## Reading the Comics, February 21, 2014: Circumferences and Monkeys Edition

And now to finish off the bundle of mathematic comics that I had run out of time for last time around. Once again the infinite monkeys situation comes into play; there’s also more talk about circumferences than average.

Brian and Ron Boychuk’s The Chuckle Brothers (February 13) does a little wordplay on how “circumference” sounds like it could kind of be a knightly name, which I remember seeing in a minor Bugs Bunny cartoon back in the day. “Circumference” the word derives from the Latin, “circum” meaning around and “fero” meaning “to carry”; and to my mind, the really interesting question is why do we have the words “perimeter” and “circumference” when it seems like either one would do? “Circumference” does have the connotation of referring to just the boundary of a circular or roughly circular form, but why should the perimeter of circular things be so exceptional as to usefully have its own distinct term? But English is just like that, I suppose.

Paul Trapp’s Thatababy (February 13) brings back the infinite-monkey metaphor. The infinite monkeys also appear in John Deering’s Strange Brew (February 20), which is probably just a coincidence based on how successfully tossing in lots of monkeys can produce giggling. Or maybe the last time Comic Strip Master Command issued its orders it sent out a directive, “more infinite monkey comics!”

Ruben Bolling’s Tom The Dancing Bug (February 14) delivers some satirical jabs about Biblical textual inerrancy by pointing out where the Bible makes mathematical errors. I tend to think nitpicking the Bible mostly a waste of good time on everyone’s part, although the handful of arithmetic errors are a fair wedge against the idea that the text can’t have any errors and requires no interpretation or even forgiveness, with the Ezra case the stronger one. The 1 Kings one is about the circumference and the diameter for a vessel being given, and those being incompatible, but it isn’t hard to come up with a rationalization that brings them plausibly in line (you have to suppose that the diameter goes from outer wall to outer wall, while the circumference is that of an inner wall, which may be a bit odd but isn’t actually ruled out by the text), which is why I think it’s the weaker.

Bill Whitehead’s Free Range (February 16) uses a blackboard full of mathematics as a generic “this is something really complicated” signifier. The symbols as written don’t make a lot of sense, although I admit it’s common enough while working out a difficult problem to work out weird bundles of partly-written expressions or abuses of notation (like on the middle left of the board, where a bracket around several equations is shown as being less than a bracket around fewer equations), just because ideas are exploding faster than they can be written out sensibly. Hopefully once the point is proven you’re able to go back and rebuild it all in a form which makes sense, either by going into standard notation or by discovering that you have soem new kind of notation that has to be used. It’s very exciting to come up with some new bit of notation, even if it’s only you and a couple people you work with who ever use it. Developing a good way of writing a concept might be the biggest thrill in mathematics, even better than proving something obscure or surprising.

Jonathan Lemon’s Rabbits Against Magic (February 18) uses a blackboard full of mathematics symbols again to give the impression of someone working on something really hard. The first two lines of equations on 8-Ball’s board are the time-dependent Schrödinger Equations, describing how the probability distribution for something evolves in time. The last line is Euler’s formula, the curious and fascinating relationship between pi, the base of the natural logarithm e, imaginary numbers, one, and zero.

Todd Clark’s Lola (February 20) uses the person-on-an-airplane setup for a word problem, in this case, about armrest squabbling. Interesting to me about this is that the commenters get into a squabble about how airplane speeds aren’t measured in miles per hour but rather in nautical miles, although nobody not involved in air traffic control really sees that. What amuses me about this is that what units you use to measure the speed of the plane don’t matter; the kind of work you’d do for a plane-travelling-at-speed problem is exactly the same whatever the units are. For that matter, none of the unique properties of the airplane, such as that it’s travelling through the air rather than on a highway or a train track, matter at all to the problem. The plane could be swapped out and replaced with any other method of travel without affecting the work — except that airplanes are more likely than trains (let’s say) to have an armrest shortage and so the mock question about armrest fights is one in which it matters that it’s on an airplane.

Bill Watterson’s Calvin and Hobbes (February 21) is one of the all-time classics, with Calvin wondering about just how fast his sledding is going, and being interested right up to the point that Hobbes identifies mathematics as the way to know. There’s a lot of mathematics to be seen in finding how fast they’re going downhill. Measuring the size of the hill and how long it takes to go downhill provides the average speed, certainly. Working out how far one drops, as opposed to how far one travels, is a trigonometry problem. Trying the run multiple times, and seeing how the speed varies, introduces statistics. Trying to answer questions like when are they travelling fastest — at a single instant, rather than over the whole run — introduce differential calculus. Integral calculus could be found from trying to tell what the exact distance travelled is. Working out what the shortest or the fastest possible trips introduce the calculus of variations, which leads in remarkably quick steps to optics, statistical mechanics, and even quantum mechanics. It’s pretty heady stuff, but I admit, yeah, it’s math.

## I Know Nothing Of John Venn’s Diagram Work

My Dearly Beloved, the professional philosopher, mentioned after reading the last comics review that one thing to protest in the Too Much Coffee Man strip — showing Venn diagram cartoons and Things That Are Funny as disjoint sets — was that the Venn diagram was drawn wrong. In philosophy, you see, they’re taught to draw a Venn diagram for two sets as two slightly overlapping circles, and then to black out any parts of the diagram which haven’t got any elements. If there are three sets, you draw three overlapping circles of equal size and again black out the parts that are empty.

I granted that this certainly better form, and indispensable if you don’t know anything about what sets, intersections, and unions have any elements in them, but that it was pretty much the default in mathematics to draw the loops that represent sets as not touching if you know the intersection of the sets is empty. That did get me to wondering what the proper way of doing things was, though, and I looked it up. And, indeed, according to MathWorld, I have been doing it wrong for a very long time. Per MathWorld (which is as good a general reference for this sort of thing as I can figure), to draw a Venn diagram reflecting data for N sets, the rules are:

1. Draw N simple, closed curves on the plane, so that the curves partition the plane into 2N connected regions.
2. Have each subset of the N different sets correspond to one and only one region formed by the intersection of the curves.

Partitioning the plane is pretty much exactly what you might imagine from the ordinary English meaning of the world: you divide the plane into parts that are in this group or that group or some other group, with every point in the plane in exactly one of these partitions (or on the border between them). And drawing circles which never touch mean that I (and Shannon Wheeler, and many people who draw Venn diagram cartoons) are not doing that first thing right: two circles that have no overlap the way the cartoon shows partition the plane into three pieces, not four.

I can make excuses for my sloppiness. For one, I learned about Venn diagrams in the far distant past and never went back to check I was using them right. For another, the thing I most often do with Venn diagrams is work out probability problems. One approach for figuring out the probability of something happen is to identify the set of all possible outcomes of an experiment — for a much-used example, all the possible numbers that can come up if you throw three fair dice simultaneously — and identify how many of those outcomes are in the set of whatever you’re interested in — say, rolling a nine total, or rolling a prime number, or for something complicated, “rolling a prime number or a nine”. When you’ve done this, if every possible outcome is equally likely, the probability of the outcome you’re interested in is the number of outcomes that satisfy what you’re looking for divided by the number of outcomes possible.

If you get to working that way, then, you might end up writing a list of all the possible outcomes and drawing a big bubble around the outcomes that give you nine, and around the outcomes that give you a prime number, and those aren’t going to touch for the reasons you’d expect. I’m not sure that this approach is properly considered a Venn diagram anymore, though, although I’d introduced it in statistics classes as such and seen it called that in the textbook. There might not be a better name for it, but it is doing violence to the Venn diagram concept and I’ll try to be more careful in future.

The Mathworld page, by the way, provides a couple examples of Venn diagrams for more than three propositions, down towards the bottom of the page. The last one that I can imagine being of any actual use is the starfish shape used to work out five propositions at once. That shows off 32 possible combinations of sets and I can barely imagine finding that useful as a way to visualize the relations between things. There are also representations based on seven sets, which have 128 different combinations, and for 11 propositions, a mind-boggling 2,048 possible combinations. By that point the diagram is no use for visualizing relationships of sets and is simply mathematics as artwork.

Something else I had no idea bout is that if you draw the three-circle Venn diagram, and set it so that the intersection of any two circles is at the center of the third, then the innermost intersection is a Reuleaux triangle, one of those oddball shapes that rolls as smoothly as a circle without actually being a circle. (MathWorld has an animated gif showing it rolling so.) This figure, it turns out, is also the base for something called the Henry Watt square drill bit. It can be used as a spinning drill bit to produce a (nearly) square hole, which is again pretty amazing as I make these things out, and which my father will be delighted to know I finally understand or have heard of.

In any case, the philosophy department did better teaching Venn diagrams properly than whatever math teacher I picked them up from did, or at least, my spouse retained the knowledge better than I did.

## Reading the Comics, February 11, 2014: Running Out Pi Edition

I’d figured I had enough mathematics comic strips for another of these entries, and discovered during the writing that I had much more to say about one than I had anticipated. So, although it’s no longer quite the 11th, or close to it, I’m going to exile the comics from after that date to the next of these entries.

Melissa DeJesus and Ed Power’s My Cage (February 6, rerun) makes another reference to the infinite-monkeys-with-typewriters scenario, which, since it takes place in a furry universe allows access to the punchline you might expect. I’ve written about that before, as the infinite monkeys problem sits at a wonderful intersection of important mathematics and captivating metaphors.

Gene Weingarten, Dan Weingarten, and David Clark’s Barney and Clyde (starting February 10) (and when am I going to make a macro for that credit and title?) has Cynthia given a slightly baffling homework lesson: to calculate the first ten digits of pi. The story continues through the 11th, the 12th, the 13th, finally resolving on the the 14th, in the way such stories must. I admit I’m not sure why exactly calculating the digits of π would be a suitable homework assignment; I can see working out division problems until the numbers start repeating, or doing a square root or something by hand until you’ve found enough digits.

π, though … well, there’s the question of why it’d be an assignment to start with, but also, what formula for generating π could be plausibly appropriate for an elementary school class. The one that seems obvious to me — π is equal to four times (1/1 minus 1/3 plus 1/5 minus 1/7 plus 1/9 minus 1/11 and so on and so on) — also takes way too long to work. If a little bit of coding is right, it takes something like 160 terms to get just the first two digits of π correct and that isn’t even stable. (The first 160 terms add to 3.135; the first 161 terms to 3.147.) Getting it to ten digits would take —

Well, I thought it might be as few was 10,000 terms, because it turns out the sum of the first ten thousand terms in that series is 3.1414926536, which looks dead-on until you notice that π is 3.1415926536. That’s a neat coincidence, though.

Anyway, obviously, that formula wouldn’t do, and we see on the strip of the 14th that Lucretia isn’t using that. There are a great many formulas that generate the value of π, any of which might be used for a project like this; some of them get the digits right quite rapidly, usually at a cost of being very complicated. The formula shown in the strip of the 14th, though, doesn’t seem to be right. Lucretia’s work uses the formula $\pi = \sqrt{12} \cdot \sum_{k = 0}^{\infty} \frac{(-3)^{-k}}{2k + 1}$, which takes only about 21 terms to get to the demanded ten digits of accuracy. I don’t want to guess how many pages of work it would take to get to 13,908 places.

If I don’t miss my guess the formula used here is one by Abraham Sharp, an astronomer and mathematician who worked for the Royal Observatory at Greenwich and set a record by calculating π to 72 decimal digits. He was also an instrument-maker, of rather some skill, and I found a page purporting to show his notes of how to cut some complicated polyhedrons out of a block of wood, so, if my father wants to carve a 120-sided figure, here’s his chance. Sharp seems to have started with Leibniz’s formula (yes, that Leibniz) — that the arctangent of a number x is equal to x minus one-third x cubed plus one-fifth x to the fifth power minus one-seventh x to the seventh power, et cetera — with the knowledge that the arctangent of the square root of one-third is equal to one-sixth π and produced this series that looks a lot like the one we started with, but which gets digits correct so very much more quickly.

Darrin Bell’s Candorville (February 13) is primarily a bit of guys insulting friends, but what do you know and π makes a cameo appearance here.

Shannon Wheeler’s Too Much Coffee Man (February 10) is a Venn Diagram cartoon in the service of arguing that Venn Diagram cartoons aren’t funny. Putting aside the smoke and sparks popping out of the Nomad space probe which Kirk and Spock are rushing to the transporter room, I don’t think it’s quite fair: the ease the Venn diagram gives to grouping together concepts and showing how they relate helps organize one’s understanding of concepts and can be a really efficient way to set up a joke. Granting that, perhaps Wheeler’s seen too many Venn Diagram cartoons that fail, a complaint I’m sympathetic to.

Bill Amend’s FoxTrot (February 11, rerun) was one of those strips trying to be taped to the math teacher’s door, with the pun-based programming for the Math Channel.

## Reading the Comics, February 1, 2014

For today’s round of mathematics-themed comic strips a little deeper pattern turns out to have emerged: π, that most popular of the transcendental numbers, turns up quite a bit in the comics that drew my attention the past couple weeks. Let me explain.

Dan Thompson’s Brevity (January 23) returns to the anthropomorphic numbers racket, with the kind of mathematics puns designed to get the strip pasted to the walls of the teacher’s lounge. I wonder how that’s going for him.

Greg Evans’s Luann Againn (January 25, rerun from 1986) has Luann not understanding how to work out an arithmetic problem until it’s shown how to do it: use the calculator. This is a joke that’s probably going to be with us as long as there are practical, personal calculating devices, because it is a good question why someone should bother learning arithmetic when a device will do it faster and better by every reasonable measure. I admit not being sure there is much point to learning arithmetic, other than as a way to practice a particular way of learning how to apply algorithms. I suppose it also stands as a way to get people who are really into mathematics to highlight themselves: someone who memorizes the times tables is probably interested in the kinds of systematic thought that mathematics depends on. But that’s a weak reason to demand it of every student. I suppose arithmetic is very testable, but that’s an even worse reason to make students go through it.

Mind you, I am quite open to the idea that arithmetic drills are useful for students. That I don’t know a particular reason why I should care whether a seventh-grader can divide 391 by 17 by hand doesn’t mean that I don’t think there is one.

## January 2014’s Statistics

So how does the first month of 2014 compare to the last month of 2013, in terms of popularity? The raw numbers are looking up: I went from 176 unique visitors looking at 352 pages in December up to 283 unique visitors looking at 498 pages. If WordPress’s statistics are to be believed that’s my greatest number of page views since June of 2013, and the greatest number of visitors since February. This hurt the ratio of views per visitor a little, which dropped from 2.00 to 1.76, but we can’t have everything unless I write stuff that lots of people want to read and they figure they want to read a lot more based on that, which is just crazy talk. The most popular articles, though, were:

1. Something Neat About Triangles, this delightful thing about forming an equilateral triangle starting from any old triangle.
2. How Many Trapezoids I Can Draw, with my best guess for how many different kinds of trapezoids there are (and despite its popularity I haven’t seen a kind not listed here, which surprises me).
3. Factor Finding, linking over to IvaSallay’s quite interesting blog with a great recreational mathematics puzzle (or educational puzzle, depending on how you came into it) that drove me and a friend crazy with this week’s puzzles.
4. What’s The Worst Way To Pack? in which I go looking for the least-efficient packing of spheres and show off these neat Mystery Science Theater 3000 foam balls I got.
5. Reading The Comics, December 29, 2013, the old year’s last bunch of mathematics-themed comic strips.

The countries sending me readers the most often were the United States (281), Canada (52), the United Kingdom (25), and Austria (23). Sending me just a single reader each this past month were a pretty good list:
Bulgaria, France, Greece, Israel, Morocco, the Netherlands, Norway, Portugal, Romania, Russia, Serbia, Singapore, South Korea, Spain, and Viet Nam. Returning on that list from last month are Norway, Romania, Spain, and Viet Nam, and none of those were single-country viewers back in November 2013.

## Reading the Comics, January 20, 2014

I’m getting to wonder whether cartoonists really do think about mathematics only when schools are in session; there was a frightening lull in mathematics-themed comic strips this month and I was getting all ready to write about something meaningful like how Gaussian integration works instead. But they came around, possibly because the kids went back to school and could resist answering word problems about driving places so they can divide apples again.

Carla Ventresca and Henry Beckett’s On A Claire Day (January 3) really just name-drops mathematics, as a vaguely unpleasant thing intruding on a conversation, even though Paul’s just dropped in a bit of silliness that, if I’m not reading it wrongly, is tautological anyway. There’s a fair question at work here, though: can “how good” a person is be measured? Obviously, it can’t if nobody tries; but could they succeed at all?

It sounds a bit silly, but then, measuring something like the economic state of a nation was not even imagined until surprisingly recently: most of the economic measures we have postdate World War II. One can argue whether they’re measuring what they are supposed to represent well, but there’s not much dispute about the idea that economic health could be measured anymore. When Assistant Secretary of State for the Truman administration, James Webb — later famous for managing NASA during the bulk of the Space Race — tried to get foreign relations measured in a similar way, though the idea was mocked as ridiculous (the joke was apparently something along the lines of a person rushing in to announce “Bulgaria is down two points!”, which is probably funnier if you haven’t grown up playing Civilization-style grand strategy games), and he gave up on that fight in favor of completing a desperately needed reorganization of the department.

I don’t know how I would measure a person’s goodness, but I could imagine a process of coming up with some things that could be measured, and trying them out, and seeing how well the measurements match what it feels they should be measuring. This is all probably too much work for a New Year’s Resolution, but it might get someone their thesis project.

Steve Moore’s In The Bleachers (January 14) comes back with a huge pile of equations standing as a big, complicated explanation for something. It doesn’t look to me like the description has much to do with describing balls bouncing, however, which is a bit of a disappointment given previous strips that name-drop Lev Landau or pull up implicit differentiation when they don’t need even need it. Maybe Moore wasn’t able to find something that looked good before deadline.

Bill Hinds’s Cleats (January 16, rerun) is just the sort of straightforward pun I actually more expect out of FoxTrot (see below).

Nate Frakes’s Break of Day (January 19) shows an infant trying to count sheep and concluding she’s too young to. Interesting to me is that the premise of the joke might actually be wrong: humans appear to have at least a rough sense of numbers, at least for things like counting and addition, from a surprisingly early age. This is a fascinating thing to learn about, both because it’s remarkable that humans should have a natural aptitude for arithmetic, and because of how difficult it is to come up with tests for understanding quantity and counting and addition that work on people with whom you can’t speak and who can’t be given instruction on how to respond to a test. Stanislas Dehaene’s The Number Sense: How The Mind Creates Mathematics describes some of this, although I’m hesitant to recommend it uncritically because I know I’m not well-read in the field. It’s somewhere to start learning, though.

Chip Sansom’s The Born Loser (January 20) could be the start of a word problem in translating from percentiles to rankings, and, for that matter, vice-versa. It’s convenient to switch a ranking to percentiles because that makes it easier to compare groups of different sizes. But many statistical tools, particularly the z-score, might be considered to be ways of meaningfully comparing the order of groups of different sizes that are nevertheless similar.

Bill Amend’s FoxTrot (January 20, rerun) is the reliable old figure-eight ice skating gag. I hope people won’t think worse of me for feeling that Droopy did it better.

T Lewis and Michael Fry’s Over The Hedge (January 20) uses a spot of the Fundamental Theorem of Calculus (rendered correctly) to stand in for “a really hard thought”. Calculus is probably secure in having that reputation: it’s about the last mathematics that the average person might be expected to take, and it introduces many new symbols and concepts that can be staggering (even the polymath Isaac Asimov found he just couldn’t grasp the subject), and so many of its equations are just beautiful to look at. The integral sign seems to me to have some graphic design sense that, for example, matrices or the polynomial representations of knots just don’t manage.

## December 2013’s Statistics

There’s a hopeful trend in my readership statistics for December 2013 around these parts: according to WordPress, my number of readers grew from 308 in November to 352 and the number of unique visitors grew from 158 to 176. Even the number of views per visitor grew, from 1.95 to 2.00. None of these are records, but the fact of improvement is a good one.

I can’t figure exactly how to get the report on most popular articles for the exact month of December, and was too busy with other things to check the past-30-day report on New Year’s Eve, but at least the most popular articles for the 30 days ending today were:

The countries sending me the most readers were the United States, Canada, Denmark and Austria (tied, and hi again, Elke), and the United Kingdom. Sending me just one viewer each were a slew of nations: Bangladesh, Cambodia, India, Japan, Jordan, Malaysia, Norway, Romania, Slovenia, South Africa, Spain, Sweden, Turkey, and Viet Nam. On that list last month were Jordan and Slovenia, so I’m also marginally interesting to a different group of people this time around.

This has all caused me to realize that I failed to promote my string of articles inspired by Arthur Christmas and getting to the recurrence theorem and the existential dread of the universe’s end during the Christmas season. Maybe next year, then.

## Reading the Comics, December 12, 2013

It’s a bit of a shame there weren’t quite enough comics to run my little roundup on the 11th of December, for that nice 11/12/13 sequence, but I’m not in charge of arranging these things. For this week’s gathering of mathematically themed comic strips there’s not any deeper theme than they mention mathematic points, but at least the first couple of them have some real meat to the subject matter. (It feels to me like if one of the gathered comics inspires an essay, it’s usually one of the first couple in a collection. That might indicate that I get tired while writing these out, or it might reflect a biased recollection of when I do break out an essay.)

John Allen’s Nest Heads (December 5) is built around a kid not understanding a probability distribution: how many days in a row does it take to get the chance of snow to be 100 percent? The big flaw here is the supposition that the chance of snow is a (uhm) cumulative thing, so that if the snow didn’t happen yesterday or the day before it’s the more likely to happen today or tomorrow. As we actually use weather forecasts, though, they’re … well, I’m not sure I’d say they’re independent, that yesterday’s 30 percent chance of snow has nothing to do with today’s 25 percent chance, since it seems to me plausible that whether it snowed yesterday affects whether it snows today. But they don’t just add up until we get a 100 percent chance of snow when things start to drop.

## Reading the Comics, December 3, 2013

It’s been long enough for a couple more mathematics-themed comics to gather, so, let me share them with you. The comics easily available to me may be increasing, too, as dailyink.com has indicated they’re looking to make it easier for people who aren’t subscribers to their service to look at the daily strips. I’d be glad to include them back in my roundup of mathematics strips, at least when I see them making mathematics jokes; there’ve been surprisingly few of them lately. Maybe the King Features Syndicate artists know it’s generally too much effort for me to feature them for a joke about how silly word problems are and have been saving us both the trouble.

Frank Page’s Bob the Squirrel began a sequence November 20 with the kid Lauren doing her math homework and Bob the Squirrel, one of multiple imaginary squirrels which I follow on Twitter, helping. It starts with percentages, a concept I admit that other people find harder than I ever did, probably because the “per cent” just made it clear to me at a young age what the whole thing was about. On the 21st Bob claims to have known a squirrel named Algebra, which wouldn’t be the strangest name for a squirrel. “Algebra”, the word, isn’t drawn from anyone’s name; it’s instead drawn from the title of the book Hisab al-jabr w’al-muqabala, Kitab al-Jabr wa-l-Muqabala (“The Compendious Book On Calculation By Completion and Balancing”), written by Muḥammad ibn Mūsā al-Khwārizmī, whose name did give us the word “algorithm”, which is the kind of successful word-generating power that you usually expect only from obscure Swedish towns. Bob closes things off with your standard breaking-the-word-problem sort of joke.

## November 2013’s Statistics

Hi again. I was hesitant to look at this month’s statistics, as I pretty much fell off the face of the earth for a week there, but I didn’t have the chance to do the serious thinking that’s needed for mathematics writing. The result’s almost exactly the dropoff in readership I might have predicted: from 440 views in October down to 308, and from 220 unique visitors down to 158. That’s almost an unchanged number of views per visitor, 2.00 dropping to 1.95, so at least the people still interested in me are sticking around.

The countries sending me the most viewers were as ever the United States, then Austria (hi, Elke, and thank you), the United Kingdom and then Canada. Sending me a single visitor each were Bulgaria, Cyprus, Czech Republic, Ethiopia, France, Jordan, Lebanon, Nepal, New Zealand, Russia, Singapore, Slovenia, Switzerland, and Thailand. This is also a drop in the number of single-viewer countries, although stalwarts Finland and the Netherlands are off the list. Slovenia’s the only country making a repeat appearance from last month, in fact.

The most popular articles the past month were:

And I apologize for not having produced many essays the past couple weeks, and can only fault myself for being more fascinated by some problems in my day job that’ve been taking up time and mental energy and waking me in the middle of the night with stuff I should try. I’ll be back to normal soon, I’m sure. Don’t tell my boss.

## Reading the Comics, November 13, 2013

For this week’s round of comic strips there’s almost a subtler theme than “they mention math in some way”: several have got links to statistical mechanics and the problem of recurrence. I’m not sure what’s gotten into Comic Strip Master Command that they sent out instructions to do comics that I can tie to the astounding interactions of infinities and improbable events, but it makes me wonder if I need to write a few essays about it.

Gene Weingarten, Dan Weingarten, and David Clark’s Barney and Clyde (October 30) summons the classic “infinite monkeys” problem of probability for its punch line. The concept — that if you had something producing strings of letters at random (taken to be monkeys because, I suppose, it’s assumed they would hit every key without knowing what sensibly comes next), it would, given enough time, produce any given result. The idea goes back a long way, and it’s blessed with a compelling mental image even though typewriters are a touch old-fashioned these days.

It seems to have gotten its canonical formulation in Émile Borel’s 1913 article “Statistical Mechanics and Irreversibility”, as you might expect since statistical mechanics brings up the curious problem of entropy. In short: every physical interaction, say, when two gases — let’s say clear air and some pink smoke as 1960s TV shows used to knock characters out — mix, is time-reversible. Look at the interaction of one clear-gas molecule and one pink-gas molecule and you can’t tell whether it’s playing forward or backward. But look at the entire room and it’s obvious whether they’re mixing or unmixing. How can something be time-reversible at every step of every interaction but not in whole?

The idea got a second compelling metaphor with Jorge Luis Borges’s Library of Babel, with a bit more literary class and in many printings fewer monkeys.

## October 2013’s Statistics

It’s been a month since I last looked over precisely how not-staggeringly-popular I am, so it’s time again.
For October 2013 I had 440 views, down from September’s 2013. These came from 220 distinct viewers, down again from the 237 that September gave me. This does mean there was a slender improvement in views per visitor, from 1.97 up to 2.00. Neither of these are records, although given that I had a poor updating record again this month that’s all tolerable.

The most popular articles from the past month are … well, mostly the comics, and the trapezoids come back again. I’ve clearly got to start categorizing the other kinds of polygons. Or else plunge directly into dynamical systems as that’s the other thing people liked. October 2013’s top hits were:

The country sending me the most readers again was the United States (226 of them), with the United Kingdom coming up second (37). Austria popped into third for, I think, the first time (25 views), followed by Denmark (21) and at long last Canada (18). I hope they still like me in Canada.

Sending just the lone reader each were a bunch of countries: Bermuda, Chile, Colombia, Costa Rica, Finland, Guatemala, Hong Kong, Laos, Lebanon, Malta, Mexico, the Netherlands, Oman, Romania, Saudi Arabia, Slovenia, Sweden, Turkey, and Ukraine. Finland and the Netherlands are repeats from last month, and the Netherlands is going on at least three months like this.

## Reading the Comics, October 26, 2013

And once again while I wasn’t quite looking we got a round of eight comic strips with mathematical themes to review. Some of them aren’t even about kids not understanding fractions, if you can imagine.

Jason Chatfield’s Ginger Meggs (October 14) does the usual confused-student joke. It’s a little unusual in having the subject be different ways to plot data, though, with line graphs, bar graphs, and scatter graphs being shown off. I think remarkable about this is that line graphs and bar graphs were both — well, if not invented, then at least popularized — by one person, William Playfair, who’s also to be credited for making pie charts a popular tool. Playfair, an engineer and economist of the late 18th and early 19th century, and I do admire him for developing not just one but multiple techniques for making complicated information easier to see.

Eric the Circle (October 16) breaks through my usual reluctance to include it — just having a circle doesn’t seem like it’s enough — because it does a neat bit of mathematical joking, in which a cube looks “my dual” in an octahedron. Duals are one of the ways mathematicians transform one problem into another, that usually turns out to be equivalent; what’s surprising is that often a problem that’s difficult for the original is easy, or at least easier, for the dual.

## Reading the Comics, October 8, 2013

As promised, I’ve got a fresh round of mathematics-themed comic strips to discuss, something that’s rather fun to do because it offers such an easy answer to the question of what to write about today. Once you have the subject and a deadline the rest of the writing isn’t so very hard. So here’s some comics with all the humor safely buried in exposition:

Allison Barrows’s PreTeena (September 24, Rerun) brings the characters to “Performance Camp” and a pun on one of the basic tools of trigonometry. The pun’s routine enough, but I’m delighted to see that Barrows threw in a (correct) polynomial expression for the sine of an angle, since that’s the sort of detail that doesn’t really have to be included for the joke to read cleanly but which shows that Barrows made the effort to get it right.

Polynomial expansions — here, a Taylor series — are great tools to have, because, generally, polynomials are nice and well-behaved things. They’re easy to compute, they’re easy to analyze, they’re pretty much easy to do whatever you might want to do. Being able to shift a complicated or realistic function into a polynomial, even a polynomial with infinitely many terms, is often a big step towards making a complicated problem easy.

## September 2013’s Statistics

And as it’s the start of the month I have a fresh round of reviewing the statistics for readership around here. I have seen a nice increase in both views — from 367 to about 466 total views — and in visitors — from 175 to 236 — which maybe reflects the resumption of the school year (in the United States, anyway) and some more reliable posting (of original articles and of links to other people’s) on my part. (Maybe. If I’m reading this rightly I actually only posted nine new things in September, which is the same as in August. I’m surprised that WordPress’s statistics page doesn’t seem to report how many new articles there were in the month, though.) My contrarian nature forces me to note this means my views-per-reader ratio has dropped to 1.97, down from 2.10. I suppose as long as the views-per-reader statistic stays above 1.00 I’m not doing too badly.

The most popular articles the past month were:

1. From ElKement: Space Balls, Baywatch, and the Geekiness of Classical Mechanics, which is really just pointing and slightly setting up ElKement’s start to a series on quantum field theory which you can too understand;
2. How Many Trapezoids I Can Draw, which is a persistent favorite and makes me suspect that I’ve hit on something that teachers ask students about. If I could think of a couple other nice little how-many-of-these-things problems there are I’d post them gladly, although that might screw up some people’s homework assignments;
3. Reading the Comics, September 11, 2012, which is another persistent favorite and I can’t imagine that it’s entirely about the date (although the similar Reading the Comics entry for September 11 of 2013 just missed being one of the top articles this month so perhaps the subject lines are just that effective a bit of click-baiting);
4. What Is Calculus I Like?, about my own realization that I never took a Calculus I course in the conditions that most people who take it do. I’d like more answers to the question of what experiences in intro-to-calculus courses are like, since I’m assuming that I will someday teach it again and while I think I can empathize with students, I would surely do better at understanding what they don’t understand if I knew better what people in similar courses went through;
5. Some Difficult Math Problems That You Understand, which is again pointing to another blog — here, Maths In A Minute — with a couple of mathematics problems that pretty much anyone can understand on their first reading. The problems are hard ones, each of which has challenged the mathematical community for generations, so you aren’t going to solve them; but, thinking about them and trying to solve them is probably a great exercise and likely to lead you to discovering something you didn’t know.

I got the greatest number of readers from the United States again (271), with Canada (31) once more in second place. The United Kingdom’s climbed back into the top three (21), while August’s number-three, Denmark, dropped out of the top ten and behind both Singapore and the Philippines. I got a mass of single-reader countries this time, too: Azerbaijan, Bangladesh, Belgium, Cambodia, the Czech Republic, Indonesia, Israel, Italy, Mexico, Norway, Poland, Qatar, Spain, Sri Lanka, Switzerland, and Thailand. Bangladesh and Sri Lanka are repeats from last month, but my Estonian readership seems to have fled entirely. At least India and New Zealand still like me.

## Reading the Comics, September 21, 2013

It must have been the summer vacation making comic strip artists take time off from mathematics-themed jokes: there’s a fresh batch of them a mere ten days after my last roundup.

John Zakour and Scott Roberts’s Maria’s Day (September 12) tells the basic “not understanding fractions” joke. I suspect that Zakour and Roberts — who’re pretty well-steeped in nerd culture, as their panel strip Working Daze shows — were summoning one of those warmly familiar old jokes. Well, Sydney Harris got away with the same punch line; why not them?

Brett Koth’s Diamond Lil (September 14) also mentions fractions, but as an example of one of those inexplicably complicated mathematics things that’ll haunt you rather than be useful or interesting or even understandable. I choose not to be offended by this insult of my preferred profession and won’t even point out that Koth totally redrew the panel three times over so it’s not a static shot of immobile talking heads.

## Reading the Comics, September 11, 2013

I may need to revise my seven-or-so-comic standard for hosting one of these roundups of mathematics-themed comic strips, at least during the summer vacation. We’ll see how they go as the school year picks up and cartoonists return to the traditional jokes of students not caring about algebra and kids giving wiseacre responses to word problems.

Jan Eliot’s Stone Soup began a sequence on the 26th of August in which Holly, the teenager, has to do flash cards to improve her memorization of the multiplication tables. It’s a baffling sequence to me, at least, since I can’t figure why a high schooler needs to study the times tables (on the 27th, Grandmom says it’s because it will make mathematics easier the more arithmetic she can do in her head). It’s also a bit infuriating because I can’t see a way to make sure Holly sees mathematics as tedious drudge work more than getting drilled by flash cards through summer vacation, particularly as she’s at an age where she ought to be doing algebra or trigonometry or geometry.

Steve Moore’s In The Bleachers (September 1) uses a bit of mathematics as a throwaway “something complicated to be thinking about” bit. I do like that the mathematics shown at least parses. I’m not sure offhand what problem the pitcher is trying to solve, that is, but the steps in it are done correctly, and even show off a nice bit of implicit differentiation. That’s a bit of differential calculus where you’ll find the rate of change of one variable with respect to another depends on the value of the variable, which isn’t actually hard to do if you follow the rules correctly but which, as I remember it, produces a vague sense of unease at its introduction. Probably it feels vaguely illicit to have a function defined in, in parts, in terms of itself.

## Reading the Comics, August 18, 2013

I’m sorry to have fallen silent so long; I was away from home and thought I’d be able to put up a couple of short pieces along the way, and turned out to be rather busy doing other things instead. It’s given me at least one nice problem with dramatic photographs to use in a near-future entry, though, so not all is lost (although I’m trying to think of a way to re-do the work in it that doesn’t involve quite so much algebra; I’m afraid of losing my readers and worse of making a hash of the LaTeX involved). Meanwhile, it’s been surprisingly close to a month since the last summary of comic strips with mathematical themes — I imagine the cartoonists are taking a break on Students In Classroom setups what with it being summer vacation across so much of the United States — so let me return to that.

## Professor Ludwig von Drake Explains Numerical Mathematics

The reruns of Donald Duck comics which appear at creators.com recently offered the above daily strip, featuring Ludwig von Drake and one of those computers of the kind movies and TV shows and comic strips had before anybody had computers of their own and, of course, the classic IBM motto that maybe they still have but I never hear anyone talking about it except as something from the distant and musty past. (Unfortunately, creators.com doesn’t note the date a strip originally ran, so all I can say is the strip first ran sometime after September of 1961 and before whenever Disney stopped having original daily strips drawn; I haven’t been able to find any hint of when that was other than not earlier than 1969 when cartoonist Al Taliaferro retired from it.)

## Reading the Comics, July 22, 2013

This is a shorter than usual entry for my roundup of comic strips mentioning mathematical topics, because I anticipate being a bit too busy to present this later in the week.

Ruben Boiling’s Tom the Dancing Bug (July 12) features one of his irresistible (to me) “Super-Fun-Pak Comix”, among them, A Voice From Another Dimension, which is a neat bit of Flatland-inspired fun between points in space. Edwin Abbot Abbot’s Flatland is one of those rare advanced-mathematical concepts that got firmly lodged into the pop culture, probably because it is a supremely accessible introduction to the concept of multidimensional space. People love learning about things which go against their everyday intuition, and the book (eventually) made the new notions of general relativity feel like they could be understood by anyone.

## Reading the Comics, July 5, 2013

I’m surprised to discover it’s been over a month since I had a roster of mathematics-themed comic strips to share, but that’s how things happen to happen. It’s also been a month with repeated references to “finding square roots”, I suppose because that sounds like a really math-y thing to do. It’s certainly computationally challenging; the task of finding such is even a (very minor) moment in Isaac Asimov’s magnificent short story about arithmetic, “The Feeling Of Power”. I remember reading the procedure for finding them when I was a kid, and finding that with considerable effort, I was able to, though I’d probably refuse to do more than give a rough estimate of such a root nowadays.

Bill Watterson’s Calvin and Hobbes (June 4, rerun) is another entry in the long string of jokes about “why bother studying mathematics”, but Watterson’s craft lifts it above average. Admire that fourth panel: that’s every resistant student in one pose.

## The Rare Days

The subject doesn’t quite feel right for my occasional roundups of mathematics-themed comic strips, but I noticed this month that the bit about “what is so rare as a day in June?” is coming up … well, twice, so it’s silly to call that “a lot” just yet, but it’s coming up at all. First was back on June 10th, with Jef Mallet’s Frazz (which actually enlightened me as I didn’t know where the line came from, and yes, it’s the Lowell family that also produced Percival), and then John Rose’s Barney Google and Snuffy Smith repeated the question on the 13th.

The question made me immediately think of an installment of Walt Kelly’s Pogo, where Pogo (I believe) asked the question and Porky Pine immediately answered “any day in February”. But it got me wondering whether the question could be answered more subtly, that is, more counter-intuitively.

## Reading the Comics, June 1, 2013

I’ve got a fresh batch of comics strips with mathematical themes. Actually, something I realized only as I was putting the list of them together, they’re word problem themes: there’s not any anthropomorphized numerals or puns on Wilhelm Leibniz’s name or anything like that. I can conjure easily reasons why word problems are good starting points for comic strip writers: they’re familiar to the reader, they don’t require any careful integration into character or storyline, and they can be designed to set up any punch line the cartoonist has in mind. Jason Chatfield’s Ginger Meggs and Gary Brookins’ and Susie MacNelly’s Shoe have running jokes in which Ginger or Skyler are asked for the collective name for a group of things, and some appropriately pun-like construct is given, and this is accepted though I don’t know why anyone would suppose there to be a collective name for a group of grocery store clerks or DSL technicians or whatnot.

Mason Mastroianni’s B.C. (May 23) sets things off with the classic form of a high school algebra word problem. I have wondered how long train-leaving-the-station problems are going to linger as example algebra problems, given that people (in the United States) really don’t take the trains for long distances if they can help it. The service is there; I just don’t believe it’s part of the common experience of students, which makes it a bit baffling as a word problem source. But the problems can be rewritten easily as airplane travel or cars on highways, if you want to salvage the question. (I’d also like to mention I generally like how Mastroianni has revitalized B.C. since Johnny Hart’s death. Particularly, the strip’s doing more of the comic anachronism that built the strip up in the first place, and this particular example contains a demonstration of that.)

## Reading the Comics, 16 May 2013

It’s a good time for another round of comic strip reading, particularly I haven’t had the time to think in detail about all the news in number theory that’s come out this past week, and that I’m not sure whether I should go into explaining arc lengths after I trapped at least one friend into trying to work out the circumference of an ellipse (you can’t do it either, but there are a lot of curves you could). I also notice I’m approaching that precious 10,000th blog hit here, so I can get back to work verifying that law about random data starting with the digit 1.

Berkeley Breathed’s Bloom County (May 2, rerun) throws up a bunch of mathematical symbols with the intention of producing a baffling result, so that Milo can make a clean getaway from Freida. The splendid thing to me, though, is that Milo’s answer — “log 10 times 10 to the derivative of 10,000” — actually does parse, if you read it a bit charitably. The “log 10” bit we can safely suppose to mean the logarithm base 10, because the strip originally ran in 1981 or so when there was still some use for the common logarithm. These days, we have calculators, and “log” is moving over to be the “natural logarithm”, base e, what was formerly denoted as “ln”.

## Reading the Comics, April 5, 2013

Before getting to the next round of comic strips that mention mathematics stuff, I’d like to do a bit of self-promotion. Freshly published is the book Oh, Sandy: An Anthology Of Humor For A Serious Purpose, edited by Lynn Beighley, Peter Barlow, Andrea Donio, and A J Fader. This is a collection of humorous bits, written out of a sense of needing to do something useful after the Superstorm. I have an essay in there, based on the strange feelings I had of being remote (and quite safe) while seeing my home state — and particularly the piers at Seaside Heights, New Jersey — being battered by a storm. The book is available also through CreateSpace.

Jenny Campbell’s Flo and Friends (March 23) mentions π, and what’s really a fairly indistinct question for a tutor to ask the student. “Explain pi” is more open-ended than I think could be useful to answer: you could write books trying to describe what it’s used for, never mind the history of studying it. After all, it’s the only transcendental number with enough pop cultural cachet to appear routinely in newspaper comic strips; what constitutes an explanation of it? Alas, the strip just goes for the easiest pi pun to be made.

Scott Hilburn’s The Argyle Sweater (March 25) returns to the gimmick of anthropomorphized numerals. It’s a cute enough joke; it’s also apparently a different pair of 1 and 2 from earlier in the month. I do wonder what, in this panel’s continuity, subtraction might mean. Still, Hilburn is obviously never far from thinking of anthropomorphized numbers, as he came back to the setting on April 3, with another 2 putting in an appearance.

## Reading the Comics, January 16, 2013

I was beginning to wonder whether my declaration last time that I’d post a comics review every time I had seven to ten strips to talk about was going to see the extinguishing of math-themed comic strips. It only felt like it. The boom-and-bust cycle continues, though; it took better than two weeks to get six such strips, and then three more came in two days. But that’s the fun of working on relatively rare phenomena. Let me get to the most recent installment of math-themed comics, mostly from Gocomics.com:

## Reading The Comics, December 15, 2012

I’ve been trying to balance how often I do the comics reviews with how often I do other essays; I admit the comics feel like particular fun to write, but the other essays are less reactive. This leaves me feeling like after I’ve done a comics roundup I should do a couple in which I come up with the topic, the exposition, and all the supplementary matter for a while, but that encourages pileups in the comics. I’m thinking of shifting over to some kind of rule less dependent on my feeling, such as writing a comics article whenever I have (say) seven to ten features to show off. We’ve got that more than that now, it turns out, so let me start out with some that came across my desktop since the last comics review.

## Reading the Comics, November 21, 2012

It’s been long enough since my last roundup of mathematics-themed comics to host a new one. I’m also getting stirred to try tracking how many of these turn up per day, because they certainly feel like they run in a feast-or-famine pattern. There’d be no point to it, besides satisfying my vague feelings that everything can be tracked, but there’s data laying there all ready to be measured, isn’t there?

## Reading the Comics, November 11, 2012

Since Scott Adams’s Dilbert hasn’t done anything to deserve my scrutiny let me carry on my quest to identify all the comic strips that mention some mathematical thing. I’m leaving a couple out; for example, today (the 11th) Rob Harrell’s Adam @ Home and Bill Amend’s FoxTrot mentioned the alignment of digits in the date’s representation in numerals, but that seems too marginal, and yet here I am talking about it. I can’t be bothered coming up with rules I can follow for my own amusement here, can I?

## Reading the Comics, October 25, 2012

As before, this is going to be the comics other than those run through King Features Syndicate, since I haven’t found a solution I like for presenting their mathematics-themed comic strips for discussion. But there haven’t been many this month that I’ve seen either, so I can stick with gocomics.com strips for today at least. (I’m also a little irked that Comics Kingdom’s archives are being shut down — it’s their right, of course, but I don’t like having so many dead links in my old articles.) But on with the strips I have got.

## Reading the Comics, October 13, 2012

I suppose it’s been long enough to resume the review of math-themed comic strips. I admit there are weeks I don’t have much chance to write regular articles and then I feel embarrassed that I post only comic strips links, but I do enjoy the comics and the comic strip reviews. This one gets slightly truncated because King Features Syndicate has indeed locked down their Comics Kingdom archives of its strips, making it blasted inconvenient to read and nearly impossible to link to them in any practical, archivable way. They do offer a service, DailyInk.com, with their comic strips, but I can hardly expect every reader of mine to pay up over there just for the odd day when Mandrake the Magician mentions something I can build a math problem from. Until I work out an acceptable-to-me resolution, then, I’ll be dropping to gocomics.com and a few oddball strips that the Houston Chronicle carries.

## Reading the Comics, September 26, 2012

I haven’t time to write a short piece today so let me go through a fresh batch of math-themed comic strips instead. There might be a change coming to these features soon, both in the strips I read and in how I present them, since Comics Kingdom, which provides the King Features Syndicate comic strips, has shown signs that they’re tightening up web access to their strips.

I can’t blame them for wanting to make sure people go through paths they control — and, pay for, at least in advertising clicks — but I can fault them for doing a rotten job of it. They’re just not very good web masters, and end up serving strips — you may have seen them if you’ve gone to the comics page of your local newspaper — that are tiny, which kills plot-heavy features like The Phantom or fine-print heavy features like Slylock Fox Sunday pages, and loaded with referrer-based and cookie-based nonsense that makes it too easy to fail to show a comic altogether or to screw up hopelessly loading up several web browser tabs with different comics in them.

For now that hasn’t happened, at least, but I’m warning that if it does, I might not necessarily read all the King Features strips — their advertising claims they have the best strips in the world, but then, they also run The Katzenjammer Kids which, believe it or not, still exists — and might not be able to comment on them. We’ll see. On to the strips for the middle of September, though:

## Reading the Comics, September 11, 2012

Since the last installment of these mathematics-themed comic strips there’s been a steady drizzle of syndicated comics touching on something mathematical. This probably reflects the back-to-school interests that are naturally going to interest the people drawing either Precocious Children strips or Three Generations And A Dog strips.

## Reading the Comics, August 27, 2012

I’m also surprised to find it’s been about a month since my last roundup of mathematics-themed comic strips, but that’s about how it worked out. There was a long stretch of not many syndicated comics touching on any subjects at all and then a rush as cartoonists noticed that summer vacation is on the verge of ending. (I understand in some United States school districts it already has ended, but I grew up in a state where school simply never started before Labor Day, so the idea of school in August feels fundamentally implausible.)

## Reading the Comics, July 28, 2012

I intend to be back to regular mathematics-based posts soon. I had a fine idea for a couple posts based on Sunday’s closing of the Diaster Transport roller coaster ride at Cedar Point, actually, although I have to technically write them first. (My bride and I made a trip to the park to get a last ride in before its closing, and that lead to inspiration.) But reviews of math-touching comic strips are always good for my readership, if I’m readin the statistics page here right, so let’s see what’s come up since the last recap, going up to the 14th of July.