Now at last I turn to last week’s mathematically-themed comic strips. They weren’t very deeply mathematical, I think. But I always think that right before I turn out a 2,000-word essay about some kid giving a snarky answer to an arithmetic problem.
Rick Detorie’s One Big Happy for the 12th has Ruthie try to teach her brother about number words. What Ruthie seems to be struggling with is the difference between a number and the name we give a number. The distinction between a thing and the name of a thing can be a tricky one, and I do remember being confused at the difference between the word “four” and the concept “four”. What I don’t remember, to my regret, is what thought I had which made the difference clear.
It was another pretty quiet week for mathematically-themed comic strips. Most of what did mention my subject just presented it as a subject giving them homework or quizzes or exams. But let’s look over what is here.
Ted Shearer’s Quincy for the 5th is the most interesting strip of the week, since it suggests an actual answerable mathematics problem. How much does a professional basketball player earn per dribble? The answer requires a fair bit of thought, like, what do you mean by “a professional basketball player”? There’s many basketball leagues around the world; even if we limit the question to United States-and-Canada leagues, there’s a fair number of minor leagues. If we limit it to the National Basketball Association there’s the question of whether the salary is the minimum union contract guarantee, or the mean salary, or the median salary. It’s exciting to look at the salary of the highest-paid players, too, of course.
Working out the number of dribbles per year is also a fun estimation challenge. Even if we pick a representative player there’s no getting an exact count of how many dribbles they’ve made over a year, even if we just consider “dribbling during games” to be what’s paid for. (And any reasonable person would have to count all the dribbling done during warm-up and practice as part of what’s being paid for.) But someone could come up with an estimate of, for example, about how long a typical player has the ball for a game, and how much of that time is spent moving the ball or preparing for a free throw or other move that calls for dribbling. How long a dribble typically takes. How many games a player typically plays over the year. The estimate you get from this will never, ever, be exactly right. But it should be close enough to give an idea how much money a player earns in the time it takes to dribble the ball once. So occasionally the comics put forth a good story problem after all.
Quincy on the 7th is again worrying about his mathematics and spelling tests. It’s a cute coincidence that these are the subjects worried about in Wee Pals too.
Paul Gilligan’s Pooch Cafe for the 7th is part of a string of jokes about famous dogs. This one’s a riff on Albert Einstein, mentioned here because Albert Einstein has such strong mathematical associations.
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.
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.
One thing to worry about during an A To Z sequence is how busy Comic Strip Master Command will decide I need to be. I’m glad to say that this first week, it wasn’t too overly busy. Even the comic strips that are most on topic are not ones that need too much explanation. They’re also all reruns from their original publication, although I don’t know the dates that any of these first ran. A casual search doesn’t find that I said anything about these in their previous appearances.
Mac King and Bill King’s Magic in a Minute for the 1st is a rerun printed without the editor reading the thing. If they had, they’d have edited the 13 to be a 19. As the explanation at the bottom of the page almost makes clear, the ‘magic number’ produced by this will be the last two digits of the current year. After all, your age (at the end of this year) will be this year minus the year of your birth.
That this can be used for a magic trick relies on two things. One is that while, yes, anyone who thinks about it sees the relationship between their birth year, their age, and the current year, the magic trick is done before they can do that thinking. They’re too busy calculating, and then counting out cards and trying to see where this is going. Calculating without thinking about why this calculation is dangerous for mathematics. But it allows for some recreational fun. the other thing this trick depends on is showmanship: the purpose of the calculation is meant to be surprising enough, and delightful enough, that people won’t care to deconstruct its logic.
Morrie Turner’s Wee Pals rerun for the 1st is your basic joke about the kid subverting a word problem. But it also shows a bit why mathematicians get trained to make as explicit as possible their assumptions. This saves us from dumb mistakes, but at the cost of putting a prologue to anything we do want to ask. But it’s a legitimate part of mathematics to look at the questions someone else has asked and find their unstated assumptions, the things that could be true and would make their claims wrong.
Ryan North’s Dinosaur Comics rerun for the 2nd presents Utahraptor struggling with a mathematics problem. This is in character for him and for the comic. The particular problem is a classic recreational mathematics puzzle. Given a balance that can only give relative weights, and that you can use up to three times, find the one ball out of twelve which is of a different weight. It’s also a classic information theory problem. We know we can solve it, though. Each weighing gives us information about which of the twelve balls might, or might not, be abnormal. There is enough information in these three weighings to pick out which ball is the unusual one.
Granted, though, just knowing three weighings are enough doesn’t tell us what to weigh, or in what order. I haven’t looked at the GoComics comments. But there are likely at least three people who’ve explained some way to do it. It’s worth playing with the problem a while to see if you have any good ideas. You can use coins if you want to play with possibilities.
Ryan North’s Dinosaur Comics rerun for the 6th is, as the last panels suggest, a sequel to a comic rerun in mid-August. The question of whether the word ‘heterological’ is itself heterological is a recasting of one of Eubulides’s paradoxes. It’s the problem of working out whether a self-referential statement can be true. Or false. It shouldn’t surprise us that common language statements can defy being called true or false. But definitions are so close to logical structures that it’s hard to see why these refuse to fit. The problem is silly, but why it’s silly is hard to say.
I hoped I’d get a Reading the Comics post in for Tuesday, and even managed it. With this I’m all caught up to the syndicated comic strips which, last week, brought up some mathematics topic. I’m open for nominations about what to publish here Thursday. Write in quick.
Hilary Price’s Rhymes With Orange for the 30th is a struggling-student joke. And set in summer school, so the comic can be run the last day of June without standing out to its United States audience. It expresses a common anxiety, about that point when mathematics starts using letters. It superficially seems strange that this change worries students. Students surely had encountered problems where some term in an equation was replaced with a blank space and they were expected to find the missing term. This is the same work as using a letter. Still, there are important differences. First is that a blank line (box, circle, whatever) has connotations of “a thing to be filled in”. A letter seems to carry meaning in to the problem, even if it’s just “x marks the spot”. And a letter, as we use it in English, always stands for the same thing (or at least the same set of things). That ‘x’ may be 7 in one problem and 12 in another seems weird. I mean weird even by the standards of English orthography.
A letter might represent a number whose value we wish to know; it might represent a number whose value we don’t care about. These are different ideas. We usually fall into a convention where numbers we wish to know are more likely x, y, and z, while those we don’t care about are more likely a, b, and c. But even that’s no reliable rule. And there may be several letters in a single equation. It’s one thing to have a single unknown number to deal with. To have two? Three? I don’t blame people fearing they can’t handle that.
Mark Leiknes’s Cow and Boy for the 30th has Billy and Cow pondering the Prisoner’s Dilemma. This is one of the first examples someone encounters in game theory. Game theory sounds like the most fun part of mathematics. It’s the study of situations in which there’s multiple parties following formal rules which allow for gains or losses. This is an abstract description. It means many things fit a mathematician’s idea of a game.
The Prisoner’s Dilemma is described well enough by Billy. It’s built on two parties, each — separately and without the ability to coordinate — having to make a choice. Both would be better off, under interrogation, to keep quiet and trust that the cops can’t get anything significant on them. But both have the temptation that if they rat out the other, they’ll get off free while their former partner gets screwed. And knowing that their partner has the same temptation. So what would be best for the two of them requires them both doing the thing that maximizes their individual risk. The implication is unsettling: everyone acting in their own best interest is supposed to produce the best possible result for society. And here, for the society of these two accused, it breaks down entirely.
Exponents have been written as numbers in superscript following a base for a long while now. The notation developed over the 17th century. I don’t know why mathematicians settled on superscripts, as opposed to the many other ways a base and an exponent might fit together. It’s a good mnemonic to remember, say, “z raised to the 10th” is z with a raised 10. But I don’t know the etymology of “raised” in a mathematical context well enough. It’s plausible that we say “raised” because that’s what the notation suggests.
The proof of the Pythagorean Theorem is one of the very many known to humanity. This one is among the family of proofs that are wordless. At least nearly wordless. You can get from here to with very little prompting. If you do need prompting, it’s this: there are two expressions for how much area of the square with sides a-plus-b. One of these expressions uses only terms of a and b. The other expression uses terms of a, b, and c. If this doesn’t get a bit of a grin out of you, don’t worry. There’s, like, 2,037 other proofs we already know about. We might ask whether we need quite so many proofs of the Pythagorean theorem. It doesn’t seem to be under serious question most of the time.
And then a couple comic strips last week just mentioned mathematics. Morrie Turner’s Wee Pals for the 1st of July has the kids trying to understand their mathematics homework. Could have been anything. Mike Thompson’s Grand Avenue for the 5th started a sequence with the kids at Math Camp. The comic is trying quite hard to get me riled up. So far it’s been the kids agreeing that mathematics is the worst, and has left things at that. Hrmph.
Rick Kirkman and Jerry Scott’s Baby Blues for the 23rd has a neat variation on story problems. Zoe’s given the assignment to make her own. I don’t remember getting this as homework, in elementary school, but it’s hard to see why I wouldn’t. It’s a great exercise: not just set up an arithmetic problem to solve, but a reason one would want to solve it.
Composing problems is a challenge. It’s a skill, and you might be surprised that when I was in grad school we didn’t get much training in it. We were just taken to be naturally aware of how to identify a skill one wanted to test, and to design a question that would mostly test that skill, and to write it out in a question that challenged students to identify what they were to do and how to do it, and why they might want to do it. But as a grad student I wasn’t being prepared to teach elementary school students, just undergraduates.
Mastroianni and Hart’s B.C. for the 23rd is a joke in the funny-definition category, this for “chaos theory”. Chaos theory formed as a mathematical field in the 60s and 70s, and it got popular alongside the fractal boom in the 80s. The field can be traced back to the 1890s, though, which is astounding. There was no way in the 1890s to do the millions of calculations needed to visualize any good chaos-theory problem. They had to develop results entirely by thinking.
Wiley’s definition is fine enough about certain systems being unpredictable. Wiley calls them “advanced”, although they don’t need to be that advanced. A compound pendulum — a solid rod that swings on the end of another swinging rod — can be chaotic. You can call that “advanced” if you want but then people are going to ask if you’ve had your mind blown by this post-singularity invention, the “screw”.
What makes for chaos is not randomness. Anyone knows the random is unpredictable in detail. That’s no insight. What’s exciting is when something’s unpredictable but deterministic. Here it’s useful to think of continental divides. These are the imaginary curves which mark the difference in where water runs. Pour a cup of water on one side of the line, and if it doesn’t evaporate, it eventually flows to the Pacific Ocean. Pour the cup of water on the other side, it eventually flows to the Atlantic Ocean. These divides are often wriggly things. Water may mostly flow downhill, but it has to go around a lot of hills.
So pour the water on that line. Where does it go? There’s no unpredictability in it. The water on one side of the line goes to one ocean, the water on the other side, to the other ocean. But where is the boundary? And that can be so wriggly, so crumpled up on itself, so twisted, that there’s no meaningfully saying. There’s just this zone where the Pacific Basin and the Atlantic Basin merge into one another. Any drop of water, however tiny, dropped in this zone lands on both sides. And that is chaos.
Neatly for my purposes there’s even a mountain at a great example of this boundary. Triple Divide Peak, in Montana, rests on the divides between the Atlantic and the Pacific basins, and also on the divide between the Atlantic and the Arctic oceans. (If one interprets the Hudson Bay as connecting to the Arctic rather than the Atlantic Ocean, anyway. If one takes Hudson Bay to be on the Atlantic Ocean, then Snow Dome, Alberta/British Columbia, is the triple point.) There’s a spot on this mountain (or the other one) where a spilled cup of water could go to any of three oceans.
John Graziano’s Ripley’s Believe It Or Not for the 23rd mentions one of those beloved bits of mathematics trivia, the birthday problem. That’s finding the probability that no two people in a group of some particular size will share a birthday. Or, equivalently, the probability that at least two people share some birthday. That’s not a specific day, mind you, just that some two people share a birthday. The version that usually draws attention is the relatively low number of people needed to get a 50% chance there’s some birthday pair. I haven’t seen the probability of 70 people having at least one birthday pair before. 99.9 percent seems plausible enough.
The birthday problem usually gets calculated something like this: Grant that one person has a birthday. That’s one day out of either 365 or 366, depending on whether we consider leap days. Consider a second person. There are 364 out of 365 chances that this person’s birthday is not the same as the first person’s. (Or 365 out of 366 chances. Doesn’t make a real difference.) Consider a third person. There are 363 out of 365 chances that this person’s birthday is going to be neither the first nor the second person’s. So the chance that all three have different birthdays is . Consider the fourth person. That person has 362 out of 365 chances to have a birthday none of the first three have claimed. So the chance that all four have different birthdays is . And so on. The chance that at least two people share a birthday is 1 minus the chance that no two people share a birthday.
As always happens there are some things being assumed here. Whether these probability calculations are right depends on those assumptions. The first assumption being made is independence: that no one person’s birthday affects when another person’s is likely to be. Obvious, you say? What if we have twins in the room? What if we’re talking about the birthday problem at a convention of twins and triplets? Or people who enjoyed the minor renown of being their city’s First Babies of the Year? (If you ever don’t like the result of a probability question, ask about the independence of events. Mathematicians like to assume independence, because it makes a lot of work easier. But assuming isn’t the same thing as having it.)
The second assumption is that birthdates are uniformly distributed. That is, that a person picked from a room is no more likely to be born the 13th of February than they are the 24th of September. And that is not quite so. September births are (in the United States) slightly more likely than other months, for example, which suggests certain activities going on around New Year’s. Across all months (again in the United States) birthdates of the 13th are slightly less likely than other days of the month. I imagine this has to be accounted for by people who are able to select a due date by inducing delivery. (Again if you need to attack a probability question you don’t like, ask about the uniformity of whatever random thing is in place. Mathematicians like to assume uniform randomness, because it akes a lot of work easier. But assuming it isn’t the same as proving it.)
Do these differences mess up the birthday problem results? Probably not that much. We are talking about slight variations from uniform distribution. But I’ll be watching Ripley’s to see if it says anything about births being more common in September, or less common on 13ths.
There’s three more comics from last week I want to talk about. To ease my workload I’m going to put those off until Saturday. This is not an attempt to inflate the number of posts I make so that I can do a post-a-day-for-a-month again, as has happened in previous A-to-Z series. I already missed yesterday anyway. I just didn’t have time to think of things to write about six comics yesterday.
Morrie Turner’s Wee Pals for the 3rd has an interesting description of a circle. Definitions are a big part of mathematical work. This is especially so as we tend to think of mathematical objects as things that relate to one another in different ways. You want a definition that includes the relationships that are important, and excludes the ones you don’t want.
Nipper’s definition of a circle … well, eh. I wouldn’t say that captures a circle. A ‘closed smooth curve’, yes. It’s closed because the ends join up. It’s smooth because there aren’t any corners, any kinks in it. It’s a curve because … well, there you go. There are many interesting shapes that are closed smooth curves. You can find some by tossing a rubber band in the air and seeing what it looks like when it lands. But I think what most people find important about circles are ideas like all the points on a curve being the same distance from some single “center” point. Nipper would probably realize his definition didn’t work by experimenting. Try drawing shapes that meet the rule he set out, but that aren’t what he thinks a circle ought to be.
This can be fruitful. It can develop a sharper idea of what a definition ought to have. Or it might force you to accept, in order to get the cases you want included, that something which seems wrong has to count too. This mathematicians faced in the late 19th and early 20th centuries. We learned that the best definition we’ve had for an idea like “a continuous function” means we have to allow weird conclusions, like that it’s possible to have a function continuous at a single point and nowhere else. But any other definition rules out things we absolutely have to call continuous, so, what’s there to do?
Jenny Campbell’s Flo and Friends for the 4th presents algebra as one of the burdens of youth. And one that’s so harsh that it makes old age more pleasant. I get the unpleasantness of being stuck in a class one doesn’t understand or like. But my own slight experience with that thing where you wake up, and a thing hurts, and there’s no good reason but eventually it either goes away or you get so used to it you don’t realize it still actually hurts? I would take the boring class, most of the time.
Ryan North’s Dinosaur Comics for the 18th is based on Hilbert’s Hotel. This is a construct very familiar to eager young mathematicians. It’s an almost unavoidable pop-mathematics introduction to infinitely large sets. It’s a great introduction because the model is so mundane as to be easily imagined. But you can imagine experiments with intuition-challenging results. T-Rex describes one of the classic examples in the third through fifth panels.
The strip made me wonder about the origins of Hilbert’s Hotel. Everyone doing pop mathematics uses the example, but who created it? And the startling result is, David Hilbert, kind of. My reference here is Helge Kragh’s paper The True (?) Story of Hilbert’s Infinite Hotel. Apparently in a 1924-25 lecture series in Göttingen, Hilbert encouraged people to think of a hotel with infinitely many rooms. He apparently did not use it for so many examples as pop mathematicians would. He just used the question of how to accommodate a single new guest after the infinitely many rooms were first filled. And then went to imagine an infinite dance party. I don’t remember ever seeing the dance party in the wild; perhaps it’s a casualty of modern rave culture.
Hilbert’s Hotel seems to have next seen print in George Gamow’s One, Two Three … Infinity. Gamow summoned the hotel back from the realms of forgotten pop mathematics with a casual, jokey tone that fooled Kragh into thinking he’d invented the model and whimsically credited Hilbert with it. (Gamow was prone to this sort of lighthearted touch.) He came back to it in The Creation Of The Universe, less to make readers consider the modern understanding of infinitely large sets than to argue for a universe having infinitely many things in it.
And then it disappeared again, except for cameo appearances trying to argue that the steady-state universe would be more bizarre than what we actually see. The philosopher Pamela Huby seems to have made Hilbert’s Hotel a thing to talk about again, as part of a debate about whether a universe could be infinite in extent. William Lane Craig furthered using the hotel, as part of the theological debate about whether there could be an infinite temporal regress of events. Rudy Rucker and Eli Maor wrote descriptions of the idea in the 1980s, with vague ideas about whether Hilbert actually had anything to do with the place. And since then it’s stayed, a famous fictional hotel.
David Hilbert was born in 1862; T-Rex misspoke.
Ernie Bushmiller’s Nancy Classics for the 20th gets me out of my Olivia Jaimes rut. We could probably get a good discussion going about whether giving an example of a sphere is an adequate description of a sphere. Granted that a bubble-gum bubble won’t be perfectly spherical; neither will any example that exists in reality. We always trust that we can generalize to an ideal example of this thing.
I did get to wondering, in Sluggo’s description of the octagon, why the specification of eight sides and eight angles. I suspect it’s meant to avoid calling an octagon something that, say, crosses over itself, thus having more angles than sides. Not sure, though. It might be a phrasing intended to make sure one remembers that there are sides and there are angles and the polygon can be interesting for both sets of component parts.
John Atkinson’s Wrong Hands for the 20th is the Venn Diagram joke for the week. The half-week anyway. Also a bunch of other graph jokes for the week. Nice compilation of things. I love the paradoxical labelling of the sections of the Venn Diagram.
Tom II Wilson’s Ziggy for the 20th is a plaintive cry for help from a despairing soul. Who’s adding up four- and five-digit numbers by hand for some reason. Ziggy’s got his projects, I guess is what’s going on here.
Glenn McCoy and Gary McCoy’s The Duplex for the 21st is set up as an I-hate-word-problems joke. The cop does ask something people would generally like to know, though: how much longer would it take, going 60 miles per hour rather than 70? It turns out it’s easy to estimate what a small change in speed does to arrival time. Roughly speaking, reducing the speed one percent increases the travel time one percent. Similarly, increasing speed one percent decreases travel time one percent. Going about five percent slower should make the travel time a little more than five percent longer. Going from 70 to 60 miles per hour reduces the speed about fifteen percent. So travel time is going to be a bit more than 15 percent longer. If it was going to be an hour to get there, now it’ll be an hour and ten minutes. Roughly. The quality of this approximation gets worse the bigger the change is. Cutting the speed 50 percent increases the travel time rather more than 50 percent. But for small changes, we have it easier.
There are a couple ways to look at this. One is as an infinite series. Suppose you’re travelling a distance ‘d’, and had been doing it at the speed ‘v’, but now you have to decelerate by a small amount, ‘s’. Then this is something true about your travel time ‘t’, and I ask you to take my word for it because it has been a very long week and I haven’t the strength to argue the proposition:
‘d’ divided by ‘v’ is how long your travel took at the original speed. And, now, — the fraction of how much you’ve changed your speed — is, by assumption, small. The speed only changed a little bit. So is tiny. And is impossibly tiny. And is ridiculously tiny. You make an error in dropping these squared and cubed and forth-power and higher terms. But you don’t make much of one, not if s is small enough compared to v. And that means your estimate of the new travel time is:
Or, that is, if you reduce the speed by (say) five percent of what you started with, you increase the travel time by five percent. Varying one important quantity by a small amount we know as “perturbations”. Working out the approximate change in one quantity based on a perturbation is a key part of a lot of calculus, and a lot of mathematical modeling. It can feel illicit; after a lifetime of learning how mathematics is precise and exact, it’s hard to deliberately throw away stuff you know is not zero. It gets you to good places, though, and fast.
Morrie Turner’s Wee Pals for the 21st shows Wellington having trouble with partitions. We can divide any counting number up into the sum of other counting numbers in, usually, many ways. I can kind of see his point; there is something strange that we can express a single idea in so many different-looking ways. I’m not sure how to get Wellington where he needs to be. I suspect that some examples with dimes, quarters, and nickels would help.
Three of these essays in a row now that Jef Mallett’s Frazz has done something worth responding to. You know, the guy lives in the same metro area. He could just stop in and visit sometime. There’s a pinball league in town and everything. He could view it as good healthy competition.
Bill Hinds’s Cleats for the 1st is another instance of the monkeys-on-typewriters metaphor. The metaphor goes back at least as far as 1913, when Émile Borel wrote a paper on statistical mechanics and the reversibility problem. Along the way it was worth thinking of the chance of impossibly unlikely events, given enough time to happen. Monkeys at typewriters formed a great image for a generator of text that knows no content or plan. Given enough time, this random process should be able to produce all the finite strings of text, whatever their content. And the metaphor’s caught people’s fancy I guess there’s something charming and Dadaist about monkeys doing office work. Borel started out with a million monkeys typing ten hours a day. Modern audiences sometimes make this an infinite number of monkeys typing without pause. This is a reminder of how bad we’re allowing pre-revolutionary capitalism get.
Sometimes it’s cut down to a mere thousand monkeys, as in this example. Often it’s Shakespeare, but sometimes it’s other authors who get duplicated. Dickens seems like a popular secondary choice. In joke forms, the number of monkeys and time it would take to duplicate something is held as a measure of the quality of the original work. This comes from people who don’t understand. Suppose the monkeys and typewriters are producing truly random strings of characters. Then the only thing that affects how long it takes them to duplicate some text is the length of the original text. How good the text is doesn’t enter into it.
Jef Mallett’s Frazz for the 1st is about the comfort of knowing about things one does not know. And that’s fine enough. Frazz cites Fermat’s Last Theorem as a thing everyone knows of but doesn’t understand. And that choice confuses me. I’m not sure what there would be to Fermat’s Last Theorem that someone who had heard of it would not understand. The basic statement of it — if you have three positive whole numbers a, b, and c, then there’s no whole number n larger than 2 so that equals — has it.
But “understanding” is a flexible concept. He might mean that people don’t know why the Theorem is true. Fair enough. Andrew Wiles and Richard Taylor’s proof is a long thing that goes deep into a field of mathematics that even most mathematicians don’t study. Why it should be true can be an interesting question, and one that’s hard to ever satisfyingly answer. What is the difference between a proof that something is true and an explanation for why it’s true? And before you say there’s not one, please consider that many mathematicians do experience a difference between seeing something proved and understanding why something is true.
And Frazz might also mean that nobody knows what use Fermat’s Last Theorem is. This is a fair complaint too. I’m not aware offhand of any interesting results which follow from its truth, nor of anything neat that would come about had it been false. It’s just one of those things that happens to be true, and that we’ve found to be pretty, perhaps because it is easy to ask whether it’s true and hard to answer. I don’t know.
Morrie Turner’s Wee Pals for the 2nd has a kid looking for a square root. We all have peculiar hobbies. His friends speak of it as though it’s a lost physical object. This is a hilarious misunderstanding until it strikes you that we speak about stuff like square roots “existing”. Indeed, the language of mathematics would be trashed if we couldn’t speak about numerical constructs “existing” somewhere to be “found”. But try to put “four” in a box and see what you get. That we mostly have little trouble understanding what we mean by showing some mathematical construct exists, and what we hope to do by looking for it, suggests we roughly know what we mean by the phrases. All right then; what is that, in terms a kid could understand?
There are many ways to numerically compute a square root, if you have to do it by hand and it isn’t a perfect square. My preference is for iterative methods, in which you start with a rough guess and try to improve things. One good enough method for we call the Babylonian method, reflecting how old we think it is. Start with your number S whose square root you want. And start with a number x0, a first guess for what the square root is. This can be anything. The great thing about iterative methods is even if you start with a garbage answer, you get to a good answer soon enough. Still, if you have a suspicion of what the square root should be, start there.
Your first iteration, the first guess for a better answer, is to calculate the number . Typically, x1 will be closer to the square root of S than will x0 be. And in any case, we can get closer still. Use x1 to calculate a new number. This is . And then x3 and x4 and x5 and so on. In theory, you never finish; you’re stuck finding an infinitely long sequence of better approximations to the square root. In practice, you finish; you find that you’re close enough to the square root. Well, the square root of a whole number is either a whole number (if it was a perfect square to start) or is an irrational number. You were going to stop on an approximation sooner or later.
The method requires doing division. Long division, too, after the first couple steps. I don’t know a way around that which doesn’t divert into something less pleasant, such as logarithms and exponentials. Or maybe into trigonometric functions. This can be tedious to do by hand. Great thing, though, is if you make a mistake? That’s kind of all right. The next iteration will (usually) correct for it. That’s the glory of iterative methods. They tend to be forgiving of numerical error, whatever its source. Another iteration reduces, or even eliminates, the mistake of the previous iteration.
For more of these Reading the Comics posts please follow this link. If you’re only interested in Reading the Cleats strips, please use this link instead. But Cleats is a new tag this essay, so for now, there aren’t others. If you’re hoping to see all my Reading the Comics posts about Frazz, try this link. If you’d like more of my essays which mention Wee Pals, you can use this link. And if you’d like more Reading the Comics posts that mention Harley, use this link. That’s another new tag, but I believe Dan Thompson is still making new examples of the strip. So it may appear again.
Comic Strip Master Command spent most of February making sure I could barely keep up. It didn’t slow down the final week of the month either. Some of the comics were those that I know are in eternal reruns. I don’t think I’m repeating things I’ve already discussed here, but it is so hard to be sure.
Bill Amend’s FoxTrot for the 24th of February has a mathematics problem with a joke answer. The approach to finding the area’s exactly right. It’s easy to find areas of simple shapes like rectangles and triangles and circles and half-circles. Cutting a complicated shape into known shapes, finding those areas, and adding them together works quite well, most of the time. And that’s intuitive enough. There are other approaches. If you can describe the outline of a shape well, you can use an integral along that outline to get the enclosed area. And that amazes me even now. One of the wonders of calculus is that you can swap information about a boundary for information about the interior, and vice-versa. It’s a bit much for even Jason Fox, though.
Jef Mallett’s Frazz for the 25th is a dispute between Mrs Olsen and Caulfield about whether it’s possible to give more than 100 percent. I come down, now as always, on the side that argues it depends what you figure 100 percent is of. If you mean “100% of the effort it’s humanly possible to expend” then yes, there’s no making more than 100% of an effort. But there is an amount of effort reasonable to expect for, say, an in-class quiz. It’s far below the effort one could possibly humanly give. And one could certainly give 105% of that effort, if desired. This happens in the real world, of course. Famously, in the right circles, the Space Shuttle Main Engines normally reached 104% of full throttle during liftoff. That’s because the original specifications for what full throttle would be turned out to be lower than was ultimately needed. And it was easier to plan around running the engines at greater-than-100%-throttle than it was to change all the earlier design documents.
Matt Janz’s Out of the Gene Pool rerun for the 25th tosses off a mention of “New Math”. It’s referenced as a subject that’s both very powerful but also impossible for Pop, as an adult, to understand. It’s an interesting denotation. Usually “New Math”, if it’s mentioned at all, is held up as a pointlessly complicated way of doing simple problems. This is, yes, the niche that “Common Core” has taken. But Janz’s strip might be old enough to predate people blaming everything on Common Core. And it might be character, that the father is old enough to have heard of New Math but not anything in the nearly half-century since. It’s an unusual mention in that “New” Math is credited as being good for things. (I’m aware this strip’s a rerun. I had thought I’d mentioned it in an earlier Reading the Comics post, but can’t find it. I am surprised.)
So I was travelling last week, and this threw nearly all my plans out of whack. We stayed at one of those hotels that’s good enough that its free Internet is garbage and they charge you by day for decent Internet. So naturally Comic Strip Master Command sent a flood of posts. I’m trying to keep up and we’ll see if I wrap up this past week in under three essays. And I am not helped, by the way, by GoComics.com rejiggering something on their server so that My Comics Page won’t load, and breaking their “Contact Us” page so that that won’t submit error reports. If someone around there can break in and turn one of their servers off and on again, I’d appreciate the help.
Hy Eisman’s Katzenjammer Kids for the 21st of January is a curiously-timed Tax Day joke. (Well, the Katzenjammer Kids lapsed into reruns a dozen years ago and there’s probably not much effort being put into selecting seasonally appropriate ones.) But it is about one of the oldest and still most important uses of mathematics, and one that never gets respect.
Morrie Turner’s Wee Pals rerun for the 21st gets Oliver the reputation for being a little computer because he’s good at arithmetic. There is something that amazes in a person who’s able to calculate like this without writing anything down or using a device to help.
Mark Anderson’s Andertoons for the 22nd is the Mark Anderson’s Andertoons for the week. Well, for Monday, as I write this. It’s got your classic blackboard full of equations for the people in over their head. The equations look to me like gibberish. There’s a couple diagrams of aromatic organic compounds, which suggests some quantum-mechanics chemistry problem, if you want to suppose this could be narrowed down.
Greg Evans’s Luann Againn for the 22nd has Luann despair about ever understanding algebra without starting over from scratch and putting in excessively many hours of work. Sometimes it feels like that. My experience when lost in a subject has been that going back to the start often helps. It can be easier to see why a term or a concept or a process is introduced when you’ve seen it used some, and often getting one idea straight will cause others to fall into place. When that doesn’t work, trying a different book on the same topic — even one as well-worn as high school algebra — sometimes helps. Just a different writer, or a different perspective on what’s key, can be what’s needed. And sometimes it just does take time working at it all.
Richard Thompson’s Richard’s Poor Almanac rerun for the 22nd includes as part of a kit of William Shakespeare paper dolls the Typing Monkey. It’s that lovely, whimsical figure that might, in time, produce any written work you could imagine. I think I’d retired monkeys-at-typewriters as a thing to talk about, but I’m easily swayed by Thompson’s art and comic stylings so here it is.
Darrin Bell and Theron Heir’s Rudy Park for the 18th throws around a lot of percentages. It’s circling around the sabermetric-style idea that everything can be quantified, and measured, and that its changes can be tracked. In this case it’s comments on Star Trek: Discovery, but it could be anything. I’m inclined to believe that yeah, there’s an astounding variety of things that can be quantified and measured and tracked. But it’s also easy, especially when you haven’t got a good track record of knowing what is important to measure, to start tracking what amounts to random noise. (See any of my monthly statistics reviews, when I go looking into things like views-per-visitor-per-post-made or some other dubiously meaningful quantity.) So I’m inclined to side with Randy and his doubts that the Math Gods sanction this much data-mining.
It’s too many comics to call this a famine edition, after last week’s feast. But there’s not a lot of theme to last week’s mathematically-themed comic strips. There’s a couple that include vintage comic strips from before 1940, though, so let’s run with that as a title.
Bill Hinds’s Tank McNamara for the 4th is built on mathematics’ successful invasion and colonization of sports management. Analytics, sabermetrics, Moneyball, whatever you want to call it, is built on ideas not far removed from the quality control techniques that changed corporate management so. Look for patterns; look for correlations; look for the things that seem to predict other things. It seems bizarre, almost inhuman, that we might be able to think of football players as being all of a kind, that what we know about (say) one running back will tell us something about another. But if we put roughly similarly capable people through roughly similar training and set them to work in roughly similar conditions, then we start to see why they might perform similarly. Models can help us make better, more rational, choices.
Morrie Turner’s Wee Pals rerun for the 4th is another word-problem resistance joke. I suppose it’s also a reminder about the unspoken assumptions in a problem. It also points out why mathematicians end up speaking in an annoyingly precise manner. It’s an attempt to avoid being shown up like Oliver is.
Which wouldn’t help with Percy Crosby’s Skippy for the 7th of April, 1930, and rerun the 5th. Skippy’s got a smooth line of patter to get out of his mother’s tutoring. You can see where Percy Crosby has the weird trait of drawing comics in 1930 that would make sense today still; few pre-World-War-II comics do.
Niklas Eriksson’s Carpe Diem for the 7th is a joke about mathematics anxiety. I don’t know that it actually explains anything, but, eh. I’m not sure there is a rational explanation for mathematics anxiety; if there were, I suppose it wouldn’t be anxiety.
George Herriman’s Krazy Kat for the 15th of July, 1939, and rerun the 8th, extends that odd little faintly word-problem-setup of the strips I mentioned the other day. I suppose identifying when two things moving at different speeds will intersect will always sound vaguely like a story problem.