The title of this installment has nothing to do with anything. My love and I just got to talking about Reader’s Digest Condensed Books and I learned moments ago that they’re still being made. I mean, the title of the series changed from “Condensed Books” to “Select Editions” in 1997, but they’re still going on, as far as anyone can tell. This got us wondering things like how they actually do the abridging. And got me wondering whether any abridged book ended up being better than the original. So I have reasons for only getting partway through last week’s mathematically-themed comics. I don’t say they’re good reasons.
Scott Hilburn’s The Argyle Sweater for the 13th is the Roman Numerals joke for the week, the first one of those in like five days. Also didn’t know that there were still sidewalk theaters that still showed porn movies. I thought they had all been renovated into either respectable neighborhood-revitalization projects that still sometimes show Star Wars films or else become incubator space for startup investment groups.
Corey Pandolph’s The Elderberries for the 13th is a joke about learning fractions. They don’t see to be having much fun thinking about them. Fair enough, I suppose. Once you’ve got the hang of basic arithmetic here come fractions to follow rules for addition and subtraction that are suddenly way more complicated. Multiplication isn’t harder, at least, although it is longer. Same with division. Without a clear idea why this is anything you want to do, yeah, it seems to be unmotivated complicating of stuff.
Dave Whamond’s Reality Check for the 13th is trying to pick a fight with me. I’m not taking the bait. Although by saying ‘likelihood’ the question seems to be setting up a probability question. Those tend to use ‘p’ and ‘q’ as a generic variable name, rather than ‘x’. I bet you imagine that ‘p’ gets used to represent a possibly-unknown ‘probability’ because, oh yeah, first letter. Well … so far as I know that’s why. I’m away from my references right now so I can’t look them over and find no quite satisfactory answer. But that sure seems like it. ‘q’ gets called in if you need a second probability, and don’t want to deal with subscripts, then it’s a nice convenient letter close to ‘p’ in the alphabet. Again, so far as I know.
The other half of last week’s mathematically-themed comics were on familiar old themes. I’ll see what I can do with them anyway.
Scott Hilburn’s The Argyle Sweater for the 9th is the anthropomorphic numerals joke for the week. I’m curious why the Middletons would need multiple division symbols, but I suppose that’s their business. It does play on the idea that “division” and “splitting up” are the same thing. And that fits the normal use of these words. We’re used to thinking, say, of dividing a desired thing between several parties. While that’s probably all right in introducing the idea, I do understand why someone would get very confused when they first divide by one-half or one-third or any number between zero and one. And then negative numbers make things even more confusing.
Thaves’s Frank and Ernest for the 9th is the anthropomorphic geometric figures joke for the week. I think I can wrangle a way by which Circle’s question has deeper mathematical context. Mathematicians use the idea of “space” a lot. The use is inspired by how, you know, the geometry of a room works. Euclidean space, in the trade. A Euclidean space is a collection of points that obey a couple simple rules. You can take two points and add them, and get something in the space. You can take any scalar and multiply it by any point and get a point in the space. A scalar is something that acts like a real number. For example, real numbers. Maybe complex numbers, if you’re feeling wild.
A Euclidean space can be two-dimensional. This is the geometry of stuff you draw on paper. It can be three-dimensional. This is the geometry of stuff in the real world, or stuff you draw on paper with shading. It can be four-dimensional. This is the geometry of stuff you draw on paper with big blobby lines around it. Each of these is an equally good space, though, as legitimate and as real as any other. Context usually puts an implicit “three dimensional” before most uses of the word “space”. But it’s not required to be there. There’s many kinds of spaces out there.
And “space” describes stuff that doesn’t look anything like rooms or table tops or sheets of paper. These are spaces built of things like functions, or of sets of things, or of ways to manipulate things. Spaces built of the ways you can subdivide the integers. The details vary. But there’s something in common in all these ideas that communicates.
Mark Anderson’s Andertoons for the 11th is the Mark Anderson’s Andertoons for the week. I think we’ve all seen this joke go across our social media feed and it’s reassuring to know Mark Anderson has social media too. We do talk about solving for x, using the language of describing how we help someone get past a problem. I wonder if people might like this kind of algebra more if we talked more about finding out what values ‘x’ could have that make the equation true. Well, it won’t stop people feeling they don’t like the mathematics they learned in school. But it might help people feel like they know why they’re doing it.
Back before suddenly everything got complicated I was working on the question of who’s the most improved pinball player? This was specifically for our local league. The league meets, normally, twice a month for a four-month season. Everyone plays the same five pinball tables for the night. They get league points for each of the five tables. The points are based on how many of their fellow players their score on that table beat that night. (Most leagues don’t keep standings this way. It’s one that harmonizes well with the vengue and the league’s history.) The highest score on a game earns its player 100 league points. Second-highest earns its scorer 99 league points. Third-highest earns 98, and so on. Setting the highest score to a 100 and counting down makes the race for the top less dependent on how many people show up each night. A fantastic night when 20 people attended is as good as a fantastic night when only 12 could make it out.
Last season had a large number of new players join the league. The natural question this inspired was, who was most improved? One answer is to use linear regression. That is, look at the scores each player had each of the eight nights of the season. This will be a bunch of points — eight, in this league’s case — with x-coordinates from 1 through 8 and y-coordinates from between about 400 to 500. There is some straight line which comes the nearest to describing each player’s performance that a straight line possibly can. Finding that straight line is the “linear regression”.
A straight line has a slope. This describes stuff about the x- and y-coordinates that match points on the line. Particularly, if you start from a point on the line, and change the x-coordinate a tiny bit, how much does the y-coordinate change? A positive slope means the y-coordinate changes as the x-coordinate changes. So a positive slope implies that each successive league night (increase in the x-coordinate) we expect an increase in the nightly score (the y-coordinate).
For me, I had a slope of about 2.48. That’s a positive number, so apparently I was on average getting better all season. Good to know. And with the data on each player and their nightly scores on hand, it was easy to calculate the slopes of all their performances. This is because I did not do it. I had the computer do it. Finding the slopes of these linear regressions is not hard; it’s just tedious. It takes these multiplications and additions and divisions and you know? This is what we have computing machines for. Setting up the problem and interpreting the results is what we have people for.
And with that work done we found the most improved player in the league was … ah-huh. No, that’s not right. The person with the highest slope, T, finished the season a quite good player, yes. Thing is he started the season that way too. He’d been playing pinball for years. Playing competitively very well, too, at least when he could. Work often kept him away from chances. Now that he’s retired, he’s a plausible candidate to make the state championship contest, even if his winning would be rather a surprise. Still. It’s possible he improved over the course of our eight meetings. But more than everyone else in the league, including people who came in as complete novices and finished as competent players?
So what happened?
T joined the league late, is what happened. After the first week. So he was proleptically scored at the bottom of the league that first meeting. He also had to miss one of the league’s first several meetings after joining. The result is that he had two boat-anchor scores in the first half of the season, and then basically middle-to-good scores for the latter half. Numerically, yeah, T started the season lousy and ended great. That’s improvement. Improved the standings by about 6.79 points per league meeting, by this standard. That’s just not so.
This approach for measuring how a competitor improved is flawed. But then every scheme for measuring things is flawed. Anything actually interesting is complicated and multifaceted; measurements of it are, at least, a couple of discrete values. We hope that this tiny measurement can tell us something about a complicated system. To do that, we have to understand in what ways we know the measurements to be flawed.
So treating a missed night as a bottomed-out score is bad. Also the bottomed-out scores are a bit flaky. If you miss a night when ten people were at league, you get a score of 450. Miss a night when twenty people were at league, you get a score of 400. It’s daft to get fifty points for something that doesn’t reflect anything you did except spread false information about what day league was.
Still, this is something we can compensate for. We can re-run the linear regression, for example, taking out the scores that represent missed nights. This done, T’s slope drops to 2.57. Still quite the improvement. T was getting used to the games, apparently. But it’s no longer a slope that dominates the league while feeling illogical. I’m not happy with this decision, though, not least because the same change for me drops my slope to -0.50. That is, that I got appreciably worse over the season. But that’s sentiment. Someone looking at the plot of my scores, that anomalous second week aside, would probably say that yeah, my scores were probably dropping night-to-night. Ouch.
Or does it drop to -0.50? If we count league nights as the x-coordinate and league points as the y-coordinate, then yeah, omitting night two altogether gives me a slope of -0.50. What if the x-coordinate is instead the number of league nights I’ve been to, to get to that score? That is, if for night 2 I record, not a blank score, but the 472 points I got on league night number three? And for night 3 I record the 473 I got on league night number four? If I count by my improvement over the seven nights I played? … Then my slope is -0.68. I got worse even faster. I had a poor last night, and a lousy league night number six. They sank me.
And what if we pretend that for night two I got an average-for-me score? There are a couple kinds of averages, yes. The arithmetic mean for my other nights was a score of 468.57. The arithmetic mean is what normal people intend when they say average. Fill that in as a provisional night two score. My weekly decline in standing itself declines, to only -0.41. The other average that anyone might find convincing is my median score. For the rest of the season that was 472; I put in as many scores lower than that as I did higher. Using this average makes my decline worse again. Then my slope is -0.62.
You see where I’m getting more dissatisfied. What was my performance like over the season? Depending on how you address how to handle a missed night, I either got noticeably better, with a slope of 2.48. Or I got noticeably worse, with a slope of -0.68. Or maybe -0.61. Or I got modestly worse, with a slope of -0.41.
There’s something unsatisfying with a study of some data if handling one or two bad entries throws our answers this far off. More thought is needed. I’ll come back to this, but I mean to write this next essay right away so that I actually do.
There are times I feel like my writing here collapses entirely to Reading the Comics posts. It’s a temptation to just give up doing anything else. They’re easy to write, since the comics give me the subjects to discuss. And it offers a nice, accessible mix of same-old topics with the occasional oddball. It’s fun. But sometimes Comic Strip Master Command decides I’ve been doing enough of that. This is one of those weeks; I only found six comics in my normal reading that were on point enough to discuss. So here’s half of them.
Bill Rechin’s Crock for the 6th is … hm. Well, let’s call it a fractions joke. I’m curious exactly what the clerk’s joke is supposed to mean. Is it intended to suggest an impossibility, putting into something far more than it can hold? Or is it just meant to suggest gross overabundance? And deep down I suspect Rechin didn’t have any specific meaning; it’s just a good-sounding insult.
Hector D Cantu and Carlos Castellanos’s Baldo for the 7th is … hm. Well, let’s call it a wordplay joke. It works by “strength” having multiple meanings, and “numbers” having multiple meanings. And there being a convenient saying to link one to the other. If this were a busier week I wouldn’t even bring it up, but I hate going without anything around here.
They’ve been phasing Roman Numerals out for a long while. Arabic numerals got their grand introduction to the (Western) Roman Empire’s territories in 1202 by Leonardo of Pisa, known now as “Fibonacci”. His Liber Abaci (Book of Calculation) laid out the Arabic numerals scheme and place values, and how to use them. By 1228 he published an edition comparing Roman numerals to Arabic numerals.
This wasn’t the first anyone in western Europe had heard of them, mind. (It never is; anyone telling you anything was the first is simplifying.) Spanish monks in the 10th century studied Arabic texts, and wrote about what they found. But after Leonardo of Pisa, Arabic numerals started displacing Roman numerals at least in specialized trades. Florence, in what is now Italy, prohibited merchants from using Arabic numerals in 1299; they could use Roman numerals or write them out in words. This, presumably, to prevent cheating by use of strange, unfamiliar calculus. Arabic numerals escaped being tools of specialists in the 16th century, thanks in large part to the German mathematician Adam Ries, who explained the scheme in terms apprentices could understand.
Still, these days, a Roman numeral is mostly an affectation. Useful for bit of style; not for serious mathematics. Good for watches.
And finally, at last, there’s a couple of comics left over from last week and that all ran the same day. If I hadn’t gone on forever about negative Kelvin temperatures I might have included them in the previous essay. That’s all right. These are strips I expect to need relatively short discussions to explore. Watch now as I put out 2,400 words explaining Wavehead misinterpreting the teacher’s question.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 4th is driving me slightly crazy. The equation on the board looks like an electrostatics problem to me. The ‘E’ is a common enough symbol for the strength of an electric field. And the funny-looking K’s look to me like the Greek kappa. This often represents the dielectric constant. That measures how well an electric field can move through a material. The upside-down triangles, known in the trade as Delta, describe — well, that’s getting complicated. By themselves, they describe measuring “how much the thing right after this changes in different directions”. When there’s a x symbol between the Delta and the thing, it measures something called the “curl”. This roughly measures how much the field inspires things caught up in it to turn. (Don’t try passing this off to your thesis defense committee.) The Delta x Delta x E describes the curl of the curl of E. Oh, I don’t like visualizing that. I don’t blame you if you don’t want to either.
Anyway. So all this looks like it’s some problem about a rod inside an electric field. Fine enough. What I don’t know and can’t work out is what the problem is studying exactly. So I can’t tell you whether the equation, so far as we see it, is legitimately something to see in class. Envisioning a rod that’s infinitely thin is a common enough mathematical trick, though. Three-dimensional objects are hard to deal with. They have edges. These are fussy to deal with. Making sure the interior, the boundary, and the exterior match up in a logically consistent way is tedious. But a wire? A plane? A single point? That’s easy. They don’t have an interior. You don’t have to match up the complicated stuff.
For real world problems, yeah, you have to deal with the interior. Or you have to work out reasons why the interiors aren’t important in your problem. And it can be that your object is so small compared to the space it has to work in that the fact it’s not infinitely thin or flat or smooth just doesn’t matter. Mathematical models, such as give us equations, are a blend of describing what really is there and what we can work with.
Mike Shiell’s The Wandering Melon for the 4th is a probability joke, about two events that nobody’s likely to experience. The chance any individual will win a lottery is tiny, but enough people play them that someone wins just about any given week. The chance any individual will get struck by lightning is tiny too. But it happens to people. The combination? Well, that’s obviously impossible.
In July of 2015, Peter McCathie had this happen. He survived a lightning strike first. And then won the Atlantic Lotto 6/49. This was years apart, but the chance of both happening the same day, or same week? … Well, the world is vast and complicated. Unlikely things will happen.
So I’m going to have a third Reading the Comics essay for last week’s strips. This happens sometimes. Two of the four strips for this essay mention percentages. But one of the others is so important to me that it gets naming rights for the essay. You’ll understand when I’m done. I hope.
Angie Bailey’s Texts From Mittens for the 2nd talks about percentages. That’s a corner of arithmetic that many people find frightening and unwelcoming. I’m tickled that Mittens doesn’t understand how easy it is to work out a percentage of 100. It’s a good, reasonable bit of characterization for a cat.
John Graziano’s Ripley’s Believe It Or Not for the 2nd is about a subject close to my heart. At least a third of it is. The mention of negative Kelvin temperatures set off a … heated … debate on the comments thread at GoComics.com. Quite a few people remember learning in school that the Kelvin temperature scale. It starts with the coldest possible temperature, which is zero. And that’s that. They have taken this to denounce Graziano as writing obvious nonsense. Well.
Something you should know about anything you learned in school: the reality is more complicated than that. This is true for thermodynamics. This is true for mathematics. This is true for anything interesting enough for humans to study. This also applies to stuff you learned as an undergraduate. Also to grad school.
So what are negative temperatures? At least on an absolute temperature scale, where the answer isn’t an obvious and boring “cold”? One clue is in the word “absolute” there. It means a way of measuring temperature that’s in some way independent of how we do the measurement. In ordinary life we measure temperatures with physical phenomena. Fluids that expand or contract as their temperature changes. Metals that expand or contract as their temperatures change. For special cases like blast furnaces, sample slugs of clays that harden or don’t at temperature. Observing the radiation of light off a thing. And these are all fine, useful in their domains. They’re also bound in particular physical experiments, though. Is there a definition of temperature that … you know … we can do mathematically?
Of course, or I wouldn’t be writing this. There are two mathematical-physics components to give us temperature. One is the internal energy of your system. This is the energy of whatever your thing is, less the gravitational or potential energy that reflects where it happens to be sitting. Also minus the kinetic energy that comes of the whole system moving in whatever way you like. That is, the energy you’d see if that thing were in an otherwise empty universe. The second part is — OK, this will confuse people. It’s the entropy. Which is not a word for “stuff gets broken”. Not in this context. The entropy of a system describes how many distinct ways there are for a system to arrange its energy. Low-entropy systems have only a few ways to put things. High-entropy systems have a lot of ways to put things. This does harmonize with the pop-culture idea of entropy. There are many ways for a room to be messy. There are few ways for it to be clean. And it’s so easy to make a room messier and hard to make it tidier. We say entropy tends to increase.
So. A mathematical physicist bases “temperature” on the internal energy and the entropy. Imagine giving a system a tiny bit more energy. How many more ways would the system be able to arrange itself with that extra energy? That gives us the temperature. (To be precise, it gives us the reciprocal of the temperature. We could set this up as how a small change in entropy affects the internal energy, and get temperature right away. But I have an easier time thinking of going from change-in-energy to change-in-entropy than the other way around. And this is my blog so I get to choose how I set things up.)
This definition sounds bizarre. But it works brilliantly. It’s all nice clean mathematics. It matches perfectly nice easy-to-work-out cases, too. Like, you may kind of remember from high school physics how the temperature of a gas is something something average kinetic energy something. Work out the entropy and the internal energy of an ideal gas. Guess what this change-in-entropy/change-in-internal-energy thing gives you? Exactly something something average kinetic energy something. It’s brilliant.
In ordinary stuff, adding a little more internal energy to a system opens up new ways to arrange that energy. It always increases the entropy. So the absolute temperature, from this definition, is always positive. Good stuff. Matches our intuition well.
So in 1956 Dr Norman Ramsey and Dr Martin Klein published some interesting papers in the Physical Review. (Here’s a link to Ramsey’s paper and here’s Klein’s, if you can get someone else to pay for your access.) Their insightful question: what happens if a physical system has a maximum internal energy? If there’s some way of arranging the things in your system so that no more energy can come in? What if you’re close to but not at that maximum?
It depends on details, yes. But consider this setup: there’s one, or only a handful, of ways to arrange the maximum possible internal energy. There’s some more ways to arrange nearly-the-maximum-possible internal energy. There’s even more ways to arrange not-quite-nearly-the-maximum-possible internal energy.
Look at what that implies, though. If you’re near the maximum-possible internal energy, then adding a tiny bit of energy reduces the entropy. There’s fewer ways to arrange that greater bit of energy. Greater internal energy, reduced entropy. This implies the temperature is negative.
So we have to allow the idea of negative temperatures. Or we have to throw out this statistical-mechanics-based definition of temperature. And the definition works so well otherwise. Nobody’s got an idea nearly as good for it. So mathematical physicists shrugged, and noted this as a possibility, but mostly ignored it for decades. If it got mentioned, it was because the instructor was showing off a neat weird thing. This is how I encountered it, as a young physics major full of confidence and not at all good on wedge products. But it was sitting right there, in my textbook, Kittel and Kroemer’s Thermal Physics. Appendix E, four brisk pages before the index. Still, it was an enchanting piece.
And a useful one, possibly the most useful four-page aside I encountered as an undergraduate. My thesis research simulated a fluid-equilibrium problem run at different temperatures. There was a natural way that this fluid would have a maximum possible internal energy. So, a good part — the most fascinating part — of my research was in the world of negative temperatures. It’s a strange one, one where entropy seems to work in reverse. Things build, spontaneously. More heat, more energy, makes them build faster. In simulation, a shell of viscosity-free gas turned into what looked for all the world like a solid shell.
All right, but you can simulate anything on a computer, or in equations, as I did. Would this ever happen in reality? … And yes, in some ways. Internal energy and entropy are ideas that have natural, irresistible fits in information theory. This is the study of … information. I mean, how you send a signal and how you receive a signal. It turns out a lot of laser physics has, in information theory terms, behavior that’s negative-temperature. And, all right, but that’s not what anybody thinks of as temperature.
Well, these ideas happen still. They usually need some kind of special constraint on the things. Atoms held in a magnetic field so that their motions are constrained. Vortices locked into place on a two-dimensional surface (a prerequisite to my little fluids problems). Atoms bound into a lattice that keeps them from being able to fly free. All weird stuff, yes. But all exactly as the statistical-mechanics temperature idea calls on.
And notice. These negative temperatures happen only when the energy is extremely high. This is the grounds for saying that they’re hotter than positive temperatures. And good reason, too. Getting into what heat is, as opposed to temperature, is an even longer discussion. But it seems fair to say something with a huge internal energy has more heat than something with slight internal energy. So Graziano’s Ripley’s claim is right.
(GoComics.com commenters, struggling valiantly, have tried to talk about quantum mechanics stuff and made a hash of it. As a general rule, skip any pop-physics explanation of something being quantum mechanics.)
If you’re interested in more about this, I recommend Stephen J Blundell and Katherine M Blundell’s Concepts in Thermal Physics. Even if you’re not comfortable enough in calculus to follow the derivations, the textbook prose is insightful.
John Hambrock’s The Brilliant Mind of Edison Lee for the 3rd is a probability joke. And it’s built on how impossible putting together a particular huge complicated structure can be. I admit I’m not sure how I’d go about calculating the chance of a heap of Legos producing a giraffe shape. Imagine working out the number of ways Legos might fall together. Imagine working out how many of those could be called giraffe shapes. It seems too great a workload. And figuring it by experiment, shuffling Legos until a giraffe pops out, doesn’t seem much better.
This approaches an argument sometimes raised about the origins of life. Grant there’s no chance that a pile of Legos could be dropped together to make a giraffe shape. How can the much bigger pile of chemical elements have been stirred together to make an actual giraffe? Or, the same problem in another guise. If a monkey could go at a typewriter forever without typing any of Shakespeare’s plays, how did a chain of monkeys get to writing all of them?
And there’s a couple of explanations. At least partial explanations. There is much we don’t understand about the origins of life. But one is that the universe is huge. There’s lots of stars. It looks like most stars have planets. There’s lots of chances for chemicals to mix together and form a biochemistry. Even an impossibly unlikely thing will happen, given enough chances.
And another part is selection. A pile of Legos thrown into a pile can do pretty much anything. Any piece will fit into any other piece in a variety of ways. A pile of chemicals are more constrained in what they can do. Hydrogen, oxygen, and a bit of activation energy can make hydrogen-plus-hydroxide ions, water, or hydrogen peroxide, and that’s it. There can be a lot of ways to arrange things. Proteins are chains of amino acids. These chains can be about as long as you like. (It seems.) (I suppose there must be some limit.) And they curl over and fold up in some of the most complicated mathematical problems anyone can even imagine doing. How hard is it to find a set of chemicals that are a biochemistry? … That’s hard to say. There are about twenty amino acids used for proteins in our life. It seems like there could be a plausible life with eighteen amino acids, or 24, including a couple we don’t use here. It seems plausible, though, that my father could have had two brothers growing up; if there were, would I exist?
Jason Chatfield’s Ginger Meggs for the 3rd is a story-problem joke. Familiar old form to one. The question seems to be a bit mangled in the asking, though. Thirty percent of Jonson’s twelve apples is a nasty fractional number of apples. Surely the question should have given Jonson ten and Fitzclown twelve apples. Then thirty percent of Jonson’s apples would be a nice whole number.
There’s really only the one strip that I talk about today that gets into non-Euclidean geometries. I was hoping to have the time to get into negative temperatures. That came up in the comics too, and it’s a subject close to my heart. But I didn’t have time to write that and so must go with what I did have. I’ve surely used “Non-Euclidean Geometry Edition” as a name before too, but that name and the date of August 2, 2018? Just as surely not.
Mark Anderson’s Andertoons for the 29th is the Mark Anderson’s Andertoons for the week, at last. Wavehead gets to be disappointed by what a numerator and denominator are. Common problem; there are many mathematics things with great, evocative names that all turn out to be mathematics things.
Both “numerator” and “denominator”, as words, trace to the mid-16th century. They come from Medieval Latin, as you might have guessed. “Denominator” parses out roughly as “to completely name”. As in, break something up into some number of equal-sized pieces. You’d need the denominator number of those pieces to have the whole again. “Numerator” parses out roughly as “count”, as in the count of how many denominator-sized pieces you have. So for all that numerator and denominator look like one another, with with the meat of the words being the letters “n-m–ator”, their centers don’t have anything to do with one another. (I would believe a claim that the way the words always crop up together encouraged them to harmonize their appearances.)
Johnny Hart’s Back to BC for the 29th is a surprisingly sly joke about non-Euclidean geometries. You wouldn’t expect that given the reputation of the comic the last decade of Hart’s life. And I did misread it at first, thinking that after circumnavigating the globe Peter had come back to have what had been the right line touch the left. That the trouble was his stick wearing down I didn’t notice until I re-read.
But Peter’s problem would be there if his stick didn’t wear down. “Parallel” lines on a globe don’t exist. One can try to draw a straight line on the surface of a sphere. These are “great circles”, with famous map examples of those being the equator and the lines of longitude. They don’t keep a constant distance from one another, and they do meet. Peter’s experiment, as conducted, would be a piece of proof that they have to live on a curved surface.
And this gets at one of those questions that bothers mathematicians, cosmologists, and philosophers. How do we know the geometry of the universe? If we could peek at it from outside we’d have some help, but that is a big if. So we have to rely on what we can learn from inside the universe. And we can do some experiments that tell us about the geometry we’re in. Peter’s line example would be one; he can use that to show the world’s curved in at least one direction. A couple more lines and he’d be confident the world was a sphere. If we could make precise enough measurements we could do better, with geometric experiments smaller than the circumference of the Earth. (Or universe.) Famously, the sum of the interior angles of a triangle tell us something about the space the triangle’s inscribed in. There are dangers in going from information about one point, or a small area, to information about the whole. But we can tell some things.
Phil Dunlap’s Ink Pen for the 29th is another use of arithmetic as shorthand for intelligence. Might be fun to ponder how Captain Victorious would know that he was right about two plus two equalling four, if he didn’t know that already. But we all are in the same state, for mathematical truths. We know we’ve got it right because we believe we have a sound logical argument for the thing being true.
Brian Boychuk and Ron Boychuk’s Chuckle Brothers for the 30th is a riff on the story of Isaac Newton and the apple. The story of Newton starting his serious thinking of gravity by pondering why apples should fall while the Moon did not is famous. And it seems to trace to Newton. We have a good account of it from William Stukeley, who in the mid-18th century wrote Memoirs of Sir Isaac Newton’s Life. Stukeley knew Newton, and claimed to get the story right from him. He also told it to his niece’s husband, John Conduitt. Whether this is what got Newton fired with the need to create such calculus and physics, or whether it was a story he composed to give his life narrative charm, is beyond my ability to say. It’s an important piece of mathematics history anyway.
If you’d like more Reading the Comics essays you can find them at this link. Some of the many essays to mention Andertoons are at this link. Other essays mentioning B.C. (vintage and current) are at this link. The comic strip Ink Pen gets its mentions at this link, although I’m surprised to learn it’s a new tag today. And the Chuckle Brothers I discuss at this link. Thank you.
July 2018 was another month in which stuff got in the way of my plans. I know it seems like I’m always apologizing for that. But I tell you truly: stuff keeps getting in the way of my plans. I could keep up the most essential stuff, the Reading the Comics posts. But bigger projects — I may as well stop being coy; I’m hoping to do another A to Z this year — kept getting lost under daily stuff. This includes pet health problems. I’ll leave it at that because they were sad ones.
But let’s see what these strained circumstances did for my readership, such as it was.
OK. Spent another month at above a thousand page views, which is a nice threshold. There were 1,058 pages looked at around here in July, down from June’s 1,077 and May’s 1,274. This is three months in a row I’ve had twelve posts, most of them Reading the Comics stuff. This goes to support the hypothesis that the thing most in my control that affects my readership is the number of things I write. There were 668 unique visitors, down a little from June’s 681 and a fair bit from May’s 837. The number of likes plummeted once again, to a mere 37. It had been 94 in June and 73 in May. But that’s still a drop.
And it hurts a bit. I think I’m doing much better Reading the Comics posts than I used to. I credit the discovery that GoComics.com links aren’t as secure as I had thought. If I’m going to include the image of every comic strip I talk about, I want to have a more substantive discussion. Reprinting strips that I don’t have the copyright to is fair use, of course, in that I’m using them for educational purpose. But making sure that I have a deeper discussion based on the strip makes me feel more secure in my use.
At least the comments held steady, with 28 of them over the month. That’s down from June’s 30, but that’s not a real difference. May saw only 17 comments. And it’s pretty good to have that many comments. Pet issues and other obligations had me spend a week and a half just checking that nothing had exploded. (It didn’t. I’ve never had something explode around here.)
And what was popular around here? One perennial, some comic strip stuff, and a post I am delighted got some attention:
There were a large number of views of the record-grooves post. But they weren’t particularly concentrated any one day or week. I think it might have reached that point where it’s Google-ranked highly enough to turn up as an answer to people’s query. I’m always embarrassed when my self-examination posts are among the most popular stuff I write. But if I view them as concentrating the stuff my readers think is particularly important, well, that’s all right then. Maybe I should do more regular recaps of what’s been popular lately. Could fill that late-in-the-week content hole.
But the important thing is I’m delighted people are reading about my prosthaphaeretic rule for finding square roots. I’m sure that it’s an old trick. And it’s not at all practical, not anymore. But I did notice it sitting there, waiting for me to uncover. That was fun.
Now to the list of countries sending me readers: will it include the United States up top?
Hong Kong SAR China
Macau SAR China
There were 52 countries sending me readers in July, down from 55 in June and 58 in May. There were 16 single-reader countries, down from 19 in June and 22 in May. Slovakia’s been on that list two months in a row. Brunei and South Korea three months now. Serbia’s on its fifth month in a row on the single-reader list. I hope they like me a little bit enough. It’s a rare month to have no countries with an & in their name, like Trinidad & Tobago. Hm.
If the Insights panel is correct, I started August viewed 64,958 times by 31,688 logged unique visitors. I’d finished with 90 posts on the year, gathering a total of 232 comments and 572 likes. That’s an average of 2.6 comments and 6.4 likes per post. I reached 83,083 total words published, an average of 923.1 words per post. At the end of June I was averaging 885.3 words per post. I don’t know how I got so much more longwinded so fast. But it does credit me with 14,032 words published in July, and that over only twelve posts. No wonder I’m tired.
One of the comics from the last half of last week is here mostly because Roy Kassinger asked if I was going to include it. Which one? Read on and see.
Scott Metzger’s The Bent Pinky for the 24th is the anthropomorphic-numerals joke for the week. It’s pretty easy to learn, or memorize, or test small numbers for whether they’re prime. The bigger a number gets the harder it is. Mostly it takes time. You can rule some numbers out easily enough. If they’re even numbers other than 2, for example. Or if their (base ten) digits add up to a multiple of three or nine. But once you’ve got past a couple easy filters … you don’t have to just try dividing them by all the prime numbers up to their square root. Comes close, though. Would save a lot of time if the numerals worked that out ahead of time and then kept the information around, in case it were needed. Seems a little creepy to be asking that of other numbers, really. Certainly to give special privileges to numbers for accidents of their creation.
Wiley Miller’s Non Sequitur for the 25th is an Einstein joke. In a rare move for the breed this doesn’t have “E = mc2” in it, except in the implication that it was easier to think of than squirrel-proof bird feeders would be. Einstein usually gets acclaim for mathematical physics work. But he was also a legitimate inventor, with patents in his own right. He and his student Leó Szilárd developed a refrigerator that used no moving parts. Most refrigeration technology requires the use of toxic chemicals to actually do the cooling. Einstein and Szilárd hoped to make something less likely to leak these toxins. The design never saw widespread use. Ordinary refrigerators, using freon (shockingly harmless biologically, though dangerous to the upper atmosphere) got reliable enough that the danger of leaks got tolerable. And the electromagnetic pump the machine used instead made noise that at least some reports say was unbearable. The design as worked out also used a potassium-sodium alloy, not the sort of thing easy to work with. Now and then there’s talk of reviving the design. Its potential, as something that could use any heat source to provide refrigeration, seems neat. And everybody in this side of science and engineering wants to work on something that Einstein touched.
Mort Walker and Greg Walker’s Beetle Bailey for the 26th is here by special request. I wasn’t sure it was on-topic enough for my usual rigorous standards. But there is some social-aspects-of-mathematics to it. The assumption that ‘five’ is naturally better than ‘four’ for example. There is the connotation that some numbers are better than others. Yes, there are famously lucky numbers like 7 or unlucky ones like 13 (in contemporary Anglo-American culture, anyway; others have different lucks). But there’s also the sense that a larger number is of course better than a smaller one.
Except when it’s not. A first-rate performance is understood to be better than a third-rate one. A star of the first magnitude is more prominent than one of the fourth. This whether we mean celebrities or heavenly bodies. We have mixed systems. One at least respects the heritage of ancient Greek astronomers, who rated the brightest of stars as first magnitude and the next bunch as second and so on. In this context, if we take brightness to be a good thing, we understand lower numbers to be better. Another system regards the larger numbers as being more of what we’re assumed to want, and therefore be better.
Nasty confusions will happen when the schemes of thought collide. Is a class three hurricane more or less of a threat than a class four? Ought we be more worried if the United States Department of Defense declares it’s moved from Defence Condition four to Defcon 3? In October 1966, the Fermi 1 fission reactor near Detroit suffered a “Class 1 emergency”. Does that mean the city was at the highest or the lowest health risk from the partial meltdown? (In this case, this particular term reflects the lowest actionable level of radiation was detected. I am not competent to speak on how great the risk to the population was.) It would have been nice to have unambiguous language on this point.
On to the joke’s logic, though. Wouldn’t General Halftrack be accustomed to thinking of lower numbers as better? Getting to the green in three strokes is obviously preferable to four, and getting there in five would be a disaster.
Darby Conley’s Get Fuzzy for the 28th is an applied-probability strip. The calculating of odds is rich with mathematical and psychological influences. With some events it’s possible to define quite precisely what the odds should be. If there are a thousand numbers each equally likely to be the daily lottery winner, and only one that will be, we can go from that to saying what the chance of 254 being the winner is. But many events are impossible to forecast that way. We have to use other approaches. If something has happened several times recently, we can say it’s probably rather likely. Fluke events happen, yes. But we can do fairly good work by supposing that stuff is mostly normal, and that the next baseball season will look something like the last one.
As to how to bet wisely — well, many have tried to work that out. One of the big questions in financial mathematics is how to hedge bets. I write financial mathematics, but it applies to sports betting and really anything else with many outcomes. One of the common goals is to simply avoid catastrophe, to make sure that whatever happens you aren’t too badly off. This involves betting various amounts on many outcomes. More on the outcomes you think likely, but also some on the improbable outcomes. Long shots do sometimes happen, and pay out well; it can be worth putting a little money on that just in case. Judging the likelihood of those events, especially in complicated problems that can’t be reduced to logic, is one of the hard parts. If it could be made into a system we wouldn’t need people to do it. But it does seem that knowing what you bet on helps make better bets.
I apologize for a post rougher than my norm. It has not been a gentle week. I am carrying on as best I can, but then, who isn’t? There is a common element to three of the strips featured this time around, so I have a meaningful name.
Steve McGarry’s KidTown for the 22nd of July is a kids-information panel. It’s a delivery system for some neat trivia about numbers. I’d never encountered the bit about the factorial of 10 (10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) being as many seconds as there are in six weeks. I’m curious how I missed that. But it’s definitely one of those slightly useful bits of calendar mathematics to keep around. Some other useful ones are that three years is about 1100 days, and that a century is about three billion seconds. That line about 12 + 3 – 4 + 5 + 67 + 8 + 9 is probably a useful answer to some mathematics riddle such as might beset Nancy.
John Zakour and Scott Roberts’s Maria’s Day for the 23rd depicts Maria misunderstanding what it is to be bad at mathematics. The Star Wars movie episode numbers show a quirky indexing scheme, yes. But the numbers in this case are mostly nominal variables. If we spoke of the movies only by their titles … well, it would be harder to guess whether The Empire Strikes Back or Return of the Jedi came first. All the names suggest is that they ought to follow on something else happening beforehand. And people would likely use numbers for shorthand anyway. Star Trek fans talk still about the odd- and even-numbered movies, even though no Star Trek movie’s had a number attached to it since 1991.
A nominal variable is as the … er … name suggests. It’s a way to reference something, but the value doesn’t mean very much. We see these, often with numbers attached, often enough to not notice it. We start to realize it when we have those moments of thinking, isn’t it odd that the office building starts numbering rooms from 101, rather than, say, 1? Or that there’s no numbers between (say) 129 and 201? Using a number carries some information, in that it suggests we think there is a preferred order for things. But your neighborhood would be no different if all the building addresses were 1000 higher, and the Star Wars movies would be no different if the one from 1977 came to be dubbed Episode 14 instead.
(I am open to an argument that the Star Wars episode numbers are ordinal variables. This is why I hedged by calling them “mostly” nominal. An ordinal variable describes some preferred order for the things. The difference between numbers isn’t particularly meaningful, just the relationship between them. And, yeah, it would be peculiar if The Empire Strikes Back had a higher episode number than did Return of the Jedi. Viewing the movies in that order would create several apparent continuity errors. But there are differences between internal chronology and production order and other ways one might watch the movies. But it seems to me the ordinary use for the numbers, if someone uses them at all, is as a label.)
Mell Lazarus’s Momma for the 23rd is another strip built on people being bad at mathematics. Arithmetic, anyway. I’m not sure this quite counts as an arithmetic joke. Granting the (correct) assumption that an episode of 60 Minutes is ordinarily 60 minutes long, is not recognizing how long the show will take really a use of mathematics? Isn’t it more reading comprehension? … And to be fair to the ever-beleaguered Francis, it’s rather more likely 60 Minutes just had one segment about grown men incapable of doing arithmetic. Asking how long that is likely to take is a fair question.
Adrian Raeside’s The Other Coast for the 23rd is another strip conflating arithmetic skill with intelligence. And intelligence with fitness. It’s flattering stuff, at least for people who are good at arithmetic and who feel flattered to be called intelligent. But there’s a lot of presumption here. And a common despicable attitude: merry little eugenicists (they’re always cheery about it, aren’t they?) always conclude they are fit ones.
Ryan North’s Dinosaur Comics for the 18th is based on Hilbert’s Hotel. This is a construct very familiar to eager young mathematicians. It’s an almost unavoidable pop-mathematics introduction to infinitely large sets. It’s a great introduction because the model is so mundane as to be easily imagined. But you can imagine experiments with intuition-challenging results. T-Rex describes one of the classic examples in the third through fifth panels.
The strip made me wonder about the origins of Hilbert’s Hotel. Everyone doing pop mathematics uses the example, but who created it? And the startling result is, David Hilbert, kind of. My reference here is Helge Kragh’s paper The True (?) Story of Hilbert’s Infinite Hotel. Apparently in a 1924-25 lecture series in Göttingen, Hilbert encouraged people to think of a hotel with infinitely many rooms. He apparently did not use it for so many examples as pop mathematicians would. He just used the question of how to accommodate a single new guest after the infinitely many rooms were first filled. And then went to imagine an infinite dance party. I don’t remember ever seeing the dance party in the wild; perhaps it’s a casualty of modern rave culture.
Hilbert’s Hotel seems to have next seen print in George Gamow’s One, Two Three … Infinity. Gamow summoned the hotel back from the realms of forgotten pop mathematics with a casual, jokey tone that fooled Kragh into thinking he’d invented the model and whimsically credited Hilbert with it. (Gamow was prone to this sort of lighthearted touch.) He came back to it in The Creation Of The Universe, less to make readers consider the modern understanding of infinitely large sets than to argue for a universe having infinitely many things in it.
And then it disappeared again, except for cameo appearances trying to argue that the steady-state universe would be more bizarre than what we actually see. The philosopher Pamela Huby seems to have made Hilbert’s Hotel a thing to talk about again, as part of a debate about whether a universe could be infinite in extent. William Lane Craig furthered using the hotel, as part of the theological debate about whether there could be an infinite temporal regress of events. Rudy Rucker and Eli Maor wrote descriptions of the idea in the 1980s, with vague ideas about whether Hilbert actually had anything to do with the place. And since then it’s stayed, a famous fictional hotel.
David Hilbert was born in 1862; T-Rex misspoke.
Ernie Bushmiller’s Nancy Classics for the 20th gets me out of my Olivia Jaimes rut. We could probably get a good discussion going about whether giving an example of a sphere is an adequate description of a sphere. Granted that a bubble-gum bubble won’t be perfectly spherical; neither will any example that exists in reality. We always trust that we can generalize to an ideal example of this thing.
I did get to wondering, in Sluggo’s description of the octagon, why the specification of eight sides and eight angles. I suspect it’s meant to avoid calling an octagon something that, say, crosses over itself, thus having more angles than sides. Not sure, though. It might be a phrasing intended to make sure one remembers that there are sides and there are angles and the polygon can be interesting for both sets of component parts.
John Atkinson’s Wrong Hands for the 20th is the Venn Diagram joke for the week. The half-week anyway. Also a bunch of other graph jokes for the week. Nice compilation of things. I love the paradoxical labelling of the sections of the Venn Diagram.
Tom II Wilson’s Ziggy for the 20th is a plaintive cry for help from a despairing soul. Who’s adding up four- and five-digit numbers by hand for some reason. Ziggy’s got his projects, I guess is what’s going on here.
Glenn McCoy and Gary McCoy’s The Duplex for the 21st is set up as an I-hate-word-problems joke. The cop does ask something people would generally like to know, though: how much longer would it take, going 60 miles per hour rather than 70? It turns out it’s easy to estimate what a small change in speed does to arrival time. Roughly speaking, reducing the speed one percent increases the travel time one percent. Similarly, increasing speed one percent decreases travel time one percent. Going about five percent slower should make the travel time a little more than five percent longer. Going from 70 to 60 miles per hour reduces the speed about fifteen percent. So travel time is going to be a bit more than 15 percent longer. If it was going to be an hour to get there, now it’ll be an hour and ten minutes. Roughly. The quality of this approximation gets worse the bigger the change is. Cutting the speed 50 percent increases the travel time rather more than 50 percent. But for small changes, we have it easier.
There are a couple ways to look at this. One is as an infinite series. Suppose you’re travelling a distance ‘d’, and had been doing it at the speed ‘v’, but now you have to decelerate by a small amount, ‘s’. Then this is something true about your travel time ‘t’, and I ask you to take my word for it because it has been a very long week and I haven’t the strength to argue the proposition:
‘d’ divided by ‘v’ is how long your travel took at the original speed. And, now, — the fraction of how much you’ve changed your speed — is, by assumption, small. The speed only changed a little bit. So is tiny. And is impossibly tiny. And is ridiculously tiny. You make an error in dropping these squared and cubed and forth-power and higher terms. But you don’t make much of one, not if s is small enough compared to v. And that means your estimate of the new travel time is:
Or, that is, if you reduce the speed by (say) five percent of what you started with, you increase the travel time by five percent. Varying one important quantity by a small amount we know as “perturbations”. Working out the approximate change in one quantity based on a perturbation is a key part of a lot of calculus, and a lot of mathematical modeling. It can feel illicit; after a lifetime of learning how mathematics is precise and exact, it’s hard to deliberately throw away stuff you know is not zero. It gets you to good places, though, and fast.
Morrie Turner’s Wee Pals for the 21st shows Wellington having trouble with partitions. We can divide any counting number up into the sum of other counting numbers in, usually, many ways. I can kind of see his point; there is something strange that we can express a single idea in so many different-looking ways. I’m not sure how to get Wellington where he needs to be. I suspect that some examples with dimes, quarters, and nickels would help.
Some of the comics last week don’t leave me much to talk about. Well, there should be another half-dozen comics under review later in the week. You’ll stick around, won’t you please?
Anthony Blades’s Bewley for the 16th is a rerun, and an old friend. It’s appeared the 14th of August, 2016, and in April 2015 and in May 2013. Maybe it’s time I dropped the strip from my reading. The scheme by which the kids got the right answer out of their father is a variation on the Clever Hans trick. Clever Hans was a famous example of animal perception: the horse appeared to be able to do arithmetic, tapping his hoof to signal a number. Brilliant experimental design found what was going on. Not that the horse was clever enough to tell (to make up an example) 18 divided by 3. But that the horse was clever enough to recognize the slight change in his trainer’s expression when he had counted off six. Animals (besides humans) do have some sense of numbers, but not that great a sense.
Jeff Stahler’s Moderately Confused for the 16th is the old joke told about accountants and lawyers when they encounter mathematics, recast to star the future disgraced former president. The way we normally define ‘two’ and ‘plus’ and ‘two’ and ‘equals’ and ‘four’ there’s not room for quibbling about their relationship. Not without just lying, anyway. Thus this satisfies the rules of joke formation.
Olivia Jaimes’s Nancy for the 16th is, I think, the point that Jaimes’s Nancy has appeared in my essays more than Guy Gilchrist’s ever did. Well, different artists have different interests. This one depicts Nancy getting the motivation she needed to excel in arithmetic. I’m not convinced of the pedagogical soundness of the Nancy comic strip. But it’s not as though people won’t practice things for rewards.
The title doesn’t mean anything. My laptop’s random-draw of pictures pulled up one from the county fair last year is all. I’m just working too close to deadline to have a good one. Pet rabbit has surgery scheduled and we are hoping that turns out well for everyone involved.
Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney for the 12th has the blackboard of mathematical symbols. Familiar old shorthand of conflating mathematics ability with genius, or at least intelligence. The blackboard isn’t particularly full of expressions, possibly because Caulfield and Rouillard’s art might not be able to render too much detail clearly. It’s also got a sort-of appearance of Einstein’s most famous equation. Although with perhaps an extra joke to it. Suppose we’re to take ‘E’ and ‘M’ and ‘C’ to mean what they do in Einstein’s use. Then has to equal zero. And there are many things you can safely do with zero. Dividing by it, though, isn’t one. I shan’t guess whether Caulfield and Rouillard were being that sly, though.
Marty Links’s Emmy Lou rerun for the 13th tries to be a paradox. How can one like mathematics without liking figures? But arithmetic is just one part of mathematics. Surely the most-used part, if we go by real-world utility. But not everything. Arithmetic is often useful, yes. But you can do good work in (say) logic or knot theory or geometry with only a slight ability to add or subtract or multiply. There’s not enough emphasis put on that in early education. I suppose it reflects the reasonable feeling that people do need to be competent at arithmetic, which is useful. But it gives one a distorted view of what mathematics can be.
Mark Parisi’s Off The Markfor the 13th is the anthropomorphic numerals joke for the week. And it presents being multiplied by zero as a terrifying fate for other numbers. This seems to reflect the idea that being multiplied by zero is equivalent to being made into nothing. That it’s being killed. Zero enjoys this dual meaning, culturally, representing both a number and the concept of a thing that doesn’t exist and the concept of non-existence. If being turned from one number to another is a numeral murder, then a 2 sneaking in with a + sign would be at least as horrifying. But that joke wouldn’t work, and I know that too.
Olivia Jaimes’s Nancy for the 14th is another recreational-mathematics puzzle. I know nothing of Jaimes’s background but apparently it involves a keen interest in that kind of play that either makes someone love or hate mathematics. (Myself, I’m only slightly interested in these kinds of puzzles, most of the time.) This one — add one line to ‘fix’ the equation 5 + 5 + 5 + 5 = 555 — I hadn’t encountered before. Took some fuming to work it out. The obvious answer, of course, is to add a slash across the = sign so that it means “does not equal”.
But that answer’s dull. What mathematicians like are statements that are true and interesting. There are many things that 5 + 5 + 5 + 5 does not equal. Why single out 555 from that set? So negating the equals sign meets the specifications of the problem, slightly better than Nancy does herself. It doesn’t have the surprise of the answer Nancy’s teacher wants.
If you don’t get how to do it, highlight over the paragraph below for a hint.
There are actually three ways to add the stroke to make this equation true. The three ways are equivalent, though. Notice that the symbols on the board comprise strokes and curves and consider that the meaning of the symbol can be changed by altering the composition of those strokes and curves.
Ted Shearer’s Quincy for the 21st of May, 1979, and rerun the 14th is a joke about making mathematics problems relevant. And, yeah, I’ll give Mrs Glover credit for making problems that reflect stuff students know they’re going to have to deal with. Also that they may have already dealt with and so have some feeling for what plausible answers will be. It’s tough to find many problems like that which don’t repeat themselves too much. (“If your pants need a new patch every two months how many would you have in three years?”).
If we suppose the number whose square we want is then we can find . The calculation on the right-hand side of this is easy; double your number and subtract one. Then to the lookup table; find the angle whose cosine is that number. That angle is two times θ. So divide that angle in two. Cosine of that is, well, and most people would agree that’s a square root of without any further work.
Why can’t I do the same thing with a triple-angle formula? … Well, here’s my choices among the normal trig functions:
Yes, I see you in the corner, hopping up and down and asking about the cosecant. It’s not any better. Trust me.
So you see the problem here. The number whose cube root I want has to be the . Or the cube of the sine of theta, or the cube of the tangent of theta. Whatever. The trouble is I don’t see a way to calculate cosine (sine, tangent) of 3θ, or 3 times the cosine (etc) of θ. Nor to get some other simple expression out of that. I can get mixtures of the cosine of 3θ plus the cosine of θ, sure. But that doesn’t help me figure out what θ is.
Can it be worked out? Oh, sure, yes. There’s absolutely approximation schemes that would let me find a value of θ which makes true, say,
But: is there a way takes less work than some ordinary method of calculating a cube root? Even if you allow some work to be done by someone else ahead of time, such as by computing a table of trig functions? … If there is, I don’t see it. So there’s another point in favor of logarithms. Finding a cube root using a logarithm table is no harder than finding a square root, or any other root.
If you’re using trig tables, you can find a square root, or a fourth root, or an eighth root. Cube roots, if I’m not missing something, are beyond us. So are, I imagine, fifth roots and sixth roots and seventh roots and so on. I could protest that I have never in my life cared what the seventh root of a thing is, but it would sound like a declaration of sour grapes. Too bad.
If I have missed something, it’s probably obvious. Please go ahead and tell me what it is.
The first half of last week’s comics are mostly ones from Comics Kingdom and Creators.com. That’s unusual. GoComics usually far outranks the other sites. Partly for sheer numbers; they have an incredible number of strips, many of them web-only, that Comics Kingdom and Creators.com don’t match. I think the strips on GoComics are more likely to drift into mathematical topics too. But to demonstrate that would take so much effort. Possibly any effort at all. Hm.
Bill Holbrook’s On the Fastrack for the 8th of July is premised on topographic maps. These are some of the tools we’ve made to understand three-dimensional objects with a two-dimensional representation. When topographic maps come to the mathematics department we tend to call them “contour maps” or “contour plots”. These are collections of shapes. They might be straight lines. They might be curved. They often form a closed loop. Each of these curves is called a “contour curve” or a “contour line” (even if it’s not straight). Or it’s called an “equipotential curve”, if someone’s being all fancy, or pointing out the link between potential functions and these curves.
Their purpose is in thinking of three-dimensional surfaces. We can represent a three-dimensional surface by putting up some reasonable coordinate system. For the sake of simplicity let’s suppose the “reasonable coordinate system” is the Cartesian one. So every point in space has coordinates named ‘x’, ‘y’, and ‘z’. Pick a value for ‘x’ and ‘y’. There’s at most one ‘z’ that’ll be on the surface. But there might be many sets of values of ‘x’ and ‘y’ together which have that height ‘z’. So what are all the values of ‘x’ and ‘y’ which match the same height ‘z’? Draw the curve, or curves, which match that particular value of ‘z’.
Topographical maps are a beloved example of this, to mathematicians, because we imagine everyone understands them. A particular spot on the ground at some given latitude and longitude is some particular height above sea level. OK. Imagine the slice of a hill representing all the spots that are exactly 10 feet above sea level, or whatever. That’s a curve. Possibly several curves, but we just say “a curve” for simplicity.
A topographical map will often include more than one curve. Often at regular intervals, say with one set of curves representing 10 feet elevation, another 20 feet, another 30 feet, and so on. Sometimes these curves will be very near one another, where a hill is particularly steep. Sometimes these curves will be far apart, where the ground is nearly level. With experience one can learn to read the lines and their spacing. One can see where extreme values are, and how far away they might be.
Topographical maps date back to 1789. These sorts of maps go back farther. In 1701 Edmond Halley, of comet fame, published maps showing magnetic compass variation. He had hopes that the difference between magnetic north and true north would offer a hint at how to find longitude. (The principle is good. But the lines of constant variation are too close to lines of latitude for the method to be practical. And variation changes over time, too.) And that shows how the topographical map idea can be useful to visualize things that aren’t heights. Weather maps include “isobars”, contour lines showing where the atmospheric pressure is a set vale. More advanced ones will include “isotherms”, each line showing a particular temperature. The isobar and isotherm lines can describe the weather and how it can be expected to change soon.
This idea, rendering three-dimensional information on a two-dimensional surface, is a powerful one. We can use it to try to visualize four-dimensional objects, by looking at the contour surfaces they would make in three dimensions. We can also do this for five and even more dimensions, by using the same stuff but putting a note that “D = 16” or the like in the corner of our image. And, yes, if Cartesian coordinates aren’t sensible for the problem you can use coordinates that are.
If you need a generic name for these contour lines that doesn’t suggest lines or topography or weather or such, try ‘isogonal curves’. Nobody will know what you mean, but you’ll be right.
Ted Key’s Hazel for the 9th is a joke about the difficulties in splitting the bill. It is archetypical of the sort of arithmetic people know they need to do in the real world. Despite that at least people in presented humor don’t get any better at it. I suppose real-world people don’t either, given some restaurants now list 15 and 20 percent tips on the bill. Well, at least everybody has a calculator on their phone so they can divide evenly. And I concede that, yeah, there isn’t really specifically a joke here. It’s just Hazel being competent, like the last time she showed up here.
Mark Anderson’s Andertoons for the 11th is the Mark Anderson’s Andertoons for the week. And it’s a bit of geometry wordplay, too. Also about how you can carry a joke over well enough even without understanding it, or the audience understanding it, if it’s delivered right.
Rick DeTorie’s One Big Happy for the 11th is another strip about arithmetic done in the real world. I’m also amused by Joe’s attempts to distract from how no kid that age has ever not known precisely how much money they have, and how much of it is fairly won.
Bill Griffith’s Zippy the Pinhead for the 11th is another example of using understanding algebra as a show of intelligence. And it follows that up with undrestanding quantum physics as a show of even greater intelligence. One can ask what’s meant by “understanding” quantum physics. Someday someone might even answer. But it seems likely that the ability to do calculations based on a model has to be part of fully understanding it.
I apologize for not having a more robust introduction here. My week’s been chopped up by concern with the health of the older of our rabbits. Today’s proved to be less alarming than we had feared, but it’s still a lot to deal with. I appreciate your kind thoughts. Thank you.
Meanwhile the comics from last week have led me to discover something really weird going on with the Mutt and Jeff reruns.
Charles Schulz’s Peanuts Classics for the 6th has the not-quite-fully-formed Lucy trying to count the vast. She’d spend a while trying to count the stars and it never went well. It does inspire the question of how to count things when doing a simple tally is too complicated. There are many mathematical approaches. Most of them are some kind of sampling. Take a small enough part that you can tally it, and estimate the whole based on what your sample is. This can require ingenuity. For example, when estimating our goldfish population, it was impossible to get a good sample at one time. When tallying the number of visible stars in the sky, we have the problem that the Galaxy has a shape, and there are more stars in some directions than in others. This is why we need statisticians.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 6th looks initially like it’s meant for a philosophy blog’s Reading the Comics post. It’s often fruitful in the study of ethics to ponder doing something that is initially horrible, but would likely have good consequences. Or something initially good, but that has bad effects. These questions challenge our ideas about what it is to do good or bad things, and whether transient or permanent effects are more important, and whether it is better to be responsible for something (or to allow something) by action or inaction.
It comes to mathematics in the caption, though, and with an assist from the economics department. Utilitarianism seems to offer an answer to many ethical problems. It posits that we need to select a primary good of society, and then act so as to maximize that good. This does have an appeal, I suspect even to people who don’t thrill of the idea of finding the formula that describes society. After all, if we know the primary good of society, why should we settle for anything but the greatest value of that good? It might be difficult in practice, say, to discount the joy a musician would bring over her lifetime with her performances fairly against the misery created by making her practice the flute after school when she’d rather be playing. But we can imagine working with a rough approximation, at least. Then the skilled thinkers point out even worse problems and we see why utilitarianism didn’t settle all the big ethical questions, even in principle.
The mathematics, though. As Weinersmith’s caption puts it, we can phrase moral dilemmas as problems of maximizing evil. Typically we pose them as ones of maximizing good. Or at least of minimizing evil. But if we have the mechanism in place to find where evil is maximized, don’t we have the tools to find where good is? If we can find the set of social parameters x, y, and z which make E(x, y, z) as big as possible, can’t we find where -E(x, y, z) is as big, too? And isn’t that then where E(x, y, z) has to be smallest?
And, sure. As long as the maximum exists, or the minimum exists. Maybe we can tell whether or not there is one. But this is why when you look at the mathematics of finding maximums you realize you’re also doing minimums, or vice-versa. Pretty soon you either start referring to what you find as extremums. Or you stop worrying about the difference between a maximum and a minimum, at least unless you need to check just what you have found. Or unless someone who isn’t mathematically expert looks at you wondering if you know the difference between positive and negative numbers.
Bud Fisher’s Mutt and Jeff for the 7th has run here before. Except that was before they redid the lettering; it was a roast beef in earlier iterations. I was thinking to drop Mutt and Jeff from my Reading the Comics routine before all these mysteries in the lettering turned up. Anyway. The strip’s joke starts with a work-rate problems. Given how long some people take to do a thing, how long does it take a different number of people to do a thing? These are problems that demand paying attention to units, to the dimensions of a thing. That seems to be out of fashion these days, which is probably why these questions get to be baffling. But if eating a ham takes 25 person-minutes to do, and you have ten persons eating, you can see almost right away how long to expect it to take. If the ham’s the same size, anyway.
Olivia Jaimes’s Nancy for the 7th is built on a spot of recreational mathematics. Also on the frustration one can have when a problem looks like it’s harmless innocent fun and turns out to take just forever and you’re never sure you have the answers just right. The commenters on GoComics.com have settled on 18. I’m content with that answer.
Sunday’s comics post got me thinking about ways to calculate square roots besides using the square root function on a calculator. I wondered if I could find my own little approach. Maybe something that isn’t iterative. Iterative methods are great in that they tend to forgive numerical errors. All numerical calculations carry errors with them. But they can involve a lot of calculation and, in principle, never finish. You just give up when you think the answer is good enough. A non-iterative method carries the promise that things will, someday, end.
And I found one! It’s a neat little way to find the square root of a number between 0 and 1. Call the number ‘S’, as in square. I’ll give you the square root from it. Here’s how.
First, take S. Multiply S by two. Then subtract 1 from this.
Next. Find the angle — I shall call it 2A — whose cosine is this number 2S – 1.
You have 2A? Great. Divide that in two, so that you get the angle A.
Now take the cosine of A. This will be the (positive) square root of S. (You can find the negative square root by taking minus this.)
Let me show it in action. Let’s say you want the square root of 0.25. So let S = 0.25. And then 2S – 1 is two times 0.25 (which is 0.50) minus 1. That’s -0.50. What angle has cosine of -0.50? Well, that’s an angle of 2 π / 3 radians. Mathematicians think in radians. People think in degrees. And you can do that too. This is 120 degrees. Divide this by two. That’s an angle of π / 3 radians, or 60 degrees. The cosine of π / 3 is 0.5. And, indeed, 0.5 is the square root of 0.25.
I hear you protesting already: what if we want the square root of something larger than 1? Like, how is this any good in finding the square root of 81? Well, if we add a little step before and after this work, we’re in good shape. Here’s what.
So we start with some number larger than 1. Say, 81. Fine. Divide it by 100. If it’s still larger than 100, divide it again, and again, until you get a number smaller than 1. Keep track of how many times you did this. In this case, 81 just has to be divided by 100 the one time. That gives us 0.81, a number which is smaller than 1.
Twice 0.81 minus 1 is equal to 0.62. The angle which has 0.81 as cosine is roughly 0.90205. Half this angle is about 0.45103. And the cosine of 0.45103 is 0.9. This is looking good, but obviously 0.9 is no square root of 81.
Ah, but? We divided 81 by 100 to get it smaller than 1. So we balance that by multiplying 0.9 by 10 to get it back larger than 1. If we had divided by 100 twice to start with, we’d multiply by 10 twice to finish. If we had divided by 100 six times to start with, we’d multiply by 10 six times to finish. Yes, 10 is the square root of 100. You see what’s going on here.
(And if you want the square root of a tiny number, something smaller than 0.01, it’s not a bad idea to multiply it by 100, maybe several times over. Then calculate the square root, and divide the result by 10 a matching number of times. It’s hard to calculate with very big or with very small numbers. If you must calculate, do it on very medium numbers. This is one of those little things you learn in numerical mathematics.)
So maybe now you’re convinced this works. You may not be convinced of why this works. What I’m using here is a trigonometric identity, one of the angle-doubling formulas. Its heart is this identity. It’s familiar to students whose Intro to Trigonometry class is making them finally, irrecoverably hate mathematics:
Here, I let ‘S’ be the squared number, . So then anything I do to find gets me the square root. The algebra here is straightforward. Since ‘S’ is that cosine-squared thing, all I have to do is double it, subtract one, and then find what angle 2θ has that number as cosine. Then the cosine of θ has to be the square root.
Oh, yeah, all right. There’s an extra little objection. In what world is it easier to take an arc-cosine (to figure out what 2θ is) and then later to take a cosine? … And the answer is, well, any world where you’ve already got a table printed out of cosines of angles and don’t have a calculator on hand. This would be a common condition through to about 1975. And not all that ridiculous through to about 1990.
This is an example of a prosthaphaeretic rule. These are calculation tools. They’re used to convert multiplication or division problems into addition and subtraction. The idea is exactly like that of logarithms and exponents. Using trig functions predates logarithms. People knew about sines and cosines long before they knew about logarithms and exponentials. But the impulse is the same. And you might, if you squint, see in my little method here an echo of what you’d do more easily with a logarithm table. If you had a log table, you’d calculate instead. But if you don’t have a log table, and only have a table of cosines, you can calculate at least.
Is this easier than normal methods of finding square roots? … If you have a table of cosines, yes. Definitely. You have to scale the number into range (divide by 100 some) do an easy multiplication (S times 2), an easy subtraction (minus 1), a table lookup (arccosine), an easy division (divide by 2), another table lookup (cosine), and scale the number up again (multiply by 10 some). That’s all. Seven steps, and two of them are reading. Two of the rest are multiplying or dividing by 10’s. Using logarithm tables has it beat, yes, at five steps (two that are scaling, two that are reading, one that’s dividing by 2). But if you can’t find your table of logarithms, and do have a table of cosines, you’re set.
This may not be practical, since who has a table of cosines anymore? Who hasn’t also got a calculator that does square roots faster? But it delighted me to work this scheme out. Give me a while and maybe I’ll think about cube roots.
Three of these essays in a row now that Jef Mallett’s Frazz has done something worth responding to. You know, the guy lives in the same metro area. He could just stop in and visit sometime. There’s a pinball league in town and everything. He could view it as good healthy competition.
Bill Hinds’s Cleats for the 1st is another instance of the monkeys-on-typewriters metaphor. The metaphor goes back at least as far as 1913, when Émile Borel wrote a paper on statistical mechanics and the reversibility problem. Along the way it was worth thinking of the chance of impossibly unlikely events, given enough time to happen. Monkeys at typewriters formed a great image for a generator of text that knows no content or plan. Given enough time, this random process should be able to produce all the finite strings of text, whatever their content. And the metaphor’s caught people’s fancy I guess there’s something charming and Dadaist about monkeys doing office work. Borel started out with a million monkeys typing ten hours a day. Modern audiences sometimes make this an infinite number of monkeys typing without pause. This is a reminder of how bad we’re allowing pre-revolutionary capitalism get.
Sometimes it’s cut down to a mere thousand monkeys, as in this example. Often it’s Shakespeare, but sometimes it’s other authors who get duplicated. Dickens seems like a popular secondary choice. In joke forms, the number of monkeys and time it would take to duplicate something is held as a measure of the quality of the original work. This comes from people who don’t understand. Suppose the monkeys and typewriters are producing truly random strings of characters. Then the only thing that affects how long it takes them to duplicate some text is the length of the original text. How good the text is doesn’t enter into it.
Jef Mallett’s Frazz for the 1st is about the comfort of knowing about things one does not know. And that’s fine enough. Frazz cites Fermat’s Last Theorem as a thing everyone knows of but doesn’t understand. And that choice confuses me. I’m not sure what there would be to Fermat’s Last Theorem that someone who had heard of it would not understand. The basic statement of it — if you have three positive whole numbers a, b, and c, then there’s no whole number n larger than 2 so that equals — has it.
But “understanding” is a flexible concept. He might mean that people don’t know why the Theorem is true. Fair enough. Andrew Wiles and Richard Taylor’s proof is a long thing that goes deep into a field of mathematics that even most mathematicians don’t study. Why it should be true can be an interesting question, and one that’s hard to ever satisfyingly answer. What is the difference between a proof that something is true and an explanation for why it’s true? And before you say there’s not one, please consider that many mathematicians do experience a difference between seeing something proved and understanding why something is true.
And Frazz might also mean that nobody knows what use Fermat’s Last Theorem is. This is a fair complaint too. I’m not aware offhand of any interesting results which follow from its truth, nor of anything neat that would come about had it been false. It’s just one of those things that happens to be true, and that we’ve found to be pretty, perhaps because it is easy to ask whether it’s true and hard to answer. I don’t know.
Morrie Turner’s Wee Pals for the 2nd has a kid looking for a square root. We all have peculiar hobbies. His friends speak of it as though it’s a lost physical object. This is a hilarious misunderstanding until it strikes you that we speak about stuff like square roots “existing”. Indeed, the language of mathematics would be trashed if we couldn’t speak about numerical constructs “existing” somewhere to be “found”. But try to put “four” in a box and see what you get. That we mostly have little trouble understanding what we mean by showing some mathematical construct exists, and what we hope to do by looking for it, suggests we roughly know what we mean by the phrases. All right then; what is that, in terms a kid could understand?
There are many ways to numerically compute a square root, if you have to do it by hand and it isn’t a perfect square. My preference is for iterative methods, in which you start with a rough guess and try to improve things. One good enough method for we call the Babylonian method, reflecting how old we think it is. Start with your number S whose square root you want. And start with a number x0, a first guess for what the square root is. This can be anything. The great thing about iterative methods is even if you start with a garbage answer, you get to a good answer soon enough. Still, if you have a suspicion of what the square root should be, start there.
Your first iteration, the first guess for a better answer, is to calculate the number . Typically, x1 will be closer to the square root of S than will x0 be. And in any case, we can get closer still. Use x1 to calculate a new number. This is . And then x3 and x4 and x5 and so on. In theory, you never finish; you’re stuck finding an infinitely long sequence of better approximations to the square root. In practice, you finish; you find that you’re close enough to the square root. Well, the square root of a whole number is either a whole number (if it was a perfect square to start) or is an irrational number. You were going to stop on an approximation sooner or later.
The method requires doing division. Long division, too, after the first couple steps. I don’t know a way around that which doesn’t divert into something less pleasant, such as logarithms and exponentials. Or maybe into trigonometric functions. This can be tedious to do by hand. Great thing, though, is if you make a mistake? That’s kind of all right. The next iteration will (usually) correct for it. That’s the glory of iterative methods. They tend to be forgiving of numerical error, whatever its source. Another iteration reduces, or even eliminates, the mistake of the previous iteration.
For more of these Reading the Comics posts please follow this link. If you’re only interested in Reading the Cleats strips, please use this link instead. But Cleats is a new tag this essay, so for now, there aren’t others. If you’re hoping to see all my Reading the Comics posts about Frazz, try this link. If you’d like more of my essays which mention Wee Pals, you can use this link. And if you’d like more Reading the Comics posts that mention Harley, use this link. That’s another new tag, but I believe Dan Thompson is still making new examples of the strip. So it may appear again.
The last half of last week was not entirely the work of Chuckle Brothers and Saturday Morning Breakfast Cereal. It seemed like it, though. Let’s review.
Patrick Roberts’s Todd the Dinosaur for the 28th is a common sort of fear-of-mathematics joke. In this case the fear of doing arithmetic even when it is about something one would really like to know. I think the question got away from Todd, though. If they just wanted to know whether they had enough money, well, they need twelve dollars and have seven. Subtracting seven from twelve is only needed if they want to know how much more they need. Which they should want to know, but wasn’t part of the setup.
Brian Boychuk and Ron Boychuk’s The Chuckle Brothers for the 28th uses mathematics as the sine qua non of rocket science. As in, well, the stuff that’s hard and takes some real genius to understand. It’s not clear to me that the equations are actually rocket science. There seem to be a shortage of things in exponentials to look quite right to me. But I can’t zoom in on the art, so, who knows just what might be in there.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 28th is a set theory joke. Or a logic joke, anyway. It refers to some of the mathematics/logic work of Bertrand Russell. Among his work was treating seriously the problems of how to describe things defined in reference to themselves. These have long been a source of paradoxes, sometimes for fun, sometimes for fairy-tale logic, and sometimes to challenge our idea of what we mean by definitions of things. Russell made a strong attempt at describing what we mean when we describe a thing by reference to itself. The iconic example here was the “set of all sets not members of themselves”.
Russell started out by trying to find some way to prove Georg Cantor’s theorems about different-sized infinities wrong. He worked out a theory of types, and what kinds of rules you can set about types of things. Most mathematicians these days prefer to solve the paradox with a particular organization of set theory. But Russell’s type theory still has value, particularly as part of the logic behind lambda calculus. This is an approach to organizing relationships between things that can do wonderful things, including in computer programming. It lets one write code that works extremely efficiently and can never be explained to another person, modified, or debugged ever. I may lack the proper training for the uses I’ve made of it.
Brian Boychuk and Ron Boychuk’s The Chuckle Brothers for the 29th is a lottery joke. It does happen that more than one person wins a drawing; sometimes three or even four people do, for the larger prizes. The chance that there’s a million winners? Frightfully unlikely unless something significant went wrong with the lottery mechanism.
So what are the chances of a million lottery winners? If I’m not mistaken the only way to do this is to work out a binomial distribution. The binomial distribution is good for cases where you have many attempts at doing a thing, where each thing can either succeed or fail, and the likelihood of success or failure is independent of all the other attempts. In this case each lottery ticket is an attempt; it winning is success and it losing is failure. Each ticket has the same chance of winning or losing, and that chance doesn’t depend on how many wins or losses there are. What is that chance? … Well, if each ticket has one chance in a million of winning, and there are a million tickets out there, the chance of every one of them winning is about one-millionth raised to the millionth power. Which is so close to zero it might as well be nothing. … And yet, for all that it’s impossible, there’s not any particular reason it couldn’t happen. It just won’t.
Jef Mallet’s Frazz for the 29th is a less dire take on what-you-learned-this-year. In this case it’s trivia, but it’s a neat sort of trivia. Once you understand how it works you can understand how to make all sorts of silly little divisibility rules. The threes rule — and the nines rule — work by the same principle. Suppose you have a three-digit number. Let me call ‘a’ the digit in the hundreds column, ‘b’ the digit in the tens column, and ‘c’ the digit in the ones column. Then the number is equal to . And, well, that’s equal to . Which is . 99 times any whole number is a multiple of 9, and also of 3. 9 times any whole number is a multiple of 9, and also of 3. So whether the original number is divisible by 9, or by 3, depends on whether is. And that’s why adding the digits up tells you whether a number is a whole multiple of three.
This has only proven anything for three-digit numbers. But with that proof in mind, you probably can imagine what the proof looks like for two- or four-digit numbers, and would believe there’s one for five- and for 500-digit numbers. Or, for that matter, the proof for an arbitrarily long number. So I’ll skip actually doing that. You can fiddle with it if you want a bit of fun yourself.
Also maybe it’s me, or the kind of person who gets into mathematics. But I find silly little rules like this endearing. It’s a process easy to understand that anyone can do and it tells you something not obvious from when you start. It feels like getting let in on a magic trick. That seems like the sort of thing that endears people to mathematics.
Mike Thompson’s Grand Avenue for the 29th is trying to pick its fight with me again. I can appreciate someone wanting to avoid kids losing their mathematical skills over summer. It’s just striking how Thompson has consistently portrayed their grandmother as doing this in a horrible, joy-crushing manner.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 29th gets into a philosophy-of-mathematics problem. Also a pure philosophy problem. It’s a problem of what things you can know independently of experience. There are things it seems as though are true, and that seem independent of the person who is aware of them, and what culture that person comes from. All right. Then how can these things be relevant to the specifics of the universe that we happen to be in just now? If ‘2’ is an abstraction that means something independent of our universe, how can there be two books on the table? There’s something we don’t quite understand yet, and it’s taking our philosophers and mathematicians a long while to work out what that is.
I’ve broken the habit of watching my WordPress readership statistics day-to-day. This is good. It’s too easy to read random fluctuations as significant changes. And to go from that to supposing that everyone’s decided they hate me now. I do still check monthly. And I try to think what I can learn from that data. Not too hard, and not enough to change what I do. But to where I might think I learned something.
I had another 12-post month. As seems to keep happening I started out with an ambitious program of the weekly Reading the Comics posts, finishing up a couple of open-ended essay threads, and then a few drop-ins as I ran across something interesting. And then my days got all busy and stuff demanded my attention and all I had time for was the comics posts after all. It turned out mostly all right, though. Here’s just how all right:
So for the sixth month running I beat a thousand page views. Came in at 1,077. It’s my thinnest margin since back in February when there were a mere 1,062 page views. Still, I had a more than this large comfortable round number of page views. The number of unique visitors dropped also, to 681. That’s my lowest number of visitors since February again. But that just seems to reflect there being less traffic overall in June; the number of views per visitor was 1.58, basically the same as May’s 1.52 and April’s 1.53. No archive-divers here, seems.
There were 94 things liked in June; that’s up from April’s and May’s 73, and down from March’s 142. There were 30 comments posted in June, up from May’s 17 and April’s 13, but down from March’s 53. All respectable enough; none exactly suggesting I know how to write stuff people love to share or comment on. Which is on me, of course; no reader’s got the job of responding to stuff they don’t care to.
The popular posts were nearly what I would have guessed: the Buggles and some comics stuff. But there were surprises even in the top five:
So I’m surprised that last month’s readership review post would be among the most popular. I guess it shows the value of having any picture at all, however marginally interesting, in a post. Still seems dangerously self-absorbed. The non-Euclidean geometry one also surprises me, since it was only up for two days and still got as many readers as anything else posted in June. The lesson here, I suppose, is that people love seeing me not know stuff that’s obvious to people familiar with a topic. This is promising for future essays, though, since there are so many obvious things I don’t know.
Then there’s the list of countries that sent me readers to include, since that’s apparently a thing people like:
Hong Kong SAR China
United Arab Emirates
There were 55 countries sending me any readers, down from 58 for three months in a row. There were 19 single-reader countries, down from 22 in May, up from 14 in April. Brunei and South Korea were single-reader countries two months in a row. Serbia’s had a single reader for me four months in a row now.
The Insights panel tells me July started with this blog having had 63,897 total page views, from an admitted 31,020 unique visitors. It logs for the year 2018 a total of 78 posts that attracted, to that point, 196 comments. And that there had been 535 total likes given to something over the year so far. This comes to an average of 2.5 comments per post, and 6.8 likes per posting. By the end of May I had gotten only 2.4 comments and 6.7 likes per post, so, at least I’ve got something figured out.
By the end of June I had posted 69,051 words as WordPress logs things; that’s 13,374 words over June, a bit more than I posted over May despite June being the shorter month. I’m up to an average of 885.3 words per post; at the end of May I was at a mere 843.6 words per post. The trend is obvious; by the end of the year I’ll just never stop writing things. You’ll just see a continuous feed of me putting more heaps of words onto this pile. You’ll be shocked how many times and how many different ways I can type ‘that’ wrong and correct it. Or how often an ‘of course’ creeps into my writing and I have to edit that out.
As ever, I encourage you to read this post and more like it. You can add this page to your WordPress reader by using the button at the upper-right corner of this page. This link is the RSS feed, which gets all my posts as they’re posted, and which you can add to your RSS reader without my ever knowing about. I’m @Nebusj on Twitter. If you see me on Tumblr you’ve found a hoax, since I’m not on Tumblr and every time I look at it I feel helpless and confused.