## Reading the Comics, February 16, 2019: The Rest And The Rejects

I’d promised on Sunday the remainder of last week’s mathematically-themed comic strips. I got busy with house chores yesterday and failed to post on time. That’s why this is late. It’s only a couple of comics here, but it does include my list of strips that I didn’t think were on-topic enough. You might like them, or be able to use them, yourself, though.

Niklas Eriksson’s Carpe Diem for the 14th depicts a kid enthusiastic about the abilities of mathematics to uncover truths. Suppressed truths, in this case. Well, it’s not as if mathematics hasn’t been put to the service of conspiracy theories before. Mathematics holds a great promise of truth. Answers calculated correctly are, after all, universally true. They can also offer a hypnotizing precision, with all the digits past the decimal point that anyone could want. But one catch among many is whether your calculations are about anything relevant to what you want to know. Another is whether the calculations were done correctly. It’s easy to make a mistake. If one thinks one has found exciting results it’s hard to imagine even looking for one.

You can’t use shadow analysis to prove the Moon landings fake. But the analysis of shadows can be good mathematics. It can locate things in space and in time. This is a kind of “inverse problem”: given this observable result, what combinations of light and shadow and position would have caused that? And there is a related problem. Johannes Vermeer produced many paintings with awesome, photorealistic detail. One hypothesis for how he achieved this skill is that he used optical tools, including a camera obscura and appropriate curved mirrors. So, is it possible to use the objects shown in perspective in his paintings to project where the original objects had to be, and where the painter had to be, to see them? We can calculate this, at least. I am not well enough versed in art history to say whether we have compelling answers.

Art Sansom and Chip Sansom’s The Born Loser for the 16th is the rare Roman Numerals joke strip that isn’t anthropomorphizing the numerals. Or a play on how the numerals used are also letters. But yeah, there’s not much use for them that isn’t decorative. Hindu-Arabic numerals have great advantages in compactness, and multiplication and division, and handling fractions of a whole number. And handling big numbers. Roman numerals are probably about as good for adding or subtracting small numbers, but that’s not enough of what we do anymore.

And past that there were three comic strips that had some mathematics element. But they were slight ones, and I didn’t feel I could write about them at length. Might like them anyway. Gordon Bess’s Redeye for the 10th of February, and originally run the 24th of September, 1972, has the start of a word problem as example of Pokey’s homework. Mark Litzler’s Joe Vanilla for the 11th has a couple scientist-types standing in front of a board with some mathematics symbols. The symbols don’t quite parse, to me, but they look close to it. Like, the line about $l(\omega) = \int_{-\infty}^{\infty} l(x) e$ is close to what one would write for the Fourier transformation of the function named l. It would need to be more like $L(\omega) = \int_{-\infty}^{\infty} l(x) e^{\imath \omega x} dx$ and even then it wouldn’t be quite done. So I guess Litzler used some actual reference but only copied as much as worked for the composition. (Which is not a problem, of course. The mathematics has no role in this strip beyond its visual appeal, so only the part that looks good needs to be there.) The Fourier transform’s a commonly-used trick; among many things, it lets us replace differential equations (hard, but instructive, and everywhere) with polynomials (comfortable and familiar and well-understood). Finally among the not-quite-comment-worthy is Pascal Wyse and Joe Berger’s Berger And Wyse for the 12th, showing off a Venn Diagram for its joke.

Next Sunday should be a fresh Reading the Comics post, which like all its kind, should appear at this link.

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## Reading the Comics, February 13, 2019: Light Geometry Edition

Comic Strip Master Command decided this would be a light week, with about six comic strips worth discussing. I’ll go into four of them here, and in a day or two wrap up the remainder. There were several strips that didn’t quite rate discussion, and I’ll share those too. I never can be sure what strips will be best taped to someone’s office door.

Alex Hallatt’s Arctic Circle for the 10th was inspired by a tabular iceberg that got some attention in October 2018. It looked surprisingly rectangular. Smoother than we expect natural things to be. My first thought about this strip was to write about crystals. The ways that molecules can fit together may be reflected in how the whole structure looks. And this gets us to studying symmetries.

But I got to another thought. We’re surprised to see lines in nature. We know what lines are, and understand properties of them pretty well. Even if we don’t specialize in geometry we can understand how we expect them to work. I don’t know how much of this is a cultural artifact: in the western mathematics tradition lines and polygons and circles are taught a lot, and from an early age. My impression is that enough different cultures have similar enough geometries, though. (Are there any societies that don’t seem aware of the Pythagorean Theorem?) So what is it that has got so many people making perfect lines and circles and triangles and squares out of crooked timbers?

Russell Myers’s Broom Hilda for the 13th is a lottery joke. Also, really, an accounting joke. Most of the players of a lottery will not win, of course. Nearly none of them will win more than they’ve paid into the lottery. If they didn’t, there would be an official inquiry. So, yes, nearly all people, even those who win money at the lottery, would have had more money if they skipped playing altogether.

Where it becomes an accounting question is how much did Broom Hilda expect to have when the week was through? If she planned to spend \$20 on lottery tickets, and got exactly that? It seems snobbish to me to say that’s a dumber way to spend twenty bucks than, say, buying twenty bucks worth of magazines that you’ll throw away in a month would be. Or having dinner at a fast-casual place. Or anything else that you like doing even though it won’t leave you, in the long run, any better off. Has she come out ahead? That depends where she figures she should be.

Eric the Circle for the 13th, this one by Alabama_Al, is a plane- and solid-geometry joke. This gets it a bit more solidly on-topic than usual. But it’s still a strip focused on the connotations of mathematically-connected terms. There’s the metaphorical use of the ‘plane’ as in the thing people perceive as reality. There’s conflation between the idea of a ‘higher plane’ and ‘higher dimensions’. Also somewhere in here is the idea that ‘higher’ and ‘more’ dimensions of space are the same thing. ‘Transcendental’ here is used in the common English sense of surpassing something. ‘Transcendental’ has a mathematical definition too. That one relates to polynomials, because everything in mathematics is about polynomials. And, of course, one of the two numbers we know to be transcendental, and that people have any reason to care about, is π, which turns up all over circles.

Larry Wright’s Motley for the 13th riffs on the form of a story problem. Joey’s mother does ask something that seems like a plausible addition problem. I’m a bit surprised he hadn’t counted all the day’s cookies already, but perhaps he doesn’t dwell on past snacks.

This and all my Reading the Comics posts should appear at this link. Thanks for looking at my comments.

## Proving That Disturbing Triangle Theorem That Isn’t Morley’s Somehow

I couldn’t leave people just hanging on that triangle theorem from the other day. Tthis was a compass-and-straightedge method to split a triangle into two shapes of equal area. The trick was you could split it along any point on one of the three legs of the triangle.

The theorem unsettled me, yes. But proving that it does work is not so bad and I thought to do that today.

The process: start with a triangle ABC. Pick a point P on one of the legs. We’ll say it’s on leg AB. Draw the line segment from the other vertex, C, to point P.

Now from the median point S on leg AB, draw the line parallel to PC and that intersects either leg AC or leg BC. Label that point R. The line segment RP cuts the triangle ABC into one triangle and another shape, normally a quadrilateral. Both shapes have the same area, half that of the original triangle.

To prove it nicely will involve one extra line, and the identification of one point. Construct the line SC. Lines SC and PC intersect at some point; call that Q. I’ve actually made a diagram of this, just below. I’ve put the intersection point R on the leg AC. All that would change if the point R were on BC instead would be some of the labels.

Here’s how the proof will go. I want to show triangle APR has half the area of triangle ABC. The area of triangle ARP has to be equal to the area of triangle ASC, plus the area of triangle SPQ, minus the area of triangle QCR. So the first step is proving that triangle ASC has half the area of triangle ABC. The second step is showing triangle SPQ has the same area as does triangle QCR. When that’s done, we know triangle APR has the same area as triangle ASC, which is half that of triangle ABC.

First. That ASC has half the area of triangle ABC. The area of a triangle is one-half times the length of a base times its height. The base is any of the three legs which connect two points. The height is the perpendicular distance from the third point to the line that first leg is on. Here, take the base of triangle ABC to be the line segment AC. Also take the base of triangle ASC to be the line segment AC. They have the same base. Point S is the median of the line segment AB. So point S is half as far from the base AC as the point B is. Triangle ASC has half the height of triangle ABC. Same base, half the height. So triangle ASC has half the area of triangle ABC.

Second. That triangle SPQ has the same area as triangle QCR. This is going to be most easily done by looking at two other triangles, SPC and PCR. They’re relevant to triangles SPQ and QCR. Triangle SPC has the same area as triangle PCR. Take as the base for both of them the leg PC. Point S and point R are both on the line SR. SR was created parallel to the line PC. So the perpendicular distance from point S to line PC has to be the same as the perpendicular distance from point R to the line PC. Triangle SPC has the same base and same height as does triangle PCR. So they have the same area.

Now. Triangle SPC is made up of two smaller triangles: triangle SPQ and triangle PCQ. Its area is split, somehow, between those two. Triangle PCR is also made of two smaller triangles: triangle PCQ and triangle QCR. Its area is split between those two.

The area of triangle SPQ plus the area of triangle PCQ is the same as the area of triangle SPC. This is equal to the area of triangle PCR. The area of triangle PCR is the area of triangle PCQ plus the area of triangle QCR.

And that all adds up only if the area of triangle SPQ is the same as the area of triangle QCR.

So. We had that area of triangle APR is equal to the area of triangle ASC plus the area of triangle SPQ minus the area of triangle QCR. That’s the area of triangle ASC plus zero. And that’s half the area of triangle ABC. Whatever shape is left has to have the remaining area, half the area of triangle ABC.

It’s still such a neat result.

Morley’s theorem, by the way, says this: take any triangle. Trisect each of its three interior angles. That is, for each vertex, draw the two lines that cut the interior angle into three equal spans. This creates six lines. Take the three points where these lines for adjacent angles intersect. (That is, draw the obvious intersection points.) This creates a new triangle. It’s equilateral. What business could an equilateral triangle possibly have in all this? Exactly.

## In Which I Am Disturbed By A Triangle Theorem That Isn’t Morley’s Somehow

I’ve been reading Alfred S Posamentier and Ingmar Lehmann’s The Secrets of Triangles: A Mathematical Journey. It is exactly what you’d think: 365 pages, plus endnotes and an index, describing what we as a people have learned about triangles. It’s almost enough to make one wonder if we maybe know too many things about triangles. I admit letting myself skim over the demonstration of how, using straightedge and compass, to construct a triangle when you’re given one interior angle, the distance from another vertex to its corresponding median point, and the radius of the triangle’s circumscribed circle.

But there are a bunch of interesting theorems to find. I wanted to share one. When I saw it I felt creeped out. The process seemed like a bit of dark magic, a result starting enough that it seemed to come from nowhere. Here it is.

Start with any old triangle ABC. Without loss of generality, select a point along the leg AB (other than the vertices). Call that point P. (This same technique would work if you put your point on another leg, but I would have to change the names of the vertices and line segments from here on. But it doesn’t matter what the names of the vertices are. So I can suppose that I was lucky enough that whatever leg you put your point P on I happened to name AB.)

Now. Pick the midpoint of the leg AB. This median is a point we’ll label S.

Draw the line PC.

Draw the line parallel to the line PC and which passes through S. This will intersect either the line segment BC or the line segment AC. Whichever it is, label this point of intersection R.

Draw the line from R to P.

The line RP divides the triangle ABC into two shapes, a triangle and (unless your P was the median point S) a quadrilateral.

The punch line: both shapes have half the area of the original triangle.

I usually read while eating. This was one of those lines that made me put the fork down and stare, irrationally angry, until I could work through the proof. It didn’t help that you can use a technique like this to cut the triangle into any whole number you like of equal-area wedges.

I’m sure this is old news to a fair number of readers. I don’t care. I haven’t noticed this before. And yes, it’s not as scary weird magic as Morley’s Theorem. But I’ve seen that one before, long enough ago I kind of accept it.

## Reading the Comics, February 9, 2019: Garfield Outwits Me Edition

Comic Strip Master Command decreed that this should be a slow week. The greatest bit of mathematical meat came at the start, with a Garfield that included a throwaway mathematical puzzle. It didn’t turn out the way I figured when I read the strip but didn’t actually try the puzzle.

Jim Davis’s Garfield for the 3rd is a mathematics cameo. Working out a problem is one more petty obstacle in Jon’s day. Working out a square root by hand is a pretty good tedious little problem to do. You can make an estimate of this that would be not too bad. 324 is between 100 and 400. This is worth observing because the square root of 100 is 10, and the square root of 400 is 20. The square of 16 is 256, which is easy for me to remember because this turns up in computer stuff a lot. But anyway, numbers from 300 to 400 have square roots that are pretty close to but a little less than 20. So expect a number between 17 and 20.

But after that? … Well, it depends whether 324 is a perfect square. If it is a perfect square, then it has to be the square of a two-digit number. The first digit has to be 1. And the last digit has to be an 8, because the square of the last digit is 4. But that’s if 324 is a perfect square, which it almost certainly is … wait, what? … Uh .. huh. Well, that foils where I was going with this, which was to look at a couple ways to do square roots.

One is to start looking at factors. If a number is equal to the product of two numbers, then its square root is the product of the square roots of those numbers. So dividing your suspect number 324 by, say, 4 is a great idea. The square root of 324 would be 2 times the square root of whatever 324 &div; 4 is. Turns out that’s 81, and the square root of 81 is 9 and there we go, 18 by a completely different route.

So that works well too. If it had turned out the square root was something like $2\sqrt{82}$ then we get into tricky stuff. One response is to leave the answer like that: $2\sqrt{82}$ is exactly the square root of 328. But I can understand someone who feels like they could use a numerical approximation, so that they know whether this is bigger than 19 or not. There are a bunch of ways to numerically approximate square roots. Last year I worked out a way myself, one that needs only a table of trigonometric functions to work out. Tables of logarithms are also usable. And there are many methods, often using iterative techniques, in which you make ever-better approximations until you have one as good as your situation demands.

Anyway, I’m startled that the cheese doodles price turned out to be a perfect square (in cents). Of course, the comic strip can be written to have any price filled in there. The joke doesn’t depend on whether it’s easy or hard to take the square root of 324. But that does mean it was written so that the problem was surprisingly doable and I’m amused by that.

Ryan North’s Dinosaur Comics for the 4th goes in some odd directions. But it’s built on the wonder of big numbers. We don’t have much of a sense for how big truly large numbers. We can approach pieces of that, such as by noticing that a billion seconds is a bit more than thirty years. But there are a lot of truly staggeringly large numbers out there. Our basic units for things like distance and mass and quantity are designed for everyday, tabletop measurements. The numbers don’t get outrageously large. Had they threatened to, we’d have set the length of a meter to be something different. We need to look at the cosmos or at the quantum to see things that need numbers like a sextillion. Or we need to look at combinations and permutations of things, but that’s extremely hard to do.

Tom Horacek’s Foolish Mortals for the 4th is a marginal inclusion for this week’s strips, but it’s a low-volume week. The intended joke is just showing off a “tube sock” and an “inner tube sock”. But it happens to depict these as a cylinder and a torus and those are some fun shapes to play with. Particularly, consider this: it’s easy to go from a flat surface to a cylinder. You know this because you can roll a piece of paper up and get a good tube. And it’s not hard to imagine going from a cylinder to a torus. You need the cylinder to have a good bit of give, but it’s easy to imagine stretching it around and taping one end to the other. But now you’ve got a shape that is very different from a sheet of paper. The four-color map theorem, for example, no longer holds. You can divide the surface of the torus so it needs at least seven colors.

Mastroianni and Hart’s B.C. for the 5th is a bit of wordplay. As I said, this was a low-volume week around here. The word “logarithm” derives, I’m told, from the modern-Latin ‘logarithmus’. John Napier, who advanced most of the idea of logarithms, coined the term. It derives from ‘logos’, here meaning ‘ratio’, and ‘re-arithmos’, meaning ‘counting number’. The connection between ratios and logarithms might not seem obvious. But suppose you have a couple of numbers, and we’ll reach deep into the set of possible names and call them a, b, and c. Suppose a &div; b equals b &div; c. Then the difference between the logarithm of a and the logarithm of b is the same as the difference between the logarithm of b and the logarithm of c. This lets us change calculations on numbers to calculations on the ratios between numbers and this turns out to often be easier work. Once you’ve found the logarithms. That can be tricky, but there are always ways to do it.

Bill Rechin’s Crock for the 8th is not quite a bit of wordplay. But it mentions fractions, which seem to reliably confuse people. Otis’s father is helpless to present a concrete, specific example of what fractions mean. I’d probably go with change, or with slices of pizza or cake. Something common enough in a child’s life.

And I grant there have been several comic strips here of marginal mathematics value. There was still one of such marginal value. Mark Parisi’s Off The Mark for the 7th has anthropomorphized numerals, in service of a temperature joke.

These are all the mathematically-themed comic strips for the past week. Next Sunday, I hope, I’ll have more. Meanwhile please come around here this week to see what, if anything, I think to write about.

## Reading the Comics, February 2, 2019: Not The February 1, 2019 Edition

The last burst of mathematically-themed comic strips last week nearly all came the 1st of the month. But the count fell just short. I can only imagine what machinations at Comic Strip Master Command went wrong, that we couldn’t get a full four comics for the same day. Well, life is messy and things will happen.

Stephen Bentley’s Herb and Jamaal for the 1st is a rerun. I discussed it last time I noticed it too. I’d previously taken Herb to be gloating about not using the calculus he’d studied. I may be reading too much into what seems like a smirk in the final panel, though. Could be he’s thinking of the strangeness that something which, at the time, is challenging and difficult and all-consuming turns out to not be such a big deal. Which could be much of high school.

But my first instinct is still to read this as thinking of the “uselessness” of calculus. It betrays the terrible attitude that education is about job training. It should be about letting people be literate in the world’s great thoughts. Mathematics seems to get this attitude a lot, but I’m aware I may feel a confirmation bias. If I had become a French major perhaps I’d pay attention to all the comic strips where someone giggles about how they never use the foreign languages they learned in high school either.

Jon Rosenberg’s Scenes from a Multiverse for the 1st is set in a “Mathpinion City”, showing people arguing about mathematical truths. It seems to me a political commentary, about the absurdity of rejecting true things over perceived insults. The 1+1=3 partisans aren’t even insisting they’re right, just that the other side is obnoxious. Arithmetic here serves as good source for things that can’t be matters of opinion, at least provided we’ve agreed on what’s meant by ideas like ‘1’ and ‘3’.

Mathematics is a human creation, though. What we decide to study, and what concepts we think worth interesting, are matters of opinion. It’s difficult to imagine people who think 1+1=2 a statement so unimportant they don’t care whether it’s true or false. At least not ones who reason anything like we do. But that is our difficulty, not a constraint on what life could think.

Neil Kohney’s The Other End for the 1st has a mathematics cameo. It’s the subject of a quiz so difficult that the kid begs for God’s help sorting it out. The problems all seem to be simplifying expressions. It’s a skill worth having. There are infinitely many ways to write the same quantity. Some of them are more convenient than others. Brief expressions, for example, are often easier to understand. But a longer expression might let us tease out relationships that are good to know. Many analysis proofs end up becoming simpler when you multiply by one — that is, multiplying by and dividing by the same quantity, but using the numerator to reduce one part of the expression and the denominator to reduce some other. Or by adding zero, in which you add and subtract a quantity and use either side to simplify other parts of the expression. So, y’know, just do the work. It’s better that way.

Mark Anderson’s Andertoons for the 2nd is the Mark Anderson’s Andertoons for the week. Wavehead’s learning about invertible operations: that a particular division can undo a multiplication. Or, presumably, that a particular multiplication can undo a division. Fair to wonder why you’d want to do that, though. Most of the operations we use in arithmetic have inverses, or come near it. (There’s one thing you can multiply by which you can’t divide out.) The term used in group theory for this is to say the real numbers are a “field”. This is a ring in which not just does addition have an inverse, but so does multiplication. And the operations commute; dividing by four and multiplying by four is as good as multiplying by for and dividing by four. You can build interesting mathematical structures that don’t have some of these properties. Elementary-school division, where you might describe (say) 26 divided by 4 as “6 with a remainder of 2” is one of them.

And that covers the comic strips. Come Sunday should be the next of this series, and it should be at this link.

## Reading the Comics, January 30, 2019: Interlude Edition

I think there are just barely enough comic strips from the past week to make three essays this time around. But one of them has to be a short group, only three comics. That’ll be for the next essay when I can group together all the strips that ran in February. One strip that I considered but decided not to write at length about was Ed Allison’s dadaist Unstrange Phenomena for the 28th. It mentions Roman Numerals and the idea of sneaking message in through them. But that’s not really mathematics. I usually enjoy the particular flavor of nonsense which Unstrange Phenomena uses; you might, too.

John McPherson’s Close to Home for the 29th uses an arithmetic problem as shorthand for an accomplished education. The problem is solvable. Of course, you say. It’s an equation with quadratic polynomial; it can hardly not be solved. Yes, fine. But McPherson could easily have thrown together numbers that implied x was complex-valued, or had radicals or some other strange condition. This is one that someone could do in their heads, at least once they practiced in mental arithmetic.

I feel reasonably confident McPherson was just having a giggle at the idea of putting knowledge tests into inappropriate venues. So I’ll save the full rant. But there is a long history of racist and eugenicist ideology that tried to prove certain peoples to be mentally incompetent. Making an arithmetic quiz prerequisite to something unrelated echoes that. I’d have asked McPherson to rework the joke to avoid that.

(I’d also want to rework the composition, since the booth, the swinging arm, and the skirted attendant with the clipboard don’t look like any tollbooth I know. But I don’t have an idea how to redo the layout so it’s more realistic. And it’s not as if that sort of realism would heighten the joke.)

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 29th riffs on the problem of squaring the circle. This is one of three classical problems of geometry. The lecturer describes it just fine: is it possible to make a square that’s got the same area as a given circle, using only straightedge and compass? There are shapes it’s easy to do this for, such as rectangles, parallelograms, triangles, and (why not?) this odd crescent-moon shaped figure called the lune. Circles defied all attempts. In the 19th century mathematicians found ways to represent the operations of classical geometry with algebra, and could use the tools of algebra to show squaring the circle was impossible. The squaring would be equivalent to finding a polynomial, with integer coefficients, that has $\sqrt{\pi}$ as a root. And we know from the way algebra works that this can’t be done. So squaring the circle can’t be done.

The lecturer’s hack, modifying the compass and straightedge, lets you in principle do whatever you want. The hack isn’t new either. Modifying the geometric tools changes what you can and can’t do. The Ancient Greeks recognized that adding some specialized tools would make the problem possible. But that falls outside the scope of the problem.

Which feeds to the secondary joke, of making the philosophers sad. Often philosophy problems test one’s intuition about an idea by setting out a problem, often with unpleasant choices. A common problem with students that I’m going ahead and guessing are engineers is then attacking the setup of the question, trying to show that the problem couldn’t actually happen. You know, as though there were ever a time significant numbers of people were being tied to trolley tracks. (By the way, that thing about silent movie villains tying women to railroad tracks? Only happened in comedies spoofing Victorian melodramas. It’s always been a parody.) Attacking the logic of a problem may make for good movie drama. But it makes for a lousy student and a worse class discussion.

Ted Shearer’s Quincy rerun for the 30th uses a bit of mathematics and logic talk. It circles the difference between the feeling one can have about the rational meaning of a situation and how the situation feels to someone. It seems like a jump that Quincy goes from being asked about logic to talking about arithmetic. Possibly Quincy’s understanding of logic doesn’t start from the sort of very abstract concept that makes arithmetic hard to get to, though.

There should be another Reading the Comics post this week. It should be here, when it appears. There should also be one on Sunday, as usual.

## How January 2019 Treated My Mathematics Blog

It seems like about fifteen minutes ago I was looking over how 2018 treated my mathematics blog. But who am I to argue with the calendar? I have a hard enough time convincing the calendar that 1998 was at most eight years ago. My arguments are useless. Look, I clearly remember watching Star Trek Nemesis, opening weekend, alone except for the friend I talked into seeing this with, and there is no possible way that this was one minute more than six years ago. Well, here’s what I can say about my readership and how much blame I can take for it, within the scope of the first month of 2019.

I posted twelve things in January, and two of them were looks at what was popular previously. Considering that, though, people were interested. I suspect it’s spillover of the A-To-Z posts. There were 1,375 pages viewed, down a little from December’s 1,409 and November’s 1,611. Considering how much less effort January was, this seems like a great tradeoff. There were 856 unique visitors, compared to December’s 875 and November’s 847. In November I had 23 posts and December 17, so I’m at least being very efficient, per post, at drawing readers. I hadn’t had a 12-post month since July, when there were 1,058 page views and 668 unique visitors. Probably people were hanging around hoping to see more A-To-Z grade stuff.

The number of items liked dropped to 63. There had been 82 in December and 85 in November. Again, per post, that’s a pretty good rate of growth. There were 22 comments, up from December’s 17, down from November’s 36, and still pretty close to nothing when you consider I try to answer every comment, so half of all that writing is just me.

There was an outright surprise among the most popular posts of the month. Do you see which one doesn’t seem to belong here? And can you spot in which one I originally wrote ‘2018’ in the subject line, and corrected it, but it’s too much trouble to correct a WordPress URL for me to bother with?

Well, I’m delighted to see interest in the Five-Color Map theorem. It’s not so famous as the (correct) Four-Color Map theorem. But it’s one with a proof a normal mortal can follow.

The Insights panel tells me there were an average of 1.9 comments per post, through the end of January. 4.8 average likes per post, too. There were a meager 11,588 words posted in January, but that still averages to 966 words per post. That’s down from the 2018 average. It’s still my second-highest word count, though. It’s all right. I’ve thought of some things I could post that would be amusing and quite short to write, and that require I do calculations that might be fun in my spare time. This supposes that I have spare time.

How about the running of the countries?

Country Readers
United States 835
Canada 84
United Kingdom 61
India 60
Philippines 54
Denmark 26
Italy 21
American Samoa 18
Macedonia 18
Slovenia 18
Germany 17
Australia 13
Poland 11
Netherlands 10
Ireland 9
Singapore 9
South Africa 7
Brazil 6
Croatia 6
Sweden 6
United Arab Emirates 6
France 5
Malaysia 5
New Zealand 5
Czech Republic 4
Indonesia 4
Israel 4
Mexico 4
Turkey 4
European Union 3
Pakistan 3
Romania 3
Russia 3
Spain 3
Taiwan 3
Bangladesh 2
Nepal 2
Norway 2
Argentina 1 (*)
Austria 1
Bosnia & Herzegovina 1
Chile 1
Colombia 1 (*****)
Ecuador 1
Finland 1
Georgia 1
Greece 1
Hong Kong SAR China 1
Iraq 1 (*)
Jamaica 1
Japan 1 (*)
Jordan 1 (*)
Kazakhstan 1
Lithuania 1
Morocco 1
Saudi Arabia 1 (**)
Switzerland 1
Thailand 1
Ukraine 1

There were 59 countries listed as sending me any readers in January. That’s way down from December’s 68 and November’s 70. 19 of them were single-reader countries, up from December’s 17 and November’s 13. Argentina, Iraq, Japan, and Jordan were single-reader countries last month too. Saudi Arabia’s been a single-reader country for three months now. Colombia’s on a six-month streak now. I could swear Colombia has done this before, too, although good luck my finding the time when. Searching for ‘Colombia’ in my archives is not as helpful as you might imagine. Oh, I can find a time in late 2015 through early 2016 when I had a single European Union reader each month, six months in a row. Maybe that’s what I was thinking of.

I began the month having ever had 74,290 page views, from an acknowledged 37,119 unique readers. I’m glad you’re one of them, at least right now. You can put my posts in your RSS reader and enjoy them at your convenience. If you do that, you won’t show up in any of my statistics. I’ll never know you read any of this unless you comment. And that’s fine by me. I’m happy with people keeping their privacy. If you’re using WordPress regularly you can also add me to your WordPress Reader. That’s from the button at the upper-right corner of the page. On Twitter I’m @Nebusj. and at least once a week I go around Reading the Comics for the mathematics stuff. Thanks for being here.

## Reading the Comics, January 28, 2019: Stock Subjects Edition

There are some subjects that seem to come up all the time in these Reading the Comics posts. Lotteries. Roman numerals. Venn Diagrams. The New Math. Kids not doing arithmetic well, or not understanding when they do it. This is the slate of comics for today’s discussion.

Olivia Jaimes’s Nancy for the 27th is the Roman Numerals joke for the week. I am not certain there is a strong consensus about the origins of Roman numerals. It’s hard to suppose that the first several numerals, though, are all that far from tally marks. Adding serifs just makes the numerals probably easier to read, if harder to write. I’ll go along with Nancy’s excuse of using the weights to represent work with a lesser weight.

Joe Martin’s Mr Boffo for the 27th is a lottery joke. And a probability joke, comparing the chances of being struck by lightning to those of winning the lottery. This gives me an excuse to link back to The Wandering Melon joke about the person who suffered both. And that incident in which a person did both win the lottery and get struck by lightning, albeit several years apart.

Rick DeTorie’s One Big Happy for the 28th has the kid, Joe, impressed by something that he ought to have already expected. Grandpa uses this to take a crack at “that new, new math”, as though there were a time people weren’t amazed by what they should have deduced. Or a level of person who’s not surprised by the implications. One of Richard Feynman’s memoirs recounts him pranking people who have taken calculus by pointing out how whatever way you hold a French curve, the lowest point on it has a horizontal slope. This is true of the drafting instrument; but it’s also true of any curve that hasn’t got a corner or discontinuity.

There aren’t comments (so far as I’m aware) on Creators.com, which hosted this strip. So there weren’t any cracks about Common Core. But I am curious whether DeTorie wrote Grandpa as mentioning the New Math because the character would, plausibly, have seen that educational reform movement come and go. Or did DeTorie just riff on the New Math because that’s been a reliable punching bag since the mid-60s?

Liniers’s Macanudo for the 28th is the Venn Diagram joke for the week. And it commits to its Venn-ness. This did make me wonder whether John Venn did marry. Well, he’d taught at Cambridge in the 19th century. Sometimes marrying was forbidden. He married Sussanna Carnegie Edmonstone in 1867, and they had one child. I know nothing about whether he ever had a significant marital problem.

This past week was much busier for mathematically-themed comic strips. There’s going to be at least one more essay this week. There might be two. They’ll appear here, along with all the other Reading the Comics posts.

## What I Learned Doing My 2018 Mathematics A To Z

I have a tradition at the end of an A To Z sequence of looking back and considering what I learned doing it. Sometimes this is mathematics I’ve learned. At the risk of spoiling the magic, I don’t know as much as I present myself as knowing. I’ll often take an essay topic and study up before writing, and hope that I look competent enough that nobody seriously questions me. Yes, I thought I was a pretty good student journalist back in the day, and harbored fantasies of doing that for a career. This before I went on to fantasize about doing mathematics. Still; I also learn things about writing in doing a big writing project like this. And now I’ve had some breathing space to sit and think. I can try finding out what I thought.

The major differences between this A To Z and those of past years was scheduling. Through 2017 I’d done A To Z’s posted three days a week. This is a thrilling schedule. It makes it easy to have full weeks, even full months, with some original posting every day. I can tell myself that the number of posts doesn’t matter. It’s the quality that does. It doesn’t work. I tend toward compulsive behavior and a-post-a-day is so gratifying.

But I also knew that the last quarter of 2018 would be busy and I had to cut something down. Some of that is Reading the Comics posts. Including pictures of each comic I discuss adds considerably to the production time. This is not least because I feel I can’t reasonably claim Fair Use of the comics without writing something of more substance. Moving files around and writing alt-text for the images also takes time. But the time can be worth it. Doing that has sometimes made me think longer, or even better, about the comics.

So switching to two-a-week seemed the thing to do. It would spread the A To Z over three months, but that’s not bad. I figured to prove out whether that schedule worked. If I could find one that let me do possibly several A To Z sequences in a year that’d be wonderful too. I’ve only done that once, so far, in 2016, but I think the exercise is always good if exhausting for me.

This completely failed to save time. I had fewer things to write, but somehow, I only wrote more. All my writing’s getting longer as I go on, yes. Last year my average post exceeded a thousand words, though, and that counts Reading the Comics and pointers to things I’ve been reading and all that. There’s a similar steady expansion going on in my humor blog. Possibly having more time between essays encouraged me to write longer for each. Work famously expands to fit the time available, and having as many as three full days to write, rather than one, might be dangerous. For the first time in an A To Z I never got ahead of myself. I would, at best, be researching and making notes for the next essay while waiting for the current one to post. There’s value in dangerous writing. But I don’t like that as a habit.

Another scheduling change was in how I took topic suggestions. In the past I’d thrown the whole alphabet open at once. This time I broke the alphabet into a couple of pieces, and asked for about one-quarter at a time. This, overall, worked. For one, it gave me more chances to talk about the A To Z. Talking about something is one of the non-annoying ways to advertise a thing. And I think it helped a greater variety of people suggest topics. I did have more collisions this time around, letters for which several people suggested different ideas. That’s a happy situation to have. Thinking of what to write is the hard part; going on about a topic someone else named? That’s easy. So I’ll certainly keep that.

I did write some about maybe doing supplementary pieces, based on topics I didn’t use for the main line. Might yet do that, perhaps under rules where I do one a week, or limit myself to 700 words, or something like that. It might be worth doing a couple just to have a buffer against weeks that there’s no comic strips worth discussing. Or to head off gaps next time around, although that would spoil some letters for people.

Also I completely ran out of ‘X’ topics, and went with the 90s alternative of “extreme”. There are plenty of “extreme” things in mathematics I could write about. But that feels a bit chintzy to do too often.

This time around I changed focus on many of my essays. It wasn’t a conscious thing, not to start with. But I got to writing more about the meaning and significance and cultural import of topics, rather than definitions or descriptions of the use of things. This is a natural direction to go for a topic like Fermat’s Last Theorem, or the Infinite Monkey Theorem, or mathematics jokes. I liked the way those pieces turned out, though, and tried doing more of it. This likely helped the essays grow so long. Context demands space, after all. And more thinking. Thinking’s the hard part of writing, but it’s also fun, because when you’re thinking about a subject you aren’t typing any specific words.

But it’s probably a worthwhile shift. For a pop mathematics blog to describe what makes something a “smooth” function is all right. But it’s not a story unless it says why we should care. That’s more about context than about definitions, which anyone could get by typing ‘mathworld smooth’ into DuckDuckGo. For all the trouble this causes me, it’s the way to go.

Every good lesson carries its opposite along, though. One of the requests this time around was about Lord Kelvin. There’s no end of things you can write about him: he did important work in basically every field of science and mathematics as the 19th century knew it. It’s easy to start writing about his work and never stop. I did the opposite, taking one tiny and often-overlooked piece and focusing on that. I’m not sure it alone would convince anyone of Kelvin’s exhaustive greatness. But I don’t imagine anyone interested in reading a single essay on Kelvin would never read a second one. It seems to me a couple narrow-focus essays help in that context. Seeing more of one detail gives scale to the big picture.

I’ve done just the one A To Z the last two years. There’s surely an optimal rate for doing these. The sequences are usually good for my readership. My experience, tracking monthly readership figures, suggests that just posting more often is good for my readership. They’re also the thing I write that most directly solicits reader responses. They’re also exhausting. The last several letters are always a challenge. The weeks after a sequence is completed I collapse into a little recuperative bubble. So I want to do these as much as I can without burning out on the idea. Also without overloading Thomas Dye, who’s been so good as to make the snappy banners for these pieces. He has his own projects, including the web comic Projection Edge, to worry about. More than once a year is probably sustainable. I may also want to stack this with hosting the Playful Mathematics Education Blog Carnival again, if I’m able to this year.

Deep down, though, I think the best moment of my Fall 2018 A To Z might have been in the first. I wrote about asymptotes and realized I could put in ordinary words why they were a thing worth having. If I could have three insights like that a year I’d be a great mathematics writer.

I put a roster of things written up in this A To Z at this page. The Summer 2015 A To Z essays should be here. The essays from the Leap Day 2016 A To Z essays are at this link. The essays from the End 2016 A To Z essays are here. Those from the Summer 2017 A To Z sequence are at this link. And I should keep using the A-To-Z tag, so all of these, and any future A To Z essays, should appear at this link. Thank you for reading along.

## Reading the Comics, January 26, 2019: The Week Ended Early Edition

Last week started out at a good clip: two comics with enough of a mathematical theme I could imagine writing a paragraph about them each day. Then things puttered out. The rest of the week had almost nothing. At least nothing that seemed significant enough. I’ll list those, since that’s become my habit, at the end of the essay.

Jonathan Lemon and Joey Alison Sayers’s Alley Oop for the 20th is my first chance to show off the new artist and writer team. They’ve decided to make Sunday strips a side continuity about a young Alley Oop and his friends. I’m interested. The strip is built on the bit of pop anthropology that tells us “primitive” tribes will have very few counting words. That you can express concepts like one, two, and three, but then have to give up and count “many”.

Perhaps it’s so. Some societies have been found to have, what seem to us, rather few numerals. This doesn’t reflect on anyone’s abilities or intelligence or the like. And it doesn’t mean people who lack a word for, say, “forty-nine” would be unable to compute. It might take longer, but probably just from inexperience. If someone practiced much calculation on “forty-nine” they’d probably have a name for it. And folks raised in the western mathematics use, even enjoy, some vagueness about big numbers too. We might say there are “dozens” of a thing even if there are not precisely 24, 36, or 48 of the thing; “52” is close enough and we probably didn’t even count it up. “Hundred” similarly has gotten the connotation of being a precise number, but it’s used to mean “really quite a lot of a thing”. The words “thousands”, “millions”, and mock-numbers like “zillions” have a similar role. They suggest different ranges of what might be “many”.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th is a SABRmetrics joke! At least, it’s an optimization joke, built on the idea that you can find an optimum strategy for anything, whether winning baseball games or The War. The principle is hard to argue with. Nobody would doubt that different approaches to a battle affect how likely winning is. We can imagine gathering data on how different tactics affect the outcome. (We can easily imagine combat simulators running these experiments, particularly.)

The catch — well, one catch — is that this tempts one to reward a process. Once it’s taken for granted the process works, then whether it’s actually doing what you want gets forgotten. And once everyone knows what’s being measured it becomes possible to game the system. Famously, in the mid-1960s the United States tried to judge its progress in the Vietname War by counting the number of enemy soldiers killed. There was then little reason to care about who was killed, or why. And reason to not care whether actual enemy soldiers were being killed. There’s good to be said about testing whether the things you try to do work. There’s great danger in thinking that the thing you can measure guarantees success.

Mark Anderson’s Andertoons for the 21st is a bit of fun with definitions. Mathematicians rely on definitions. It’s hard to imagine a proof about something undefined. But definitions are hard to compose. We usually construct a definition because we want a common term to describe a collection of things, and to exclude another collection of things. And we need people like Wavehead who can find edge cases, things that seem to satisfy a definition while breaking its spirit. This can let us find unstated assumptions that we should pay attention to. Or force us to accept that the definition is so generally useful that we’ll tolerate it having some counter-intuitive implications.

My favorite counter-intuitive implication is in analysis. The field has a definition for what it means that a function is continuous. It’s meant to capture the idea that you could draw a curve representing the function without having to lift the pen that does it. The best definition mathematicians have settled on allows you to count a function that’s continuous at a single point in all of space. Continuity seems like something that should need an interval to happen. But we haven’t found a better way to define “continuous” that excludes this pathological case. So we embrace the weirdness in exchange for general usefulness.

Charles Brubaker’s Ask A Cat for the 21st is a guest appearance from Brubaker’s other strip, The Fuzzy Princess. It’s a rerun and I did discuss it earlier. Soap bubbles make for great mathematics. They’re easy to play with, for one thing. That’s good for capturing imagination. And the mathematics behind them is deep, and led to important results analytically and computationally. It happens when this strip first ran I’d encountered a triplet of essays about the mathematics of soap bubbles and wireframe surfaces. My introduction to those essays is here.

Benita Epstein’s Six Chix for the 25th I wasn’t sure I’d include. But Roy Kassinger asked about it, so that tipped the scales. The dog tries to blame his bad behavior on “the algorithm”, bringing up one of the better monsters of the last couple years. An algorithm is just the procedure by which you do something. Mathematically, that’s usually to solve a problem. That might be finding some interesting part of the domain or range of a function. That might be putting a collection of things in order. that might be any of a host of things. And then we go make a decision based on the results of the algorithm.

What earns The Algorithm its deserved bad name is mindlessness. The idea that once you have an algorithm that a problem is solved. Worse, that once an algorithm is in place it would be irrational to challenge it. I have seen the process termed “mathwashing”, by analogy with whitewashing, and it’s a good one. The notion that because something is done by computer it must be done correctly is absurd. We knew it was absurd before there were computers as we knew them, as see anyone for the past century who has spoken of a “Kafkaesque” interaction with a large organization. It’s impossible to foresee all the outcomes of any reasonably complicated process, much less to verify that all the outcomes are handled correctly. This is before we consider that there will always be mistakes made in the handling of data. Or in the carrying out of the process. And that’s before we consider bad actors. I’m sure there must be research into algorithms designed to handle gaming of the system. I don’t know that there are any good results yet, though. We certainly need them.

There were a couple comics that didn’t seem to be substantial enough for me to write at length about. You might like them anyway. Connie Sun’s Connie to the Wonnie for the 21st shows off a Venn Diagram. Hector D Cantú and Carlos Castellanos’s Baldo for the 23rd is a bit of wordplay about what mathematicians do. Jonathan Lemon’s Rabbits Against Magic for the 23rd similarly is a bit of wordplay built around percentages. (Lemon is the new artist for Alley Oop.) And Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips features Albert Einstein, and a joke based on one of the symmetries which make relativity such a useful explanation of the world’s workings.

I don’t plan to have another Reading the Comics post until next Sunday. But when I do, it’ll be here.

## Reading the Comics, January 19, 2019: Not Making The Cut Edition

I’m trying to be a bit more rigorous about comic strips needing mathematical content before I talk about them. So, for example, Maria Scrivan had a Half Full that’s a Venn Diagram joke. But I feel like that isn’t quite enough for me to discuss at greater length. There was a Barney Google where Jughead explains why he didn’t do his mathematics homework. And I’m trying not to bring up Randolph Itch, since I’ve been through several circuits of the short-lived strip already. But it re-ran the one that renders Randolph as a string of numerals. If you haven’t seen that before, it’s a cute bit of symbols play.

Now here’s the comics that did make the cut:

Bill Holbrook’s On The Fastrack for the 18th is an anthropomorphic numerals joke. It’s part of Holbrook’s style to draw metaphors as literal happenings. It’s also a variation on a joke Holbrook used just last month, depicting then the phrase “accepting his numbers”. What I said about “accepting numbers” transfers over naturally to “trusting numbers”. It’s not that a number itself means anything. It’s that numbers are used to represent some narrative. If we can’t believe the narrative, we don’t believe the numbers. And the numbers used to represent something can give us reasons to trust, or reject, a narrative.

Eric the Circle for the 18th I can dub an anthropomorphic geometry joke for the week. At least it brings up one of the handful of geometry facts that people remember outside school. The relationship between the circumference and the diameter (or radius, if you rather) of a circle has been known just forever. It has the advantage of going through π, supporting and being supported by that celebrity number. … I’m not quite sure about the logic of this joke, though. My experience is that guys at least are fairly good about knowing their waist size (if you don’t know, it’s 38, although a 40 can feel so comfortable, and they’re sure they can wear a 36). Radius is a harder thing to keep in mind. But maybe it’s different for circles.

Russell Myers’s Broom Hilda for the 19th is a student-and-teacher problem. One thing is that Nerwin’s not wrong. It’s just that simply saying something true isn’t enough. We want to say things that are true and interesting.

But “you add two numbers and get a number” can be interesting. It depends on context. For example, in group theory, we will start by describing groups as a collection of things and an operation which works like addition. What does it mean to work like addition? Here, it means if you add two things from the collection, you get something from the collection. The collection of things is “closed” under your operation. And mathematical operations defined this abstractly — or defined this vaguely, if you don’t like the way it goes — can be great. We’re introduced to vectors, for example, as “ordered sets of numbers”. And that definition works all right. But when you start thinking of them instead as “things you can add to vectors and get other vectors out” you gain new power. You can use the mechanism developed for ordered sets of numbers to describe many things, including matrices and functions and shapes. But when we do that we’re saying things about how addition works, rather than what this particular addition is.

You know, on reflection, I’m not sure that Eric the Circle was more worthy of discussion than that Barney Google was. Hm.

And I should be back with more comics on Sunday. They should appear at this link when it’s all ready.

## Reading the Comics, January 16, 2019: Young People’s Mathematics Edition

Today’s quartet of mathematically-themed comic strips doesn’t have an overwhelming theme. There’s some bits about the mathematics that young people do, so, that’s enough to separate this from any other given day’s comics essay.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 14th is built on a bit of mathematical folklore. As Weinersmith’s mathematician (I don’t remember that we’ve been given her name) mentions, there is a belief that “revolutionary” mathematics is done by young people. That isn’t to say that older mathematicians don’t do great work. But the stereotype is that an older mathematician will produce masterpieces in already-established fields. It’s the young that establish new fields. Indeed, one of mathematics’s most prestigious awards, the Fields Medal, is only awarded to mathematicians under the age of forty. I was cheated of mine. Long story.

There’s intuitive appeal in the idea that revolutions in thinking are for the young. We think that people get set in their ways as they develop their careers. We have a couple dramatic examples, most notably Évariste Galois, who developed what we now see as foundations of group theory and died at twenty. While the idea is commonly held, I don’t know that it’s actually true. That is, that it holds up to scrutiny. It seems hard to create a definition for “revolutionary mathematics” that could be agreed upon by two people. So it would be difficult to test at what age people do their most breathtaking work, and whether it is what they do when young or when experienced.

Is there harm to believing an unprovable thing? If it makes you give up on trying, yes. My suspicion is that true revolutionary work happens when a well-informed, deep thinker comes to a field that hasn’t been studied in that way before. And when it turns out to be a field well-suited to study that way. That doesn’t require youth. It requires skill in one field, and an understanding that there’s another field ready to be studied that way.

Will Henry’s Wallace the Brave for the 14th is a mathematics anxiety joke. Wallace tries to help by turning an abstract problem into a concrete one. This is often a good way to approach a problem. Even in more advanced mathematics, one can often learn the way to solve a general problem by trying a couple of specific examples. It’s almost as though there’s only a certain amount of abstraction people can deal with, and you need to re-cast problems so they stay within your limits.

Yes, the comments turn to complaining about Common Core. I’m not sure what would help Spud work through this problem (or problems in general). But thinking of alternate problems that estimated or approached what he really wanted might help. If he noticed, for example, that 10 + 12 has to be a little more than 10 + 10, and he found 10 + 10 easy, then he’d be close to a right answer. If he noticed that 10 + 12 had to be 10 + 10 + 2, and he found 10 + 10 easy, then he might find 20 + 2 easy as well. Maybe Spud would be better off thinking of ways to rewrite a problem without changing the result.

Wiley Miller’s Non Sequitur for the 15th mentions calculus. It’s more of a probability joke. To speak of a calculated risk is to speak of doing something that’s not certain, but that has enough of a payoff to be worth the cost of failure. But one problem with this attitude is that people are very, very bad at estimating probabilities. We have terrible ideas of how likely losses are and how uncertain rewards can be. But even if we allow that the risks and rewards are calculated right, there’s a problem with things you only do once. Or only can do once. You can get into a good debate about whether there’s even a meaningful idea of probability for things that happen only the one time. Life’s among them.

Bob Weber Sr’s Moose and Molly for the 16th is a homework joke. It does actually depend on being mathematics homework, though, or there’d be no grounds for Moose’s kid to go to the savings and loan clerk who’ll help with “money problems”.

I think there’s one more batch of comic strips to discuss this week. When I’ve published it, you should find the essay at this link. And then there’ll be Sunday again.

## Reading the Comics, January 13, 2019: January 13, 2019 Edition

I admit I’m including a fairly marginal strip in this, just so I can have the fun of another single-day edition. What can I say? I can be easily swayed by silly things. Also, somehow, all four strips today have circumstances where one might mistake them for reruns. Let’s watch.

Bill Amend’s FoxTrot for the 13th is wordplay, mashing up ‘cell division’ with ‘long division’. As you might expect from Bill Amend — who loves sneaking legitimate mathematics and physics in where it’s not needed — Paige’s long cell division is a legitimate one. If you’d like a bit of recreational mathematics fun, you can figure out which microscopic organisms correspond to which numerals. The answer is also the Featured Comment on the page, at least as I write this. So if you need an answer, or you want to avoid having the answer spoiled, know what’s there.

Greg Evans’s Luann Againn for the 13th is the strip of most marginal relevance here. Part of Luann’s awful ay is a mathematics test. The given problems are nothing particularly meaningful. There is the sequence ‘mc2’ in the problem, although written as $m^c 2$. There’s also a mention of ‘googleplex’, which when the strip was first published in 1991 was nothing more than a misspelling of the quite large number. (‘Googol’ is the number; ‘Google’ a curious misspelling. Or perhaps a reversion. The name was coined in 1938 by Milton Sirotta. Sirotta was seven years old at the time. I accept that it is at least possible Sirotta was thinking of the then-very-popular serial-comic strip Barney Google, and that his uncle Edward Kasner, who brought the name to mathematics, wrote it down wrong.) And that carries with it the connotation that big numbers are harder than small numbers. This is … kind of true. At least, long numbers are more tedious than short numbers. But you don’t really do different work, dividing 1428 by 7, than you do dividing 147 by 7. It’s just longer. “Hard” is a flexible idea.

Mac King and Bill King’s Magic in a Minute for the 13th felt like a rerun to me. It took a bit of work to find, but yeah, it was. The strip itself, as presented, is new. But the same neat little modular-arithmetic coincidence was used the 31st of July, 2016.

Mathematics on clock faces is often used as a way to introduce modular arithmetic, a variation on arithmetic with only finitely many integers. This can help, if you’re familiar with clock faces. Like regular arithmetic, modular arithmetic can form a group and a ring. Clock faces won’t give you a group or ring, not unless you replace the number before ‘1’ with a ‘0’. To be a group, you need a collection of items, and a binary operation on the items. This operation we often think of as either addition or multiplication, depending on what makes sense for the problem. To be a ring, you need two binary operations, which interact by a distributive law. So the operations are often matched to addition and multiplication. Modular arithmetic is fun, yes. It’s also useful, not just as a way to do something like arithmetic that’s different. Many schemes for setting up checksums, quick and easy tests against data entry errors, rely on modular arithmetic on the data. And many schemes for generating ‘random’ numbers are built on finding multiplicative inverses in modular arithmetic. This isn’t truly random, of course. But you can look at a string of digits and not see any clear patterns. This is often as close to random as you need.

Rick DeTorie’s One Big Happy for the 13th is mostly a bunch of complaints the old always have against the young. Well, the complaint about parallel parking I haven’t seen before. But the rest are common enough. Featured in it is a complaint that the young can’t do arithmetic. I’m not sure there was ever a time that the older generation thought the young were well-trained in arithmetic. Nor that there was ever a time that the current educational vogue wasn’t blamed for destroying a generation’s ability to calculate. I’m sure there are better and worse ways to teach calculation. But I suspect any teaching method will fall short of addressing a couple issues. One is that people over-rate their own competence and under-rate other’s competence. So the older generation will see itself as having got the best possible arithmetic education and anything that’s different is a falling away. And another is that people get worse at stuff they don’t think is enjoyable or don’t have to do a lot. If you haven’t got a use for the fact, or an appreciation for the beauty in it, three times six is a bit of trivia, and not one that inspires much conversation when shared.

There’s more comics with something of a mathematical theme that got published last week. When I get to them the essays should be at this link.

## How All Of 2018 Treated My Mathematics Blog

It’s looking as though WordPress has really and permanently discontinued its year-in-review posts. That’s a shame. They had this animation that presented your year as a set of fireworks, one for each post, paced the same way your posts for the year were. The size of the fireworks explosion corresponded to how much it was liked or drew comments or something. Great stuff. Haven’t seen it in a couple of years. The web washes away everything whimsical.

I can do it manually, at least, looking at the summaries for yearly readership and all that. It’s just a bit different from the monthly reviews. And then I can see what lessons I draw from that, and go on to ignore them all. My impression of 2018 had been that I’d had a mildly better-read year than I had in 2017, but that my comments and likes had cratered. That is, people might find something they wanted to read, but saw no reason to stick around and chat with me, which I understand. But here’s what the data says.

And, for the sake of convenience, let me put things since 2012 — my first full year — in a coherent table.

Year Posts Published Page Views Unique Visitors Likes Comments
2012 6,094 180 275* 97 190
2013 106 5,729 2,905 262 161
2014 129 7,020 3,382 1,045 308
2015 188 11,241 5,159 3,273 822
2016 213 12,851 7,168 2,163 474
2017 164 12,214 7,602 1,094 301
2018 182 16,597 9,769 1,016 386

The 2012 visitors count doesn’t; they only started keeping track of those numbers (where they’d admit to us) partway through the year.

2015 you can see was a busy year. That’s the first year I did an A-To-Z sequence, and that got a fantastic response. In 2016 I tried two over the year and while neither was as well-received, it did turn out nicely. 2017 and 2018 had a single A-To-Z sequence each. I’m surprised how nearly I track to a post every other day over seven years straight. And I’m surprised that my page-view count grew by about one-third from 2017 to 2018. And that unique visitors grew by about the same amount, and has been except for 2016-to-2017. I’m certainly not doing much to be better about promoting myself, so something else is at work. The evaporating number of likes and comments I can’t explain. It’s looking like 2015 and 2016 were exceptional years, but what was the exception?

I can say what’s popular: posts that tell you how to do something. And, of course, my participation in the Playful Mathematics Education Blog Carnival. I hope to do that again this year. The ten most popular things from 2018 were:

Fascinating, to me, is that only one piece (the Playful Mathematics Education Blog Carnival) was posted in 2018. But overall it suggests I should start more pieces with the tag “How to … ”.

122 of the world’s countries sent me any readers at all in 2018. Here they are, and how many came from each, as WordPress organizes them and thinks dubious things like the “European Union” or the “United Kingdom” are countries:

Country Readers
United States 10,545
Philippines 803
United Kingdom 737
Canada 641
India 635
Australia 285
Singapore 246
Denmark 199
Turkey 148
Germany 122
South Africa 114
Sweden 106
Brazil 105
Slovenia 105
France 85
Italy 83
Netherlands 72
Spain 71
Hong Kong SAR China 70
Puerto Rico 67
European Union 66
Switzerland 63
Poland 62
Austria 53
Indonesia 53
New Zealand 50
Mexico 45
Ireland 44
Pakistan 43
Belgium 41
Norway 39
Malaysia 37
Greece 36
South Korea 35
Russia 29
Algeria 28
Romania 27
Israel 25
Argentina 24
Kenya 22
Japan 21
Czech Republic 20
Finland 20
United Arab Emirates 20
Thailand 19
Egypt 18
Vietnam 16
Ghana 15
Peru 15
Portugal 14
Bangladesh 13
Nigeria 13
Croatia 12
Lithuania 12
Ukraine 12
Taiwan 11
Bulgaria 10
Bhutan 9
Brunei 9
Chile 9
Serbia 9
Hungary 8
Nepal 8
Saudi Arabia 8
Slovakia 8
Belize 7
China 7
Kazakhstan 7
Venezuela 7
Afghanistan 6
Morocco 6
Qatar 6
Sri Lanka 6
American Samoa 5
Colombia 5
Iraq 5
Kuwait 5
Lebanon 5
Macau SAR China 5
Mongolia 5
Albania 4
Estonia 4
Georgia 4
Jamaica 4
Jordan 4
Uruguay 4
Barbados 3
Costa Rica 3
Guernsey 3
Iceland 3
Latvia 3
Mauritius 3
Palestinian Territories 3
Panama 3
Cambodia 2
Cyprus 2
Ecuador 2
Laos 2
Libya 2
Luxembourg 2
Namibia 2
St. Kitts & Nevis 2
Tanzania 2
Trinidad & Tobago 2
Armenia 1
Bahamas 1
Bahrain 1
Botswana 1
El Salvador 1
Ethiopia 1
Fiji 1
Gibraltar 1
Guam 1
Kyrgyzstan 1
Macedonia 1
Malta 1
Mozambique 1
Myanmar (Burma) 1
Oman 1
Senegal 1
Sint Maarten 1
Tunisia 1

I’m quite surprised to have so many readers from the Philippines and wonder if some peculiar event happened, like a teacher told the school to look at my piece about the number of grooves on a record. I figured to appeal more to countries where English is a primary language, and know I have a strong United States cultural bias. (Quick, name a non-American comic strip that’s ever got into a Reading The Comics post. Time’s up! You were trying to think of Sandra Bell-Lundy’s Between Friends.) But the gap in readers per capita between, say, the United States and Canada seems more than I should have expected.

In all, in 2018, I posted 182 things. They came out to 186,612 words overall, for an average of 1,025 words per post. On average posts attracted 5.3 likes, and 2.8 comments. Seems as though I could do more. I don’t really know what.

## Reading the Comics, January 12, 2019: A Edition

As I said Sunday, last week was a slow one for mathematically-themed comic strips. Here’s the second half of them. They’re not tightly on point. But that’s all right. They all have titles starting with ‘A’. I mean if you ignore the article ‘the’, the way we usually do when alphabetizing titles.

Tony Cochran’s Agnes for the 11th is basically a name-drop of mathematics. The joke would be unchanged if the teacher asked Agnes to circle all the adjectives in a sentence, or something like that. But there are historically links between religious thinking and mathematics. The Pythagoreans, for example, always a great and incredible starting point for any mathematical topic or just some preposterous jokes that might have nothing to do with their reality, were at least as much a religious and philosophical cult. For a long while in the Western tradition, the people with the time and training to do advanced mathematics work were often working for the church. Even as people were more able to specialize, a mystic streak remained. It’s easy to understand why. Mathematics promises to speak about things that are universally true. It encourages thinking about the infinite. It encourages thinking about the infinitely tiny. It courts paradoxes as difficult as any religious Mystery. It’s easy to snark at someone who takes numerology seriously. But I’m not sure the impulse that sees magic in arithmetic is different to the one that sees something supernatural in a “transfinite” item.

Scott Hilburn’s The Argyle Sweater for the 11th is another mistimed Pi Day joke. π is, famously, an irrational number. But so is every number, except for a handful of strange ones that we’ve happened to find interesting. That π should go on and on follows from what an irrational number means. It’s a bit surprising the 4 didn’t know all this before they married.

I appreciate the secondary joke that the marriage counselor is a “Hugh Jripov”, and the counselor’s being a ripoff is signaled by being a &div; sign. It suggests that maybe successful reconciliation isn’t an option. I’m curious why the letters ‘POV’ are doubled, in the diploma there. In a strip with tighter drafting I’d think it was suggesting the way a glass frame will distort an image. But Hilburn draws much more loosely than that. I don’t know if it means anything.

Mark Anderson’s Andertoons for the 12th is the Mark Anderson’s Andertoons for the essay. I’m so relieved to have a regular stream of these again. The teacher thinks Wavehead doesn’t need to annotate his work. And maybe so. But writing down thoughts about a problem is often good practice. If you don’t know what to do, or you aren’t sure how to do what you want? Absolutely write down notes. List the things you’d want to do. Or things you’d want to know. Ways you could check your answer. Ways that you might work similar problems. Easier problems that resemble the one you want to do. You find answers by thinking about what you know, and the implications of what you know. Writing these thoughts out encourages you to find interesting true things.

And this was too marginal a mention of mathematics even for me, even on a slow week. But Georgia Dunn’s Breaking Cat News for the 12th has a cat having a nightmare about mathematics class. And it’s a fun comic strip that I’d like people to notice more.

And that’s as many comics as I have to talk about from last week. Sunday, I should have another Reading the Comics post and it’ll be at this link.

## Reading the Comics, January 9, 2018: I Go On About Johnny Appleseed Edition

This was a slow week for mathematically-themed comic strips. Such things happen. I put together a half-dozen that see on-topic enough to talk about, but I stretched to do it. You’ll see.

Mark Anderson’s Andertoons for the 6th mentions addition as one of the things you learn in an average day of elementary school. I can’t help noticing also the mention of Johnny Appleseed, who’s got a weird place in my heart as he and I share a birthday. He got to it first. Although Johnny Appleseed — John Champan — is legendary for scattering apple seeds, that’s not what he mostly did. He would more often grow apple-tree nurseries, from which settlers could buy plants and demonstrate they were “improving” their plots. He was also committed to spreading the word of Emanuel Swedenborg’s New Church, one of those religious movements that you somehow don’t hear about. But there was this like 200-year-long stretch where a particular kind of idiosyncratic thinker was Swedenborgian, or at least influenced by that. I don’t know offhand of any important Swedenborgian mathematicians, I admit, but I’m glad to hear if someone has news.

Justin Thompson’s MythTickle rerun for the 9th mentions “algebra” as something so dreadful that even being middle-aged is preferable. Everyone has their own tastes, yes, although it would be the same joke if it were “gym class” or something. (I suppose that’s not one word. “Dodgeball” would do, but I never remember playing it. It exists just as a legendarily feared activity, to me.) Granting, though, that I had a terrible time with the introduction to algebra class I had in middle school.

Tom Wilson’s Ziggy for the 9th is a very early Pi Day joke, so, there’s that. There’s not much reason a take-a-number dispenser couldn’t give out π, or other non-integer numbers. What the numbers are doesn’t matter. It’s just that the dispensed numbers need to be in order. It should be helpful if there’s a clear idea how uniformly spaced the numbers are, so there’s some idea how long a wait to expect between the currently-serving number and whatever number you’ve got. But that only helps if you have a fair idea of how long an order should on average take.

I’ll close out last week’s comics soon. The next Reading the Comics post, like all the earlier ones, should be at this link.

## Reading the Comics, January 5, 2019: Start of the Year Edition

With me wrapping up the mathematically-themed comic strips that ran the first of the year, you can see how far behind I’m falling keeping everything current. In my defense, Monday was busier than I hoped it would be, so everything ran late. Next week is looking quite slow for comics, so maybe I can catch up then. I will never catch up on anything the rest of my life, ever.

Scott Hilburn’s The Argyle Sweater for the 2nd is a bit of wordplay about regular and irregular polygons. Many mathematical constructs, in geometry and elsewhere, come in “regular” and “irregular” forms. The regular form usually has symmetries that make it stand out. For polygons, this is each side having the same length, and each interior angle being congruent. Irregular is everything else. The symmetries which constrain the regular version of anything often mean we can prove things we otherwise can’t. But most of anything is the irregular. We might know fewer interesting things about them, or have a harder time proving them.

I’m not sure what the teacher would be asking for in how to “make an irregular polygon regular”. I mean if we pretend that it’s not setting up the laxative joke. I can think of two alternatives that would make sense. One is to draw a polygon with the same number of sides and the same perimeter as the original. The other is to draw a polygon with the same number of sides and the same area as the original. I’m not sure of the point of either. I suppose polygons of the same area have some connection to quadrature, that is, integration. But that seems like it’s higher-level stuff than this class should be doing. I hate to question the reality of a comic strip but that’s what I’m forced to do.

Bud Fisher’s Mutt and Jeff rerun for the 4th is a gambler’s fallacy joke. Superficially the gambler’s fallacy seems to make perfect sense: the chance of twelve bad things in a row has to be less than the chance of eleven bad things in a row. So after eleven bad things, the twelfth has to come up good, right? But there’s two ways this can go wrong.

Suppose each attempted thing is independent. In this case, what if each patient is equally likely to live or die, regardless of what’s come before? And in that case, the eleven deaths don’t make it more likely that the next will live.

Suppose each attempted thing is not independent, though. This is easy to imagine. Each surgery, for example, is a chance for the surgeon to learn what to do, or not do. He could be getting better, that is, more likely to succeed, each operation. Or the failures could reflect the surgeon’s skills declining, perhaps from overwork or age or a loss of confidence. Impossible to say without more data. Eleven deaths on what context suggests are low-risk operations suggest a poor chances of surviving any given surgery, though. I’m on Jeff’s side here.

Mark Anderson’s Andertoons for the 5th is a welcome return of Wavehead. It’s about ratios. My impression is that ratios don’t get much attention in themselves anymore, except to dunk on stupid Twitter comments. It’s too easy to jump right into fractions, and division. Ratios underlie this, at least historically. It’s even in the name, ‘rational numbers’.

Wavehead’s got a point in literally comparing apples and oranges. It’s at least weird to compare directly different kinds of things. This is one of those conceptual gaps between ancient mathematics and modern mathematics. We’re comfortable stripping the units off of numbers, and working with them as abstract entities. But that does mean we can calculate things that don’t make sense. This produces the occasional bit of fun on social media where we see something like Google trying to estimate a movie’s box office per square inch of land in Australia. Just because numbers can be combined doesn’t mean they should be.

Larry Wright’s Motley rerun for the 5th has the form of a story problem. And one timely to the strip’s original appearance in 1987, during the National Football League players strike. The setup, talking about the difference in weekly pay between the real players and the scabs, seems like it’s about the payroll difference. The punchline jumps to another bit of mathematics, the point spread. Which is an estimate of the expected difference in scoring between teams. I don’t know for a fact, but would imagine the scab teams had nearly meaningless point spreads. The teams were thrown together extremely quickly, without much training time. The tools to forecast what a team might do wouldn’t have the data to rely on.

The at-least-weekly appearances of Reading the Comics in these pages are at this link.

## Reading the Comics, January 1, 2019: New Year’s Day Edition

It’s a new year. That doesn’t mean I’m not going to keep up some of my old habits. One of them is reading the comics for the mathematics bits. For example …

Johnny Hart’s Back To BC for the 30th presents some curious use of mathematics. At least the grammar of mathematics. It’s a bunch of statements that are supposed to, taken together, overload … I’m going to say BC’s … brain. (I’m shaky on which of the characters is Peter and which is BC. Their difference in hair isn’t much of a visual hook.) Certainly mathematics inspires that feeling that one’s overloaded one’s brain. The long strings of reasoning and (ideally) precise definitions are hard to consider. And the proofs mathematicians find the most fun are, often, built cleverly. That is, going about their business demonstrating things that don’t seem relevant, and at the end tying them together. It’s hard to think.

But then … Peter … isn’t giving a real mathematical argument. He’s giving nonsense. And obvious nonsense, rather than nonsense because the writer wanted something that sounded complicated without caring what was said. Talking about a “four-sided triangle” or a “rectangular circle” has to be Peter trying to mess with BC’s head. Confidently-spoken nonsense can sound as if it’s deeper wisdom than the listener has. Which, fair enough: how can you tell whether an argument is nonsense or just cleverer than you are? Consider the kind of mathematics proof I mentioned above, where the structure might almost be a shaggy dog joke. If you can’t follow the logic, is it because the argument is wrong or because you haven’t worked out why it is right?

I believe that … Peter … is just giving nonsense and trusting that … BC … won’t know the difference, but will wear himself out trying to understand. Pranks.

Tim Lachowski’s Get A Life for the 31st just has some talk about percentages and depreciation and such. It’s meant to be funny that we might think of a brain depreciating, as if anatomy could use the same language as finance. Still, one of the virtues of statistics is the ability to understand a complicated reality with some manageable set of numbers. If we accept the convention that some number can represent the value of a business, why not the convention that some number could represent the health of a brain? So, it’s silly, but I can imagine a non-silly framing for it.

Tony Cochran’s Agnes for the 1st is about calendars. The history of calendars is tied up with mathematics in deep and sometimes peculiar ways. One might imagine that a simple ever-increasing index from some convenient reference starting time would do. And somehow that doesn’t. Also, the deeper you go into calendars the more you wonder if anyone involved in the project knew how to count. If you ever need to feel your head snapping, try following closely just how the ancient Roman calendar worked. Especially from the era when they would occasionally just drop an extra month in to the late-middle of February.

The Julian and Gregorian calendars have a year number that got assigned proleptically, that is, with the year 1 given to a set of dates that nobody present at the time called the year 1. Which seems fair enough; not many people in the year 1 had any idea that something noteworthy was under way. Calendar epochs dated to more clear events, like the reign of a new emperor or the revolution that took care of that whole emperor problem, will more reliably start with people aware of the new numbers. Proleptic dating has some neat side effects, though. If you ever need to not impress someone, you can point out that the dates from the 1st of March, 200 to the 28th of February, 300 both the Julian and the Gregorian calendar dates exactly matched.

Niklas Eriksson’s Carpe Diem for the 2nd is a dad joke about mathematics. And uses fractions as emblematic of mathematics, fairly enough. Introducing them and working with them are the sorts of thing that frustrate and confuse. I notice also the appearance of “37” here. Christopher Miller’s fascinating American Cornball: A Laffopedic Guide to the Formerly Funny identifies 37 as the current “funniest number”, displacing the early 20th century’s preferred 23 (as in skidoo). Among other things, odd numbers have a connotation of seeming more random than even numbers; ask someone to pick a whole number from 1 to 50 and you’ll see 37’s and 33’s more than you’ll see, oh, 48’s. Why? Good question. It’s among the mysteries of psychology. There’s likely no really deep reason. Maybe a sense that odd numbers are, well, odd as in peculiar, and that a bunch of peculiarities will be funny. Now let’s watch the next decade make a food of me and decide the funniest number is 64.

I’m glad to be back on schedule publishing Reading the Comics posts. I should have another one this week. It’ll be at this link when it’s ready. Thanks for reading.

## How December 2018 Treated My Mathematics Blog

With the end of December it’s the time to see what was popular around here, and just how popular it was. I keep figuring I’ll learn something useful from these explorations. Now and then I come to conclusions and one of these months I’ll even act on them.

December was an exhausting month. The last couple weeks of any A To Z sequence always are. These sequences are great fun, of course, or I wouldn’t keep doing them. But fatigue sets in, especially as I discover I’m not getting as far ahead of deadline as I imagined I would be and I get to the difficult letters of the alphabet’s end. And the Fall 2018 A To Z made up for being less frequent than past glossaries — two, rather than three, essays a week — with being crazily longer. So I was exhausted by that. And then the mathematically-themed comics for my Reading the Comics posts completely dried up. Add to that real-life obligations that I would not skip — being with family, and going to pinball events — and I ended up posting 17 things in December. Which is more than usual, yes. A typical month is 12 to 14 posts. But it’s down from the 23 of October and November, and I’m as convinced as I can be without evidence that the number of posts determines how many page views I get.

So October 2018, aided in part by my hosting the Playful Mathematics Education Blog Carnival, had 2,010 page views from 1,063 unique visitors. November I got in 1,611 page views from 847 visitors. December, well, that saw 1,409 page views from 875 visitors.

There were 82 things liked in December. It’s a slight drop from November’s 85, and October’s 94. I suppose it’s a rise in likes per page view, at least. The number of comments utterly collapsed, which probably reflects the end of the A To Z project. In October and November I had appeals for suggested topics; December didn’t have time for them. So what was 60 comments in October and 36 in November dropped to 17 for December. It’s not my least talkative period of the last year, but it’s up there. I need, seriously, to work on opening my posts to more comments. I’d ask people for suggestions how to do that, but who would answer?

The most popular essays around here in December were two perennials, two A To Z pieces, and some comics:

What countries sent me page views, and in what quantity? These, and these:

Country Readers
United States 819
Canada 77
United Kingdom 68
India 46
Turkey 39
Philippines 31
Singapore 31
Australia 27
Denmark 23
Indonesia 15
Slovenia 14
South Africa 13
Italy 11
New Zealand 11
Sweden 10
Greece 9
Netherlands 9
South Korea 9
Bangladesh 8
Ireland 8
Lithuania 8
Germany 7
Brazil 6
Hong Kong SAR China 6
Poland 6
American Samoa 5
European Union 5
Mexico 5
Russia 5
Switzerland 5
Finland 4
France 4
Norway 4
Belgium 3
Czech Republic 3
Ghana 3
Hungary 3
Israel 3
Kenya 3
Pakistan 3
Serbia 3
Austria 2
Chile 2
Egypt 2
Libya 2
Romania 2
Spain 2
Taiwan 2
Ukraine 2
Vietnam 2
Albania 1
Algeria 1
Argentina 1
Botswana 1
China 1
Colombia 1 (****)
Iceland 1
Iraq 1
Japan 1
Jordan 1
Kuwait 1
Latvia 1
Palestinian Territories 1 (*)
Portugal 1
Puerto Rico 1
Saudi Arabia 1 (*)
Sint Maarten 1
Slovakia 1

That was 68 different countries sending readers, down from November’s 70 and October’s 74. There were 17 single-reader countries, up from November’s 13 and down from October’s 23. The Palestinian Territories and Saudi Arabia were single-reader countries in November too. Colombia’s been a single-reader country five months now. I don’t see how China can be a single-reader country, even given that English isn’t a primary language there. More than one person has to stumble across here just by accident. There’s something going on there.

According to Insights, I start the month and year with 72,915 total page views, from an admitted 36,260 unique visitors.

I published something like 18,587 words here in December, which is a drop from the 26,644 of November. I write “something like” because I don’t know how WordPress tallies stuff like words in captions, and I don’t think it counts words in comments. And the idea of a “word” in a count like this is difficult to make precise and indisputable. So don’t be fooled by the digits into thinking there’s any precision there. Also it’s still 1,093 words per post, which is a bit down from the 1,158 in an average November essay but still.

For the year-to-date, by the end of December, I was writing an average of 1,025 words per post. That is, posts for the whole of the year, rather than just in December. That’s down from the end-of-November average of 1,108 words per post. I averaged 5.2 likes per post, down from the end-of-November average of 5.3. And 2.8 comments per post, up from the end-of-November average of 2.7. That’s certainly not a significant change.

I’m glad to have you as a reader, even if it’s just for this sort of self-preening post. You can put my posts in your RSS reader and enjoy them at your convenience. If you’re using WordPress regularly you can also add me to your WordPress Reader. That’s from the button at the upper-right corner of the page. On Twitter I’m @Nebusj. And just about every Sunday plus, usually, some other day of the week I’ll be Reading the Comics for the mathematics stuff. Thanks for being here.