It’s mostly there in the subject line. My plan, for what plans are worth this year, is to start publishing my A-to-Z essays the week of the 15th of June. So I’d like to know if you, the reader, would like to see me explaining anything. This then is the first of many appeals for topics that I’ll be putting out.

The rules are pretty loose. Suggest something mathematical that starts with the relevant letter. Whatever topic I think I can be most interesting about, I’ll write up. This might be something I pick myself, too.

But please leave suggestions in comments here. Also, if you have a blog, a Twitter feed, a YouTube channel, any project worth mention, please let me know. I try to credit the people who generously give me suggestions I can write about, and want to share their projects with people.

Right now, I’m looking just for the latter ‘A’. In the past years I’ve written these essays about a-level subjects:

This was a week of few mathematically-themed comic strips. I don’t mind. If there was a recurring motif, it was about parents not doing mathematics well, or maybe at all. That’s not a very deep observation, though. Let’s look at what is here.

Liniers’s Macanudo for the 18th puts forth 2020 as “the year most kids realized their parents can’t do math”. Which may be so; if you haven’t had cause to do (say) long division in a while then remembering just how to do it is a chore. This trouble is not unique to mathematics, though. Several decades out of regular practice they likely also have trouble remembering what the 11th Amendment to the US Constitution is for, or what the rule is about using “lie” versus “lay”. Some regular practice would correct that, though. In most cases anyway; my experience suggests I cannot possibly learn the rule about “lie” versus “lay”. I’m also shaky on “set” as a verb.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 18th shows a mathematician talking, in the jargon of first and second derivatives, to support the claim there’ll never be a mathematician president. Yes, Weinersmith is aware that James Garfield, 20th President of the United States, is famous in trivia circles for having an original proof of the Pythagorean theorem. It would be a stretch to declare Garfield a mathematician, though, except in the way that anyone capable of reason can be a mathematician. Raymond Poincaré, President of France for most of the 1910s and prime minister before and after that, was not a mathematician. He was cousin to Henri Poincaré, who founded so much of our understanding of dynamical systems and of modern geometry. I do not offhand know what presidents (or prime ministers) of other countries have been like.

Weinersmith’s mathematician uses the jargon of the profession. Specifically that of calculus. It’s unlikely to communicate well with the population. The message is an ordinary one, though. The first derivative of something with respect to time means the rate at which things are changing. The first derivative of a thing, with respect to time being positive means that the quantity of the thing is growing. So, that first half means “things are getting more bad”.

The second derivative of a thing with respect to time, though … this is interesting. The second derivative is the same thing as the first derivative with respect to time of “the first derivative with respect to time”. It’s what the change is in the rate-of-change. If that second derivative is negative, then the first derivative will, in time, change from being positive to being negative. So the rate of increase of the original thing will, in time, go from a positive to a negative number. And so the quantity will eventually decline.

So the mathematician is making a this-is-the-end-of-the-beginning speech. The point at which the the second derivative of a quantity changes sign is known as the “inflection point”. Reaching that is often seen as the first important step in, for example, disease epidemics. It is usually the first good news, the promise that there will be a limit to the badness. It’s also sometimes mentioned in economic crises or sometimes demographic trends. “Inflection point” is likely as technical a term as one can expect the general public to tolerate, though. Even that may be pushing things.

Gary Wise and Lance Aldrich’s Real Life Adventures for the 19th has a father who can’t help his son do mathematics. In this case, finding square roots. There are many ways to find square roots by hand. Some are iterative, in which you start with an estimate and do a calculation that (typically) gets you a better estimate of the square root you want. And then repeat the calculation, starting from that improved estimate. Some use tables of things one can expect to have calculated, such as exponentials and logarithms. Or trigonometric tables, if you know someone who’s worked out lots of cosines and sines already.

Julie Larson’s The Dinette Set rerun for the 21st fusses around words. Along the way Burl mentions his having learned that two negatives can make a positive, in mathematics. Here it’s (most likely) the way that multiplying or dividing two negative numbers will produce a positive number.

One natural question is: does the order matter? Are you better off going first, second, or third? Contestants don’t get to choose order; they’re ranked by how much they’ve won on the show already. (I believe this includes the value of their One-Bids, the item-up-for-bid that gets them on stage. This lets them rank contestants when all three lost their pricing games.) The first contestant always has a choice of whether to spin once or twice. The second and third contestants don’t necessarily get to choose what to do. Is that an advantage or a disadvantage?

In this paper, published 2002, Tenorio and Cason look at the game-theoretical logic. And compare it to how people actually play the game, on the show and in laboratory experiments. (The advantage of laboratory experiments, besides that you can get more than two each day, is that participants’ behavior won’t be thrown off by the thoughts of winning a thousand or more dollars for a good spin.) They also look some at how the psychology of risk affects people’s play.

(I’m compelled — literally, I can’t help myself — to note they make some terminology errors. They mis-label the Showcase Showdown as the bit at the end of the show, where two contestants put up bids for showcases. It’s a common mistake, and probably reflects that “showdown” has connotations of being one-on-one. But that segment is simply the Showcase Round. The Showcase Showdown is the spinning-the-big-wheel part.)

Their research, anyway, suggests that if every contestant played perfectly — achieving a “Nash equilibrium”, in which nobody can pick a better strategy given the choices other players make — going later does, indeed, give a slight advantage. The first contestant would win about 31% of the time, the second about 33%, and the third about 36% of the time. In watching the show to see what happens they found the first contestant won about 30% of the time, the second about 34%, and the third about 36% of the time. That’s no big difference.

The article includes more fascinating statistical breakdowns, answering questions such as “are spins on the wheel uniformly distributed?” That is, are you as likely to spin $1.00 on the first spin as you are to spin 0.05? Or 0.50? They have records of what people actually do. Or what prize payouts would be expected, from theoretical perfect play, and how they compare to actual play.

The paper is written for an academic audience, particularly one versed in game theory. If you are somehow not, it can be tough going. It’s all right to let your eye zip past a paragraph of jargon, or of calculations, to get back to the parts that read as English. Real mathematicians do that too, as a way of understanding the point. They can come back around later to learn how the authors got to the point.

The end of last week offered just a few more comic strips, and some pretty casual mathematics content. Let me wrap that up.

Daniel Beyer’s Long Story Short for the 13th has the “math department lavatory” represented as a door labelled . It’s an interesting joke in that it reads successfully, but doesn’t make sense. To match the references to the commonly excreted substances they’d want .

On funny labels, though, I did once visit a mathematics building in which the dry riser had the label N Bourbaki. Nicholas Bourbaki was not a member of that college’s mathematics department, of course. This is why the joke was correctly formed and therefore funny.

Gary Larson’s The Far Side strips for the 14th includes the famous one of Albert Einstein coming so close to working out . The usual derivations for don’t start with that and then explore whether it makes sense, which is what Einstein seems to be doing here. Instead they start from some uncontroversial premises and find that they imply this business. Dimensional analysis would also let you know that, if c is involved, it’s probably to the second power rather than anything else.

But that doesn’t mean we can’t imagine Einstein assuming there must be a relationship between energy and mass, finding one that makes sense, and then finding a reason it’s that rather than something else. That’s a common enough pattern of mathematical discovery. Also, a detail I hadn’t noticed before, is that Einstein tried out , rejected it, and then tried it again. This is also a common pattern of discovery.

Niklas Eriksson’s Carpe Diem for the 15th depicts a couple ancient Greek deep-thinkers. A bit of mathematics, specifically geometry, is used as representative of that deep thinking.

The past week had a fair number of comic strips mentioning some aspect of mathematics. One of them is, really, fairly slight. But it extends a thread in the comic strip that I like and so that I will feature here.

Sam Hurt’s Eyebeam for the 11th uses heaps of mathematical expressions, graphs, charts, and Venn diagrams to represent the concept of “data”. It’s spilled all over to represent “sloppy data”. Usually by the term we mean data that we feel is unreliable. Measurements that are imprecise, or that are unlikely to be reliable. Precision is, roughly, how many significant digits your measurement has. Reliability is, roughly, if you repeated the measurement would you get about the same number?

Ryan North’s Dinosaur Comics for the 12th talks about immortality. And what the probability of events means when there are infinitely many opportunities for a thing to happen.

We’re accustomed in probability to thinking of the expectation value. This is the chance that something will happen, given some number N opportunities to happen, if at each opportunity it has the probability p of happening. Let me assume the probability is always the same number. If it’s not, our work gets harder, although it’s basically the same kind of work. But, then, the expectation value, the number of times we’d expect to see the thing happen, is N times p. Which, as Utahraptor points out, we can expect has to be at least 1 for any event, however unlikely, given enough chances. So it should be.

But, then, to take Utahraptor’s example: what is the probability that an immortal being never trips down the stairs? At least not badly enough to do harm? Why should we think that’s zero? It’s not as if there’s a physical law that compels someone to go to stairs and then to fall down them to their death. And, if there’s any nonzero chance of someone not dying this way? Then, if there are enough immortals, there’s someone who will go forever without falling down stairs.

That covers just the one way to die, of course. But the same reasoning holds for every possible way to die. If there’s enough immortals, there’s someone who would not die from falling down stairs and from never being struck by a meteor. And someone who’d never fall down stairs and never be struck by a meteor and never fall off a cliff trying to drop an anvil on a roadrunner. And so on. If there are infinitely many people, there’s at least one who’d avoid all possible accidental causes of death.

More. If there’s infinitely many immortals, then there are going to be a second and a third — indeed, an infinite number — of people who happen to be lucky enough to never die from anything. Infinitely many immortals die of accidents, sure, but somehow not all of them. We can’t even say that more immortals die of accidents than don’t.

My point is that probability gets really weird when you try putting infinities into it. Proceed with extreme caution. But the results of basic, incautious, thinking can be quite heady.

Bill Amend’s FoxTrot Classics for the 12th has Paige cramming for a geometry exam. Don’t cram for exams; it really doesn’t work. It’s regular steady relaxed studying that you need. That and rest. There is nothing you do that you do better for being sleep-deprived.

Bob Weber Jr and Jay Stephens’s Oh Brother for the 12th has Lily tease her brother with a story problem. I believe the strip’s a rerun, but it had been gone altogether for more than a year. It’s nice to see it returned anyway.

At last I buckled down and took care of a nagging small task that, somehow, I kept putting off dedicating the time to doing. And that’s why my e-mail has gone from 300 things to a scant … wait, 26? That can’t be right. I must have deleted something I shouldn’t have.

But. The other thing I did was finally tag the first couple rounds of my Mathematics A-to-Z glossaries correctly. Now, so far as I am aware, every one of the essays I’ve written in the past six rounds appears on this roster of essays. There’s 181 posts listed there. But about two dozen of them are solicitations for essay topics or index pages or similar things, important but not explaining any particular mathematical topic.

That I have taken care of this hints at my next announcement. I am planning to start the 2020 A-to-Z. My plan is to start soliciting topics soon, and to begin publishing essays sometime in June.

I am going to do things a little different from previous years, though. The main thing is publishing schedule. I plan to publish just one essay per week.

In past years I’ve done two or three essays per week, and that is exciting to do. It’s also a lot of work and, I have to tell you, I’m not up for that much labor. A major part of my coping mechanism for the pandemic has been creative outlets. That, and building my daily walk up to where I should soon be able to hike to Ontario and back in an afternoon.

Still. I grant don’t know just what my limits are. But I feel certain doing more than glossary-type essay per week is beyond my limits. As it is I’m thinking of how I might adapt my self-imposed workload so that it stays a constructive force.

The most important things should be unchanged, though. I’ll ask for mathematical topics to write about, for each letter, and write about the ones that I think I can be interesting about. This might be my own choice if I feel I’m too ignorant of the suggested topics to do any of them competently. I’ll post the essays previously done for various letters, but am open to revisiting a topic, if I feel I can do a much better job given a second try.

Should be exciting, anyway, however it all turns out.

There were a couple more comic strips in the block of time I want to write about. Only one’s got some deeper content and, I admit, I had to work to find it.

Olivia Jaimes’s Nancy for the 8th has Nancy and Sluggo avoiding mathematics homework. Or, “practice”, anyway. There’s more, though; Nancy and Sluggo are doing some analysis of viewing angles. That’s actual mathematics, certainly. Computer-generated imagery depends on it, just like you’d imagine. There are even fun abstract questions that can give surprising insights into numbers. For example: imagine that space were studded, at a regular square grid spacing, with perfectly reflective marbles of uniform size. Is there, then, a line of sight between any two points outside any marbles? Even if it requires tens of millions of reflections; we’re interested in what perfect reflections would give us.

Using playing cards as a makeshift protractor is a creative bit of making do with what you have. The cards spread in a fanfold easily enough and there’s marks on the cards that you can use to keep your measurements reasonably uniform. Creating ad hoc measurement tools like this isn’t mathematics per se. But making a rough tool is a first step to making a precise tool. And you can use reason to improve your estimates.

It’s not on-point, but I did want to share the most wondrous ad hoc tool I know of: You can use an analog clock hand, and the sun, as a compass. You don’t even need a real clock; you can draw the time on a sheet of paper and use that. It’s not a precise measure, of course. But if you need some help, here you go. You’ve got it.

Last week saw a modest number of mathematically-themed comic strips. Then it threw in a bunch of them all on Thursday. I’m splitting the week partway through that, since it gives me some theme to this collection.

Tim Rickard’s Brewster Rockit for the 3rd of May is a dictionary joke, with Brewster naming each kind of chart and making a quick joke about it. The comic may help people who’ve had trouble remembering the names of different kinds of graphs. I doubt people are likely to confuse a pie chart with a bar chart, admittedly. But I could imagine thinking a ‘line graph’ is what we call a bar chart, especially if the bars are laid out horizontally as in the second panel here.

The point of all these graphs is to understand data geometrically. We have fair intuitions about relatives lengths and areas. Bar charts represent relative magnitudes in lengths. Pie charts and bubble charts represent magnitudes in area. We have okay skills in noticing structures in complex shapes. Line graphs and scatter plots use that skill. So these pictures can help us understand some abstraction or something we can’t sense using a sense we do have. It’s not necessarily great; note that I said our intuitions were ‘fair’ and ‘okay’. But we hope to use reason helped by intuition to better understand what we are doing.

And, yes, in the greater scheme of things, any homework or classwork problem is trivial. It’s meant to teach how to calculate things we would like to know. The framing of the story is meant to give us a reason to want to know a thing. But they are practice, and meant to be practice. One practices on something of no consequence, where errors in one’s technique can be corrected without breaking anything.

It happens a round of story problems broke out among my family. My sister’s house has some very large trees. There turns out to be a poorly-organized process for estimating the age of these trees from their circumference. This past week saw a lot of chatter and disagreement about what the ages of these trees might be.

Michael Fry’s Committed rerun for the 7th finally gets us to golf. The Lazy Parent tries to pass off watching golf as educational, with working out the distance to the pin as a story problem. Structurally this is just fine, though: a golfer would be interested to know how far the ball has yet to go. All the information needed is given. It’s the question of whether anyone but syndicated cartoonists cares about golf that’s a mystery.

Bill Amend’s FoxTrot Classics for the 7th is another golf and mathematics joke. Jason has taken the homonym of ‘fore’ for ‘four’, and then represented ‘four’ in a needlessly complicated way. Amend does understand how nerd minds work. The strip originally ran the 21st of May, 1998.

That’s enough comics for me for today. I should have the rest of last week’s in a post at this link soon. Thank you.

Yes, I feel a bit weird looking at the past month’s readership this early in the month too. I was tempted to go back and look at March’s figures all over again just so I stay tardy. But, no sense putting it off further, especially as I’m thinking to over-commit myself again already.

In April I managed to publish 15 things. This amazes me given that my spirits are about like everyone’s spirits are. I did not repeat having 2,000 readers this past month. But it came surprisingly close. Here’s a look at the readership figures.

There were 1,959 pages viewed over the course of April. This is a bit under the twelve-month running average of 2,127.1. But I’m going to be under the twelve-month running average at least until that October 2019 spike fades into the background. I’m all right with that. There were 1,314 unique visitors, which again is under the running average of 1,440.2 unique visitors in a month.

The measures that I think of as showing engagement were poor, as they usually are. There were nine comments received over the month, down from the 15.3 average. More surprisingly there were only 44 likes given over the month, noticeably below the 60.4 average.

Everything looks a bit better when pro-rated per posting. The 130.6 views per posting are above even the twelve-month average for that of 120.8 views per posting. The 87.6 unique visitors per posting beats the average of 81.1. It’s still 0.6 comments per posting, below the average of 1.0. And only 2.9 likes per posting, below the average of 3.8. Can’t have everything, I suppose. But I may be doing something to affect that pattern.

There were, counting my home page, 265 postings that got any kind of views in April. That’s up from the 255 of March and 210 of February. 134 of them got more than one view, down from March’s 145 but up fro February’s 108. 36 of them got at least ten views, compared to 35 in March and 25 in February. And what got the most page views? About what you’d expect:

The most popular thing I published in April was Rjlipton’s thoughts on the possible ABC Conjecture proof, which is pretty good performance for a post that just says someone else wrote a thing. I don’t know why my headsup posts like that are so reliably popular. But I suppose if people trust my judgement about stuff that’s almost as good as people trusting my prose.

73 countries or country-like things sent me readers in April. 12 of them were single-view countries. This is down from the 78 countries in march, but up from the 67 in February. There had been 30 single-view countries in Marc and 19 in February, so I guess people are doing more archive-reading, though. Here’s the details for that:

Country

Readers

United States

1,160

Canada

128

India

105

United Kingdom

102

Australia

34

Singapore

31

Germany

29

Poland

21

Romania

21

Austria

15

Brazil

15

Philippines

15

Finland

14

Netherlands

14

China

13

Italy

13

Ireland

12

Kazakhstan

10

South Korea

10

Thailand

10

American Samoa

9

Japan

9

Saudi Arabia

9

South Africa

9

France

8

Spain

8

Hong Kong SAR China

7

United Arab Emirates

7

Albania

6

Belgium

6

Indonesia

6

Portugal

6

Turkey

6

Kenya

5

Israel

4

Malaysia

4

Slovenia

4

Sweden

4

Switzerland

4

Argentina

3

Croatia

3

Egypt

3

European Union

3

Greece

3

New Zealand

3

Russia

3

Trinidad & Tobago

3

Uruguay

3

Vietnam

3

Bangladesh

2

Czech Republic

2

Denmark

2

Dominican Republic

2

Estonia

2

Greenland

2

Mexico

2

Norway

2

Peru

2

Puerto Rico

2

Serbia

2

Taiwan

2

Bahrain

1

Bosnia & Herzegovina

1

Bulgaria

1

Hungary

1

Kyrgyzstan

1

Lithuania

1 (**)

Malawi

1

Nigeria

1

Pakistan

1

Seychelles

1

Sri Lanka

1

St. Lucia

1

Lithuania has given me a single view each of the last three months. No other countries are on a similar streak.

WordPress says I published a mere 8,566 words in April. That’s my most laconic month since January. With 15 posts, that gives me an average of just under 571.1 words per posting, which is my shortest of the year. It brings my average words per posting for the year down to 691; it had been 721 at the start of April. As of the start of May I’d published 50 posts and 34,536 words since the start of the year.

As of the start of May I’ve posted 1,454 pieces altogether. They’ve drawn 104,439 views from 57,501 acknowledged unique visitors.

I’d be glad, always, to have you as a regular reader. You can put this blog in your WordPress reader by using the “Follow Nebusresearch” button on the upper right corner of the page. If you prefer to use an RSS reader, the feed is available to you. If you don’t have an RSS reader, you can get one as your friends page with a free Dreamwidth or Livejournal account. And, excitingly, my Twitter account @Nebusj has resumed working. I’ll maybe even get back into the habit of using it. I am also on the mathematics-themed Mastodon instance @nebusj@mathstodon.xyz, and trying to figure a good habit for that. We are all on journeys of discovering our new habits, though.

Thank you for reading this. I hope you read more, and maybe comment some. Please take care.

The program is three people, plus host Melvyn Bragg, talking about the life and work of Gauss. Gauss is one of those figures hard to exaggerate. He was extremely prolific and insightful. It is an exaggeration to say that he did foundational work in every field of mathematics, but only a slight exaggeration. (He compares to Leonhard Euler that way.) I’d imagine that anyone reading a pop mathematics blog knows something of Gauss. But you may learn something new, or a new perspective on something familiar.

The past week was a light one for mathematically-themed comic strips. So let’s see if I can’t review what’s interesting about them before the end of this genially dumb movie (1940’s Hullabaloo, starring Frank Morgan and featuring Billie Burke in a small part). It’ll be tough; they’re reaching a point where the characters start acting like they care about the plot either, which is usually the sign they’re in the last reel.

Jenny Campbell’s Flo and Friends for the 26th is a joke about fumbling a bit of practical mathematics, in this case, cutting a recipe down. When I look into arguments about the metric system, I will sometimes see the claim that English traditional units are advantageous for cutting down a recipe: it’s quite easy to say that half of “one cup” is a half cup, for example. I doubt that this is much more difficult than working out what half of 500 ml is, and my casual inquiries suggest that nobody has the faintest idea what half of a pint would be. And anyway none of this would help Ruthie’s problem, which is taking two-fifths of a recipe meant for 15 people. … Honestly, I would have just cut it in half and wonder who’s publishing recipes that serve 15.

Ed Bickford and Aaron Walther’s American Chop Suey for the 28th uses a panel of (gibberish) equations to represent deep thinking. It’s in part of a story about an origami competition. This interests me because there is serious mathematics to be done in origami. Most of these are geometry problems, as you might expect. The kinds of things you can understand about distance and angles from folding a square may surprise. For example, it’s easy to trisect an arbitrary angle using folded squares. The problem is, famously, impossible for compass-and-straightedge geometry.

Origami offers useful mathematical problems too, though. (In practice, if we need to trisect an angle, we use a protractor.) It’s good to know how to take a flat, or nearly flat, thing and unfold it into a more interesting shape. It’s useful whenever you have something that needs to be transported in as few pieces as possible, but that on site needs to not be flat. And this connects to questions with pleasant and ordinary-seeming names like the map-folding problem: can you fold a large sheet into a small package that’s still easy to open? Often you can. So, the mathematics of origami is a growing field, and one that’s about an accessible subject.

Nate Fakes’s Break of Day for the 29th is the anthropomorphic-symbols joke for the week, with an x talking about its day job in equations and its free time in games like tic-tac-toe.

Bill Holbrook’s On The Fastrack for the 2nd of May also talks about the use of x as a symbol. Curt takes eagerly to the notion that a symbol can represent any number, whether we know what it is or not. And, also, that the choice of symbol is arbitrary; we could use whatever symbol communicates. I remember getting problems to work in which, say, 3 plus a box equals 8 and working out what number in the box would make the equation true. This is exactly the same work as solving 3 + x = 8. Using an empty box made the problem less intimidating, somehow.

Dave Whamond’s Reality Check for the 2nd is, really, a bit baffling. It has a student asking Siri for the cosine of 174 degrees. But it’s not like anyone knows the cosine of 174 degrees off the top of their heads. If the cosine of 174 degrees wasn’t provided in a table for the students, then they’d have to look it up. Well, more likely they’d be provided the cosine of 6 degrees; the cosine of an angle is equal to minus one times the cosine of 180 degrees minus that same angle. This allows table-makers to reduce how much stuff they have to print. Still, it’s not really a joke that a student would look up something that students would be expected to look up.

… That said …

If you know anything about trigonometry, you know the sine and cosine of a 30-degree angle. If you know a bit about trigonometry, and are willing to put in a bit of work, you can start from a regular pentagon and work out the sine and cosine of a 36-degree angle. And, again if you know anything about trigonometry, you know that there are angle-addition and angle-subtraction formulas. That is, if you know the cosine of two angles, you can work out the cosine of the difference between them.

So, in principle, you could start from scratch and work out the cosine of 6 degrees without using a calculator. And the cosine of 174 degrees is minus one times the cosine of 6 degrees. So it could be a legitimate question to work out the cosine of 174 degrees without using a calculator. I can believe in a mathematics class which has that as a problem. But that requires such an ornate setup that I can’t believe Whamond intended that. Who in the readership would think the cosine of 174 something to work out by hand? If I hadn’t read a book about spherical trigonometry last month I wouldn’t have thought the cosine of 6 a thing someone could reasonably work out by hand.

I didn’t finish writing before the end of the movie, even though it took about eighteen hours to wrap up ten minutes of story. My love came home from a walk and we were talking. Anyway, this is plenty of comic strips for the week. When there are more to write about, I’ll try to have them in an essay at this link. Thanks for reading.

Comic Strip Master Command decided I should have a week to catch up on things, and maybe force me to write something original. Of all the things I read there were only four strips that had some mathematics content. And three of them are such glancing mentions that I don’t feel it proper to include the strip. So let me take care of this.

Mark Anderson’s Andertoons for the 20th is the Mark Anderson’s Andertoons for the week. Wavehead apparently wants to know whether or is the better of these equivalent forms. I understand the impulse. Rarely in real life do we see two things that are truly equivalent; there’s usually some way in which one is better than the other. There may be two ways to get home for example, both taking about the same time to travel. One might have better scenery, though, or involve fewer difficult turns or less traffic this time of day. This is different, though: or are two ways to describe the same number. Which one is “better”?

The only answer is, better for what? What do you figure to do with this number afterwards? I admit, and suppose most people have, a preference for . But that’s trained into us, in large part, by homework set to reduce fractions to “lowest terms”. There’s honest enough reasons behind that. It seems wasteful to have a factor in the numerator that’s immediately divided out by the denominator.

If this were 25 years ago, I could ask how many of you have written out a check for twenty-two and 3/4 dollars, then, rather than twenty-two and 75/100 dollars? The example is dated but the reason to prefer an equivalent form is not. If I know that I need the number represented by , and will soon be multiplying it by eight, then may save me the trouble of thinking what three times two is. Or if I’ll be adding it to , or something like that. If I’m measuring this for a recipe I need to cut in three, because the original will make three dozen cookies and I could certainly eat three dozen cookies, then may be more convenient than . What is the better depends on what will clarify the thing I want to do.

A significant running thread throughout all mathematics, not just arithmetic, is finding equivalent forms. Ways to write the same concept, but in a way that makes some other work easier. Or more likely to be done correctly. Or, if the equivalent form is more attractive, more likely to be learned or communicated. It’s of value.

Jan Eliot’s Stone Soup Classics rerun for the 20th is a joke about how one can calculate what one is interested in. In this case, going from the number of days left in school to the number of hours and minutes and even seconds left. Personally, I have never had trouble remembering there are 24 hours in the day, nor that there are 86,400 seconds in the day. That there are 1,440 minutes in the day refuses to stick in my mind. Your experiences may vary.

Harry Bliss’s Bliss for the 23rd speaks of “a truck driver with a PhD in mathematical logic”. It’s an example of signifying intelligence through mathematics credentials. (It’s also a bit classicist, treating an intelligent truck driver as an unlikely thing.)

I’m caught up! This coming Sunday I hope to start discussingthis week’s comics in a post at this link. And for this week? I don’t know; maybe I’ll figure something to write. We’ll see. Thanks for reading.

And now let me close out a week ago, in the comics. It was a slow week and it finished on a bunch of casual mentions of mathematical topics.

Gary Larson’s The Far Side compilation “Hands Off My Bunsen Burner” features this panel creating a model of how to get rights out of wrongs. The material is a joke, but trying to find a transformation from one mathematical object to another is a reasonable enough occupation.

Ted Shearer’s Quincy rerun for the 15th is one in the lineage of strips about never using mathematics in later life. Quincy challenges us to think of a time a reporter asks the President how much is 34 times 587.

That’s an unpleasant multiplication to do. But I can figure some angles on it. 34 is just a bit over one-third of 100. 587 is just a bit under 600. So, 34 times 587 has to be tolerably near one-third of 100 times 600. So it should be something around 20,000. To get it more exact: 587 is 13 less than 600. So, 587 times one-third of a hundred will be 600 times one-third of a hundred minus 13 times one-third of a hundred. That’s one-third of 130, which is about 40. So the product has to be something close to 19,960. And the product has be some number which ends in an 8, what with 4 times 7 being 28. So the answer has to be one of 19,948, 19,958, or 19,968. And, indeed, it’s 19,958. I doubt I could do that so well during a press conference, I’ll admit. (If I wanted to be sure about that second digit, I’d have worked out: the tens unit in 34 times the ones in 587 is three times seven which is 21; the ones unit in 34 times the tens unit in 587 is four times eight which is 32; and the 4 times 7 being 28 gives me a 2 in the tens unit. So, 1 plus 2 plus 2 is 5, and there we go.)

Brian Anderson’s Dog Eat Doug for the 15th uses blackboards full of equations to represent deep thinking. I can’t make out what the symbols say. They look quite good, though, and seem to have the form of legitimate expressions.

Terri Liebenson’s The Pajama Diaries for the 17th imagines creating a model for the volume of a laundry pile. The problem may seem trivial, but it reflects an important kind of work. Many processes are about how something that’s always accumulating will be handled. There’s usually a hard limit to the rate at which whatever it is gets handled. And there’s usually very little reserve, in either capacity or time. This will cause, for example, a small increase in traffic in a neighborhood to produce great jams, or how a modest rain can overflow the whole city’s sewer systems. Or how a day of missing the laundry causes there to be a week’s backlog of dirty clothes.

And a little final extra comic strip. I don’t generally mention web comics here, except for those that have fallen in with a syndicator like GoComics.com. (This is not a value judgement against web comics. It’s that I have to stop reading sometime.) But Kat Swenski’s KatRaccoon Comics recently posted this nice sequence with a cat facing her worst fear: a calculus date.

Now at last I turn to last week’s mathematically-themed comic strips. They weren’t very deeply mathematical, I think. But I always think that right before I turn out a 2,000-word essay about some kid giving a snarky answer to an arithmetic problem.

Rudolph Dirks’s Katzenjammer Kids vintage rerun for the 12th is a slightly hard-to-read joke about the association between rabbits and multiplication and reproduction. There is a neat reference in the first panel to being smart enough to do multiplication without a slide rule.

Rick Detorie’s One Big Happy for the 12th has Ruthie try to teach her brother about number words. What Ruthie seems to be struggling with is the difference between a number and the name we give a number. The distinction between a thing and the name of a thing can be a tricky one, and I do remember being confused at the difference between the word “four” and the concept “four”. What I don’t remember, to my regret, is what thought I had which made the difference clear.

I don’t just read comic strips around here. It seems like it, I grant. But there’s other things that catch my interest and that you might also like.

The first: many people have talked about what great thinkers did during their quarantine-induced disruptions to their lives. Isaac Newton is held up as a great example. While avoiding the Plague, after all, he had that great year of discovering calculus, gravity, optics, and an automatic transmission that doesn’t fail after eight years of normal driving. It’s a great story. The trouble is that real thing is always more ambiguous, more hesitant, and less well-defined than the story. The Renaissance Mathematicus discusses, in detail, something closer to the reality of Newton’s accomplishments during that plague year. This is not to say that his work was not astounding. But it was not as much, or as intense, or as superhuman as inspirational tweets would like.

If you do decide the quarantine is a great chance to revolutionize academia, good luck. You need some reference material, though. Springer publishing has put out several hundred of its textbooks as free PDFs or eBooks. A list of 408 of them (the poster claims) is here on Reddit. This is not only a list of mathematics and mathematics-related topics, and I not undrestand the poster’s organization scheme. But there are a lot of books here, including at least two Introduction to Partial Differential Equations texts. There’s something of note there. This could finally be the thing that gets me to learn the mathematical-statistics programming language R. (It will not get me to learn the mathematical-statistics programming language R.)

And, finally, the disruption to everything has messed up academic departments’ routines. Some of those routines are seminars, in which people share the work they’re doing. Fortunately, many of these seminars are moving to online presentations. And then you can join in, and at least listen, without needing even to worry about being the stranger hanging around the mathematics department. Mathseminars.org has a list of upcoming seminars, with links to what the sessions are about and how to join them. The majority are in English, but there are listed seminars in Spanish, Russian, and French.

This is closing out a busy week’s worth of comic strips mentioning some mathematics theme. Three of these are of extremely slight mathematical content, but I’ll carry on anyway.

Reza Farazmand’s Poorly Drawn Lines for the 8th has a bear admit the one thing which frightens him still is mathematics. It adds to it a joke showing that he’s not very good at mathematics, by making a mistake with percentages.

Dana Simpson’s Ozy and Millie rerun for the 9th is part of a sequence of Ozy being home-schooled. The joke puts the transient nature of knowledge up against the apparent permanent of arithmetic. The joke does get at one of those fundamental questions in the philosophy of mathematics: is mathematics created or discovered? The expression of mathematics is unmistakably created. There is nothing universal in declaring “six times eight is forty-eight” and if you wish to say there is, then ask someone who speaks only Tamil and not a word of English whether they agree with exactly that proposition.

But, grant that while we may have different representations of the concept, it is the case that “eight” exists, right? We get right back into trouble if we follow up by asking, all right, will “eight” fit in my hand? Is “eight” larger than the weather? Is “eight” more or less red than nominalism? I chose nouns that made those questions obviously ridiculous. But if we want to talk about a mathematical construct existing, someone’s going to ask what traits that existence implies. It’s convenient for mathematicians, and good publicity, for us to think that we work on things that exist independently of the accidental facts of the universe. But then we’re stuck when we’re asked how we, stuck in the universe, can have anything to do with a thing that’s not part of it.

Not mentioned in this particular Ozy and Millie strip is that the characters are Buddhist. The (American) pop culture interpretation of Buddhism includes an emphasis on understanding the transient nature of … everything … which would seem to include mathematical knowledge. Still, there is a long history of great mathematical work done by Buddhist scholars; the oldest known manuscript of Indian mathematics is written in a Buddhist Hybrid Sanskrit. The author of that manuscript is unknown, but it’s not as if that were the lone piece of mathematical writing.

My limited understanding is that Indian mathematics used an interesting twist on the problem of the excluded middle. This is a question important to proofs. Can we take every logical proposition as being either true or false? If we can, then we are able to prove statements by contradiction: suppose the reverse of what we want to prove and show that implies nonsense. This is common in western mathematics. But there is a school of thought that we should not do this, and only allow as true statements we have directly proven to be true. My understanding is that at least one school of Indian mathematics allowed proof by contradiction if it proved that a thing did not exist. It would not be used to show that a thing existed. So, for example, it would allow the ordinary proof that the square root of two can’t be a rational number; it would not allow an indirect proof that, say, a kind of mapping must have a fixed point. (It would allow a proof that showed you how to find that point, though.) It’s an interesting division, and a reminder that even what counts as a logical derivation is a matter of custom.

That’s finally taken care of a week. Time to take care of another week! When I have some of last week’s comic strips written up I will post the essay at this link. Thanks for reading.

I’m again falling behind the comic strips; I haven’t had the writing time I’d like, and that review of last month’s readership has to go somewhere. So let me try to dig my way back to current. The happy news is I get to do one of those single-day Reading the Comics posts, nearly.

Harley Schwadron’s 9 to 5 for the 7th strongly implies that the kid wearing a lemon juicer for his hat has nearly flunked arithmetic. At the least it’s mathematics symbols used to establish this is a school.

Jef Mallett’s Frazz for the 7th has kids thinking about numbers whose (English) names rhyme. And that there are surprisingly few of them, considering that at least the smaller whole numbers are some of the most commonly used words in the language. It would be interesting if there’s some deeper reason that they don’t happen to rhyme, but I would expect that it’s just, well, why should the names of 6 and 8 (say) have anything to do with each other?

There are, arguably, gaps in Evan and Kevyn’s reasoning, and on the 8th one of the other kids brings them up. Basically, is there any reason to say that thirteen and nineteen don’t rhyme? Or that twenty-one and forty-one don’t? Evan writes this off as pedantry. But I, admittedly inclined to be a pedant, think there’s a fair question here. How many numbers do we have names for? Is there something different between the name we have for 11 and the name we have for 1100? Or 2011?

There isn’t an objectively right or wrong answer; at most there are answers that are more or less logically consistent, or that are more or less convenient. Finding what those differences are can be interesting, and I think it bad faith to shut down the argument as “pedantry”.

Dave Whamond’s Reality Check for the 7th claims “birds aren’t partial to fractions” and shows a bird working out, partially with diagrams, the saying about birds in the hand and what they’re worth in the bush.

The narration box, phrasing the bird as not being “partial to fractions”, intrigues me. I don’t know if the choice is coincidental on Whamond’s part. But there is something called “partial fractions” that you get to learn painfully well in Calculus II. It’s used in integrating functions. It turns out that you often can turn a “rational function”, one whose rule is one polynomial divided by another, into the sum of simpler fractions. The point of that is making the fractions into things easier to integrate. The technique is clever, but it’s hard to learn. And, I must admit, I’m not sure I’ve ever used it to solve a problem of interest to me. But it’s very testable stuff.

And now I can close my books on March 2020. Late? Yes, so it’s late. You know what it’s been like. It was a month full of changes of fate, not least because on the 10th I volunteered to tape the empty slot hosting Denise Gaskins’s Playful Math Education Blog Carnival, and right after that the world ended. Hosting such an event I can expect to bring in new readers, although the trouble organizing things meant I didn’t post until the last day of the month. Still, I could hope to see some readership bump. How did that all turn out?

In March I posted 15 things, which is about as busy as I could hope to manage for a month that’s not eaten up by an A-to-Z sequence. And that for a month when I didn’t feel I could point out my series on information theory as explained by the March Madness basketball tournament. I believe the frequency of my own posting is the one variable in my control that affects my readership numbers. And this looks to be true. There were 2,049 page views here in March. This is a bit below the twelve-month running average of 2,072.3 views, but remember, that figure has the October 2019 spike in it. Take October out of it and the running average was a mere 1,472.7 page views.

There were 1,267 unique visitors in March. That’s again below the running average of 1,414.1, but again, the October spike throws that off. Without the October spike the running average was 964.3. 1,267 unique visitors is still my fourth-greatest number of unique visitors on record.

There were 61 likes given to any of my posts in March, essentially tied with the running average of 63.4 likes for a month. There were 21 comments, a nice boost from my running average of 13.9.

Per posting, my averages look pretty good. There were 136.6 views per posting in March, above the running average of 117.7. There were 84.5 visitors per posting, above the average 79.7. There were 4.1 likes per posting, above the average of 4.0 for the first time in ages. And there were even 1.4 comments per posting, well above the 0.9 comments per posting average, and my highest average there since January 2019.

So what all was particularly popular? The Playful Math Education Blog Carnival, alas, posted too late to take the top spot, although it’s looking good to place in April. The top five postings last month in order were:

I assume the popularity of that March 11 Reading the Comics post came from people looking for Pi Day strips. Why they ultimately found the 2016 Pi Day comics, rather than another year’s, I don’t know. I think the 2016 was a good year for strips, so maybe that’s what drew people in.

Counting my home page, 255 pages got any views at all in March. That’s up from the 210 of February and 218 of January. 145 of them got more than one page view, up from 108 in February and 102 in January. 35 posts got at least ten views, up from 25 in February and 27 in January.

There were 78 countries or country-like entities sending me readers in March. Hey, one for each episode of the Original Star Trek, nice. That’s up from 67 in February and 63 in January. But this time there were 30 single-view countries, well above February’s 19 and January’s 18. Here’s the list of them:

Country

Readers

United States

1,244

Philippines

125

Thailand

80

United Kingdom

75

Canada

69

India

60

Germany

53

Singapore

35

Australia

27

Puerto Rico

26

Italy

17

Finland

16

France

14

Taiwan

12

Turkey

11

Brazil

10

Spain

10

Indonesia

9

Israel

8

China

7

Greece

7

Malaysia

7

South Africa

7

Denmark

6

Pakistan

6

Belgium

5

Hong Kong SAR China

5

Sweden

5

Switzerland

5

United Arab Emirates

5

European Union

4

Mexico

4

Netherlands

4

Saudi Arabia

4

Sri Lanka

4

Bulgaria

3

Croatia

3

Czech Republic

3

Nigeria

3

Norway

3

Qatar

3

Romania

3

Fiji

2

Hungary

2

Luxembourg

2

New Zealand

2

Oman

2

Serbia

2

American Samoa

1 (***)

Bahamas

1

Bangladesh

1

Bermuda

1

Cambodia

1 (**)

Colombia

1

Costa Rica

1

Cyprus

1

Egypt

1 (*)

Georgia

1

Guam

1

Ireland

1 (*)

Jamaica

1

Kenya

1

Latvia

1

Lebanon

1

Lithuania

1 (*)

Macau SAR China

1

Malta

1

Nepal

1

Nicaragua

1

Panama

1

Russia

1

Rwanda

1

Slovenia

1

South Korea

1 (**)

Trinidad & Tobago

1

Ukraine

1

Uruguay

1

Vietnam

1

Egypt, Ireland, and Lithuania were single-reader countries two months in a row. Cambodia and South Korea are single-reader countries three months in a row now. American Samoa is in its fourth month of a single reader for me.

In March I published 10,113 words by WordPress’s counter. This was 674.2 words per posting. So while that’s about five hundred more words than I wrote in February the average post shrank by nearly two hundred words. For the year to date I’m averaging now 721 words per post, down from 755.1 at the end of February.

As of the start of April I had collected 102,481 views from 56,182 logged unique visitors, over the course of 1,439 postings.

If you’d like to be among my regular readers, please do. You can use the “Follow Nebusresearch” button on the upper right corner of the page to add this to your WordPress reader. Or you can use the RSS feed https://nebusresearch.wordpress.com/feed/ to read my stuff without showing up in any of my statistics. If you need an RSS reader, get a free account on Dreamwidth or Livejournal, which still exists. You can use their Friends pages as RSS readers. I’m still officially on Twitter as @Nebusj, and sort-of on the mathematics-themed Mastodon instance as @nebusj@mathstodon.xyz, although I haven’t really got the hang of what to do there yet. We’ll see. Thank you for reading.

As much as everything is still happening, and so much, there’s still comic strips. I’m fortunately able here to focus just on the comics that discuss some mathematical theme, so let’s get started in exploring last week’s reading. Worth deeper discussion are the comics that turn up here all the time.

Lincoln Peirce’s Big Nate for the 5th is a casual mention. Nate wants to get out of having to do his mathematics homework. This really could be any subject as long as it fit the word balloon.

John Hambrock’s The Brilliant Mind of Edison Lee for the 6th is a funny-answers-to-story-problems joke. Edison Lee’s answer disregards the actual wording of the question, which supposes the group is travelling at an average 70 miles per hour. The number of stops doesn’t matter in this case.

Mark Anderson’s Andertoons for the 6th is the Mark Anderson’s Andertoons for the week. In it Wavehead gives the “just use a calculator” answer for geometry problems.

Not much to talk about there. But there is a fascinating thing about perimeters that you learn if you go far enough in Calculus. You have to get into multivariable calculus, something where you integrate a function that has at least two independent variables. When you do this, you can find the integral evaluated over a curve. If it’s a closed curve, something that loops around back to itself, then you can do something magic. Integrating the correct function on the curve around a shape will tell you the enclosed area.

And this is an example of one of the amazing things in multivariable calculus. It tells us that integrals over a boundary can tell us something about the integral within a volume, and vice-versa. It can be worth figuring out whether your integral is better solved by looking at the boundaries or at the interiors.

Heron’s Formula, for the area of a triangle based on the lengths of its sides, is an expression of this calculation. I don’t know of a formula exactly like that for the perimeter of a quadrilateral, but there are similar formulas if you know the lengths of the sides and of the diagonals.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 5th depicts, fairly, the sorts of things that excite mathematicians. The number discussed here is about algorithmic complexity. This is the study of how long it takes to do an algorithm. How long always depends on how big a problem you are working on; to sort four items takes less time than sorting four million items. Of interest here is how much the time to do work grows with the size of whatever you’re working on.

The mathematician’s particular example, and I thank dtpimentel in the comments for finding this, is about the Coppersmith–Winograd algorithm. This is a scheme for doing matrix multiplication, a particular kind of multiplication and addition of squares of numbers. The squares have some number N rows and N columns. It’s thought that there exists some way to do matrix multiplication in the order of N^{2} time, that is, if it takes 10 time units to multiply matrices of three rows and three columns together, we should expect it takes 40 time units to multiply matrices of six rows and six columns together. The matrix multiplication you learn in linear algebra takes on the order of N^{3} time, so, it would take like 80 time units.

We don’t know the way to do that. The Coppersmith–Winograd algorithm was thought, after Virginia Vassilevska Williams’s work in 2011, to take something like N^{2.3728642} steps. So that six-rows-six-columns multiplication would take slightly over 51.796 844 time units. In 2014, François le Gall found it was no worse than N^{2.3728639} steps, so this would take slightly over 51.796 833 time units. The improvement doesn’t seem like much, but on tiny problems it never does. On big problems, the improvement’s worth it. And, sometimes, you make a good chunk of progress at once.