals make it easy to represent whole numbers as large as you like. But that’s at the cost of adding ten (well, I guess eight) symbols that have nothing to do with the concept represented. Not that, like, ‘J’ looks like the letter J either. (There is a folk etymology that the Arabic numerals correspond to the number of angles made if you write them out in a particular way. Or less implausibly, the number of strokes needed for the symbol. This is ingenious and maybe possibly has helped one person somewhere, ever, learn the symbols. But it requires writing, like, ‘7’ in a way nobody has ever done, and it’s ahistorical nonsense. See section 96, on page 64 of the book and 84 of the web presentation, in Florian Cajori’s History of Mathematical Notations.)
Still, in time we discovered, for example, that there were irrational numbers and those were useful to have. Negative numbers, and those are useful to have. That there are complex-valued numbers, and those are useful to have. That there are quaternions, and … I guess we can use them. And that we can set up systems that resemble arithmetic, and work a bit like numbers. Those are often quite useful. I expect Lemon and Sayers were having fun with the idea of new numbers. They are a thing that, effectively, happens.
David Malki’s Wondermark for the 27th describes engineering as “like math, but louder”, which is a pretty good line. And it uses backgrounds of long calculations to make the point of deep thought going on. I don’t recognize just what calculations are being done there, but they do look naggingly familiar. And, you know, that’s still a pretty lucky day.
Mark Anderson’s Andertoons for the 27th is the Mark Anderson’s Andertoons for the week. It depicts Wavehead having trouble figuring where to put the decimal point in the multiplication of two decimal numbers. Relatable issue. There are rules you can follow for where to put the decimal in this sort of operation. But the convention of dropping terminal zeroes after the decimal point can make that hazardous. It’s something that needs practice, or better: though. In this case, what catches my eye is that 2.95 times 3.2 has to be some number close to 3 times 3. So 9.440 is the plausible answer.
Mike Twohy’s That’s Life for the 27th presents a couple of plausible enough word problems, framed as Sports Math. It’s funny because of the idea that the workers who create events worth billions of dollars a year should be paid correspondingly.
Or they’re making it easy for me. But for another week all the comic strips mentioning mathematics have done so in casual ways. Ones that I don’t feel I can write a substantial paragraph about. And so, ones that I don’t feel I can fairly use the images of here. Here’s strips that at least said “math” somewhere in them:
Greg Cravens’s The Buckets for the 19th plays on the conflation of “zero” and “nothing”. The concepts are related, and we wouldn’t have a zero if we weren’t trying to worth with the concept of nothing. But there is a difference that’s quite hard to talk about without confusing matters.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 25th presents a sinister reading of the fad of “prove you’re human” puzzles that demanded arithmetic expressions be done. All computer programs, including, like, Facebook group messages are arithmetic operations ultimately. The steps could be translated into simple expressions like this and be done by humans. It just takes work which, I suppose, could also be translated into other expressions.
So Mark Anderson’s Andertoons for the 12th is the only comic strip of some substance that I noticed last week. You see what a slender month it’s been. It does showcase the unsettling nature of seeing notations for similar things mixed. It’s not that there’s anything which doesn’t parse about having decimals in the numerator or denominator. It just looks weird. And that can be enough to throw someone out of a problem. They might mistake the problem for one that doesn’t have a coherent meaning. Or they might mistake it for one too complicated to do. Learning to not be afraid of a problem that looks complicated is worth doing. As is learning how to tell whether a problem parses at all, even if it looks weird.
I am not my state’s pinball champion, although for the first time I did make it through the first round of play. What is important about this is that between that and a work trip I needed time for things which were not mathematics this past week. So my first piece this week will be a partial listing of comic strips that, last week, mentioned mathematics but not in a way I could build an essay around. … It’s not going to be a week with long essays, either, though. Here’s a start, though.
Henry Scarpelli’s Archie rerun for the 12th of January was about Moose’s sudden understanding of algebra, and wish for it to be handy. Well, every mathematician knows the moment when suddenly something makes sense, maybe even feels inevitably true. And then we do go looking for excuses to show it off.
Jason Chatfield’s Ginger Meggs for the 13th has Meggs fail a probability quiz, an outcome his teacher claims is almost impossible. If the test were multiple-choice (including true-or-false) it is possible to calculate the probability of a person making wild guesses getting every answer wrong (or right) and it usually is quite the feat, at least if the test is of appreciable length. For more open answers it’s harder to say what the chance of someone getting the question right, or wrong, is. And then there’s the strange middle world of partial credit.
My love does give multiple-choice quizzes occasionally and it is always a source of wonder when a student does worse than blind chance would. Everyone who teaches has seen that, though.
Ed Allison’s Unstrange Phenomenon for the 13th plays with optical illusions, which include several based on geometric tricks. Humans have some abilities at estimating relative areas and distances and lengths. But they’re not, like, smart abilities. They can be fooled, basically because their settings are circumstances where there’s no evolutionary penalty for being fooled this way. So we can go on letting the presence of arrow pointers mislead us about the precise lengths of lines, and that’s all right. There are, like, eight billion cognitive tricks going on all around us and most of them are much more disturbing.
Ryan North’s Dinosaur Comics for the 10th sees T-Rex pondering the point of solitaire. As he notes, there’s the weird aspect of solitaires that many of them can’t be won, even if you play perfectly. This comes close, without mentioning, an important event in numerical mathematics. So let me mention it.
There have always been things we could compute by random experiments. The digits of π, for example, if we’re willing to work at it. The catch is that this takes a lot of work. So we did not do much of this before we had computers, which are able to do a lot of work for the cost of electricity. There is a deep irony in this, since computers are — despite appearances — deterministic. They cannot do anything unpredictable. We have to provide random numbers, somehow. Or numbers that look enough like random numbers that we won’t make a grave error by using them.
Many of these techniques are known as Monte Carlo methods. These were developed in the 1940s. Stanislaw Ulam described convalescing from an illness, and playing a lot of solitaire. He pondered particularly the chance of winning a Canfield solitaire, a kind of game I have never heard of outside this anecdote. There seemed no way to work out this problem by reason alone. But he could imagine doing it in simulation, and with John von Neumann began calculating. Nicholas Metropolis gave it the gambling name, although something like that would be hard to resist. This is far from the only game that’s inspired useful mathematics. It is a good one, though.
It’s been another of those weeks where the comic strips mentioned mathematics but not in any substantive way. I haven’t read Saturday’s comics yet, as I write this, so perhaps the last day of the week will revolutionize things. In the meanwhile, here’s the strips you can look at and agree say ‘mathematics’ in them somewhere.
Dave Whamond’s Reality Check for the 5th of January, 2020, uses a blackboard full of arithmetic as signifier of teaching. The strip is an extended riff on Extruded Inspirational Teacher Movie product. I like the strip, but I don’t fault you if you think it’s a lot of words deployed against a really ignorable target.
Jim Meddick’s Monty for the 29th has the time-travelling Professor Xemit (get it?) show a Times Square Ball Drop of the future. The ball gets replaced with a “demihypercube”, the idea being that the future will have some more complicated geometry than a mere “ball”. There is no such thing as “a” demihypercube, in the same way there is not “a” pentagon. There is a family of shapes, all called demihypercubes. There’s a variety of ways to represent them. A reasonable one, though, is a roughly spherical shape made of pointy triangles all over. It wouldn’t look absurd. There are probably time ball drops that use something like a demihypercube already.
The start of the year brings me comic strips I can discuss in some detail. There are also some that just mention a mathematical topic, and don’t need more than a mention that the strip exists. I’ll get to those later.
Jonathan Lemon’s Rabbits Against Magic for the 2nd is another comic strip built on a very simple model of animal reproduction. We saw one late last year with a rat or mouse making similar calculations. Any calculation like this builds on some outright untrue premises, particularly in supposing that every rabbit that’s born survives, and that the animals breed as much as could do. It also builds on some reasonable simplifications. Things like an average litter size, or an average gestation period, or time it takes infants to start breeding. These sorts of exponential-growth calculations depend a lot on exactly what assumptions you make. I tried reproducing Lemon’s calculation. I didn’t hit 95 billion offspring. But I got near enough to say that Lemon’s right to footnote this as ‘true’. I wouldn’t call them “baby bunnies”, though; after all, some of these offspring are going to be nearly seven years old by the end of this span.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 3rd justifies why “mathematicians are no longer allowed to [sic] sporting events” with mathematicians being difficult. Each of the signs is mean to convey the message “We’re #1”. The notations are just needlessly inaccessible, in that way nerds will do things.
first. The bar over over a decimal like this means to repeat what is underneath the bar without limit. So this is the number represented by 0.99999… and this is another way to write the number 1. This sometimes makes people uncomfortable; the proof is to think what the difference is between 1 and the number represented by 0.999999 … . The difference is smaller than any positive number. It’s certainly not negative. So the difference is zero. So the two numbers have to be the same number.
is the controversial one here. The trouble is that there are two standard rules that clash here. One is the rule that any real number raised to the zeroth power is 1. The other is the rule that zero raised to any positive real number is 0. We don’t ask about zero raised to a negative number. These seem to clash. That we only know zero raised to positive real numbers is 0 seems to break the tie, and justify concluding the number-to-the-zero-power rule should win out. This is probably what Weinersmith, or Weinersmith’s mathematician, was thinking. If you forced me to say what I think should be, and didn’t let me refuse to commit to a value, I’d probably pick “1” too. But.
The expression exists for real-valued numbers x, and that’s fine. We can look at and that number’s 1. But what if x is a complex-valued number? If that’s the case, then this limit isn’t defined. And mathematicians need to work with complex-valued numbers a lot. It would be daft to say “real-valued is 1, but complex-valued isn’t anything”. So we avoid the obvious daftness and normally defer to saying is undefined.
The last expression is . This is that famous base of imaginary numbers, one of those numbers for which . Complex-valued numbers can be multiplied and divided and raised to powers just like real-valued numbers can. And, remarkably — it surprised me — the number is equal to . That’s the reciprocal of .
There are a couple of ways to show this. A straightforward method uses the famous Euler formula, that . This implies that . So has to equal . That’s equal to , or . If you find it weird that an imaginary number raised to an imaginary number gives you a real number — it’s a touch less than 0.208 — then, well, you see how weird even the simple things can be.
Gary Larson’s The Far Side for the 4th references Abraham Lincoln’s famous use of “four score and seven” to represent 87. There have been many ways to give names to numbers. As we’ve gotten comfortable with decimalization, though, most of them have faded away. I think only dozens and half-dozens remain in common use; if it weren’t for Lincoln’s style surely nobody today would remember “score” as a way to represent twenty. It probably avoids ambiguities that would otherwise plague words like “hundred”, but it does limit one’s prose style. The talk about carrying the one and taking away three is flavor. There’s nothing in turning eighty-seven into four-score-and-seven that needs this sort of arithmetic.
I hope later this week to list the comic strips which just mentioned some mathematical topic. That essay, and next week’s review of whatever this week is mathematical, should appear at this link. Thanks for reading.
Doug Savage’s Savage Chickens for the 26th uses a blackboard of mathematics (as part of “understanding of particle physics”) as symbolic of intelligence. I’m not versed enough in particle physics to say whether the expressions make sense. I’m inclined toward it, since the first line has an integral of the reciprocal of the distance between a point x and a point x’. That looks to me like a calculation of some potential energy-related stuff.
Dana Simpson’s Phoebe and her Unicorn for the 27th uses “memorizing multiplication tables” as the sort of challenging and tedious task that a friend would not put another one through. The strip surprised me; I would have thought Phoebe the sort of kid who’d find multiplication tables, with their symmetry and teasing hints of structure (compare any number on the upper-left-to-lower-right diagonal to the numbers just up-and-right or down-and-left to it, for example), fascinating enough to memorize on their own.
Leigh Rubin’s Rubes for the 27th has a rat-or-mouse showing off one of those exciting calculations about how many rats-or-mice could breed in a year if absolutely nothing limited their growth. These sorts of calculations are fun for getting to big numbers in pretty little time. They’re only the first, loosest pieces of a model for anything’s population, though.
Pab Sungenis’s New Adventures of Queen Victoria for the 28th gets into the question about whether the new decade starts in 2020 or 2021. I wasn’t aware people were asking the question until a few weeks ago, when my father asked me for an authoritative answer. He respects my credentials as a mathematician and a calendar freak. The only answer I can defend, though, is to say of course a new decade starts in 2020. A new decade also starts in 2021. There’s also a decade starting in 2022. There’s a new decade starting five minutes from the moment you read this sentence. If you hurry you just might make it.
If you want to make any claims about “the” new decade, you have to say what you pick “the” to signify. Complete decades from the (proleptically defined) 1st of January, 1, is a compelling choice. “Years starting the 1st of January, 2020” is also a compelling choice. Decide your preference and you’ll decide your answer.
Thank you for reading, this essay and this whole year. 2020 is, of course, a leap year, or “bissextile year” if you want to establish your reputation as a calendar freak. Good luck.
The last full week of the year had, again, comic strips that mostly mention mathematics without getting into detail. That’s all right. I have a bit of a cold so I’m happy not to have to compose thoughts about too many of them.
Percy Crosby’s Skippy for the 23rd has Skippy and Sookie doing the sort of story problem arithmetic of working out a total bill. The strip originally ran the 11th of August, 1932.
Cy Olson’s Office Hours for the 24th, which originally ran the 14th of October, 1971, comes the nearest to having enough to talk about here. The secretary describes having found five different answers in calculating the profits and so used the highest one. The joke is on incompetent secretaries, yes. But it is respectable, if trying to understand something very complicated, to use several different models for what one wants to know. These will likely have different values, although how different they are, and how changes in one model tracks changes in another, can be valuable. We’re accustomed to this, at least in the United States, by weather forecasts: any local weather report will describe expected storms by different models. These use different ideas about how much moisture moves into the air, how fast raindrops will form (a very difficult problem), how winds will shift, that sort of thing. It’s defensible to make similar different models for reporting the health of a business, particularly if company owns things with a price that can’t be precisely stated.
Marguerite Dabaie and Tom Hart’s Ali’s House for the 24th continues a story from the week before in which a character imagines something tossing us out of three-dimensional space. A seven-dimensional space is interesting mathematically. We can define a cross product between vectors in three-dimensional space and in seven-dimensional space. Most other spaces don’t allow something like a cross product to be coherently defined. Seven-dimensional space also allows for something called the “exotic sphere”, which I hadn’t heard of before either. It’s a structure that’s topologically a sphere, but that has a different kind of structure. This isn’t unique to seven-dimensional space. It’s not known whether four-dimensional space has exotic spheres, although many spaces higher than seven dimensions have them.
And here’s the other half of last week’s comic strips that name-dropped mathematics in such a way that I couldn’t expand it to a full paragraph. We’ll likely be back to something more normal next week.
David Malki’s Wondermark for the 20th is built on the common idiom of giving more than 100%. I’m firmly on the side of allowing “more than 100%” in both literal and figurative uses of percent, so there’s not much more to say.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th uses Big Numbers as the sort of thing that need a down-to-earth explanation. The strip is about explanations that don’t add clarity. It shows my sense of humor that I love explanations that are true but explain nothing. The more relevant and true without helping the better. Right up until it’s about something I could be explaining instead.
Tom Batiuk’s vintage Funky Winkerbean for the 21st is part of a week of strips from the perspective of a school desk. It includes a joke about football players working mathematics problems. The strip originally ran the 8th of February, 1974, looks like.
As I referenced on Sunday while there were a good number of comic strips mentioning mathematics last week, there weren’t many touching deeply enough for me to make real essays about them. But you may enjoy seeing the strips anyway. So here’s the first half of this roster.
As will sometimes happen I write this without having read Saturday’s comic strips. Press of time and all that. But it has been a week of only casual mentions of mathematics, not enough to need much detail. There were a lot of strips with this kind of casual mention. But one is of special interest.
So, yes. Gary Larson’s The Far Side has an official online home, and is reprinting strips from the classic 80s-to-early-90s comic strip. I’m glad for this, not just to reacquaint myself with an old friend. The strip was a pioneer in the good sort of nerd humor. Jokes about topics of narrow, specific interest, but — generally — not told in an exclusionary way. One might not understand why a particular joke should be funny, but only because you don’t happen to know something in the background. I’m thinking here of a desert-island strip that Larson, in one of his collections, said went over almost everybody’s head. The characters remarked on their good luck that the island was covered with mussels (or something), so at least they wouldn’t get hungry. The thing that makes this funny is that the mussels (or whatever) only grow places that get covered in water every day; that is, the island sinks with the tides.
Anyway, the first official online Far Side is, as you can see, your generic mathematics anxiety joke, using a story problem — with trains leaving stations, even — as the premise. And I admit this particular strip might not convince a young reader to today that The Far Side was anything special. This is the fate of many pioneers. If you look at it and think, well, that could run in Bizarro or The Argyle Sweater or Brevity or F Minus or Non Sequitur a dozen other comics, it’s because those are comic strips that want to be like this.
I’m sorry to say that, as best I can tell, there isn’t a lasting archive of strips on the new site. This particular rerun was one of the selection printed the 19th of December, but when I go to the link that should have shown that day’s strips I get bounced to the front page. This is vexing to someone who hopes to use the strips to lead conversations about mathematics topics. I’ll have to deal with that in one way or another.
And here’s the rest of last week’s mathematically-themed comic strips. On reflection, none of them are so substantially about the mathematics they mention for me to go into detail. Again, Comic Strip Master Command is helping me rebuild my energies after the A-to-Z wrapped up. I appreciate it, folks, but would like, you know, two or three strips a week I can sink my teeth into.
Charles Schulz’s Peanuts rerun for the 11th sees Sally Brown working out metric system unit conversions. The strip originally ran the 13th of December, 1972, a year when people in the United States briefly thought there might ever be a reason to use the prefix “deci-” for something besides decibels. “centi-” for anything besides “centimeter” is pretty dodgy too.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 13th is titled “Do Not Date A Mathematician”. This seems personal. The point here is the mathematician believing her fiancee has “demonstrated a poor understanding of probability” by declaring his belief in soulmates. The joke seems to be missing some key points, though. Just declaring a belief in soulmates doesn’t say anything about his understanding of probability. If we suppose that he believed every person had exactly one soulmate, and that these soulmates were uniformly distributed across the world’s population, and that people routinely found their soulmates. But if those assumptions aren’t made then you can’t say that the fiancee is necessarily believing in something improbable.
Comic Strip Master Command decided to respect my need for a writing break. At least a break around here. So here’s the first half of last week’s comic strips that mention mathematics. None of them get into material substantial enough that I feel justified including pictures. Some of them are even repeats, at least to my Reading the Comics essays.
You know, I had picked these comic strips out as the ones that, last week, had the most substantial mathematics content. And on preparing this essay I realize there’s still not much. Maybe I could have skipped out on the whole week instead.
Bill Amend’s FoxTrot for the 1st is mostly some wordplay. Jason’s finding ways to represent the counting numbers with square roots. The joke plays more tightly than one might expect. Root beer was, traditionally, made with sassafras root, hence the name. (Most commercial root beers don’t use actual sassafras anymore as the safrole in it is carcinogenic.) The mathematical term root, meanwhile, derives from the idea that the root of a number is the thing which generates it. That 2 is the fourth root of 16, because four 2’s multiplied together is 16. That idea. This draws on the metaphor of the roots of a plant being the thing which lets the plant grow. This isn’t one of those cases where two words have fused together into one set of letters.
Jef Mallett’s Frazz for the 1st is set up with an exponential growth premise. The kid — I can’t figure out his name — promises to increase the number of push-ups he does each day by ten percent, with exciting forecasts for how many that will be before long. As Frazz observes, it’s not especially realistic. It’s hard to figure someone working themselves up from nothing to 300 push-ups a day in only two months.
Also much else of the kid’s plan doesn’t make sense. On the second day he plans to do 1.1 push-ups? On the third 1.21 push-ups? I suppose we can rationalize that, anyway, by taking about getting a fraction of the way through a push-up. But if we do that, then, I make out by the end of the month that he’d be doing about 15.863 push-ups a day. At the end of two months, at this rate, he’d be at 276.8 push-ups a day. That’s close enough to three hundred that I’d let him round it off. But nobody could be generous enough to round 15.8 up to 90.
An alternate interpretation of his plans would be to say that each day he’s doing ten percent more, and round that up. So that, like, on the second day he’d do 1.1 rounded up to 2 push-ups, and on the third day 2.2 rounded up to 3 push-ups, and so on. Then day thirty looks good: he’d be doing 94. But the end of two months is a mess as by then he’d be doing 1,714 push-ups a day. I don’t see a way to fit all these pieces together. I’m curious what the kid thought his calculation was. Or, possibly, what Jef Mallett thought the calculation was.
Zach Weinersmith’s for the 2nd has a kid rejecting accounting in favor of his art. But, wanting to do that art with optimum efficiency … ends up doing accounting. It’s a common story. A common question after working out that someone can do a thing is how to do it best. Best has many measures, yes. But the logic behind how to find it stays the same. Here I admit my favorite kinds of games tend to have screen after screen of numbers, with the goal being to make some number as great as possible considering. If they ever made Multiple Entry Accounting Simulator none of you would ever hear from me again.
Which may be some time! Between Reading the Comics, A to Z, recap posts, and the occasional bit of filler I’ve just finished slightly over a hundred days in a row posting something. That is, however, at its end. I don’t figure to post anything tomorrow. I may not have anything before Sunday’s Reading the Comics post, at this link. I’ll be letting my typing fingers sleep in instead. Thanks for reading.
Although I’m out of the A to Z sequence, I like the habit of posting just the comic strips that name-drop mathematics for the Sunday post. It frees up so much of my Saturday, at the cost of committing my Sunday. So here’s last week’s casual mentions of some mathematics topic.
Bill Holbrook’s On The Fastrack for the 5th has the CEO of Fastrack, Inc, disappointed in what analytics can do. Analytics, here, is the search for statistical correlations, traits that are easy to spot and that indicate greater risks or opportunities. The desire to find these is great and natural. Real data is, though, tantalizingly not quite good enough to answer most interesting questions.
Tauhid Bondia’s Crabgrass for the 6th uses a background panel of calculus work as part of illustrating deep thinking about something, in this case, how to fairly divide chocolate. One of calculus’s traditional strengths is calculating the volumes of interesting figures.
Joe Martin’s Mr Boffo for the 6th is a cute joke on one of the uses of numbers, that of being a convenient and inexhaustible index. The strip ran on Friday and I don’t know how to link to the archives in a stable way. This is why I’ve put the comic up here.
And that’s enough comics for just now. Later this week I’ll get to the comics that inspire me to write more.
See if you can spot where I discover my having made a big embarrassing mistake. It’s fun! For people who aren’t me!
Lincoln Peirce’s Big Nate for the 24th has boy-genius Peter drawing “electromagnetic vortex flow patterns”. Nate, reasonably, sees this sort of thing as completely abstract art. I’m not precisely sure what Peirce means by “electromagnetic vortex flow”. These are all terms that mathematicians, and mathematical physicists, would be interested in. That specific combination, though, I can find only a few references for. It seems to serve as a sensing tool, though.
No matter. Electromagnetic fields are interesting to a mathematical physicist, and so mathematicians. Often a field like this can be represented as a system of vortices, too, points around which something swirls and which combine into the field that we observe. This can be a way to turn a continuous field into a set of discrete particles, which we might have better tools to study. And to draw what electromagnetic fields look like — even in a very rough form — can be a great help to understanding what they will do, and why. They also can be beautiful in ways that communicate even to those who don’t undrestand the thing modelled.
Megan Dong’s Sketchshark Comics for the 25th is a joke based on the reputation of the Golden Ratio. This is the idea that the ratio, (roughly 1:1.6), is somehow a uniquely beautiful composition. You may sometimes see memes with some nice-looking animal and various boxes superimposed over it, possibly along with a spiral. The rectangles have the Golden Ratio ratio of width to height. And the ratio is kind of attractive since is about 1.618, and is about 0.618. It’s a cute pattern, and there are other similar cute patterns.. There is a school of thought that this is somehow transcendently beautiful, though.
It’s all bunk. People may find stuff that’s about one-and-a-half times as tall as it is wide, or as wide as it is tall, attractive. But experiments show that they aren’t more likely to find something with Golden Ratio proportions more attractive than, say, something with proportions, or , or even to be particularly consistent about what they like. You might be able to find (say) that the ratio of an eagle’s body length to the wing span is something close to . But any real-world thing has a lot of things you can measure. It would be surprising if you couldn’t find something near enough a ratio you liked. The guy is being ridiculous.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 26th builds on the idea that everyone could be matched to a suitable partner, given a proper sorting algorithm. I am skeptical of any “simple algorithm” being any good for handling complex human interactions such as marriage. But let’s suppose such an algorithm could exist.
This turns matchmaking into a problem of linear programming. Arguably it always was. But the best possible matches for society might not — likely will not be — the matches everyone figures to be their first choices. Or even top several choices. For one, our desired choices are not necessarily the ones that would fit us best. And as the punch line of the comic implies, what might be the globally best solution, the one that has the greatest number of people matched with their best-fit partners, would require some unlucky souls to be in lousy fits.
Although, while I believe that’s the intention of the comic strip, it’s not quite what’s on panel. The assistant is told he’ll be matched with his 4,291th favorite choice, and I admit having to go that far down the favorites list is demoralizing. But there are about 7.7 billion people in the world. This is someone who’ll be a happier match with him than 6,999,995,709 people would be. That’s a pretty good record, really. You can fairly ask how much worse that is than the person who “merely” makes him happier than 6,999,997,328 people would
I like this scheme where I use the Sunday publication slot to list comics that mention mathematics without inspiring conversation. I may need a better name for that branch of the series, though. But, nevertheless, here are comic strips from last week that don’t need much said about them.
John Deering’s Strange Brew for the 24th features Pythagoras, here being asked about his angles. I’m not aware of anything actually called a Pythagorean Angle, but there’s enough geometric things with Pythagoras’s name attached for the joke to make sense.
Maria Scrivan’s Half Full for the 25th is a Venn Diagram joke for the week. It doesn’t quite make sense as a Venn Diagram, as it’s not clear to me that “invasive questions” is sensibly a part of “food”. But it’s a break from every comic strip doing a week full of jokes about turkeys preferring to not be killed.
Tony Carrillo’s F Minus for the 26th is set in mathematics class. And talks about how the process of teaching mathematics is “an important step on the road to hating math”, which is funny because it’s painfully true.
There were just a handful of comic strips that mentioned mathematical topics I found substantial. Of those that did, computational science came up a couple times. So that’s how we got to here.
Rick Detorie’s One Big Happy for the 17th has Joe writing an essay on the history of computing. It’s basically right, too, within the confines of space and understandable mistakes like replacing Pennsylvania with an easier-to-spell state. And within the confines of simplification for the sake of getting the idea across briefly. Most notable is Joe explaining ENIAC as “the first electronic digital computer”. Anyone calling anything “the first” of an invention is simplifying history, possibly to the point of misleading. But we must simplify any history to have it be understandable. ENIAC is among the first computers that anyone today would agree is of a kind with the laptop I use. And it’s certainly the one that, among its contemporaries, most captured the public imagination.
Incidentally, Heman Hollerith was born on Leap Day, 1860; this coming year will in that sense see only his 39th birthday.
Ryan North’s Dinosaur Comics for the 18th is based on the question of whether P equals NP. This is, as T-Rex says, the greatest unsolved problem in computer science. These are what appear to be two different kinds of problems. Some of them we can solve in “polynomial time”, with the number of steps to find a solution growing as some polynomial function of the size of the problem. Others seem to be “non-polynomial”, meaning the number of steps to find a solution grows as … something not a polynomial.
You see one problem. Not knowing a way to solve a problem in polynomial time does not necessarily mean there isn’t a solution. It may mean we just haven’t thought of one. If there is a way we haven’t thought of, then we would say P equals NP. And many people assume that very exciting things would then follow. Part of this is because computational complexity researchers know that many NP problems are isomorphic to one another. That is, we can describe any of these problems as a translation of another of these problems. This is the other part which makes this joke: the declaration that ‘whether God likes poutine’ is isomorphic to the question ‘does P equal NP’.
We tend to assume, also, that if P does equal NP then NP problems, such as breaking public-key cryptography, are all suddenly easy. This isn’t necessarily guaranteed. When we describe something as polynomial or non-polynomial time we’re talking about the pattern by which the number of steps needed to find the solution grows. In that case, then, an algorithm that takes one million steps plus one billion times the size-of-the-problem to the one trillionth power is polynomial time. An algorithm that takes two raised to the size-of-the-problem divided by one quintillion (rounded up to the next whole number) is non-polynomial. But for most any problem you’d care to do, this non-polynomial algorithm will be done sooner. If it turns out P does equal NP, we still don’t necessarily know that NP problems are practical to solve.
Bil Keane and Jeff Keane’s The Family Circus for the 20th has Dolly explaining to Jeff about the finiteness of the alphabet and infinity of numbers. I remember in my childhood coming to understand this and feeling something unjust in the difference between the kinds of symbols. That we can represent any of those whole numbers with just ten symbols (thirteen, if we include commas, decimals, and a multiplication symbol for the sake of using scientific notation) is an astounding feat of symbolic economy.
Zach Weinersmth’s Saturday Morning Breakfast cereal for the 21st builds on the statistics of genetics. In studying the correlations between one thing and another we look at something which varies, usually as the result of many factors, including some plain randomness. If there is a correlation between one variable and another we usually can describe how much of the change in one quantity depends on the other. This is what the scientist means on saying the presence of this one gene accounts for 0.1% of the variance in eeeeevil. The way this is presented, the activity of one gene is responsible for about one-thousandth of the level of eeeeevil in the person.
As the father observes, this doesn’t seem like much. This is because there are a lot of genes describing most traits. And that before we consider epigenetics, the factors besides what is in DNA that affect how an organism develops. I am, unfortunately, too ignorant of the language of genetics to be able to say what a typical variation for a single gene would be, and thus to check whether Weinersmith has the scale of numbers right.
I’m finding it surprisingly good for my workflow to use Sundays for the comic strips which mention mathematics only casually. Tomorrow or so I’ll get to the ones with substantial material, in an essay available at this link.
Jim Meddick’s Monty for the 19th is a sudoku joke, with Monty filling in things that aren’t numerals. Many of them are commonly used mathematical symbols. The ones that I don’t recognize I suspect come from physics applications, especially particle physics. These rely heavily on differential equations and group theory and are likely where Meddick got things like the and the from.
Now let me discuss the comic strips from last week with some real meat to their subject matter. There weren’t many: after Wednesday of last week there were only casual mentions of any mathematics topic. But one of the strips got me quite excited. You’ll know which soon enough.
Mac King and Bill King’s Magic in a Minute for the 10th uses everyone’s favorite topological construct to do a magic trick. This one uses a neat quirk of the Möbius strip: that if sliced along the center of its continuous loop you get not two separate shapes but one Möbius strip of greater length. There are more astounding feats possible. If the strip were cut one-third of the way from an edge it would slice the strip into two shapes, one another Möbius strip and one a simple loop.
Or consider not starting with a Möbius strip. Make the strip of paper by taking one end and twisting it twice around, for a full loop, before taping it to the other end. Slice this down the center and what results are two interlinked rings. Or place three twists in the original strip of paper before taping the ends together. Then, the shape, cut down the center, unfolds into a trefoil knot. But this would take some expert hand work to conceal the loops from the audience while cutting. It’d be a neat stunt if you could stage it, though.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 10th uses mathematics as obfuscation. We value mathematics for being able to make precise and definitely true statements. And for being able to describe the world with precision and clarity. But this has got the danger that people hear mathematical terms and tune out, trusting that the point will be along soon after some complicated talk.
The formulas on the blackboard are nearly all legitimate, and correct, formulas for the value of π. The upper-left and the lower-right formulas are integrals, and ones that correspond to particular trigonometric formulas. The The middle-left and the upper-right formulas are series, the sums of infinitely many terms. The one in the upper right, , was roughly proven by Leonhard Euler. Euler developed a proof that’s convincing, but that assumed that infinitely-long polynomials behave just like finitely-long polynomials. In this context, he was correct, but this can’t be generally trusted to happen. We’ve got proofs that, to our eyes, seem rigorous enough now.
The center-left formula doesn’t look correct to me. To my eye, this looks like a mistaken representation of the formula
The center-right formula is interesting because, in part, it looks weird. It’s written out as
That looks at first glance like something’s gone wrong with one of those infinite-product series for π. Not so; this is a notation used for continued fractions. A continued fraction has a string of denominators that are typically some whole number plus another fraction. Often the denominator of that fraction will itself be a whole number plus another fraction. This gets to be typographically challenging. So we have this notation instead. Its syntax is that
There are many attractive formulas for π. It’s temping to say this is because π is such a lovely number it naturally has beautiful formulas. But more likely humans are so interested in π we go looking for formulas with some appealing sequence to them. There are some awful-looking formulas out there too. I don’t know your tastes, but for me I feel my heart cool when I see that π is equal to four divided by this number:
however much I might admire the ingenuity which found that relationship, and however efficiently it may calculate digits of π.
Glenn McCoy and Gary McCoy’s The Duplex for the 13th uses skill at arithmetic as shorthand for proving someone’s a teacher. There’s clearly some implicit idea that this is a school teacher, probably for elementary schools, and doesn’t have a particular specialty. But it is only three panels; they have to get the joke done, after all.
So finally I get to the mathematically-themed comic strips of last week. There were four strips which group into natural pairings. So let’s use that as the name for this edition.
Vic Lee’s Pardon My Planet for the 3rd puts forth “cookie and cake charts”, as a riff on pie charts. There’s always room for new useful visual representations of data, certainly, although quite a few of the ones we do use are more than two centuries old now. Pie charts, which we trace to William Playfair’s 1801 Statistical Breviary, were brought to the public renown by Florence Nightingale. She wanted her reports on the causes of death in the Crimean War to communicate well, and illustrations helped greatly.
Wayno and Piraro’s Bizarro for the 9th is another pie chart joke. If I weren’t already going on about pie charts this week I probably would have relegated this to the “casual mentions” heap. I love the look of the pie, though.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 5th jokes about stereotypes of mathematics and English classes. Or exams, anyway. There is some stabbing truth in the presentation of English-as-math-class. Many important pieces of mathematics are definitions or axioms. In an introductory class there’s not much you can usefully say about, oh, why we’d define a limit to be this rather than that. The book surely has its reasons and we’ll avoid confusion by trusting in them.
I dislike the stereotype of English as a subject rewarding longwinded essays that avoid the question. It seems at least unfair to what good academic writing strives for. (If you wish to argue about bad English writing, you have your blog for that, but let’s not pretend mathematics lacks fundamentally bad papers.) And writing an essay about why a thing should be true, or interesting, is certainly worthwhile. I’m reminded of a mathematical logic professor I had, who spoke of a student who somehow could not do a traditional proper-looking proof. But could write a short essay explaining why a thing should be true which convinced the professor that the student deserved an A. The professor was sad that the student was taking the course pass-fail.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 6th shows off a bit of mathematical modeling. The specific problem is silly, yes. But the approach is dead on: identify the things that affect what you’re interested in, and how they interact. Add to this estimates of the things’ values and you’ll get at least a provisional answer. You can then use that answer to guide the building of a more precise model, if you need one.
This little bugs-on-Superman problem makes note of the units everything’s measured in. Paying attention to the units is often done in dimensional analysis, a great tool for building simple models. I ought to write an essay sequence about that sometime.
Mark Anderson’s Andertoons for the 9th is the Mark Anderson’s Andertoons for the week. This one plays on the use of the same word to measure an angle and a temperature. Degree, etymologically, traces back to “a step”, like you might find in stairs. This, taken to represent a stage of progress, got into English in the 13th century. By the late 14th century “degree” was used to describe this 1/360th slice of a circle. By the 1540s it was a measure of heat. Making the degree the unit of temperature, as on a thermometer, seems to be written down only as far back as the 1720s.
And for a last strip of the week, Gary Wise and Lance Aldrich’s Real Life Adventures for the 7th mentions an advantage of being a cartoonist “instead of an engineer” is how cartooning doesn’t require math. Also I guess this means the regular guy in Real Life Adventures represents one (or both?) of the creators? I guess that makes the name Real Life Adventures make more sense. I just thought he was a generic comic strip male. And, of course, there’s nothing about mathematics that keeps one from being a cartoonist, although I don’t know of any current daily-syndicated cartoonists with strong mathematics backgrounds. Bill Amend, of FoxTrot, and Bud Grade, of The Piranha Club/Ernie, were both physics majors, which is a heavy-mathematics program.
As will sometimes happen it’s inconvenient for met to write up a paragraph or two on the particularly mathematically significant comic strips of the past week. Let me here share the comics that just mentioned mathematics, then, and save the heavy stuff for a bit later on.
And this covers things through to Friday’s comics. I write this not having had the chance to read Saturday’s yet. When I do, and when I have the whole week’s strips to discuss, I’ll have it at this link. Furthermore, this week sees the last quarter of the Fall 2019 A to Z under way. I’m excited to learn what I’m doing for the letter ‘U’ also.
I knew by Thursday this would be a brief week. The number of mathematically-themed comic strips has been tiny. I’m not upset, as the days turned surprisingly full on me once again. At some point I would have to stop being surprised that every week is busier than I expect, right?
Anyway, the week gives me plenty of chances to look back to 1936, which is great fun for people who didn’t have to live through 1936.
Elzie Segar’s Thimble Theatre rerun for the 28th of October is part of the story introducing Eugene the Jeep. The Jeep has astounding powers which, here, are finally explained as being due to it being a fourth-dimensional creature. Or at least able to move into the fourth dimension. This is amazing for how it shows off the fourth dimension being something you could hang a comic strip plot on, back in the day. (Also back in the day, humor strips with ongoing plots that might run for months were very common. The only syndicated strips like it today are Gasoline Alley, Alley Oop, the current storyline in Safe Havens where they’ve just gone and terraformed Mars, and Popeye, rerunning old daily stories.) The Jeep has many astounding powers, including that he can’t be kept inside — or outside — anywhere against his will, and he’s able to forecast the future.
Could there be a fourth-dimensional animal? I dunno, I’m not a dimensional biologist. It seems like we need a rich chemistry for life to exist. Lots of compounds, many of them long and complicated ones. Can those exist in four dimensions? I don’t know the quantum mechanics of chemical formation well enough to say. I think there’s obvious problems. Electrical attraction and repulsion would fall off much more rapidly with distance than they do in three-dimensional space. This seems like it argues chemical bonds would be weaker things, which generically makes for weaker chemical compounds. So probably a simpler chemistry. On the other hand, what’s interesting in organic chemistry is shapes of molecules, and four dimensions of space offer plenty of room for neat shapes to form. So maybe that compensates for the chemical bonds. I don’t know.
But if we take the premise as given, that there is a four-dimensional animal? With some minor extra assumptions then yeah, the Jeep’s powers fit well enough. Not being able to be enclosed follows almost naturally. You, a three-dimensional being, can’t be held against your will by someone tracing a line on the floor around you. The Jeep — if the fourth dimension is as easy to move through as the third — has the same ability.
Forecasting the future, though? We have a long history of treating time as “the” fourth dimension. There’s ways that this makes good organizational sense. But we do have to treat time as somehow different from space, even to make, for example, general relativity work out. If the Jeep can see and move through time? Well, yeah, then if he wants he can check on something for you, at least if it’s something whose outcome he can witness. If it’s not, though? Well, maybe the flow of events from the fourth dimension is more obvious than it is from a mere three, in the way that maybe you can spot something coming down the creek easily, from above, in a way that people on the water can’t tell.
Olive Oyl and Popeye use the Jeep to tease one another, asking for definite answers about whether the other is cute or not. This seems outside the realm of things that the fourth dimension could explain. In the 1960s cartoons he even picks up the power to electrically shock offenders; I don’t remember if this was in the comic strips at all.
Elzie Segar’s Thimble Theatre rerun for the 29th of October has Wimpy doing his best to explain the fourth dimension. I think there’s a warning here for mathematician popularizers here. He gets off to a fair start and then it all turns into a muddle. Explaining the fourth dimension in terms of the three dimensions we’re familiar with seems like a good start. Appealing to our intuition to understand something we have to reason about has a long and usually successful history. But then Wimpy goes into a lot of talk about the mystery of things, and it feels like it’s all an appeal to the strangeness of the fourth dimension. I don’t blame Popeye for not feeling it’s cleared anything up. Segar would come back, in this storyline, to several other attempted explanations of the Jeep’s powers, although they do come back around to, y’know, it’s a magical animal. They’re all over the place in the Popeye comic universe.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 28th of October is a riff on predictability and encryption. Good encryption schemes rely on randomness. Concealing the content of a message means matching it to an alternate message. Each of the alternate messages should be equally likely to be transmitted. This way, someone who hasn’t got the key would not be able to tell what’s being sent. The catch is that computers do not truly do randomness. They mostly rely on quasirandom schemes that could, in principle, be detected and spoiled. There are ways to get randomness, mostly involving putting in something from the real world. Sensors that detect tiny fluctuations in temperature, for example, or radio detectors. I recall one company going for style and using a wall of lava lamps, so that the rise and fall of lumps were in some way encoded into unpredictable numbers.
Robb Armstrong’s JumpStart for the 2nd of November is a riff on the Birthday “Paradox”, the thing where you’re surprised to find someone shares a birthday with you. (I have one small circle of friends featuring two people who share my birthday, neatly enough.) Paradox is in quotes because it defies only intuition, not logic. The logic is clear that you need only a couple dozen people before some pair will probably share a birthday. Marcie goes overboard in trying to guess how many people at her workplace would share their birthday on top of that. Birthdays are nearly uniformly spread across all days of the year. There are slight variations; September birthdays are a little more likely than, say, April ones; the 13th of any month is a less likely birthday than the 12th or the 24th are. But this is a minor correction, aptly ignored when you’re doing a rough calculation. With 615 birthdays spread out over the year you’d expect the average day to be the birthday of about 1.7 people. (To be not silly about this, a ten-day span should see about 17 birthdays.) However, there are going to be “clumps”, days where three or even four people have birthdays. There will be gaps, days nobody has a birthday, or even streaks of days where nobody has a birthday. If there weren’t a fair number of days with a lot of birthdays, and days with none, we’d have to suspect birthdays weren’t random here.
There were also a handful of comic strips just mentioning mathematics, that I can’t make anything in depth about. Here’s two.
Chris Browne’s Raising Duncan rerun for the 23rd has a man admitting bad mathematics skills for why he can’t count the ways he loves his wife. The strip originally ran the 27th of September, 2003. (The strip was short-lived, and is in perpetual reruns. It may be worth reading at least one time through, though, since the pairs of main characters in it are eagerly in love, without being sappy about it, and it’s pleasant seeing people enthusiastic about each other. This is the strip that had the exchange “Marry me!” “I did!” “Marry me more!” “Okay!” that keeps bringing me cheer and relationship goals.)
The past week started strong for mathematically-themed comics. Then it faded out into strips that just mentioned the existence of mathematics. I have no explanation for this phenomenon. It makes dividing up the week’s discussion material easy enough, though.
John Zakour and Scott Roberts’s Maria’s Day rerun for the 19th is a lottery joke. Maria’s come up with a scheme to certainly win the grand prize in a lottery. There’s no disputing that one could, on buying enough tickets, get an appreciable chance of winning. Even, in principle, get a certain win. There’s no guaranteeing a solo win, though. But sometimes lottery jackpots will grow large enough that even if you had to split the prize two or three ways it’d be worth it.
Tom Horacek’s Foolish Mortals for the 21st plays on the common wisdom that mathematicians’ best work is done when they’re in their 20s. Or at least their most significant work. I don’t like to think that’s so, as someone who went through his 20s finding nothing significant. But my suspicion is that really significant work is done when someone with fresh eyes looks at a new problem. Young mathematicians are in a good place to learn, and are looking at most everything with fresh eyes, and every problem is new. Still, experienced mathematicians, bringing the habits of thought that served well one kind of problem, looking at something new will recreate this effect. We just need to find ideas to think about that we haven’t worn down.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 21st has a petitioner asking god about whether P = NP. This is shorthand for a famous problem in the study of algorithms. It’s about finding solutions to problems, and how much time it takes to find the solution. This time usually depends on the size of whatever it is you’re studying. The question, interesting to mathematicians and computer scientists, is how fast this time grows. There are many classes of these problems. P stands for problems solvable in polynomial time. Here the number of steps it takes grows at, like, the square or the cube or the tenth power of the size of the thing. NP is non-polynomial problems, growing, like, with the exponential of the size of the thing. (Do not try to pass your computer science thesis defense with this description. I’m leaving out important points here.) We know a bunch of P problems, as well as NP problems.
Like, in this comic, God talks about the problem of planning a long delivery route. Finding the shortest path that gets to a bunch of points is an NP problem. What we don’t know about NP problems is whether the problem is we haven’t found a good solution yet. Maybe next year some bright young 68-year-old mathematician will toss of a joke on a Reddit subthread and then realize, oh, this actually works. Which would be really worth knowing. One thing we know about NP problems is there’s a big class of them that are all, secretly, versions of each other. If we had a good solution for one we’d have a solution for all of them. So that’s why a mathematician or computer scientist would like to hear God’s judgement on how the world is made.
Hector D. Cantú and Carlos Castellanos’s Baldo for the 22nd has Baldo asking his sister to do some arithmetic. I fancy he’s teasing her. I like doing some mental arithmetic. If nothing else it’s worth having an expectation of the answer to judge whether you’ve asked the computer to do the calculation you actually wanted.
Mike Thompson’s Grand Avenue for the 22nd has Gabby demanding to know the point of learning Roman numerals. As numerals, not much that I can see; they serve just historical and decorative purposes these days, mostly as a way to make an index look more fancy. As a way to learn that how we represent numbers is arbitrary, though? And that we can use different schemes if that’s more convenient? That’s worth learning, although it doesn’t have to be Roman numerals. They do have the advantage of using familiar symbols, though, which (say) the Babylonian sexagesimal system would not.
Gary Wise and Lance Aldrich’s Real Life Adventures for the 15th is a percentages joke. It’s really tempting to just add and subtract percentages like this, when talking about sales and interest and such. If the percentages are small, like, one or two percent, this is near enough to being right. A sale of 15 percent and interest of 22 percent? That’s not close enough to approximate like that. A 15 percent sale with 22 percent interest charge would come to about a 3.7 percent surcharge. But how long the charge stays on the credit card will affect the amount.
Ryan North’s Dinosaur Comics for the 17th has one long message turn out to encode a completely unrelated thing. This is something you can deliberately build in to a signal. You might want to, in order to confound codebreakers working on your message. It’s possible in any message to encode a second by accident. As you’d think, the longer the unintentional message the less likely it is to just turn up.