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.
I’m writing this on Thursday, because I’m expecting to be busy Friday and Saturday. It might be a good policy if I planned the deadline for all these Reading the Comics posts to be a couple days before publishing. But it’ll probably ever come to that. I am not yet begun resisting treating this blog like a professional would. Well, what’s been interesting this week so far have been comic strips presenting or about story problems. That’s enough for a theme.
Olivia Jaimes’s Nancy for the 13th makes the familiar complaint that story problems aren’t “useful”. Perhaps not; she can cut the Gordion knot for most common setups. Well, students aren’t likely to get problems for which there’s no other way to find a solution. I suppose what’s happening is that many mathematical puzzles come from questions like, what’s the least amount of information you need to deduce something? Or what’s an indirect way to find that? Mathematicians are often drawn to questions like this. At least Nancy has found there are problems she’s legitimately interested in, questions about how to do a thing she finds important.
Mort Walker and Dik Browne’s vintage Hi and Lois for the 17th has Chip’s new arithmetic book trying to be more relevant. Chip’s still bored by the problem, which chooses to be about foreign aid. (In the 50s and 60s comics discovered it was very funny that the United States would just give money to other countries and not get anything back out of it except maybe their economies staying stable or their countries not going to war too much.)
Today, I’m just listing the comics from last week that mentioned mathematics, but which didn’t raise a deep enough topic to be worth discussing. You know what a story problem looks like. I can’t keep adding to that.
Hector D. Cantú and Carlos Castellanos’s Baldo for the 10th quotes René Descartes, billing him as a “French mathematician”. Which is true, but the quote is one about living properly. That’s more fairly a philosophical matter. Descartes has some reputation for his philosophical work, I understand.
Comic Strip Master Command hoped to give me an easy week, one that would let me finally get ahead on my A-to-Z essays and avoid the last-minute rush to complete tasks. I showed them, though. I can procrastinate more than they can give me breaks. This essay alone I’m writing about ten minutes after you read it.
Eric the Circle for the 7th, by Shoy, is one of the jokes where Eric’s drawn as something besides a circle. I can work with this, though, because the cube is less far from a circle than you think. It gets to what we mean by “a circle”. If it’s all the points that are exactly a particular distance from a given center? Or maybe all the points up to that particular distance from a given center? This seems too reasonable to argue with, so you know where the trick is.
The trick is asking what we mean by distance? The ordinary distance that normal people use has a couple names. The Euclidean distance, often. Or Euclidean metric. Euclidean norm. It has some fancier names that can wait. Give two points. You can find this distance easily if you have their coordinates in a Cartesian system. (There’s infinitely many Cartesian systems you could use. You can pick whatever one you like; the distance will be the same whatever they are.) That’s that thing about finding the distance between corresponding coordinates, squaring those distances, adding that up, and taking the square root. And that’s good.
That’s not our only choice, though. We can make a perfectly good distance using other rules. For example, take the difference between corresponding coordinates, take the absolute value of each, and add all those absolute values up. This distance even has real-world application. It’s how far it is to go from one place to another on a grid of city squares, where it’s considered poor form to walk directly through buildings. There’s another. Instead of adding those absolute values up? Just pick the biggest of the absolute values. This is another distance. In it, circles look like squares. Or, in three dimensions, spheres look like cubes.
Ryan North’s Dinosaur Comics for the 9th builds on a common science fictional premise, that contact with an alien intelligence is done through mathematics first. It’s a common supposition in science fiction circles, and among many scientists, that mathematics is a truly universal language. It’s hard to imagine a species capable of communication with us that wouldn’t understand two and two adding up to four. Or about the ratio of a circle circumference to its diameter being independent of that diameter. Or about how an alternating knot for which the minimum number of crossing points is odd can’t ever be amphicheiral.
All right, I guess I can imagine a species that never ran across that point. Which is one of the things we suppose in using mathematics as a universal language. Its truths are indisputable, if we allow the rules of logic and axioms and definitions that we use. And I agree I don’t know that it’s possible not to notice basic arithmetic and basic geometry, not if one lives in a sensory world much like humans’. But it does seem to me at least some of mathematics is probably idiosyncratic. In representation at least; certainly in organization. I suspect there may be trouble in using universal and generically true things to express something local and specific. I don’t know how to go from deductive logic to telling someone when my birthday is. Well, I’m sure our friends in the philosophy department have considered that problem and have some good thoughts we can use, if there were only some way to communicate with them.
Bill Whitehead’s Free Range for the 12th is your classic blackboard-full-of-symbols. I like the beauty of the symbols used. I mean, the whole expression doesn’t parse, but many of the symbols do and are used in reasonable ways. Long trailing strings of arrows to extend one line to another are common and reasonable too. In the middle of the second line is , which doesn’t make sense, but which doesn’t make sense in a way that seems authentic to working out an idea. It’s something that could be cleaned up if the reasoning needed to be made presentable.
John Hambrock’s The Brilliant Mind of Edison Lee for the 1st of October is a calendar joke. Well, many of the months used to have names that denoted their count. Month names have changed more than you’d think. For a while there every Roman Emperor was renaming months after himself. Most of these name changes did not stick. Lucius Aurelius Commodus, who reined from 177 to 192, gave all twelve months one or another of his names.
Several of the mathematically-themed comic strips from last week featured the fine art of calculation. So that was set to be my title for this week. Then I realized that all the comics worth some detailed mention were published last Sunday, and I do like essays that are entirely one-day affairs. There are a couple of other comic strips that mentioned mathematics tangentially and I’ll list those later this week.
John Hambrock’s The Brilliant Mind of Edison lee for the 29th has Edison show off an organic computer. This is a person, naturally enough. Everyone can do some arithmetic in their heads, especially if we allow that sometimes approximate answers are often fine. People with good speed and precision have always been wonders, though. The setup may also riff on the ancient joke of mathematicians being ways to turn coffee into theorems. (I would imagine that Hambrock has heard that joke. But it is enough to suppose that he’s aware many adult humans drink coffee.)
John Kovaleski’s Daddy Daze for the 29th sees Paul, the dad, working out the calculations his son (Angus) proposed. It’s a good bit of arithmetic that Paul’s doing in his head. The process of multiplying an insubstantial thing by many, many times until you get something of moderate size happens all the time. Much of integral calculus is based on the idea that we can add together infinitely many infinitesimal numbers, and from that get something understandable on the human scale. Saving nine seconds every other day is useless for actual activities, though. You need a certain fungibility in the thing conserved for the bother to be worth it.
Dan Thompson’s Harley for the 29th gets us into some comic strips not drawn by people named John. The comic has some mathematics in it qualitatively. The observation that you could jump a motorcycle farther, or higher, with more energy, and that you can get energy from rolling downhill. It’s here mostly because of the good fortune that another comic strip did a joke on the same topic, and did it quantitatively. That comic?
Bill Amend’s FoxTrot for the 29th. Young prodigies Jason and Marcus are putting serious calculation into their Hot Wheels track and working out the biggest loop-the-loop possible from a starting point. Their calculations are right, of course. Bill Amend, who’d been a physics major, likes putting authentic mathematics and mathematical physics in. The key is making sure the car moves fast enough in the loop that it stays on the track. This means the car experiencing a centrifugal force that’s larger than that of gravity. The centrifugal force on something moving in a circle is proportional to the square of the thing’s speed, and inversely proportional to the radius of the circle. This for a circle in any direction, by the way.
So they need to know, if the car starts at the height A, how fast will it go at the top of the loop, at height B? If the car’s going fast enough at height B to stay on the track, it’s certainly going fast enough to stay on for the rest of the loop.
The hard part would be figuring the speed at height B. Or it would be hard if we tried calculating the forces, and thus acceleration, of the car along the track. This would be a tedious problem. It would depend on the exact path of the track, for example. And it would be a long integration problem, which is trouble. There aren’t many integrals we can actually calculate directly. Most of the interesting ones we have to do numerically or work on approximations of the actual thing. This is all right, though. We don’t have to do that integral. We can look at potential energy instead. This turns what would be a tedious problem into the first three lines of work. And one of those was “Kinetic Energy = Δ Potential Energy”.
But as Peter observes, this does depend on supposing the track is frictionless. We always do this in basic physics problems. Friction is hard. It does depend on the exact path one follows, for example. And it depends on speed in complicated ways. We can make approximations to allow for friction losses, often based in experiment. Or try to make the problem one that has less friction, as Jason and Marcus are trying to do.