Two of the four comic strips I mean to feature here have credits that feel unsatisfying to me. One of them is someone’s pseudonym and, yeah, that’s their business. One is Dennis the Menace, for which I find an in-strip signature that doesn’t match the credentials on Comics Kingdom’s web site, never mind Wikipedia. I’ll go with what’s signed in the comic as probably authoritative. But I don’t like it.
R Ferdinand and S Ketcham’s Dennis the Menace for the 16th is about calculation. One eternally surprising little thing about calculators and computers is that they don’t do anything you can’t do by hand. Or, for that matter, in your head. They do it faster, typically, and more reliably. They can seem magical. But the only difference between what they do and what we do is the quantity with which they do this work. You can take this as humbling or as inspirational, as fits your worldview.
Ham’s Life on Earth for the 16th is a joke about the magical powers we attribute to mathematics. It’s also built on one of our underlying assumptions of the world, that it must be logically consistent. If one has an irrefutable logical argument that something isn’t so, then that thing must not be so. It’s hard to imagine how an illogical world would work. But it is hard not to wonder if there’s some arrogance involved in supposing the world has to square with the rules of logic that we find sensible. And to wonder whether we perceive world consistent with that logic because our expectations frame what we’re able to perceive.
In any case, as we frame logic, an argument’s validity shouldn’t depend on the person making the argument. Or even whether the argument has been made. So it’s hard to see how simply voicing the argument that one doesn’t exist could have that effect. Except that mathematics has got magical connotations, and vice-versa. That’ll be good for building jokes for a while yet.
Mark Anderson’s Andertoons for the 17th is the Mark Anderson’s Andertoons for the week. It’s wordplay, built on the connotation that division is a bad thing. It seems less dire if we think of division as learning how to equally share something that’s been held in common, though. Or if we think of it as learning what to multiply a thing by to get a particular value. Most mathematical operations can be taken to mean many things. Surely division has some constructive and happy interpretations.
Paul Gilligan’s Pooch Cafe for the 17th is a variation of the monkeys-on-keyboards joke. If what you need is a string of nonsense characters then … well, a cat on the keys is at least famous for producing some gibberish. It’s likely not going to be truly random, though. If a cat’s paw has stepped on, say, the ‘O’, there’s a good chance the cat is also stepping on ‘P’ or ‘9’. It also suggests that if the cat starts from the right, they’re more likely to have a character like ‘O’ early in the string of characters and less likely at the end. A completely random string would be as likely to have an ‘O’ at the start as at the end of the string.
And even if a cat on the keyboard did produce good-quality randomness, well. How likely a randomly-generated string of characters is to match a thing depends on the length of the thing. If the meaning of the symbols doesn’t matter, then ‘Penny Lane’ is as good as ‘*2ft,2igFIt’. This is not to say you can just use, say, ‘asdfghjkl’ as your password, at least not for anything that would hurt you if it were cracked. If everyone picked all passwords with no regard for what the symbols meant, these would be. But passwords that seem easy to think get used more often than they should be. It’s not that they’re easier to guess, but that guessing them is more likely to be correct.
Later this week I’ll host this month’s Playful Mathematics Blog Carnival! If you know of any mathematics that teaches or delights or both please share it with me, and we’ll let the world know. Also this week I should finally start my 2018 Mathematics A To Z, explaining words from mathematics one at a time.
Tom Toles’s Randolph Itch, 2 am for the 13th is the Roman Numerals joke for the week. IV is a well-established way to write four, although on clock faces IIII is a quite common use. There’s not a really clear reason why this should be. I’m convinced that it’s mostly for reasons of symmetry. IIII comes nearer the length of VIII, across it on the clock face. The subtractive principle, where ‘IV’ means ‘one taken away from five’, wasn’t really a thing until the middle ages. But then neither were clocks like that.
Bill Rechin’s Crock for the 14th is a joke about being bad at arithmetic. And yeah, most instructors wouldn’t accept “a lot” as the answer to 125 times 140. But we can go from approximations to something more precise. The number’s got to be more than 10,000, for example. 125 is more than 100, and 140 is more than 100. So 125 times 140 has to be more than 100 times 100. And then I notice: 125 is a hundred plus a quarter-of-a-hundred. So, 125 times 140 is a hundred times 140 plus a quarter-of-a-hundred times 140. A hundred times 140 is easy: it’s 14,000. A quarter of that? … Is a quarter of 12,000 plus a quarter of 2,000. That’s 3,000 plus 500. So 125 times 140 has to be 14,000 plus 3,000 plus 500. 17,500. My calculator agrees, so I feel pretty good. If this all seems like an ad hoc process, well, it is. But it’s how I can do this in my head.
Yes, the comments at ComicsKingdom include a warning that “using this obscenity called new math he may never be right, but he will never be wrong either”. I mention this for fans of cranky old person comics commentary.
Ted Shearer’s Quincy for the 21st of July, 1979 was rerun the 14th. It expresses the then-common wish for a calculator, which held such promise for making mathematics easy. It does make some kinds of mathematics easy. It especially takes considerable tedium out of mathematics. And it opens up new things to discover. Especially if the calculator lets you put the last thing calculated into a formula. That makes it easy to play with all sorts of iterative processes. They let you find solutions to weird and complicated problems. Or explore beautiful fractals. Figure out what limits work like. Or just notice what’s neat about 3.302775638. They let you get into different things.
Daniel Shelton’s Ben for the 14th has Nicholas doing mathematics homework. And something that couldn’t just be any subject; arranging fractions by size is something worth learning. They do have the peculiar and hard-to-adjust-to property that making the denominator larger, without changing the numerator, makes the entire fraction represent a smaller number. I mean a number closer to zero. So I think sorting fractions a reasonable homework project. Cutting them out and pasting them down seems weird to me. But maybe there’s some benefit in making the project tactile like that.
Three of the five comic strips I review today are reruns. I think that I’ve only mentioned two of them before, though. But let me preface all this with a plea I’ve posted before: I’m hosting the Playful Mathematics Blog Carnival the last week in September. Have you run across something mathematical that was educational, or informative, or playful, or just made you glad to know about? Please share it with me, and we can share it with the world. It can be for any level of mathematical background knowledge. Thank you.
Tom Batiuk’s Funky Winkerbean vintage rerun for the 10th is part of an early storyline of Funky attempting to tutor football jock Bull Bushka. Mathematics — geometry, particularly — gets called on as a subject Bull struggles to understand. Geometry’s also well-suited for the joke because it has visual appeal, in a way that English or History wouldn’t. And, you know, I’ll take “pretty” as a first impression to geometry. There are a lot of diagrams whose beauty is obvious even if their reasons or points or importance are obscure.
Dan Collins’s Looks Good on Paper for the 10th is about everyone’s favorite non-orientable surface. The first time this strip appeared I noted that the road as presented isn’t a Möbius strip. The opossums and the car are on different surfaces. Unless there’s a very sudden ‘twist’ in the road in the part obscured from the viewer, anyway. If I’d drawn this in class I would try to save face by saying that’s where the ‘twist’ is, but none of my students would be convinced. But we’d like to have it that the car would, if it kept driving, go over all the pavement.
Bud Fisher’s Mutt and Jeff for the 10th is a joke about story problems. The setup suggests that there’s enough information in what Jeff has to say about the cop’s age to work out what it must be. Mutt isn’t crazy to suppose there is some solution possible. The point of this kind of challenge is realizing there are constraints on possible ages which are not explicit in the original statements. But in this case there’s just nothing. We would call the cop’s age “underdetermined”. The information we have allows for many different answers. We’d like to have just enough information to rule out all but one of them.
John Rose’s Barney Google and Snuffy Smith for the 11th is here by popular request. Jughead hopes that a complicated process of dubious relevance will make his report card look not so bad. Loweezey makes a New Math joke about it. This serves as a shocking reminder that, as most comic strip characters are fixed in age, my cohort is now older than Snuffy and Loweezey Smith. At least is plausibly older than them.
Anyway it’s also a nice example of the lasting cultural reference of the New Math. It might not have lasted long as an attempt to teach mathematics in ways more like mathematicians do. But it’s still, nearly fifty years on, got an unshakable and overblown reputation for turning mathematics into doubletalk and impossibly complicated rules. I imagine it’s the name; “New Math” is a nice, short, punchy name. But the name also looks like what you’d give something that was being ruined, under the guise of improvement. It looks like that terrible moment of something familiar being ruined even if you don’t know that the New Math was an educational reform movement. Common Core’s done well in attracting a reputation for doing problems the complicated way. But I don’t think its name is going to have the cultural legacy of the New Math.
Mark Anderson’s Andertoons for the 11th is another kid-resisting-the-problem joke. Wavehead’s obfuscation does hit on something that I have wondered, though. When we describe things, we aren’t just saying what we think of them. We’re describing what we think our audience should think of them. This struck me back around 1990 when I observed to a friend that then-current jokes about how hard VCRs were to use failed for me. Everyone in my family, after all, had no trouble at all setting the VCR to record something. My friend pointed out that I talked about setting the VCR. Other people talk about programming the VCR. Setting is what you do to clocks and to pots on a stove and little things like that; an obviously easy chore. Programming is what you do to a computer, an arcane process filled with poor documentation and mysterious problems. We framed our thinking about the task as a simple, accessible thing, and we all found it simple and accessible. Mathematics does tend to look at “problems”, and we do, especially in teaching, look at “finding solutions”. Finding solutions sounds nice and positive. But then we just go back to new problems. And the most interesting problems don’t have solutions, at least not ones that we know about. What’s enjoyable about facing these new problems?
I’d like to add something to my roundup up of last week’s mathematically-themed comic strips. That thing is a reminder that I’m hosting this month’s Playful Mathematics Education Carnival. It’ll post the last week of September. If you’ve recently seen pages that teach, that play games, that show any kind of mathematics that makes you smile, please, let me know. It’s worth sharing with more people.
Tom Gammill’s The Doozies for the 6th is the Venn Diagrams joke for the week. It’s only a two-circle diagram, but the comic strip hasn’t got that large a cast. And, really, would be hard to stage in a way that made the joke communicable with three or four participants.
Is that superhuman? Well, obviously, literally not. But it’s beyond what most of us could imagine doing. I admit I can’t imagine keeping anything straight in my head for nine hours. But. The basic rules of addition aren’t that exotic. Even a process like finding square roots can be done as additions and divisions and multiplications. Much of what makes this look hard is memory. How do you keep track of a hundred or so partial results each of a hundred or so digits? Much of what else is hard is persistence. How do you keep going after the seventh hour of this? And both are traits that you can develop, and practice, and at least get a little better on.
Or bypass the hard work. If asked 235 plus 747 I’d at least answer “a bit under a thousand”, which isn’t bad for an instant answer. 235 is a little under 250; 747 a little under 750; and 250 plus 750 is easy. Rewrite 235 as 250 – 15, and 747 as 750 – 3, and you have this: 235 + 747 is 250 + 750 – 15 – 3. So that’s 1000 minus 18. 982, pretty attainable. This takes practice. It amounts to learning how to spot an easy problem that looks like the question you actually have.
Greg Evans’s Luann Againn for the 7th shows a date living up to its potential as a fiasco. But it’s not a surprise Gunther finds himself comfortable talking trigonometry. The subject is not one that most people find cozy. I’d guess most people on introduction see it as some weird hybrid that fuses the impenetrable diagrams of geometry with the baffling formulas of algebra.
But there’s comfort in it, especially to a particular personality type. There are a lot of obscure things making up trigonometry. But there’s this beauty, too. All the basic trigonometry functions are tied together in neat little pairs and triplets. Formulas connect the properties of an angle with those of its half and its double. There’s a great many identities, particular calculations that have the same value for every angle.
You can say that about anything, of course. Any topic humans study has endless fascination. What makes mathematical fields comfortable? For one, that they promise this certain knowability. Trigonometry has a jillion definitions and rules and identities and all that. But that means you have a great many things of absolute reliability. They offer this certainty that even “hard” sciences like physics don’t have. Far more security than you see with the difficult sciences, like biology or sociology. And true dependability, compared to the mystifying and obscure rules of interacting with other humans. If you don’t feel you know how to be with people, and don’t feel like you could ever learn, a cosecant is at least something you can master.
For the second part of last week’s comics, there’s several strips whose authors prefer to use a single name. I’m relieved. Somehow my writing seems easier when I don’t have a long authorial credit to give. I can take writing “Zach Weinersmith” fourteen times a week. It’s all those appearances of, like, “Corey Pandolph and Phil Frank and Joe Troise” (The Elderberries) that slow me way up.
Darrin Bell’s Candorville for the 4th shows off one of the things statistics can do. Tracking some measurable thing lets one notice patterns. These patterns might signify something important. At the least they can suggest things that deserve more scrutiny. There’s dangers, of course. If you’re measuring something that’s rare, or that naturally fluctuates a lot, you might misinterpret changes. You could suppose the changes represent some big, complicated, and invariably scary pattern that isn’t actually there. You can take steps to avoid how much weight you give to little changes. For example, you could look at running averages. Instead of worrying about how often Lemont has asked for his clippers this year versus last, look at how often he’s asked for it, on average, each of the last three years, compared to the average of the three years before that. Changes in that are more likely to be meaningful. But doing this does mean that a sudden change, or a slight but persistent change, is harder to notice. There are always mistakes to be made, when analyzing data. You have to think about what kinds of mistakes you would rather make, and how likely you want to make them.
C-Dog talks about fitting Lemont’s hair growth to a curve. This means looking at the data one has as points in space. What kinds of curves will come as close as possible to including all those points? It turns out infinitely many curves will, and you can fit a curve to all the data points you have. (Unless you have some inconsistent data, like, in 2017 Lemont asked both 14 times and 18 times.) So to do an interpolation you need to make some suppositions. Suppose that the data is really a straight line, with some noise in it. Or is really a parabola. Really a sine wave. Or, drawing from a set of plausible curves, which of those best fits the data?
The Bézier Curve mentioned here is a family of shapes. They’re named for Pierre Bézier, an engineer with Renault who in the 1950s pioneered the using of these curves. There are infinitely many of them. But they’re nice to work with. You can make great-looking curves as sharply curved or as smoothly curved as you like, using them. Most modern fonts use Bézier Curves to compute the shapes of letters. If you have a drawing program, it’s got some kind of Bézier curve in there. It’s the weird tool with a bunch of little dots, most of which are nowhere near the curve they draw. But moving the dots changes the way the curve looks.
A Bézier curve can be linear; indeed, it can just be a line. C-Dog’s showing off by talking about a linear Bézier curve. Or he means something that looks a lot like a line, to the casual eye. Negative-sloped means what it would in high school algebra when you talk about lines: it’s a thing with a value that decreases as the independent variable increases. Something getting rarer in time, for example.
Thaves’s Frank and Ernest for the 5th uses arithmetic, particularly simple addition, as emblematic of the basics of life. Hard to argue that this isn’t some of the first things anyone would learn, and that mathematics as it’s taught builds from that. A mathematician might see other fields — particularly set theory and category theory — as more fundamental than arithmetic. That is, that you can explain arithmetic in terms of set theory, and set theory in terms of category theory. So one could argue that those are the more basic. But if we mean basic as in the first things anyone learns, yeah, it’s arithmetic. Definitely.
Kliban’s Kliban Cartoons for the 5th speaks of proofs. A good bit of mathematics is existence proofs, which is to say, showing that a thing with desired properties does exist. Sometimes they actually show you the thing. Such a “constructive proof” — showing how you make an example of the thing — pretty well proves the thing exists. But sometimes the best you can do is show that there is an answer. In any case, an example of a fish would convince all but the most hardcore skeptics that fish do exist.
There won’t be, this week, any mathematically-themed comic strips featuring the long-running, Carl Anderson-created character Henry. You’ll come to see why I find this worth mentioning soon enough. Not today.
Hart, Mastroianni, and Parker’s Wizard of Id for the 2nd features the blackboard full of symbols to represent the difficult and unsolved problem. And sometimes it does seem like it takes magic to solve an equation. That magic usually takes the form of a transformation. That is, we find a way to rewrite the problem as something different, and find that this different problem is solvable. And then that the solution to this altered problem can be transformed into a solution of the original. This is normal magic, the kind any trained mathematician can do, if haltingly. But sometimes it’ll be just a stroke of imaginative genius, solving a problem that seems at first to have nothing to do with the original. This is genius work, and we all hope we can find a problem on which we can do that.
I can also take the strip to represent one of those things I’m curmudgeonly about. That is that I tend to look at big special-effects-laden attempts to make mathematics look beautiful as … well, they’re nice. But I don’t think they help anyone learn how to do anything. So that the Wizard’s work doesn’t actually solve the problem feels true to me.
Mort Walker and Dik Browne’s vintage Hi and Lois for the 3rd sees Chip struggling with mathematics. His father has a noble idea, that it’ll be easier if he tries to see the problems as fun puzzles. Maybe so, but I agree with Chip: there’s not a punch line to 246 ÷ 3. Also, points to Chip for doing that division right away. Clearly he isn’t bad at arithmetic; he just doesn’t like it. We’ve all got things like that.
Hector D Cantu and Carlos Castellanos’s Baldo for the 4th is a joke about being helpless with numbers. … Actually, from the phrasing, I’m not positive that Cruz doesn’t mean he got question number 9, or maybe 19, or maybe number 10 wrong. It’s a bit sloppy to not remember which question was, but I certainly know the pain of remembering having done a problem wrong.
I figure to do something rare, and retire one of my comic strip tags after today. Which strip am I going to do my best to drop from Reading the Comics posts? Given how many of the ones I read are short-lived comics that have been rerun three or four times since I started tracking them? Read on and see!
Bill Holbrook’s On The Fastrack for the 29th of August continues the sequence of Fi talking with kids about mathematics. My understanding was that she tried to give talks about why mathematics could be fun. That there are different ways to express the same number seems like a pretty fine-grain detail to get into. But this might lead into some bigger point. That there are several ways to describe the same thing can be surprising and unsettling to discover. That you have, when calculating, the option to switch between these ways freely can be liberating. But you have to know the option is there, and where to look for it. And how to see it’ll make something simpler.
Bill Holbrook’s On The Fastrack for the 30th of August gets onto a thread about statistics. The point of statistics is to describe something complicated with something simple. So detail must be lost. That said, there are something like 2,038 different things called “average”. Each of them has a fair claim to the term, too. In Fi’s example here, 73 degrees (Fahrenheit) could be called the average as in the arithmetic mean, or average as in the median. The distribution reflects how far and how often the temperature is from 73. This would also be reflected in a quantity called the variance, or the standard deviation. Variance and standard deviation are different things, but they’re tied together; if you know one you know the other. It’s just sometimes one quantity is more convenient than the other to work with.
Bill Holbrook’s On The Fastrack for the 1st of September has Fi argue that apparent irrelevance makes mathematics boring. It’s a common diagnosis. I think I’ve advanced the claim myself. I remember a 1980s probability textbook asking the chance that two transistors out of five had broken simultaneously. Surely in the earlier edition of the textbook, it was two vacuum tubes out of five. Five would be a reasonable (indeed, common) number of vacuum tubes to have in a radio. And it would be plausible that two might be broken at the same time.
It seems obvious that wanting to know an answer makes it easier to do the work needed to find it. I’m curious whether that’s been demonstrated true. Like, it seems obvious that a reference to a thing someone doesn’t know anything about would make it harder to work on. But does it? Does it distract someone trying to work out the height of a ziggurat based on its distance and apparent angle, if all they know about a ziggurat is their surmise that it’s a thing whose height we might wish to know?
Tom Toles’s Randolph Itch, 2 am rerun for the 30th of August is an old friend that’s been here a couple times. I suppose I do have to retire the strip from my Reading the Comics posts, at least, although I’m still amused enough by it to keep reading it daily. Simon Garfield’s On The Map, a book about the history of maps, notes that the X-marks-the-spot thing is an invention of the media. Robert Louis Stevenson’s Treasure Island particularly. Stevenson’s treasure map, Garfield notes, had to be redrawn from the manuscript and the author’s notes. The original went missing in the mail to the publishers. I just mention because I think that adds a bit of wonder to the treasure map. And since, I guess, I won’t have the chance to mention this again.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 30th of August satisfies the need for a Venn Diagram joke this time around. It’s also the strange-geometry joke for the week. Klein bottles were originally described by Felix Klein. They exist in four (or more) dimensions, in much the way that M&oum;bius strips exist in three. And like the M&oum;bius strip the surface defies common sense. You can try to claim some spot on the surface is inside and some other spot outside. But you can get from your inside to your outside spot in a continuous path, one you might trace out on the surface without lifting your stylus.
If you were four-dimensional. Or more. If we were to see one in three dimensions we’d see a shape that intersects itself. As beings of only three spatial dimensions we have to pretend that doesn’t happen. It’s the same we we pretend a drawing of a cube shows six squares all of equal size and connected at right angles to one another, even though the drawing is nothing like that. The bottle-like shape Weinersmith draws is, I think, the most common representation of the Klein bottle. It looks like a fancy bottle, and you can buy one as a novelty gift for a mathematician. I don’t need one but do thank you for thinking of me. MathWorld shows another representation, a figure-eight-based one which looks to me like an advanced pasta noodle. But it doesn’t look anything like a bottle.
Eric the Circle for the 31st of August, this one by JohnG, is a spot of wordplay. The pun here is the sine of an angle in a (right) triangle. That would be the length of the leg opposite the angle divided by the length of the hypotenuse. This is still stuff relevant to circles, though. One common interpretation of the cosine and sine of an angle is to look at the unit circle. That is, a circle with radius 1 and centered on the origin. Draw a line segment opening up an angle θ from the positive x-axis. Draw it counterclockwise. That is, if your angle is a very small number, you’re drawing a line segment that’s a little bit above the positive x-axis. Draw the line segment long enough that it touches the unit circle. That point where the line segment and the circle intersect? Look at its Cartesian coordinates. The y-coordinate will be the sine of θ. The x-coordinate will be the cosine of θ. The triangle you’re looking at has vertices at the origin; at x-coordinate cosine θ, y-coordinate 0; and at x-coordinate cosine θ, y-coordinate sine θ.
Have you ever wondered how I gather comic strips for these Reading the Comics posts? Sure, why not go along with me. Well, I do it by this: I read a lot of comic strips. When I run across one that’s got some mathematical theme, I copy the URL for it over to a page of notes. Then I go back to those notes and write up a paragraph or so on each. That is, I do it exactly the way you might imagine if you weren’t very imaginative or trying hard. I explain all this to say that I made a note that I then didn’t notice. So I missed a comic strip. And opened myself up to wondering if there’s an etymological link between “note” and “notice”. Anyway, it’s here. I’m just explaining why it’s late.
Jim Toomey’s Sherman’s Lagoon for the 19th of August is the belated inclusion. It’s a probability strip. It’s built partly on how badly people estimate probability, especially of rare events. And of how badly people estimate the consequences of rare events. For anything that isn’t common, our feelings about the likelihood of events are garbage. And even for common events we’re not that good.
But then it’s hard to quantify a low-probability event too. Take the claim that a human has one chance in 3.7 million of being attacked by a shark. We’ll pretend that’s the real number; I don’t know what is. (I’m suspicious of the ‘3-7’. People picking a random two-digit number are surprisingly likely to pick 37 because, I guess, it ‘feels’ random.) Is that over their lifetime? Over a summer? In a single swimming event? In any case it’s such a tiny chance it’s not worth serious worry. But even then, a person who lives in Wisconsin and only ever swims in Lake Michigan has a considerably smaller chance of shark attack than a person from New Jersey who swims at the Shore. At least some of these things are probabilities we can affect.
So the fellow may be irrational, denying himself something he’d enjoy based on a fantastically unlikely event. But he is acting to avoid something he’s decided he doesn’t want to risk. And, you know, we all act irrationally at times, or else I couldn’t justify buying a lottery ticket every eight months or so. Also is Fillmore (the turtle) the person who needs to hear this argument?
Gary McCoy and Glenn McCoy’s The Duplex for the 26th is an accounting joke. And a cry about poverty, with the idea that one could make the adding up of one’s assets and debts work only by making mathematics logically inconsistent. Or maybe inconsistent. Arithmetic modulo a particular number could be said to make zero equal to some other number, after all, and that’s all valid. Useful, too, especially in enciphering messages and in generating random numbers. It’s less useful for accounting, though. At least it would draw attention if used unilaterally.
Steve Kelley and Jeff Parker’s Dustin for the 28th is roughly a student-resisting-the-homework problem. From the first panel I thought Hayden might be complaining that ‘x’ was used, once again, as the variable to be solved for. It is the default choice, made because we all grew up learning of ‘x’ as the first choice for a number with a not-yet-known identity. ‘y’ and ‘z’ come in as second and third choices, most likely because they’re quite close to ‘x’. Sometimes another letter stands out, usually because the problem compels it. If the framing of the problem is about when events happen then ‘t’ becomes the default choice. If the problem suggests circular motion then ‘r’ or ‘θ’ — radius and angle — become compelling. But if we know no context, and have only the one variable, then ‘x’ it is. It seems weird to do otherwise.
Bill Holbrook’s On The Fastrack for the 28th is part of a week of Fi talking about mathematics to kids. She occasionally delivers seminars meant to encourage enthusiasm about mathematics. I love the principle although I don’t know how long the effect lasts. (Although it is kind of what I’m doing here. Except I think maybe Fi gets paid.) Holbrook’s strips of this mode often include nice literal depictions of metaphors. This week didn’t offer much chance for that particular artistic play.
Now I’ve finally had the time to deal with the rest of last week’s comics. I’ve rarely been so glad that Comic Strip Master Command has taken it easy on me for this week.
Tom Toles’s Randolph Itch, 2am for the 20th is about a common daydream, that of soap bubbles of weird shapes. There’s fun mathematics to do with soap bubbles. Most of these fall into the “calculus of variations”, which is good at finding minimums and maximums. The minimum here is a surface with zero mean curvature that satisfies particular boundaries. In soap bubble problems the boundaries have a convenient physical interpretation. They’re the wire frames you dunk into soap film, and pull out again, to see what happens. There’s less that’s proven about soap bubbles than you might think. For example: we know that two bubbles of the same size will join on a flat common surface. Do three bubbles? They seem to, when you try blowing bubbles and fitting them together. But this falls short of mathematical rigor.
Parker and Hart’s Wizard of Id Classics for the 21st is a joke about the ignorance of students. Of course they don’t know basic arithmetic. Curious thing about the strip is that you can read it as an indictment of the school system, failing to help students learn basic stuff. Or you can read it as an indictment of students, refusing the hard work of learning while demanding a place in politics. Given the 1968 publication date I have a suspicion which was more likely intended. But it’s hard to tell; 1968 was a long time ago. And sometimes it’s just so easy to crack an insult there’s no guessing what it’s supposed to mean.
Gene Mora’s Graffiti for the 22nd mentions what’s probably the most famous equation after that thing with two times two in it. It does cry out something which seems true, that was there before Albert Einstein noticed it. It does get at one of those questions that, I say without knowledge, is probably less core to philosophers of mathematics than the non-expert would think. But are mathematical truths discovered or invented? There seems to be a good argument that mathematical truths are discovered. If something follows by deductive logic from the axioms of the field, and the assumptions that go into a question, then … what’s there to invent? Anyone following the same deductive rules, and using the same axioms and assumptions, would agree on the thing discovered. Invention seems like something that reflects an inventor.
But it’s hard to shake the feeling that there is invention going on. Anyone developing new mathematics decides what things seem like useful axioms. She decides that some bundle of properties is interesting enough to have a name. She decides that some consequences of these properties are so interesting as to be named theorems. Maybe even the Fundamental Theorem of the field. And there was the decision that this is a field with a question interesting enough to study. I’m not convinced that isn’t invention.
Mark Anderson’s Andertoons for the 23rd sees Wavehead — waaait a minute. That’s not Wavehead! This throws everything off. Well, it’s using mathematics as the subject that Not-Wavehead is trying to avoid. And it’s not using arithmetic as the subject easiest to draw on the board. It needs some kind of ascending progression to make waiting for some threshold make sense. Numbers rising that way makes sense.
That birds will fly in V-formation has long captured people’s imaginations. We’re pretty confident we know why they do it. The wake of one bird’s flight can make it easier for another bird to stay aloft. This is especially good for migrating birds. The fluid-dynamic calculations of this are hard to do, but any fluid-dynamic calculations are hard to do. Verifying the work was also hard, but could be done. I found and promptly lost an article about how heartbeat monitors were attached to a particular flock of birds whose migration path was well-known, so the sensors could be checked and data from them gathered several times over. (Birds take turns as the lead bird, the one that gets no lift from anyone else’s efforts.)
So far as I’m aware there’s still some mystery as to how they do it. That is, how they know to form this V-formation. A particularly promising line of study in the 80s and 90s was to look at these as self-organizing structures. This would have each bird just trying to pay attention to what made sense for itself, where to fly relative to its nearest-neighbor birds. And these simple rules created, when applied to the whole flock, that V pattern. I do not know whether this reflects current thinking about bird formations. I do know that the search for simple rules that produce rich, complicated patterns goes on. Centuries of mathematics, physics, and to an extent chemistry have primed us to expect that everything is the well-developed result of simple components.
I apologize for the ragged nature of this entry, but I’ve had a ragged sort of week and it’s all I can do to keep up. Alert calendar-watchers might have figured out I would have rather had this posted on Thursday or Friday, but I couldn’t make that work. I’m trying. Thanks for your patience.
Mark Anderson’s Andertoons for the 17th feeds rumors that I just reflexively include Mark Anderson’s Andertoons in these posts whenever I see one. But it features the name of something dear to me, so that’s worthwhile. And I love etymology, although not enough to actually learn anything substantive about it. I just enjoy trivia about where some words come from, and sometimes how they change over time. (The average English word meant the exact opposite thing about two hundred years ago, and it meant something hilariously unrelated two centuries before that.)
So I’m not sure how real word-studyers would regard the “geo” in “geometry”. The word is more or less Ancient Greek, given a bit of age and worn down into common English forms. It’s fair enough to describe it as originally meaning “land survey” or “land measure”. This might seem eccentric. But much of the early use of geometry was to figure out where things were, and how far they were from each other. It seems likely the earliest uses, for example, of the Pythagorean Theorem dealt with how to draw right angles on the surface of the Earth. And how to draw boundaries. The Greek fascination with compass-and-straightedge construction — work done without a ruler, so that you know distance only as a thing relative to other things in your figure — obscures how much of the field is about measurement.
Brett Koth’s Diamond Lil for the 17th is another geometry joke, and a much clearer one. And if there’s one thing we can say about parallel lines it’s that they don’t meet. There are some corners of geometry in which it’s convenient to say they “meet at infinity”, that is, they intersect at some point an infinite distance away. I don’t recommend bringing this up in casual conversation. I’m not sure I wanted to bring it up here.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 14th is a fractals joke. Benoit Mandelbrot became the centerpiece of the big fractals boom in pop mathematics in the 80s and 90s. This was thanks to a fascinating property of complex-valued numbers that he discovered and publicized. The Mandelbrot set is a collection of complex-valued numbers. It’s a border, properly, between two kinds of complex-valued numbers. This boundary has this fascinating shape that looks a bit like a couple kidney beans surrounded by lightning. That’s neat enough.
What’s amazing, and makes this joke anything, is what happens if you look closely at this boundary. Anywhere on it. In the bean shapes or in the lightning bolts. You find little replicas of the original shape. Not precisely the original shape. No two of these replicas are precisely identical (except for the “complex conjugate”, that is, something near the number has a mirror image near ). None of these look precisely like the original shape. But they look extremely close to one another. They’re smaller, yes, and rotated relative to the original, and to other copies. But go anywhere on this boundary and there it is: the original shape, including miniature imperfect copies, all over again.
The Mandelbrot Set itself — well, there are a bunch of ways to think of it. One is in terms of something called the Julia Set, named for Gaston Julia. In 1918 he published a massive paper about the iteration of rational functions. That is, start with some domain and a function rule; what’s the range? Now if we used that range as the domain again, and used the same rule for the function, what’s the range of that range? If we use the range-of-that-range as the domain for the same function rule, what’s the range-of-the-range-of-the-range? The particular function here has one free parameter, a single complex-valued number. Depending on what it is, the range-of-the-range-of-the-range-etc becomes a set that’s either one big blob or a bunch of disconnected blobs. The Mandelbrot Set is the locus of parameters separating the one-big-blob from the many-disconnected-blob outcomes.
By the way, yes, Julia published this in 1918. The work was amazing. It was also forgotten. You can study this stuff analytically, but it’s hard. To visualize it you need to do incredible loads of computation. So this is why so much work lay fallow until the 1970s, when Mandelbrot could let computers do incredible loads of computation, and even draw some basic pictures.
Thom Bleumel’s Birdbrains for the 15th is a calendars joke. Numbers come into play since, well, it seems odd to try tracking large numbers of dates without some sense of arithmetic. Also, likely, without some sense of geometry. Calendars are much used to forecast coming events, such as New and Full Moons or the seasons. That takes basic understanding of how to locate things in the sky to do at all. It takes sophisticated understanding of how to locate things in the sky to do well.
Scott Hilburn’s The Argyle Sweater for the 16th is the first anthropomorphic-numerals joke around here in like three days. Certainly, the scandalous thing is supposed to be these numbers multiplying out in public where anyone might see them. I wonder if any part of the scandal should be that multiplication like this has to include three partners: the 4, the 7, and the x. In algebra we get used to a convention in which we do without the ‘x’. Just placing one term next to another carries an implicit multiplication: ‘4a’ for ‘4 times a’. But that convention fails disastrously with numerals; what should we make of ’47’? We might write 4(7), or maybe (4)(7), to be clear. Or we might put a little centered dot between the two, . The ‘x’ by that point is reserved for “some variable whose value isn’t specified”. And it would be weird to write ‘4 times x times 7’. It wouldn’t be wrong; it’d just look weird. It would suggest you were trying to emphasize a point. I’ve probably done it in one of my long derivation-happy posts.
The title of this installment has nothing to do with anything. My love and I just got to talking about Reader’s Digest Condensed Books and I learned moments ago that they’re still being made. I mean, the title of the series changed from “Condensed Books” to “Select Editions” in 1997, but they’re still going on, as far as anyone can tell. This got us wondering things like how they actually do the abridging. And got me wondering whether any abridged book ended up being better than the original. So I have reasons for only getting partway through last week’s mathematically-themed comics. I don’t say they’re good reasons.
Scott Hilburn’s The Argyle Sweater for the 13th is the Roman Numerals joke for the week, the first one of those in like five days. Also didn’t know that there were still sidewalk theaters that still showed porn movies. I thought they had all been renovated into either respectable neighborhood-revitalization projects that still sometimes show Star Wars films or else become incubator space for startup investment groups.
Corey Pandolph’s The Elderberries for the 13th is a joke about learning fractions. They don’t see to be having much fun thinking about them. Fair enough, I suppose. Once you’ve got the hang of basic arithmetic here come fractions to follow rules for addition and subtraction that are suddenly way more complicated. Multiplication isn’t harder, at least, although it is longer. Same with division. Without a clear idea why this is anything you want to do, yeah, it seems to be unmotivated complicating of stuff.
Dave Whamond’s Reality Check for the 13th is trying to pick a fight with me. I’m not taking the bait. Although by saying ‘likelihood’ the question seems to be setting up a probability question. Those tend to use ‘p’ and ‘q’ as a generic variable name, rather than ‘x’. I bet you imagine that ‘p’ gets used to represent a possibly-unknown ‘probability’ because, oh yeah, first letter. Well … so far as I know that’s why. I’m away from my references right now so I can’t look them over and find no quite satisfactory answer. But that sure seems like it. ‘q’ gets called in if you need a second probability, and don’t want to deal with subscripts, then it’s a nice convenient letter close to ‘p’ in the alphabet. Again, so far as I know.
The other half of last week’s mathematically-themed comics were on familiar old themes. I’ll see what I can do with them anyway.
Scott Hilburn’s The Argyle Sweater for the 9th is the anthropomorphic numerals joke for the week. I’m curious why the Middletons would need multiple division symbols, but I suppose that’s their business. It does play on the idea that “division” and “splitting up” are the same thing. And that fits the normal use of these words. We’re used to thinking, say, of dividing a desired thing between several parties. While that’s probably all right in introducing the idea, I do understand why someone would get very confused when they first divide by one-half or one-third or any number between zero and one. And then negative numbers make things even more confusing.
Thaves’s Frank and Ernest for the 9th is the anthropomorphic geometric figures joke for the week. I think I can wrangle a way by which Circle’s question has deeper mathematical context. Mathematicians use the idea of “space” a lot. The use is inspired by how, you know, the geometry of a room works. Euclidean space, in the trade. A Euclidean space is a collection of points that obey a couple simple rules. You can take two points and add them, and get something in the space. You can take any scalar and multiply it by any point and get a point in the space. A scalar is something that acts like a real number. For example, real numbers. Maybe complex numbers, if you’re feeling wild.
A Euclidean space can be two-dimensional. This is the geometry of stuff you draw on paper. It can be three-dimensional. This is the geometry of stuff in the real world, or stuff you draw on paper with shading. It can be four-dimensional. This is the geometry of stuff you draw on paper with big blobby lines around it. Each of these is an equally good space, though, as legitimate and as real as any other. Context usually puts an implicit “three dimensional” before most uses of the word “space”. But it’s not required to be there. There’s many kinds of spaces out there.
And “space” describes stuff that doesn’t look anything like rooms or table tops or sheets of paper. These are spaces built of things like functions, or of sets of things, or of ways to manipulate things. Spaces built of the ways you can subdivide the integers. The details vary. But there’s something in common in all these ideas that communicates.
Mark Anderson’s Andertoons for the 11th is the Mark Anderson’s Andertoons for the week. I think we’ve all seen this joke go across our social media feed and it’s reassuring to know Mark Anderson has social media too. We do talk about solving for x, using the language of describing how we help someone get past a problem. I wonder if people might like this kind of algebra more if we talked more about finding out what values ‘x’ could have that make the equation true. Well, it won’t stop people feeling they don’t like the mathematics they learned in school. But it might help people feel like they know why they’re doing it.
There are times I feel like my writing here collapses entirely to Reading the Comics posts. It’s a temptation to just give up doing anything else. They’re easy to write, since the comics give me the subjects to discuss. And it offers a nice, accessible mix of same-old topics with the occasional oddball. It’s fun. But sometimes Comic Strip Master Command decides I’ve been doing enough of that. This is one of those weeks; I only found six comics in my normal reading that were on point enough to discuss. So here’s half of them.
Bill Rechin’s Crock for the 6th is … hm. Well, let’s call it a fractions joke. I’m curious exactly what the clerk’s joke is supposed to mean. Is it intended to suggest an impossibility, putting into something far more than it can hold? Or is it just meant to suggest gross overabundance? And deep down I suspect Rechin didn’t have any specific meaning; it’s just a good-sounding insult.
Hector D Cantu and Carlos Castellanos’s Baldo for the 7th is … hm. Well, let’s call it a wordplay joke. It works by “strength” having multiple meanings, and “numbers” having multiple meanings. And there being a convenient saying to link one to the other. If this were a busier week I wouldn’t even bring it up, but I hate going without anything around here.
They’ve been phasing Roman Numerals out for a long while. Arabic numerals got their grand introduction to the (Western) Roman Empire’s territories in 1202 by Leonardo of Pisa, known now as “Fibonacci”. His Liber Abaci (Book of Calculation) laid out the Arabic numerals scheme and place values, and how to use them. By 1228 he published an edition comparing Roman numerals to Arabic numerals.
This wasn’t the first anyone in western Europe had heard of them, mind. (It never is; anyone telling you anything was the first is simplifying.) Spanish monks in the 10th century studied Arabic texts, and wrote about what they found. But after Leonardo of Pisa, Arabic numerals started displacing Roman numerals at least in specialized trades. Florence, in what is now Italy, prohibited merchants from using Arabic numerals in 1299; they could use Roman numerals or write them out in words. This, presumably, to prevent cheating by use of strange, unfamiliar calculus. Arabic numerals escaped being tools of specialists in the 16th century, thanks in large part to the German mathematician Adam Ries, who explained the scheme in terms apprentices could understand.
Still, these days, a Roman numeral is mostly an affectation. Useful for bit of style; not for serious mathematics. Good for watches.
And finally, at last, there’s a couple of comics left over from last week and that all ran the same day. If I hadn’t gone on forever about negative Kelvin temperatures I might have included them in the previous essay. That’s all right. These are strips I expect to need relatively short discussions to explore. Watch now as I put out 2,400 words explaining Wavehead misinterpreting the teacher’s question.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 4th is driving me slightly crazy. The equation on the board looks like an electrostatics problem to me. The ‘E’ is a common enough symbol for the strength of an electric field. And the funny-looking K’s look to me like the Greek kappa. This often represents the dielectric constant. That measures how well an electric field can move through a material. The upside-down triangles, known in the trade as Delta, describe — well, that’s getting complicated. By themselves, they describe measuring “how much the thing right after this changes in different directions”. When there’s a x symbol between the Delta and the thing, it measures something called the “curl”. This roughly measures how much the field inspires things caught up in it to turn. (Don’t try passing this off to your thesis defense committee.) The Delta x Delta x E describes the curl of the curl of E. Oh, I don’t like visualizing that. I don’t blame you if you don’t want to either.
Anyway. So all this looks like it’s some problem about a rod inside an electric field. Fine enough. What I don’t know and can’t work out is what the problem is studying exactly. So I can’t tell you whether the equation, so far as we see it, is legitimately something to see in class. Envisioning a rod that’s infinitely thin is a common enough mathematical trick, though. Three-dimensional objects are hard to deal with. They have edges. These are fussy to deal with. Making sure the interior, the boundary, and the exterior match up in a logically consistent way is tedious. But a wire? A plane? A single point? That’s easy. They don’t have an interior. You don’t have to match up the complicated stuff.
For real world problems, yeah, you have to deal with the interior. Or you have to work out reasons why the interiors aren’t important in your problem. And it can be that your object is so small compared to the space it has to work in that the fact it’s not infinitely thin or flat or smooth just doesn’t matter. Mathematical models, such as give us equations, are a blend of describing what really is there and what we can work with.
Mike Shiell’s The Wandering Melon for the 4th is a probability joke, about two events that nobody’s likely to experience. The chance any individual will win a lottery is tiny, but enough people play them that someone wins just about any given week. The chance any individual will get struck by lightning is tiny too. But it happens to people. The combination? Well, that’s obviously impossible.
In July of 2015, Peter McCathie had this happen. He survived a lightning strike first. And then won the Atlantic Lotto 6/49. This was years apart, but the chance of both happening the same day, or same week? … Well, the world is vast and complicated. Unlikely things will happen.
So I’m going to have a third Reading the Comics essay for last week’s strips. This happens sometimes. Two of the four strips for this essay mention percentages. But one of the others is so important to me that it gets naming rights for the essay. You’ll understand when I’m done. I hope.
Angie Bailey’s Texts From Mittens for the 2nd talks about percentages. That’s a corner of arithmetic that many people find frightening and unwelcoming. I’m tickled that Mittens doesn’t understand how easy it is to work out a percentage of 100. It’s a good, reasonable bit of characterization for a cat.
John Graziano’s Ripley’s Believe It Or Not for the 2nd is about a subject close to my heart. At least a third of it is. The mention of negative Kelvin temperatures set off a … heated … debate on the comments thread at GoComics.com. Quite a few people remember learning in school that the Kelvin temperature scale. It starts with the coldest possible temperature, which is zero. And that’s that. They have taken this to denounce Graziano as writing obvious nonsense. Well.
Something you should know about anything you learned in school: the reality is more complicated than that. This is true for thermodynamics. This is true for mathematics. This is true for anything interesting enough for humans to study. This also applies to stuff you learned as an undergraduate. Also to grad school.
So what are negative temperatures? At least on an absolute temperature scale, where the answer isn’t an obvious and boring “cold”? One clue is in the word “absolute” there. It means a way of measuring temperature that’s in some way independent of how we do the measurement. In ordinary life we measure temperatures with physical phenomena. Fluids that expand or contract as their temperature changes. Metals that expand or contract as their temperatures change. For special cases like blast furnaces, sample slugs of clays that harden or don’t at temperature. Observing the radiation of light off a thing. And these are all fine, useful in their domains. They’re also bound in particular physical experiments, though. Is there a definition of temperature that … you know … we can do mathematically?
Of course, or I wouldn’t be writing this. There are two mathematical-physics components to give us temperature. One is the internal energy of your system. This is the energy of whatever your thing is, less the gravitational or potential energy that reflects where it happens to be sitting. Also minus the kinetic energy that comes of the whole system moving in whatever way you like. That is, the energy you’d see if that thing were in an otherwise empty universe. The second part is — OK, this will confuse people. It’s the entropy. Which is not a word for “stuff gets broken”. Not in this context. The entropy of a system describes how many distinct ways there are for a system to arrange its energy. Low-entropy systems have only a few ways to put things. High-entropy systems have a lot of ways to put things. This does harmonize with the pop-culture idea of entropy. There are many ways for a room to be messy. There are few ways for it to be clean. And it’s so easy to make a room messier and hard to make it tidier. We say entropy tends to increase.
So. A mathematical physicist bases “temperature” on the internal energy and the entropy. Imagine giving a system a tiny bit more energy. How many more ways would the system be able to arrange itself with that extra energy? That gives us the temperature. (To be precise, it gives us the reciprocal of the temperature. We could set this up as how a small change in entropy affects the internal energy, and get temperature right away. But I have an easier time thinking of going from change-in-energy to change-in-entropy than the other way around. And this is my blog so I get to choose how I set things up.)
This definition sounds bizarre. But it works brilliantly. It’s all nice clean mathematics. It matches perfectly nice easy-to-work-out cases, too. Like, you may kind of remember from high school physics how the temperature of a gas is something something average kinetic energy something. Work out the entropy and the internal energy of an ideal gas. Guess what this change-in-entropy/change-in-internal-energy thing gives you? Exactly something something average kinetic energy something. It’s brilliant.
In ordinary stuff, adding a little more internal energy to a system opens up new ways to arrange that energy. It always increases the entropy. So the absolute temperature, from this definition, is always positive. Good stuff. Matches our intuition well.
So in 1956 Dr Norman Ramsey and Dr Martin Klein published some interesting papers in the Physical Review. (Here’s a link to Ramsey’s paper and here’s Klein’s, if you can get someone else to pay for your access.) Their insightful question: what happens if a physical system has a maximum internal energy? If there’s some way of arranging the things in your system so that no more energy can come in? What if you’re close to but not at that maximum?
It depends on details, yes. But consider this setup: there’s one, or only a handful, of ways to arrange the maximum possible internal energy. There’s some more ways to arrange nearly-the-maximum-possible internal energy. There’s even more ways to arrange not-quite-nearly-the-maximum-possible internal energy.
Look at what that implies, though. If you’re near the maximum-possible internal energy, then adding a tiny bit of energy reduces the entropy. There’s fewer ways to arrange that greater bit of energy. Greater internal energy, reduced entropy. This implies the temperature is negative.
So we have to allow the idea of negative temperatures. Or we have to throw out this statistical-mechanics-based definition of temperature. And the definition works so well otherwise. Nobody’s got an idea nearly as good for it. So mathematical physicists shrugged, and noted this as a possibility, but mostly ignored it for decades. If it got mentioned, it was because the instructor was showing off a neat weird thing. This is how I encountered it, as a young physics major full of confidence and not at all good on wedge products. But it was sitting right there, in my textbook, Kittel and Kroemer’s Thermal Physics. Appendix E, four brisk pages before the index. Still, it was an enchanting piece.
And a useful one, possibly the most useful four-page aside I encountered as an undergraduate. My thesis research simulated a fluid-equilibrium problem run at different temperatures. There was a natural way that this fluid would have a maximum possible internal energy. So, a good part — the most fascinating part — of my research was in the world of negative temperatures. It’s a strange one, one where entropy seems to work in reverse. Things build, spontaneously. More heat, more energy, makes them build faster. In simulation, a shell of viscosity-free gas turned into what looked for all the world like a solid shell.
All right, but you can simulate anything on a computer, or in equations, as I did. Would this ever happen in reality? … And yes, in some ways. Internal energy and entropy are ideas that have natural, irresistible fits in information theory. This is the study of … information. I mean, how you send a signal and how you receive a signal. It turns out a lot of laser physics has, in information theory terms, behavior that’s negative-temperature. And, all right, but that’s not what anybody thinks of as temperature.
Well, these ideas happen still. They usually need some kind of special constraint on the things. Atoms held in a magnetic field so that their motions are constrained. Vortices locked into place on a two-dimensional surface (a prerequisite to my little fluids problems). Atoms bound into a lattice that keeps them from being able to fly free. All weird stuff, yes. But all exactly as the statistical-mechanics temperature idea calls on.
And notice. These negative temperatures happen only when the energy is extremely high. This is the grounds for saying that they’re hotter than positive temperatures. And good reason, too. Getting into what heat is, as opposed to temperature, is an even longer discussion. But it seems fair to say something with a huge internal energy has more heat than something with slight internal energy. So Graziano’s Ripley’s claim is right.
(GoComics.com commenters, struggling valiantly, have tried to talk about quantum mechanics stuff and made a hash of it. As a general rule, skip any pop-physics explanation of something being quantum mechanics.)
If you’re interested in more about this, I recommend Stephen J Blundell and Katherine M Blundell’s Concepts in Thermal Physics. Even if you’re not comfortable enough in calculus to follow the derivations, the textbook prose is insightful.
John Hambrock’s The Brilliant Mind of Edison Lee for the 3rd is a probability joke. And it’s built on how impossible putting together a particular huge complicated structure can be. I admit I’m not sure how I’d go about calculating the chance of a heap of Legos producing a giraffe shape. Imagine working out the number of ways Legos might fall together. Imagine working out how many of those could be called giraffe shapes. It seems too great a workload. And figuring it by experiment, shuffling Legos until a giraffe pops out, doesn’t seem much better.
This approaches an argument sometimes raised about the origins of life. Grant there’s no chance that a pile of Legos could be dropped together to make a giraffe shape. How can the much bigger pile of chemical elements have been stirred together to make an actual giraffe? Or, the same problem in another guise. If a monkey could go at a typewriter forever without typing any of Shakespeare’s plays, how did a chain of monkeys get to writing all of them?
And there’s a couple of explanations. At least partial explanations. There is much we don’t understand about the origins of life. But one is that the universe is huge. There’s lots of stars. It looks like most stars have planets. There’s lots of chances for chemicals to mix together and form a biochemistry. Even an impossibly unlikely thing will happen, given enough chances.
And another part is selection. A pile of Legos thrown into a pile can do pretty much anything. Any piece will fit into any other piece in a variety of ways. A pile of chemicals are more constrained in what they can do. Hydrogen, oxygen, and a bit of activation energy can make hydrogen-plus-hydroxide ions, water, or hydrogen peroxide, and that’s it. There can be a lot of ways to arrange things. Proteins are chains of amino acids. These chains can be about as long as you like. (It seems.) (I suppose there must be some limit.) And they curl over and fold up in some of the most complicated mathematical problems anyone can even imagine doing. How hard is it to find a set of chemicals that are a biochemistry? … That’s hard to say. There are about twenty amino acids used for proteins in our life. It seems like there could be a plausible life with eighteen amino acids, or 24, including a couple we don’t use here. It seems plausible, though, that my father could have had two brothers growing up; if there were, would I exist?
Jason Chatfield’s Ginger Meggs for the 3rd is a story-problem joke. Familiar old form to one. The question seems to be a bit mangled in the asking, though. Thirty percent of Jonson’s twelve apples is a nasty fractional number of apples. Surely the question should have given Jonson ten and Fitzclown twelve apples. Then thirty percent of Jonson’s apples would be a nice whole number.
There’s really only the one strip that I talk about today that gets into non-Euclidean geometries. I was hoping to have the time to get into negative temperatures. That came up in the comics too, and it’s a subject close to my heart. But I didn’t have time to write that and so must go with what I did have. I’ve surely used “Non-Euclidean Geometry Edition” as a name before too, but that name and the date of August 2, 2018? Just as surely not.
Mark Anderson’s Andertoons for the 29th is the Mark Anderson’s Andertoons for the week, at last. Wavehead gets to be disappointed by what a numerator and denominator are. Common problem; there are many mathematics things with great, evocative names that all turn out to be mathematics things.
Both “numerator” and “denominator”, as words, trace to the mid-16th century. They come from Medieval Latin, as you might have guessed. “Denominator” parses out roughly as “to completely name”. As in, break something up into some number of equal-sized pieces. You’d need the denominator number of those pieces to have the whole again. “Numerator” parses out roughly as “count”, as in the count of how many denominator-sized pieces you have. So for all that numerator and denominator look like one another, with with the meat of the words being the letters “n-m–ator”, their centers don’t have anything to do with one another. (I would believe a claim that the way the words always crop up together encouraged them to harmonize their appearances.)
Johnny Hart’s Back to BC for the 29th is a surprisingly sly joke about non-Euclidean geometries. You wouldn’t expect that given the reputation of the comic the last decade of Hart’s life. And I did misread it at first, thinking that after circumnavigating the globe Peter had come back to have what had been the right line touch the left. That the trouble was his stick wearing down I didn’t notice until I re-read.
But Peter’s problem would be there if his stick didn’t wear down. “Parallel” lines on a globe don’t exist. One can try to draw a straight line on the surface of a sphere. These are “great circles”, with famous map examples of those being the equator and the lines of longitude. They don’t keep a constant distance from one another, and they do meet. Peter’s experiment, as conducted, would be a piece of proof that they have to live on a curved surface.
And this gets at one of those questions that bothers mathematicians, cosmologists, and philosophers. How do we know the geometry of the universe? If we could peek at it from outside we’d have some help, but that is a big if. So we have to rely on what we can learn from inside the universe. And we can do some experiments that tell us about the geometry we’re in. Peter’s line example would be one; he can use that to show the world’s curved in at least one direction. A couple more lines and he’d be confident the world was a sphere. If we could make precise enough measurements we could do better, with geometric experiments smaller than the circumference of the Earth. (Or universe.) Famously, the sum of the interior angles of a triangle tell us something about the space the triangle’s inscribed in. There are dangers in going from information about one point, or a small area, to information about the whole. But we can tell some things.
Phil Dunlap’s Ink Pen for the 29th is another use of arithmetic as shorthand for intelligence. Might be fun to ponder how Captain Victorious would know that he was right about two plus two equalling four, if he didn’t know that already. But we all are in the same state, for mathematical truths. We know we’ve got it right because we believe we have a sound logical argument for the thing being true.
Brian Boychuk and Ron Boychuk’s Chuckle Brothers for the 30th is a riff on the story of Isaac Newton and the apple. The story of Newton starting his serious thinking of gravity by pondering why apples should fall while the Moon did not is famous. And it seems to trace to Newton. We have a good account of it from William Stukeley, who in the mid-18th century wrote Memoirs of Sir Isaac Newton’s Life. Stukeley knew Newton, and claimed to get the story right from him. He also told it to his niece’s husband, John Conduitt. Whether this is what got Newton fired with the need to create such calculus and physics, or whether it was a story he composed to give his life narrative charm, is beyond my ability to say. It’s an important piece of mathematics history anyway.
If you’d like more Reading the Comics essays you can find them at this link. Some of the many essays to mention Andertoons are at this link. Other essays mentioning B.C. (vintage and current) are at this link. The comic strip Ink Pen gets its mentions at this link, although I’m surprised to learn it’s a new tag today. And the Chuckle Brothers I discuss at this link. Thank you.
One of the comics from the last half of last week is here mostly because Roy Kassinger asked if I was going to include it. Which one? Read on and see.
Scott Metzger’s The Bent Pinky for the 24th is the anthropomorphic-numerals joke for the week. It’s pretty easy to learn, or memorize, or test small numbers for whether they’re prime. The bigger a number gets the harder it is. Mostly it takes time. You can rule some numbers out easily enough. If they’re even numbers other than 2, for example. Or if their (base ten) digits add up to a multiple of three or nine. But once you’ve got past a couple easy filters … you don’t have to just try dividing them by all the prime numbers up to their square root. Comes close, though. Would save a lot of time if the numerals worked that out ahead of time and then kept the information around, in case it were needed. Seems a little creepy to be asking that of other numbers, really. Certainly to give special privileges to numbers for accidents of their creation.
Wiley Miller’s Non Sequitur for the 25th is an Einstein joke. In a rare move for the breed this doesn’t have “E = mc2” in it, except in the implication that it was easier to think of than squirrel-proof bird feeders would be. Einstein usually gets acclaim for mathematical physics work. But he was also a legitimate inventor, with patents in his own right. He and his student Leó Szilárd developed a refrigerator that used no moving parts. Most refrigeration technology requires the use of toxic chemicals to actually do the cooling. Einstein and Szilárd hoped to make something less likely to leak these toxins. The design never saw widespread use. Ordinary refrigerators, using freon (shockingly harmless biologically, though dangerous to the upper atmosphere) got reliable enough that the danger of leaks got tolerable. And the electromagnetic pump the machine used instead made noise that at least some reports say was unbearable. The design as worked out also used a potassium-sodium alloy, not the sort of thing easy to work with. Now and then there’s talk of reviving the design. Its potential, as something that could use any heat source to provide refrigeration, seems neat. And everybody in this side of science and engineering wants to work on something that Einstein touched.
Mort Walker and Greg Walker’s Beetle Bailey for the 26th is here by special request. I wasn’t sure it was on-topic enough for my usual rigorous standards. But there is some social-aspects-of-mathematics to it. The assumption that ‘five’ is naturally better than ‘four’ for example. There is the connotation that some numbers are better than others. Yes, there are famously lucky numbers like 7 or unlucky ones like 13 (in contemporary Anglo-American culture, anyway; others have different lucks). But there’s also the sense that a larger number is of course better than a smaller one.
Except when it’s not. A first-rate performance is understood to be better than a third-rate one. A star of the first magnitude is more prominent than one of the fourth. This whether we mean celebrities or heavenly bodies. We have mixed systems. One at least respects the heritage of ancient Greek astronomers, who rated the brightest of stars as first magnitude and the next bunch as second and so on. In this context, if we take brightness to be a good thing, we understand lower numbers to be better. Another system regards the larger numbers as being more of what we’re assumed to want, and therefore be better.
Nasty confusions will happen when the schemes of thought collide. Is a class three hurricane more or less of a threat than a class four? Ought we be more worried if the United States Department of Defense declares it’s moved from Defence Condition four to Defcon 3? In October 1966, the Fermi 1 fission reactor near Detroit suffered a “Class 1 emergency”. Does that mean the city was at the highest or the lowest health risk from the partial meltdown? (In this case, this particular term reflects the lowest actionable level of radiation was detected. I am not competent to speak on how great the risk to the population was.) It would have been nice to have unambiguous language on this point.
On to the joke’s logic, though. Wouldn’t General Halftrack be accustomed to thinking of lower numbers as better? Getting to the green in three strokes is obviously preferable to four, and getting there in five would be a disaster.
Darby Conley’s Get Fuzzy for the 28th is an applied-probability strip. The calculating of odds is rich with mathematical and psychological influences. With some events it’s possible to define quite precisely what the odds should be. If there are a thousand numbers each equally likely to be the daily lottery winner, and only one that will be, we can go from that to saying what the chance of 254 being the winner is. But many events are impossible to forecast that way. We have to use other approaches. If something has happened several times recently, we can say it’s probably rather likely. Fluke events happen, yes. But we can do fairly good work by supposing that stuff is mostly normal, and that the next baseball season will look something like the last one.
As to how to bet wisely — well, many have tried to work that out. One of the big questions in financial mathematics is how to hedge bets. I write financial mathematics, but it applies to sports betting and really anything else with many outcomes. One of the common goals is to simply avoid catastrophe, to make sure that whatever happens you aren’t too badly off. This involves betting various amounts on many outcomes. More on the outcomes you think likely, but also some on the improbable outcomes. Long shots do sometimes happen, and pay out well; it can be worth putting a little money on that just in case. Judging the likelihood of those events, especially in complicated problems that can’t be reduced to logic, is one of the hard parts. If it could be made into a system we wouldn’t need people to do it. But it does seem that knowing what you bet on helps make better bets.
I apologize for a post rougher than my norm. It has not been a gentle week. I am carrying on as best I can, but then, who isn’t? There is a common element to three of the strips featured this time around, so I have a meaningful name.
Steve McGarry’s KidTown for the 22nd of July is a kids-information panel. It’s a delivery system for some neat trivia about numbers. I’d never encountered the bit about the factorial of 10 (10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) being as many seconds as there are in six weeks. I’m curious how I missed that. But it’s definitely one of those slightly useful bits of calendar mathematics to keep around. Some other useful ones are that three years is about 1100 days, and that a century is about three billion seconds. That line about 12 + 3 – 4 + 5 + 67 + 8 + 9 is probably a useful answer to some mathematics riddle such as might beset Nancy.
John Zakour and Scott Roberts’s Maria’s Day for the 23rd depicts Maria misunderstanding what it is to be bad at mathematics. The Star Wars movie episode numbers show a quirky indexing scheme, yes. But the numbers in this case are mostly nominal variables. If we spoke of the movies only by their titles … well, it would be harder to guess whether The Empire Strikes Back or Return of the Jedi came first. All the names suggest is that they ought to follow on something else happening beforehand. And people would likely use numbers for shorthand anyway. Star Trek fans talk still about the odd- and even-numbered movies, even though no Star Trek movie’s had a number attached to it since 1991.
A nominal variable is as the … er … name suggests. It’s a way to reference something, but the value doesn’t mean very much. We see these, often with numbers attached, often enough to not notice it. We start to realize it when we have those moments of thinking, isn’t it odd that the office building starts numbering rooms from 101, rather than, say, 1? Or that there’s no numbers between (say) 129 and 201? Using a number carries some information, in that it suggests we think there is a preferred order for things. But your neighborhood would be no different if all the building addresses were 1000 higher, and the Star Wars movies would be no different if the one from 1977 came to be dubbed Episode 14 instead.
(I am open to an argument that the Star Wars episode numbers are ordinal variables. This is why I hedged by calling them “mostly” nominal. An ordinal variable describes some preferred order for the things. The difference between numbers isn’t particularly meaningful, just the relationship between them. And, yeah, it would be peculiar if The Empire Strikes Back had a higher episode number than did Return of the Jedi. Viewing the movies in that order would create several apparent continuity errors. But there are differences between internal chronology and production order and other ways one might watch the movies. But it seems to me the ordinary use for the numbers, if someone uses them at all, is as a label.)
Mell Lazarus’s Momma for the 23rd is another strip built on people being bad at mathematics. Arithmetic, anyway. I’m not sure this quite counts as an arithmetic joke. Granting the (correct) assumption that an episode of 60 Minutes is ordinarily 60 minutes long, is not recognizing how long the show will take really a use of mathematics? Isn’t it more reading comprehension? … And to be fair to the ever-beleaguered Francis, it’s rather more likely 60 Minutes just had one segment about grown men incapable of doing arithmetic. Asking how long that is likely to take is a fair question.
Adrian Raeside’s The Other Coast for the 23rd is another strip conflating arithmetic skill with intelligence. And intelligence with fitness. It’s flattering stuff, at least for people who are good at arithmetic and who feel flattered to be called intelligent. But there’s a lot of presumption here. And a common despicable attitude: merry little eugenicists (they’re always cheery about it, aren’t they?) always conclude they are fit ones.
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.