Comic Strip Master Command decided I should have a week to catch up on things, and maybe force me to write something original. Of all the things I read there were only four strips that had some mathematics content. And three of them are such glancing mentions that I don’t feel it proper to include the strip. So let me take care of this.
Mark Anderson’s Andertoons for the 20th is the Mark Anderson’s Andertoons for the week. Wavehead apparently wants to know whether or is the better of these equivalent forms. I understand the impulse. Rarely in real life do we see two things that are truly equivalent; there’s usually some way in which one is better than the other. There may be two ways to get home for example, both taking about the same time to travel. One might have better scenery, though, or involve fewer difficult turns or less traffic this time of day. This is different, though: or are two ways to describe the same number. Which one is “better”?
The only answer is, better for what? What do you figure to do with this number afterwards? I admit, and suppose most people have, a preference for . But that’s trained into us, in large part, by homework set to reduce fractions to “lowest terms”. There’s honest enough reasons behind that. It seems wasteful to have a factor in the numerator that’s immediately divided out by the denominator.
If this were 25 years ago, I could ask how many of you have written out a check for twenty-two and 3/4 dollars, then, rather than twenty-two and 75/100 dollars? The example is dated but the reason to prefer an equivalent form is not. If I know that I need the number represented by , and will soon be multiplying it by eight, then may save me the trouble of thinking what three times two is. Or if I’ll be adding it to , or something like that. If I’m measuring this for a recipe I need to cut in three, because the original will make three dozen cookies and I could certainly eat three dozen cookies, then may be more convenient than . What is the better depends on what will clarify the thing I want to do.
A significant running thread throughout all mathematics, not just arithmetic, is finding equivalent forms. Ways to write the same concept, but in a way that makes some other work easier. Or more likely to be done correctly. Or, if the equivalent form is more attractive, more likely to be learned or communicated. It’s of value.
Jan Eliot’s Stone Soup Classics rerun for the 20th is a joke about how one can calculate what one is interested in. In this case, going from the number of days left in school to the number of hours and minutes and even seconds left. Personally, I have never had trouble remembering there are 24 hours in the day, nor that there are 86,400 seconds in the day. That there are 1,440 minutes in the day refuses to stick in my mind. Your experiences may vary.
Harry Bliss’s Bliss for the 23rd speaks of “a truck driver with a PhD in mathematical logic”. It’s an example of signifying intelligence through mathematics credentials. (It’s also a bit classicist, treating an intelligent truck driver as an unlikely thing.)
And so the Reading the Comics posts have returned to Sunday after a month in exile to Tuesdays. I’m curious whether Sunday is actually the best day to post my signature series of essays, since everybody is usually doing stuff on the weekends. Tuesdays more people are at work and looking for other things to think about. But at least for the duration of the A to Z series there’s not a good time to schedule them besides Sundays. So Sundays it is and I’ll possibly think things over again in December, if all goes well.
Ralph Hagen’s The Barn for the 27th poses a question that’s ridiculous when you look at it. Why should being twenty times as old as your newborn (sic) when you’re twenty years old imply you’d be twenty times as old as the newborn when you’re sixty? Age increases linearly. The ratios between ages, though, those decrease, in a ratio asymptotically approaching 1. So as far as that goes, this strip isn’t much of anything.
But I do like how it captures the way a mathematics puzzle can come from nowhere. Often interesting ones seem to generate themselves. You notice a pattern and wonder whether it reaches some interesting point. If you convince yourself it does, you wonder when it does. If it does not, you wonder why it can’t. This is the fun sort of mathematics, and you create it by looking at the two separate tile patterns in the kitchen or, as here, thinking about the ages of parent and child. Anything that catches the imagination of a bored mind. It’s fun being there.
Rory (the sheep) makes a common enough slip. Saying a twenty-year-old with a newborn is twenty times as old as the newborn is, implicitly, saying the newborn is one year old. This kind of error is so common it’s got a folksy name, the “fencepost error”. It has a more respectable name, for its LinkedIn profile, the “off-by-one error”. But you see the problem. Say that your birthday is the 1st of September. How many times were you alive on the 1st of September by the time you’re ten years old? Eleven times, the first one being the one you were born on, with one more counted up each year you’d lived. This was probably more clear before I explained it.
John Rose’s Barney Google and Snuffy Smith for the 27th has Mis Prunelly complimenting Jughaid’s creativity, but not wanting it in arithmetic. There is creativity in mathematics. And there is great value in calculating something in an original way. There’s value in calculating things wrong, too, if it’s an approximate calculation. Knowing whether your answer is nearer 10 or 20 is of some value, and it might be all that you in fact want. That’s being wrong in a productive way, though.
Harry Bliss and Steve Martin’s Bliss for the 27th uses a string of mathematical symbols as emblem of genius. Most of the symbols look just near enough meaningful that I wonder if Bliss and Martin got a mathematician friend of theirs to give them some scraps. Why I say mathematician rather than, say, physicist is because some of the lines look more mathematician than physicist.
The most distinctive one, to me, is right above Dumbo’s pencil and trunk there: . This is the kind of equation you’ll see all the time in group theory. It’s an important field of mathematics, the one studying sets that work like arithmetic does. This starts with groups, which have a set of things and a binary operation between those things. Think of it as either addition or multiplication. You notice that already looks like multiplication. ‘g’ and ‘h’ serve, for group theory, the roles that ‘x’ and ‘y’ do in (high school) algebra. ‘x’ and ‘y’ mean some number, whose value we might or might not care about. Similarly, ‘g’ and ‘h’ are some elements, things in the set for our group. We might or might not care which ones they are. means the identity element, the thing which won’t change the value of the other partner in an operation. The thing that works like zero for addition, or like one for multiplication. And means the inverse of : the thing which, added (or multiplied) to gives us the identity element. So if we were talking addition and were 5, then would be -5. This might not sound like very much, but we can make it complicated.
Also distinctive to me: that first line. I’m not perfectly sure I’m transcribing this right. But it looks a good deal to me like the binomial distribution. This is the probability of seeing something happen k times, if you give it n chances to happen, and every chance has the same probability p of it happening. The formula isn’t quite right. It’s missing a power on the (1 – p) term at the end. But it’s wrong in ways that make sense for the need to draw something legible.
Just under Dumbo’s pencil, too, is a line that I had to look up how to render in WordPress’s LaTeX. It’s the one about . The union symbol, the U there, speaks of set theory. It means to form a new set, one that has all the elements in the set called X or the set called Y or both. The straight vertical lines flanking these set names or descriptions are how we describe taking the norm, finding the size, of a set. This is ordinarily how many things are inside the set. If the sets X and Y have no elements in common, then the size of the union of X and Y will be the size of the set X plus the size of the set Y.
There’s other lines that come near making sense. The line about has the form of the “mapping” way to define a function. I just don’t understand what the rule here means. The final line, , first … well, this sort of e-raised-to-the-minus-something-squared form turns up all the time. But second, to end a bit of work with an exclamation point really captures the surprise and joy of having reached a goal. Mathematicians take delight in their work, like you’d expect.
Maria Scrivan’s Half Full for the 29th is a Rubik’s Cube joke. A variation of it ran back in June 2018. I hate that this time I noticed that on the right, the cubelet — with white on top, red on the lower left, and green on the lower right — is inconsistent with the ordered cube. The corresponding cubelet there has blue on top, red on the lower left, and green on the lower right. Well, maybe the cube on the right had its color stickers applied differently. This is a little thing. But it’s close to a problem that turns up all the time in representing geometry. It’s easy to say you have, say, axes going in the x, y, and z directions. But which direction is x? Which is y? Which is z? You can lay all three out so every pair makes a right angle. Whatever way you lay them out will turn out to be, up to a rotation, one of two patterns. Let’s say the x axis points east, and the y axis points north. Then the z axis can point up. Or it can point down. You can pick which one makes sense for your problem. The two choices are mirror images of the other. You get primed to notice this when you do mathematical physics. The Rubik’s Cube on the left is just this kind of representation, with (let’s say) the red face pointing in the x direction, the green face pointing in the y direction, and the blue pointing in the z direction. Which is a lot of thought to put into what was an arbitrary choice, as I’m sure the cartoonist (or whoever did the coloring) just wanted a cube that looked attractive.
Thanks for sticking around as I finally got to the past week’s comic strips. There were just enough for me to divide them into two chunks and not feel like I’m cheating anyone of my sparkling prose.
Sandra Bell-Lundy’s Between Friends for the 4th is another entry in this strip’s string of not-quite-Venn-Diagram jokes. As will happen, the point of the diagram seems clear enough even if it doesn’t quite parse. And it isn’t a proper Venn diagram, of course; a Venn diagram for five propositions has to have 31 regions, representing all the possible ways five things can combine or be excluded. They can be beautiful to look at, but start losing their value as ways to organize thought. This is again a Euclid diagram, which doesn’t need to show every possible overlap.
Michael Jantze’s The Norm 4.0 for the 5th is the other Venn Diagram joke for the week. Again properly the first one, showing the complete lack of overlap between two positions, is an Euler rather than a Venn diagram. The second, the “Amity Venn diagram on planet X”, is a Venn diagram and showing the intersection of blue and yellow regions as green is a nice way to show that. (I’m not fond of the gender stereotyping here, nor of the conflation of gender and chromosomes. But the comic strip does have to rely on shorthands or there’s just not going to be the space to compose a joke.)
Harry Bliss’s Bliss for the 6th name-checks tetrahedrons. These are the shapes the rest of us would probably call pyramids or perhaps d4. It’s a bit silly to suppose a hairball should be a tetrahedron. But natural processes will form particular shapes. The obvious example is the hexagonal prisms of honeycombs, which come about for reasons … I’m not sure biologists are completely agreed on. Hexagons do seem to be efficient ways to encompass a lot of volume with a minimum of material, at least. But even the classic hairball looks like that for reasons, related to how it’s created and how it’s expelled from the cat. They just don’t usually have corners.
Niklas Eriksson’s Carpe Diem for the 9th has you common blackboard full of symbols to represent mathematical work. It also evokes a well-worn joke that defines a mathematician as a mechanism for turning coffee into theorems. The explosion of creativity though is true to mathematicians, though. When inspiration is flowing the notes will get abundant and start going in many different wild directions. The symbols in the comic strip don’t mean anything. But that’s not inauthentic. The notes written during an inspired burst will be nonsensical. The great idea needs to be preserved. It can be cleaned up and, one hopes, made presentable later.
Of course I like all the comics. … Well, that’s not literally true; but I have at least some affection for nearly all of the syndicated comics. This essay I bring up some strips, partly, because I just like them. This is my content hole. If you want a blog not filled with comic strips, go start your own and don’t put these things on it.
Mark Anderson’s Andertoons for the 5th is the Mark Anderson’s Andertoons for the week. Also a bit of a comment on the ability of collective action to change things. Wavehead is … well, he’s just wrong about making the number four plus the number four equal to the number seven. Not based on the numbers we mean by the words “four” and “seven”, and based on the operation we mean by “plus” and the relationship we mean by “equals”. The meaning of those things is set by, ultimately, axioms and deductive reasoning and the laws of deductive reasoning and there’s no changing the results.
But. The thing we’re referring to when we say “seven”? Or when we write the symbol “7”? That is convention. That is a thing we’ve agreed on as a reference for this concept. And that we can change, if we decide we want to. We’ve done this. Look at a thousand-year-old manuscript and the symbol that looks like ‘4’ may represent the number we call five. And the names of numbers are just common words. They’re subject to change the way every other common word is. Which is, admittedly, not very subject. It would be almost as much bother to change the word ‘four’ as it would be to change the word ‘mom’. But that’s not impossible. Just difficult.
Juba’s Viivi and Wagner for the 5th is a bit of a percentage joke. The characters also come to conclude that a thing either happens or it does not; there’s no indefinite states. This principle, the “excluded middle”, is often relied upon for deductive logic, and fairly so. It gets less clear that this can be depended on for predictions of the future, or fears for the future. And real-world things come in degrees that a mathematical concept might not. Like, your fear of the home catching fire comes true if the building burns down. But it’s also come true if a quickly-extinguished frying pan fire leaves the wall scorched, embarrassing but harmless. Anyway, relaxing someone else’s anxiety takes more than a quick declaration of statistics. Show sympathy.
Harry Bliss and Steve Martin’s Bliss for the 6th is a cute little classroom strip, with arithmetic appearing as the sort of topic that students feel overwhelmed and baffled by. It could be anything, but mathematics uses the illustration space efficiently. The strip may properly be too marginal to include, but I like Bliss’s art style and want more people to see it.
Will Henry’s Wallace the Brave for the 7th puts up what Spud calls a sadistic math problem. And, well, it is a story problem happening in their real life. You could probably turn this into an actual exam problem without great difficulty.