Reading the Comics, January 21, 2017: Homework Edition

Now to close out what Comic Strip Master Command sent my way through last Saturday. And I’m glad I’ve shifted to a regular schedule for these. They ordered a mass of comics with mathematical themes for Sunday and Monday this current week.

Karen Montague-Reyes’s Clear Blue Water rerun for the 17th describes trick-or-treating as “logarithmic”. The intention is to say that the difficulty in wrangling kids from house to house grows incredibly fast as the number of kids increases. Fair enough, but should it be “logarithmic” or “exponential”? Because the logarithm grows slowly as the number you take the logarithm of grows. It grows all the slower the bigger the number gets. The exponential of a number, though, that grows faster and faster still as the number underlying it grows. So is this mistaken?

I say no. It depends what the logarithm is, and is of. If the number of kids is the logarithm of the difficulty of hauling them around, then the intent and the mathematics are in perfect alignment. Five kids are (let’s say) ten times harder to deal with than four kids. Sensible and, from what I can tell of packs of kids, correct.

'Anne has six nickels. Sue has 41 pennies. Who has more money?' 'That's not going to be easy to figure out. It all depends on how they're dressed!'
Rick Detorie’s One Big Happy for the 17th of January, 2017. The section was about how the appearance and trappings of wealth matter for more than the actual substance of wealth so everyone’s really up to speed in the course.

Rick Detorie’s One Big Happy for the 17th is a resisting-the-word-problem joke. There’s probably some warning that could be drawn about this in how to write story problems. It’s hard to foresee all the reasonable confounding factors that might get a student to the wrong answer, or to see a problem that isn’t meant to be there.

Bill Holbrook’s On The Fastrack for the 19th continues Fi’s story of considering leaving Fastrack Inc, and finding a non-competition clause that’s of appropriate comical absurdity. As an auditor there’s not even a chance Fi could do without numbers. Were she a pure mathematician … yeah, no. There’s fields of mathematics in which numbers aren’t all that important. But we never do without them entirely. Even if we exclude cases where a number is just used as an index, for which Roman numerals would be almost as good as regular numerals. If nothing else numbers would keep sneaking in by way of polynomials.

'Uh, Fi? Have you looked at the non-compete clause in your contract?' 'I wouldn't go to one of Fastrack's competitors.' 'No, but, um ... you'd better read this.' 'I COULDN'T USE NUMBERS FOR TWO YEARS???' 'Roman numerals would be okay.'
Bill Holbrook’s On The Fastrack for the 19th of January, 2017. I feel like someone could write a convoluted story that lets someone do mathematics while avoiding any actual use of any numbers, and that it would probably be Greg Egan who did it.

Dave Whamond’s Reality Check for the 19th breaks our long dry spell without pie chart jokes.

Mort Walker and Dik Browne’s Vintage Hi and Lois for the 27th of July, 1959 uses calculus as stand-in for what college is all about. Lois’s particular example is about a second derivative. Suppose we have a function named ‘y’ and that depends on a variable named ‘x’. Probably it’s a function with domain and range both real numbers. If complex numbers were involved then the variable would more likely be called ‘z’. The first derivative of a function is about how fast its values change with small changes in the variable. The second derivative is about how fast the values of the first derivative change with small changes in the variable.

'I hope our kids are smart enough to win scholarships for college.' 'We can't count on that. We'll just have to save the money!' 'Do you know it costs about $10,000 to send one child through college?!' 'That's $40,000 we'd have to save!' Lois reads to the kids: (d^2/dx^2)y = 6x - 2.
Mort Walker and Dik Browne’s Vintage Hi and Lois for the 27th of July, 1959. Fortunately Lois discovered the other way to avoid college costs: simply freeze the ages of your children where they are now, so they never face student loans. It’s an appealing plan until you imagine being Trixie.

The ‘d’ in this equation is more of an instruction than it is a number, which is why it’s a mistake to just divide those out. Instead of writing it as \frac{d^2 y}{dx^2} it’s permitted, and common, to write it as \frac{d^2}{dx^2} y . This means the same thing. I like that because, to me at least, it more clearly suggests “do this thing (take the second derivative) to the function we call ‘y’.” That’s a matter of style and what the author thinks needs emphasis.

There are infinitely many possible functions y that would make the equation \frac{d^2 y}{dx^2} = 6x - 2 true. They all belong to one family, though. They all look like y(x) = \frac{1}{6} 6 x^3 - \frac{1}{2} 2 x^2 + C x + D , where ‘C’ and ‘D’ are some fixed numbers. There’s no way to know, from what Lois has given, what those numbers should be. It might be that the context of the problem gives information to use to say what those numbers should be. It might be that the problem doesn’t care what those numbers should be. Impossible to say without the context.


Reading the Comics, August 19, 2016: Mathematics Signifier Edition

I know it seems like when I write these essays I spend the most time on the first comic in the bunch and give the last ones a sentence, maybe two at most. I admit when there’s a lot of comics I have to write up at once my energy will droop. But Comic Strip Master Command apparently wants the juiciest topics sent out earlier in the week. I have to follow their lead.

Stephen Beals’s Adult Children for the 14th uses mathematics to signify deep thinking. In this case Claremont, the dog, is thinking of the Riemann Zeta function. It’s something important in number theory, so longtime readers should know this means it leads right to an unsolved problem. In this case it’s the Riemann Hypothesis. That’s the most popular candidate for “what is the most important unsolved problem in mathematics right now?” So you know Claremont is a deep-thinking dog.

The big Σ ordinary people might recognize as representing “sum”. The notation means to evaluate, for each legitimate value of the thing underneath — here it’s ‘n’ — the value of the expression to the right of the Sigma. Here that’s \frac{1}{n^s} . Then add up all those terms. It’s not explicit here, but context would make clear, n is positive whole numbers: 1, 2, 3, and so on. s would be a positive number, possibly a whole number.

The big capital Pi is more mysterious. It’s Sigma’s less popular brother. It means “product”. For each legitimate value of the thing underneath it — here it’s “p” — evaluate the expression on the right. Here that’s \frac{1}{1 - \frac{1}{p^s}} . Then multiply all that together. In the context of the Riemann Zeta function, “p” here isn’t just any old number, or even any old whole number. It’s only the prime numbers. Hence the “p”. Good notation, right? Yeah.

This particular equation, once shored up with the context the symbols live in, was proved by Leonhard Euler, who proved so much you sometimes wonder if later mathematicians were needed at all. It ties in to how often whole numbers are going to be prime, and what the chances are that some set of numbers are going to have no factors in common. (Other than 1, which is too boring a number to call a factor.) But even if Claremont did know that Euler got there first, it’s almost impossible to do good new work without understanding the old.

Charlos Gary’s Working It Out for the 14th is this essay’s riff on pie charts. Or bar charts. Somewhere around here the past week I read that a French idiom for the pie chart is the “cheese chart”. That’s a good enough bit I don’t want to look more closely and find out whether it’s true. If it turned out to be false I’d be heartbroken.

Ryan North’s Dinosaur Comics for the 15th talks about everyone’s favorite physics term, entropy. Everyone knows that it tends to increase. Few advanced physics concepts feel so important to everyday life. I almost made one expression of this — Boltzmann’s H-Theorem — a Theorem Thursday post. I might do a proper essay on it yet. Utahraptor describes this as one of “the few statistical laws of physics”, which I think is a bit unfair. There’s a lot about physics that is statistical; it’s often easier to deal with averages and distributions than the mass of real messy data.

Utahraptor’s right to point out that it isn’t impossible for entropy to decrease. It can be expected not to, in time. Indeed decent scientists thinking as philosophers have proposed that “increasing entropy” might be the only way to meaningfully define the flow of time. (I do not know how decent the philosophy of this is. This is far outside my expertise.) However: we would expect at least one tails to come up if we simultaneously flipped infinitely many coins fairly. But there is no reason that it couldn’t happen, that infinitely many fairly-tossed coins might all come up heads. The probability of this ever happening is zero. If we try it enough times, it will happen. Such is the intuition-destroying nature of probability and of infinitely large things.

Tony Cochran’s Agnes on the 16th proposes to decode the Voynich Manuscript. Mathematics comes in as something with answers that one can check for comparison. It’s a familiar role. As I seem to write three times a month, this is fair enough to say to an extent. Coming up with an answer to a mathematical question is hard. Checking the answer is typically easier. Well, there are many things we can try to find an answer. To see whether a proposed answer works usually we just need to go through it and see if the logic holds. This might be tedious to do, especially in those enormous brute-force problems where the proof amounts to showing there are a hundred zillion special cases and here’s an answer for each one of them. But it’s usually a much less hard thing to do.

Johnny Hart and Brant Parker’s Wizard of Id Classics for the 17th uses what seems like should be an old joke about bad accountants and nepotism. Well, you all know how important bookkeeping is to the history of mathematics, even if I’m never that specific about it because it never gets mentioned in the histories of mathematics I read. And apparently sometime between the strip’s original appearance (the 20th of August, 1966) and my childhood the Royal Accountant character got forgotten. That seems odd given the comic potential I’d imagine him to have. Sometimes a character’s only good for a short while is all.

Mark Anderson’s Andertoons for the 18th is the Andertoons representative for this essay. Fair enough. The kid speaks of exponents as a kind of repeating oneself. This is how exponents are inevitably introduced: as multiplying a number by itself many times over. That’s a solid way to introduce raising a number to a whole number. It gets a little strained to describe raising a number to a rational number. It’s a confusing mess to describe raising a number to an irrational number. But you can make that logical enough, with effort. And that’s how we do make the idea rigorous. A number raised to (say) the square root of two is something greater than the number raised to 1.4, but less than the number raised to 1.5. More than the number raised to 1.41, less than the number raised to 1.42. More than the number raised to 1.414, less than the number raised to 1.415. This takes work, but it all hangs together. And then we ask about raising numbers to an imaginary or complex-valued number and we wave that off to a higher-level mathematics class.

Nate Fakes’s Break of Day for the 18th is the anthropomorphic-numerals joke for this essay.

Lachowski’s Get A Life for the 18th is the sudoku joke for this essay. It’s also a representative of the idea that any mathematical thing is some deep, complicated puzzle at least as challenging as calculating one’s taxes. I feel like this is a rerun, but I don’t see any copyright dates. Sudoku jokes like this feel old, but comic strips have been known to make dated references before.

Samson’s Dark Side Of The Horse for the 19th is this essay’s Dark Side Of The Horse gag. I thought initially this was a counting-sheep in a lab coat. I’m going to stick to that mistaken interpretation because it’s more adorable that way.

Reading the Comics, July 8, 2016: Filling Out The Week Edition

When I split last week’s mathematically-themed comics I had just supposed there’d be some more on Friday to note. Live and learn, huh? Well, let me close out last week with a not-too-long essay. Better a couple of these than a few Reading the Comics posts long enough to break your foot on.

Adrian Raeside’s The Other Coastfor the 6th uses mathematics as a way to judge the fit and the unfit. (And Daryl isn’t even far wrong.) It’s understandable and the sort of thing people figure should flatter mathematicians. But it also plays on 19th-century social-Darwinist/eugenicist ideas which try binding together mental acuity and evolutionary “superiority”. It’s a cute joke but there is a nasty undercurrent.

Wayno’s Waynovisionfor the 6th is this essay’s pie chart. Good to have.

The Vent Diagram. The overlap of Annoying Stuff At Work and Annoying Stuff At Home is the bar stool.
Hilary Price’s Rhymes With Orangefor the 7th of July, 2016. I don’t know how valid it is; I don’t use the bar stools, myself.

Hilary Price’s Rhymes With Orangefor the 7th is this essay’s Venn Diagram joke. Good to have.

Rich Powell’s Wide Open for the 7th shows a Western-style “Convolution Kid”. It’s shown here as just shouting numbers in-between a count so as to mess things up. That matches the ordinary definition and I’m amused with it as-is. Convolution is a good mathematical function, though one I don’t remember encountering until a couple years into my undergraduate career. It’s a binary operation, one that takes two functions and combines them into a new function. It turns out to be a natural way to understand signal processing. The original signal is one function. The way a processor changes a signal is another function. The convolution of the two is what actually comes out of the processing. Dividing this lets us study the behaviors of the processor separate from a particular problem.

And it turns up in other contexts. We can use convolution to solve differential equations, which turn up everywhere. We need to solve the differential equation for a special particular boundary condition, one called the Dirac delta function. That’s a really weird one. You have no idea. And it can require incredible ingenuity to find a solution. But once you have, you can find solutions for every boundary condition. You convolute the solution for the special case and the boundary condition you’re interested in, and there you go. The work may be particularly hard for this one case, but it is only the one case.

Basic Mathematics in Nature: A small mountain, a flock of birds flying in a less-than symbol, and a tall mountain.
Daniel Beyer’s Long Story Shortfor the 9th of July, 2016. The link’s probably good for a month or so. If you’re in the far future don’t worry about telling me how the link turned out, though. It’s not that important that I know.

Daniel Beyer’s Long Story Shortfor the 9th is this essay’s mathematical symbols joke. Good to have.

Reading the Comics, December 23, 2015: Richard Thompson Christmas Trees Edition

Richard Thompson’s Cul de Sac for the 19th of December (a rerun, alas, from the 18th of December, 2010) gives me a name for this Reading the Comics installment. Just as in a Richard’s Poor Almanac mentioned last time he gives us a Christmas tree occupying a non-Euclidean space. Non-Euclidean spaces do open up the possibility of many wondrous and counterintuitive phenomena. Trees probably aren’t among them, but I don’t know a better shorthand way to describe their mysteries. And if you’re not sure why so many people say this was the greatest comic strip of our still-young century, look at little Pete in the last panel. Both his expression and the composition of the panel are magnificent.

Tom Toles’s Randolph Itch, 2 am for the 21st of December is a rerun. And it’s one that’s been mentioned around here as recently as August. I don’t care. It’s still a good funny slapstick joke. The kicker at the bottom is also a solid giggle.

Richard Thompson’s Poor Richard’s Almanac for the 21st of December justifies my theme with its Platonic Fir. The Platonic Ideals of objects are, properly speaking, philosophical constructs. If they are constructs, anyway, and not the things that truly exist, and yes, we must be careful what we mean by ‘exist’ in this context. But Thompson’s diagram shows this Platonic Fir drawn as a mathematical diagram. That’s another common motif. Mathematical constructs, ideas like “triangles” and “circles” and “rotations”, do suggest Platonic Ideals quite closely. We might be a bit pressed to say what the quintessence of chair-ness is, the thing all chairs must be aspects of. But we can be pretty sure we understand what a triangle is, apart from our messy and imperfect real-world approximations of a true triangle. When mathematics enthusiasts speak of the beauty of pure mathematics it does seem like they speak of the beauty of approaching Platonic Ideals.

John Graziano’s Ripley’s Believe It or Not for the 21st of December continues its Rubik’s Cube obsession. Graziano spells Rubik correctly this time.

Don Asmussen’s Bad Reporter panel for the 23rd of December does a joke that depends on the idea of getting to be “more than infinity”. Every kid has run into the problem of trying to understand “infinity plus one”. The way we speak of “infinity” we can’t really talk about getting “more than infinity”. But we are able to think meaningfully of ways to differentiate sizes of infinity. There are some infinitely large sets that, in a sensible way, are bigger than other infinitely large sets. That’s a fun field of mathematics. You can get to interesting questions in it without needing much background or experience. It’s almost ideal for pop-mathematics essays and if you don’t believe me, then look at how many results you get googling for “Cantor’s Diagonalization Argument”. It’s not an infinite number of results, but it’ll get you quite close.

Brian and Ron Boychuk’s Chuckle Brothers for the 23rd of December is the anthropomorphic-numerals joke for this time around.

Mark Litzler’s Joe Vanilla for the 23rd of December is built on the idea that it’s absurd to develop an algorithm that could predict earning potential, hairline at 50, and fidelity. It sounds silly at first glance. But if we’ve learned anything from sabermetrics it’s that all kinds of physical traits can be studied, and modeled, and predicted. With a large and reliable enough data set, and with a mindfully developed algorithm, these models can become quite good at predicting things. The underlying property is that on average, people are average. If we know what is typical, and we have reason to think that “typical” is not changing, then we can forecast the future pretty well based on what we already see. Or if we have reason to expect that “typical” is changing in ways we understand, we can still make good forecasts.