## Reading the Comics, October 1, 2015: Big Questions Edition

I’m cutting the collection of mathematically-themed comic strips at the transition between months. The set I have through the 1st of October is long enough already. That’s mostly because the first couple strips suggested some big topics at least somewhat mathematically-based came up. Those are fun to reason about, but take time to introduce. So let’s jump into them.

Lincoln Pierce’s Big Nate: First Class for the 27th of September was originally published the 22nd of September, 1991. Nate and Francis trade off possession of the basketball, and a strikingly high number of successful shots in a row considering their age, in the infinitesimally sliced last second of the game. There’s a rather good Zeno’s-paradox-type-question to be made out of this. Suppose the game started with one second to go and Nate ahead by one point, since it is his strip. At one-half second to go, Francis makes a basket and takes a one point lead. At one-quarter second to go, Nate makes a basket and takes a one point lead. At one-eighth of a second to go, Francis repeats the basket; at one-sixteenth of a second, Nate does. And so on. Suppose they always make their shots, and suppose that they are able to make shots without requiring any more than half the remaining time available. Who wins, and why?

Tim Rickard’s Brewster Rockit for the 27th of September is built on the question of whether the universe might be just a computer simulation, and if so, how we might tell. Being a computer simulation is one of those things that would seem to explain why mathematics tells us so much about the universe. One can make a probabilistic argument about this. Suppose there is one universe, and there are some number of simulations of the universe. Call that number N. If we don’t know whether we’re in the real or the simulated universe, then it would seem we have an estimated probability of being in the real universe of one divided by N plus 1. The chance of being in the real universe starts out none too great and gets dismally small pretty fast.

But this does put us in philosophical difficulties. If we are in something that is a complete, logically consistent universe that cannot be escaped, how is it not “the real” universe? And if “the real” universe is accessible from within “the simulation” then how can they be separate? The question is hard to answer and it’s far outside my realm of competence anyway.

Mark Leiknes’s Cow and Boy Classics for the 27th of September originally ran the 15th of September, 2008. And it talks about the ideas of zero-point energy and a false vacuum. This is about something that seems core to cosmology: how much energy is there in a vacuum? That is, if there’s nothing in a space, how much energy is in it? Quantum mechanics tells us it isn’t zero, in part because matter and antimatter flutter into and out of existence all the time. And there’s gravity, which is hard to explain quite perfectly. Mathematical models of quantum mechanics, and gravity, make various predictions about how much the energy of the vacuum should be. Right now, the models don’t give us really good answers.

Some suggest that there might be more energy in the vacuum than we could ever use, and that if there were some way to draw it off — well, there’d never be a limit to anything ever again. I think this an overly optimistic projection. The opposite side of this suggests that if it is possible to draw energy out of the vacuum, that means it must be possible to shift empty space from its current state to a lower-energy state, much the way you can get energy out of a pile of rocks by making the rocks fall. But the lower-energy vacuum might have different physics in ways that make it very hard for us to live, or for us to exist. I think this an overly pessimistic projection. But I am not an expert in the fields, which include cosmology, quantum mechanics, and certain rather difficult tinkerings with the infinitely many.

Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. for the 28th of September is a joke in the form of true, but useless, word problem answers. Well, putting down a lower bound on what the answer is can help. If you knew what three times twelve was, you could get to four times twelve reliably, and that’s a help. But if you’re lost for three times twelve then you’re just stalling for time and the teacher knows it.

Paul Gilligan’s Pooch Cafe for the 28th of September uses the monkeys-on-keyboards concept. It’s shifted here to cats on a keyboard, but the principle is the same. Give a random process enough time and you can expect it to produce anything you want. It’s a matter of how long you can wait, though. And all the complications of how to make something that’s random. Cats won’t do it.

Mel Henze’s Gentle Creatures for the 29th of September is a rerun. I’m not sure when it was first printed. But it does use “ability to do mathematics” as a shorthand for “is intelligent at all”. That’s flattering to put in front of a mathematician, but I don’t think that’s really fair.

Paul Trap’s Thatababy for the 30th of September is a protest about using mathematics in real life. I’m surprised Thatababy’s Dad had an algebra teacher proclaiming differential equations would be used. Usually teachers assert that whatever they’re teaching will be useful, which is how we provide motivation.

## What People Did Like In My Mathematics Blog In September 2015

I got so busy with my self-pitying yesterday I never got around to talking about what was popular in September. Well, I mentioned: the six most popular posts in September were all Reading the Comics articles, which I’m pretty sure is the first time it’s swept the top of the charts. Also I think for the first time none of the top ten articles were reblogs of anything, nor were they trapezoid-counting.

For some reason the most popular Reading The Comics entry was one from April. The rest were all September posts, which makes more sense. Anyway, to avoid being boring I’ll skip listing the September Reading the Comics posts. I’ll jump to numbers six through ten for the popular-postings roundup:

The country sending me the greatest number of readers was, as ever, the United States, with 418 this time around, down from August’s 496. In second place this time was the Philippines, with 43, up from 26. Italy came in third, with 34, and I didn’t see that coming either. (They’d sent nine in August). Canada with 29 and Australia with 22 round out the top five and it’s kind of a relief to see them finally. Singapore sent eight page views, up from five. India sent five, down from 22. So it goes.

It was another hefty list of singe-reader countries in September: Argentina, Austria, Bangladesh, Bosnia and Herzegovina, Egypt, Greece, Indonesia, Japan, Nepal, Peru, Saudi Arabia, Senegal, Serbia, South Korea, St Lucia, Switzerland, Trinidad and Tobago, United Arab Emirates, and Uruguay.

Repeats on that list from August are Argentina, Bangladesh, Indonesia, Nepal, South Korea, and Switzerland. Nepal is on a three-month single-reader streak here.

There’s not much good in the search terms; nearly all of them were listed as “unknown”. Among the few that were known:

• foxtrot maths 8 cartoon comic
• 8 piece math joke comic strip
• foxtrot maths 9 comic cartoon
• origin is the gateway to your entire gaming universe
• james clerk maxwell theory comics
• comics strip james clerk maxwell

I feel like there’s a niche here and that I need to commission some comics about James Clerk Maxwell.

September ends with the page having had 28,350 views altogether, and some 10,346 visitors. There’s 518 people listed as WordPress followers which is an increase of one, though the Statistics Insights page says five people started following me. Well, I guess at least it’s upward, from the area code of Albany, New York, up to the area code of Lansing, Michigan. I wonder what state capitol has area code 519. There were fifteen postings in September, up from fourteen in August, down from twenty-four in July. (July had the trailing end of the Mathematics A To Z project.)

And let me encourage people again to consider the “Follow Blog via Email” link on the upper right of the page. Or if you have an RSS reader, https://nebusresearch.wordpress.com/feed/ will give you posts. My Twitter account is @Nebusj.

## How September 2015 Treated My Mathematics Blog

So, well, that was disappointing. My readership was off in September. The month saw the fewest page views since November of 2014. The number of unique visitors was only back to about what it was in June of 2015, though, which is less alarming. Still, I can’t fault WordPress’s suspect statistics, not without inconsistency. My humor blog saw its highest readership on record and if I accept that, I have to accept the other.

The humor blog readership I understand. I started explaining what the heck was going on in Apartment 3-G and it’s been really baffling for a long while now because nothing has been going on since February, maybe March at the latest. You can see how that attracts eager readers.

But here’s the sad numbers count: there were 708 views on the mathematics blog in September, down from 909 in August and 863 in July. And well done from June’s record of 1,051.

The number of unique visitors was 381, down from August’s 506 and July’s 415. June had only 367 unique visitors, but that was part of the big Summer 2015 Mathematics A To Z project. That’s probably why more people were reading, too.

I can’t even point to signs of reader engagement. The number of likes was down to 188, compared to August’s 282 and July’s 381. Extrapolating, November should see me get a negative number of likes. Comments are even worse: after three months in a row of about a hundred comments each there were only 25 in September.

So as I say, disappointing. I can think of a few things I did differently in September. The most obvious is that I didn’t have the time I needed to go around to other mathematics blogs and pay visits. I can’t fault people not coming around to me when I don’t come around to them. And I can admit that September didn’t have the richest diversity of postings. A lot of it was Reading the Comics posts, which are fun but I admit also prone to sameness. On the other hand, those are the most popular posts too. I haven’t found a new project that engages my imagination the way the A To Z did, although I think the Set Tour has promise.

I would also put some blame on WordPress’s Publicize, which they keep making worse and worse. See, Publicize announces new posts to Twitter and whatever other social media networks you have linked to it. And in the old days of, like, May, it just worked. By default it posted the name of the article and a wp.me shortened link. If you wanted to customize this you could hit an ‘edit’ button and the article name and wp.me shortened link were there at the start, and it was easy to add a short sentence to tell people what’s happening.

But in June they stopped with the wp.me shortened links; instead it shows as much of the full URL as fits in the Twitter 140-character limit after whatever text you enter. And last month they made it worse. It’ll give the article name as a suggested default publicity post, but you have to copy-paste or retype the name to get even that. The message WordPress is sending is, clearly, ‘stop using Publicize’, although what they have as a substitue is unclear.

I suspect what they mean for us to do is use the new modernized article-entry page. The trouble is, the page is awful. It might be salvageable, or something I could get used to, in time. But it’s also this very watery and Ajax-dependent thing that assumes you have fast, reliable Internet. And I don’t. I have AT&T, which has no interest in providing high-speed Internet to my neighborhood and possibly my city. They aren’t even willing to pretend they mean to bring it in anytime soon. We’d dump them happily but the only alternative right now is Kabletown and goodness knows that’s a recipe for disaster. I suspect AT&T and Kabletown have decided not to compete for the Lansing, Michigan, market and we’re stuck between awful we know and awful we know we’d flee to.

Anyway, my suspicion is that the equivalent of Publicize for the newfangled WordPress add-a-post page works better. But that is blaming WordPress for my own laziness; there’s no reason I couldn’t put in the post and wp.me link and a clear #math tag so people know what they’re getting into. It just seemed like too much work. I suppose for a week or two I should try changing just that and see if there’s an appreciable difference.

I’m sorry to turn all this into a round of crankiness, especially when I can think of easy things I should be doing to get better results. I’m just sulking. It’ll pass.

• #### scifihammy 3:31 pm on Friday, 2 October, 2015 Permalink | Reply

haha I like that – “Extrapolating should see a negative number of likes!”
I have (almost) given up on WP stats. My unique visitors and views are down, but I have the same number of likes per post, so someone is reading it, even if it no longer registers.
Last year, December and January peaked – Holiday bloggers I guess – so if you hang on, your stats ought to increase again :)
As someone who is not on fb or twitter and who persists in using the Old Dashboard to write posts, I can’t really comment on your other problems.
I just think, post what you feel like and enjoy the whole blogging experience, and let the numbers fall where they may! :)

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• #### Joseph Nebus 9:40 pm on Friday, 2 October, 2015 Permalink | Reply

I did see a marked rise in readership the last couple months of 2014, although that doesn’t seem to have been the pattern in 2013. Well, maybe Holiday Bloggers will come to my rescue after all.

I am writing primarily for the pleasure of it, and secondarily to be a better writer. But I have to admit it would be a good bit more pleasurable if I could believe that a great number of people were looking forward to my essays, and were talking about ideas they were inspired to have from them.

Liked by 1 person

• #### scifihammy 5:24 am on Saturday, 3 October, 2015 Permalink | Reply

Yes I agree. It is great to get feedback and feel you’ve made someone think a bit about your topic. However, I often find posts I’ve spent weeks perfecting, with added mines of info and links to click, pass by hardly noticed, whereas the photo of a flower I’ve rustled up because pushed for time, with the amazing caption of “Here is a flower!” gets many likes and comments!
Oh well, the vagaries of the blogging world I guess! :)

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• #### Joseph Nebus 5:38 am on Monday, 5 October, 2015 Permalink | Reply

That’s always been the most baffling thing, the utter disconnect between the amount of time spent on an essay, the amount you feel an essay represents your best work, and how much people like the essay. My big type case is over on my humor blog where I try to do a major, 700-word piece every Friday. Some of them I really love. Occasionally they’re liked, but mostly, reading about Apartment 3-G or else the little Statistics Saturday or Caption This one-shot jokes that take three sentences are the most popular. I understand the Apartment 3-G explanations being liked, since the comic strip has got really baffling, but otherwise …

Well, if I didn’t love the mysteries I wouldn’t be here still. I think.

Liked by 1 person

I’ve been having problems with the new publicize too. Sometimes my blog posts are posting as images rather than card tweets, which turns into a lot of Br Monday pics getting in the way. I had a similar drop off in August when I was too busy with the move and I’m just starting to get readership back.

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• #### Joseph Nebus 9:42 pm on Friday, 2 October, 2015 Permalink | Reply

I have wondered what’s been going on with Publicize. I haven’t seen any explanations for what the changes are supposed to do or how they’re supposed to be better. I don’t believe they are, but usually there’s at least a press release announcing vague reasons we should think they’re better.

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• #### ivasallay 6:08 pm on Friday, 2 October, 2015 Permalink | Reply

I look forward to being a part of the group that likes your posts a negative number of times in November.
How do we make that happen?

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• #### Joseph Nebus 9:44 pm on Friday, 2 October, 2015 Permalink | Reply

I’m looking forward to the spectacle myself! I don’t know how it’s going to happen but, what the heck, the discovery will be the thrilling thing.

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• #### Michelle H 3:04 pm on Saturday, 3 October, 2015 Permalink | Reply

September is a rough month for most people, a busy time of year for kids in school, many types of businesses also have more work in the last quarter… overall, less time for personal interests.

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• #### Joseph Nebus 5:39 am on Monday, 5 October, 2015 Permalink | Reply

This is true. More, it’s a transitional sort of month that spoils people’s patterns. That can be a chance to pick up people who’re looking for new things to read, although I suspect the average person has nearly as much as they can handle as it is and really needs chances to drop stuff.

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• #### Michelle H 12:28 pm on Monday, 5 October, 2015 Permalink | Reply

I realized, too, that I’ve read quite a few of your posts in part, but struggle to get back to finish. Your series on sets I need to point out to a friend who asked about set theory, as it has intersected some of his work in another field.

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## How Gibbs derived the Phase Rule

I knew I’d been forgetting something about the end of summer. I’m embarrassed again it was Peter Mander’s Carnot Cycle blog resuming its discussions of thermodynamics.

The September article is about Gibbs’s phase rule. Gibbs here is Josiah Willard Gibbs, who established much of the mathematical vocabulary of thermodynamics. The phase rule here talks about the change of a substance from one phase to another. The classic example is water changing from liquid to solid, or solid to gas, or gas to liquid. Everything does that for some combinations of pressure and temperature and available volume. It’s just a good example because we can see those phase transitions happen whenever we want.

The question that feels natural to mathematicians, and physicists, is about degrees of freedom. Suppose that we’re able take a substance and change its temperature or its volume or its pressure. How many of those things can we change at once without making the material different? And the phase rule is a way to calculate that. It’s not always the same number because at some combinations of pressure and temperature and volume the substance can be equally well either liquid or gas, or be gas or solid, or be solid or liquid. These represent phase transitions, melting or solidifying or evaporating. There’s even one combination — the triple point — where the material can be solid, liquid, or gas simultaneously.

Carnot Cycle presents the way that Gibbs originally derived his phase rule. And it’s remarkably neat and clean and accessible. The meat of it is really a matter of counting, keeping track of how much information we have and how much we want and looking at the difference between the things. I recommend reading it even if you are somehow not familiar with differential forms. Simply trust that a “d” followed by some other letter (or a letter with a subscript) is some quantity whose value we might be interested in, and you should follow the reasoning well.

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Originally posted on carnotcycle:

The Phase Rule formula was first stated by the American mathematical physicist Josiah Willard Gibbs in his monumental masterwork On the Equilibrium of Heterogeneous Substances (1875-1878), in which he almost single-handedly laid the theoretical foundations of chemical thermodynamics.

In a paragraph under the heading “On Coexistent Phases of Matter”, Gibbs gives the derivation of his famous formula in just 77 words. Of all the many Phase Rule proofs in the thermodynamic literature, it is one of the simplest and shortest. And yet textbooks of physical science have consistently overlooked it in favor of more complicated, less lucid derivations.

To redress this long-standing discourtesy to Gibbs, CarnotCycle here presents Gibbs’ original derivation of the Phase Rule in an up-to-date form. His purely prose description has been supplemented with clarifying mathematical content, and the outmoded symbols used in the single equation to which he refers have been replaced with their…

View original 863 more words

## Making Lots Of Change

John D Cook’s Algebra Fact of the Day points to a pair of algorithms about making change. Specifically these are about how many ways there are to provide a certain amount of change using United States coins. By that he, and the algorithms, mean 1, 5, 10, 25, and 50 cent pieces. I’m not sure if 50 cent coins really count, since they don’t circulate any more than dollar coins do. Anyway, if you want to include or rule out particular coins it’s clear enough how to adapt things.

What surprised me was a simple algorithm, taken from Ronald L Graham, Donald E Knuth, and Oren Patashnik’s Concrete Mathematics: A Foundation For Computer Science to count the number of ways to make a certain amount of change. You start with the power series that’s equivalent to this fraction:

$\frac{1}{\left(1 - z\right)\cdot\left(1 - z^{5}\right)\cdot\left(1 - z^{10}\right)\cdot\left(1 - z^{25}\right)\cdot\left(1 - z^{50}\right)}$

A power series is a polynomial. The power series for $\frac{1}{1 - z}$, for example, is $1 + z + z^2 + z^3 + z^4 + \cdots ...$ and carries on forever like that. But if you choose a number between minus one and positive one, and put that in for z in either $\frac{1}{1 - z}$ or in that series $1 + z + z^2 + z^3 + z^4 + \cdots ...$ you’ll get the same number. (If z is not between minus one and positive one, it doesn’t. Don’t worry about it. For what we’re doing we will never need any z.)

The power series for that big fraction with all the kinds of change in it is more tedious to work out. You’d need the power series for $\frac{1}{1 - z}$ and $\frac{1}{1 - z^5}$ and $\frac{1}{1 - z^{10}}$ and so on, and to multiply all those series together. And yes, that’s multiplying infinitely long polynomials together, which you might reasonably expect will take some time.

You don’t need to, though. All you really want is a single term in this series. To tell how many ways there are to make n cents of change, look at the coefficient, the number, in front of the zn term. That’s the number of ways. So while this may be a lot of work, it’s not going to be hard work, and it’s going to be finite. You only have to work out the products that give you a zn power. That will take planning and preparation to do correctly, but that’s all.

• #### mathcination 6:47 am on Tuesday, 29 September, 2015 Permalink | Reply

Another cool thing about this is that you can get $z^n$ term by computing the complex contour integral:
http://math.stackexchange.com/questions/338228/what-is-the-significance-of-this-theorem-coefficients-of-power-series-as-integr
And if you are not willing to do the analytic computation, you can let the computer do a Riemann sum approximation and thus get the coefficient you are looking for. As we already know that the result is an integer, an error less than .5 is sufficient.

Liked by 1 person

• #### Joseph Nebus 9:59 pm on Tuesday, 29 September, 2015 Permalink | Reply

Oh, that’s quite a good thought. And as we do know the result has to be an integer (for that matter, that it has to be a counting number) that does make numerical integration almost uniquely practical for this approach.

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## Reading the Comics, September 24, 2015: Yes, I Do So Edition

Yes, in this roundup of mathematically-themed comic strips I talk seriously about the educational techniques of the fictional Great Smokey Mountains community where the comic strip Barney Google and Snuffy Smith takes place. I accept the implications of this.

John Rose’s Barney Google And Snuffy Smith for the 23rd of September is your standard snarky-response joke. I’m a bit surprised to see that at whatever class level Jughaid’s in they’re using “x” to stand in for the not-yet-known number. I thought empty boxes or question marks were more common. But I also think Miz Prunelly’s not working most effectively by getting angry at Jughaid for not knowing what x is.

John Rose’s Barney Google And Snuffy Smith for the 23rd of September, 2015.

I would suggest trying this: can Jughaid find some possible values of x that are definitely too small? And some possible values that are certainly too big? Then what kinds of numbers are both not-too-small and not-too-big? One standard mathematician’s trick for finding an unknown quantity is to show that it can’t be smaller than some number, giving us a lower bound. And then show it can’t be larger than some number, giving us an upper bound. If the lower bound and the upper bound are the same number, we’re done. If they’re not the same number we might have to go looking, but at least we’ve got a better idea what a correct answer should look like. If the lower bound is a larger number than the upper bound, we have to go back and check whether there actually is an answer, or if we started off in the wrong direction.

Scott Adams’s Dilbert Classics for the 23rd of September (a rerun from the 30th of July, 1992) mentions “conversational geometry”. It’s built on a bit of geometry that somehow escaped into being a common allusion, and that occasionally riles up grammar nerds. The problem is trying to use “turned around 360 degrees” for “turned completely around”. 360 degrees is certainly turning something all the way around, but it leaves the thing back where it started, apparently unchanged. (Well, there are some oddball structures where you can rotate something 360 degrees and have it not back the way it started, but those only occur in abstract mathematical constructions and in some — not all! — subatomic particles. Yes, it’s weird. It’s like that.)

The grammar nerd will insist that what’s meant is to turn something 180 degrees, reversing its direction. Or maybe changed 90 degrees, looking perpendicular to whatever the original situation was. Personally I can’t get upset about a shorthand English phrase not making literal sense, because there are only about six shorthand English phrases that make even the slightest literal sense, and four of those are tapas orders. Eventually you have to stop with the rage and just say something already. And rotating 360 degrees is a different process from rotating not at all. You move, you break your focus, you break your attention. Even if you face the same things again you face them having refreshed your perceptions. You might now see something you had not before.

John Zakour and Scott Roberts’s Maria’s Day for the 23rd of September asserts that mathematics is important so that one can check one’s accountants. This is true, although it’s hardly everything mathematics is enjoyable for. And while I don’t often get to call attention to comic strip artwork, do look at the different papers; there’s some fun there.

Pab Sungenis’s New Adventures of Queen Victoria for the 24th of September — and the days around it — have seen Victoria and Nikola Tesla facing the end result of too much holiday creep: a holiday singularity. By a singularity a mathematician means a point where stuff gets weird: where a function isn’t defined, where a surface breaks off, where several independent solutions suddenly stop being independent, that sort of thing. It’ll often correspond with some measure becoming infinitely large (as a positive or a negative number), though I don’t think it’s safe to say that always happens.

We generally can’t say what’s happening at a singularity. But the existence of a singularity, and what it behaves like, can tell us something about what’s happening away from the singularity. It can happen, for example, that a singularity is removable. That is, if a function is undefined for some values, perhaps we can come up with a logically compelling definition for what it might do at those values. If you can remove a singularity then we call this a “removable singularity”. This serves to show you don’t necessarily need grad school to understand everything mathematicians are saying. Sometimes a singularity can’t be removed, and those are known as “nonremovable singularities” or “essential singularities” or sometimes some other nastier names.

Usually, if one has a singularity in a mathematical construct, then information about one side of the singularity isn’t enough to extrapolate what might be on the other side. This makes the literary use of a “singularity” as “something magical that does whatever the plot requires” justified enough. Tesla here is clearly using the idea of reaching an infinitely vast, or an infinitely dense, holiday concentration as a singularity. I grant that would be singular enough. The strip does make me think of a fun sequence in Walt Kelly’s Pogo where one year the Bun Rabbit decided to get all the holiday-celebrating done first thing in the year, to clear out the rest. He went about banging the drum and listing every holiday ever, which is what made me aware of the New Jersey Big Sea Day.

Shaenon K Garrity and Jeffrey C Wells’s Skin Horse for the 24th of September includes a sequence identified as the “Catalan Series”. I’d have said “sequence” myself. The Catalan sequence describes (among other things) how many ways you can break down a regular polygon into a particular number of triangles. A square can be broken down into two triangles just two ways (if orientation counts, which for this problem, it does). A pentagon can be broken down into three triangles in five ways. A hexagon can be broken down into four triangles in fourteen ways, and so on. (The key is you break the polygon into a number of triangles that’s two less than the number of sides. So if you had a 9-sided polygon, you’d break it up into 7 triangles. If you had a 20-sided polygon, you’d break it up into 18 triangles.) The sequence describes more stuff than that, but this is an easy-to-understand application. As the name of the sequence suggests, it comes to us from the Belgian-French mathematician Eugène Charles Catalan (1814 – 1894).

Catalan’s name also might be faintly familiar for a conjecture he posed in 1844, which was finally proven true in 2002 by Preda Mihăilescu. His conjecture is based on observing that the number 2 raised to the third power is 8, while the number 3 raised to the second power is 9, quite close together. Catalan conjectured this was the only case of consecutive powers. That is, there’s nothing like 15 to the twentieth power being one less than 12 to the twenty-fourth power or anything like that. I’m afraid I don’t know enough of this kind of mathematics, known as number theory, to say whether that’s of use for anything more than settling curiosity on the point.

## The Set Tour, Stage 2: The Real Star

For the second of my little tour of sets that get commonly used as domains and ranges I want to name the most common of them all.

## R

This is the real numbers. In text that’s written with a bold R. Written by hand, and often in text, that’s written with a capital R that has a double stroke for the main vertical line. That’s an easy-to-write way to distinguish it from a plain old civilian R. The double-vertical-stroke convention is used for many of the most common sets of numbers. It will get used for letters like I and J (the integers), or N (the counting numbers). A vertical stroke will even get added to symbols that technically don’t have any vertical strokes, like Q (the rational numbers). There it’s just put inside the loop, on the left side, far enough from the edge that the reader can notice the vertical stroke is there.

R is a big one. It’s not just a big set. It’s also a popular one. It may as well be the default domain and range. If someone fails to tell you what either set is, you can suppose she meant R and be only rarely wrong. The real numbers are familiar and popular and it feels like we know what they are. It’s a bit tricky to define them exactly, though, and you’ll notice that I’m not doing that. You know what I mean, though. It’s whole numbers, and rational numbers, and irrational numbers like the square root of pi, and for that matter pi, and a whole bunch of other boring numbers nobody looks at. Let’s leave it at that.

All the intervals I talked about last time are subsets of R. If we really wanted to, we could turn a function with domain an interval like [0, 1] into a function with a domain of R. That’s a kind of “embedding”. Let me call the function with domain [0, 1] by the name “f”. I’ll then define g, on the domain R, by the rule “whatever f(x) is, if x is from 0 to 1; and some other, harmless value, if x isn’t”. Probably the harmless value is zero. Sometimes we need to change the domain a function’s defined on, and this is a way to do it.

If we only want to talk about the positive real numbers we can denote that by putting a plus sign in superscript: R+. If we only want the negative numbers we put in a minus sign: R. Do either of these include zero? My heart tells me neither should, but I wouldn’t be surprised if in practice either did, because zero is often useful to have around. To be careful we might explicitly include zero, using the notations of set theory. Then we might write $\textbf{R}^+ \cup \left\{0\right\}$.

Sometimes the rule for a function doesn’t make sense for some values. For example, if a function has the rule $f: x \mapsto 1 / (x - 1)$ then you can’t work out a value for f(1). That would require dividing by zero and we dare not do that. A careful mathematician would say the domain of that function f is all the real numbers R except for the number 1. This exclusion gets written as “R \ {1}”. The backslash means “except the numbers in the following set”. It might be a single number, such as in this example. It might be a lot of numbers. The function $g: x \mapsto \log\left(1 - x\right)$ is meaningless for any x that’s equal to or greater than 1. We could write its domain then as “R \ { x: x ≥ 1 }”.

That’s if we’re being careful. If we get a little careless, or if we’re writing casually, or if the set of non-permitted points is complicated we might omit that. Mathematical writing includes an assumption of good faith. The author is supposed to be trying to say something interesting and true. The reader is expected to be skeptical but not quarrelsome. Spotting a flaw in the argument because the domain doesn’t explicitly rule out some points it shouldn’t have is tedious. Finding that the interesting thing only holds true for values that are implicitly outside the domain is serious.

The set of real numbers is a group; it has an operation that works like addition. We call it addition. For that matter, it’s a ring. It has an operation that works like multiplication. We call it multiplication. And it’s even more than a ring. Everything in R except for the additive identity — 0, the number you can add to anything without changing what the thing is — has a multiplicative inverse. That is, any number except zero has some number you can multiply it by to get 1. This property makes it a “field”, to people who study (abstract) algebra. This “field” hasn’t got anything to do with gravitational or electrical or baseball or magnetic fields. But the overlap in names does serve to sometimes confuse people.

But having this multiplicative inverse means that we can do something that operates like division. Divide one thing by a second by taking the first thing and multiplying it by the second thing’s multiplicative inverse. We call this division-like operation “division”.

It’s not coincidence that the algebraic “addition” and “multiplication” and “division” operations are the ones we call addition and multiplication and division. What makes abstract algebra abstract is that it’s the study of things that work kind of like the real numbers do. The operations we can do on the real numbers inspire us to look for other sets that can let us do similar things.

## Reading the Comics, September 22, 2015: Rock Star Edition

The good news is I’ve got a couple of comic strips I feel responsible including the pictures for. (While I’m confident I could include all the comics I talk about as fair use — I make comments which expand on the strips’ content and which don’t make sense without the original — Gocomics.com links seem reasonably stable and likely to be there in the future. Comics Kingdom links generally expire after a month except to subscribers and I don’t know how long Creators.com links last.) And a couple of them talk about rock bands, so, that’s why I picked that titel.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 17th of September is a subverted-fairy-tale-moral strip, naturally enough. It’s also a legitimate point, though. Unlikely events do happen sometimes, and it’s a mistake to draw too-strong conclusions from them. This is why it’s important to reproduce interesting results. It’s also why, generally, we like larger sample sizes. It’s not likely that twenty fair coins flipped will all come up tails at once. But it’s far more likely that will happen than that two hundred fair coins flipped will all come up tails. And that’s far more likely than that two thousand fair coins will. For that matter, it’s more likely that three-quarters of twenty fair coins flipped will come up tails than that three-quarters of two hundred fair coins will. And the chance that three-quarters of two thousand fair coins will come up tails is ignorable. If that happens, then something interesting has been found.

In Juba’s Viivi and Wagner for the 17th of September, Wagner announces his decision to be a wandering mathematician. I applaud his ambition. If I had any idea where to find someone who needed mathematics done I’d be doing that myself. If you hear something give me a call. I’ll be down at the City Market, in front of my love’s guitar case, multiplying things by seven. I may get it wrong, but nobody will know how to correct me.

Daniel Beyer’s Long Story Short for the 18th of September, 2015. I never actually heard of Tool before this comic.

Daniel Beyer’s Long Story Short for the 18th of September uses a page full of calculations to predict when prog-rock band Tool will release their next album. (Wikipedia indicates they’re hoping for sometime before the end of 2015, but they’ve been working on it since 2008.) Some of the symbols make a bit of sense as resembling those of quantum physics. An expression like (in the lower left of the board) $\langle \psi_1 u_1 | {H}_{\gamma} | \psi_1 \rangle$ resembles a probability distribution calculation. (There should be a ^ above the H there, but that’s a little beyond what WordPress can render in the simple mathematical LaTeX tools it has available. It’s in the panel, though.) The letter ψ stands for a probability wave, containing somehow all the information about a system. The composition of symbols literally means to calculate how an operator — a function that has a domain of functions and a range of functions — changes that probability distribution. In quantum mechanics every interesting physical property has a matching operator, and calculating this set of symbols tells us the distribution of whatever that property is. H generally suggests the total energy of the system, so the implication is this measures, somehow, what energies are more and are less probable. I’d be interested to know if Beyer took the symbols from a textbook or paper and what the original context was.

Dave Whamond’s Reality Check for the 19th of September brings in another band to this review. It uses a more basic level of mathematics, though.

Percy Crosby’s Skippy from the 19th of September — rerun from sometime in 1928 — is a clever way to get a word problem calculated. It also shows off what’s probably been the most important use of arithmetic, which is keeping track of money. Accountants and shopkeepers get little attention in histories of mathematics, but a lot of what we do has been shaped by their needs for speed, efficiency, and accuracy. And one of Gocomics’s commenters pointed out that the shopkeeper didn’t give the right answer. Possibly the shopkeeper suspected what was up.

Paul Trap’s Thatababy for the 20th of September uses a basic geometry fact as an example of being “very educated”. I don’t think the area of the circle rises to the level of “very” — the word means “truly”, after all — but I would include it as part of the general all-around awareness of the world people should have. Also it fits in the truly confined space available. I like the dad’s eyes in the concluding panel. Also, there’s people who put eggplant on pizza? Really? Also, bacon? Really?

Alex Hallatt’s Arctic Circle for the 21st of September, 2015.

Alex Hallatt’s Arctic Circle for the 21st of September is about making your own luck. I find it interesting in that it rationalizes magic as a thing which manipulates probability. As ways to explain magic for stories go that isn’t a bad one. We can at least imagine the rigging of card decks and weighting of dice. And its plot happens in the real world, too: people faking things — deceptive experimental results, rigged gambling devices, financial fraud — can often be found because the available results are too improbable. For example, a property called Benford’s Law tells us that in many kinds of data the first digit is more likely to be a 1 than a 2, a 2 than a 3, a 3 than a 4, et cetera. This fact serves to uncover fraud surprisingly often: people will try to steal money close to but not at some limit, like the \$10,000 (United States) limit before money transactions get reported to the federal government. But that means they work with checks worth nine thousand and something dollars much more often than they do checks worth one thousand and something dollars, which is suspicious. Randomness can be a tool for honesty.

Peter Maresca’s Origins of the Sunday Comics feature for the 21st of September ran a Rube Goldberg comic strip from the 19th of November, 1913. That strip, Mike and Ike, precedes its surprisingly grim storyline with a kids-resisting-the-word-problem joke. The joke interests me because it shows a century-old example of the joke about word problems being strings of non sequiturs stuffed with unpleasant numbers. I enjoyed Mike and Ike’s answer, and the subversion of even that answer.

Mark Anderson’s Andertoons for the 22nd of September tries to optimize its targeting toward me by being an anthropomorphized-mathematical-objects joke and a Venn diagram joke. Also being Mark Anderson’s Andertoons today. If I didn’t identify this as my favorite strip of this set Anderson would just come back with this, but featuring monkeys at typewriters too.

## Reading the Comics, September 16, 2015: Celebrity Appearance Edition

I couldn’t go on calling this Back To School Editions. A couple of the comic strips the past week have given me reason to mention people famous in mathematics or physics circles, and one who’s even famous in the real world too. That’ll do for a title.

Jeff Corriveau’s Deflocked for the 15th of September tells what I want to call an old joke about geese formations. The thing is that I’m not sure it is an old joke. At least I can’t think of it being done much. It seems like it should have been.

The formations that geese, or other birds, form has been a neat corner of mathematics. The question they inspire is “how do birds know what to do?” How can they form complicated groupings and, more, change their flight patterns at a moment’s notice? (Geese flying in V shapes don’t need to do that, but other flocking birds will.) One surprising answer is that if each bird is just trying to follow a couple of simple rules, then if you have enough birds, the group will do amazingly complex things. This is good for people who want to say how complex things come about. It suggests you don’t need very much to have robust and flexible systems. It’s also bad for people who want to say how complex things come about. It suggests that many things that would be interesting can’t be studied in simpler models. Use a smaller number of birds or fewer rules or such and the interesting behavior doesn’t appear.

Jeff Corriveau’s Deflocked for the 15th of September, 2015.

Scott Adams’s Dilbert Classics from the 15th and 16th of September (originally run the 22nd and 23rd of July, 1992) are about mathematical forecasts of the future. This is a hard field. It’s one people have been dreaming of doing for a long while. J Willard Gibbs, the renowned 19th century physicist who put the mathematics of thermodynamics in essentially its modern form, pondered whether a thermodynamics of history could be made. But attempts at making such predictions top out at demographic or rough economic forecasts, and for obvious reason.

The next day Dilbert’s garbageman, the smartest person in the world, asserts the problem is chaos theory, that “any complex iterative model is no better than a wild guess”. I wouldn’t put it that way, although I’m not sure what would convey the idea within the space available. One problem with predicting complicated systems, even if they are deterministic, is that there is a difference between what we can measure a system to be and what the system actually is. And for some systems that slight error will be magnified quickly to the point that a prediction based on our measurement is useless. (Fortunately this seems to affect only interesting systems, so we can still do things like study physics in high school usefully.)

Maria Scrivan’s Half Full for the 16th of September makes the Common Core joke. A generation ago this was a New Math joke. It’s got me curious about the history of attempts to reform mathematics teaching, and how poorly they get received. Surely someone’s written a popular or at least semipopular book about the process? I need some friends in the anthropology or sociology departments to tell, I suppose.

In Mark Tatulli’s Heart of the City for the 16th of September, Heart is already feeling lost in mathematics. She’s in enough trouble she doesn’t recognize mathematics terms. That is an old joke, too, although I think the best version of it was done in a Bloom County with no mathematical content. (Milo Bloom met his idol Betty Crocker and learned that she was a marketing icon who knew nothing of cooking. She didn’t even recognize “shish kebob” as a cooking term.)

Mell Lazarus’s Momma for the 16th of September sneers at the idea of predicting where specks of dust will land. But the motion of dust particles is interesting. What can be said about the way dust moves when the dust is being battered by air molecules that are moving as good as randomly? This becomes a problem in statistical mechanics, and one that depends on many things, including just how fast air particles move and how big molecules are. Now for the celebrity part of this story.

Albert Einstein published four papers in his “Annus mirabilis” year of 1905. One of them was the Special Theory of Relativity, and another the mass-energy equivalence. Those, and the General Theory of Relativity, are surely why he became and still is a familiar name to people. One of his others was on the photoelectric effect. It’s a cornerstone of quantum mechanics. If Einstein had done nothing in relativity he’d still be renowned among physicists for that. The last paper, though, that was on Brownian motion, the movement of particles buffeted by random forces like this. And if he’d done nothing in relativity or quantum mechanics, he’d still probably be known in statistical mechanics circles for this work. Among other things this work gave the first good estimates for the size of atoms and molecules, and gave easily observable, macroscopic-scale evidence that molecules must exist. That took some work, though.

Dave Whamond’s Reality Check for the 16th of September shows off the Metropolitan Museum of Symmetry. This is probably meant to be an art museum. Symmetries are studied in mathematics too, though. Many symmetries, the ways you can swap shapes around, form interesting groups or rings. And in mathematical physics, symmetries give us useful information about the behavior of systems. That’s enough for me to claim this comic is mathematically linked.

## The Set Tour, Stage 1: Intervals

I keep writing about functions. I’m not exactly sure how. I keep meaning to get to other things but find interesting stuff to say about domains and ranges and the like. These domains and ranges have to be sets. There are some sets that come up all the time in domains and ranges. I thought I’d share some of the common ones. The first family of sets is known as “intervals”.

## [ 0, 1 ]

This means all the real numbers from 0 to 1. Written with straight brackets like that means to include the matching point there — that is, 0 is in the domain, and so is 1. We don’t always want to include the ending points in a domain; if we want to omit them, we write parentheses instead. So (0, 1) would mean all the real numbers bigger than zero and smaller than one, but neither zero nor one. We can also include one but not both endpoints: [0, 1) is fine. It offends copy editors, by having its open bracket and its closed parenthesis be unmatched, but its meaning is clear enough. It’s the real numbers from zero to one, with zero allowed but one ruled out. We can include or omit either or both endpoints, and we have to keep that straight. But for most of our work it doesn’t matter what we choose, as long as we stay consistent. It changes proofs a bit, but in routine ways.

Zero to one is a popular interval. Negative 1 to 1 is another popular interval. They’re nice round numbers. And the intervals between -π and π, or between 0 and 2π, are also popular. Those match nicely with trigonometric functions such as sine and tangent. You can make an interval that runs from any number to any other number — [ a, b ] if you don’t want to pin down just what numbers you mean. But [ 0, 1 ] and [ -1, 1 ] are popular choices among mathematicians. If you can prove something interesting about a function with a domain that’s either of these intervals, you can then normally prove it’s true on whatever a function with a domain that’s whatever interval you want. (I can’t think of an exception offhand, but mathematics is vast and my statement sweeping. There may be trouble.)

Suppose we start out with a function named f that has its domain the interval [ -16, 48 ]. I’m not saying anything about its range or its rule because they don’t matter. You can make them anything you like, if you need them to be anything. (This is exactly the way we might, in high school algebra, talk about the number named `x’ without ever caring whether we find out what number that actually is.) But from this start we can talk about a related function named g, which has as its domain [ -1, 1 ]. A rule for g is $g: x \mapsto f\left(32\cdot x + 16\right)$; that is, g(0) is whatever number you get from f(16), for example. So if we can prove something’s true about g on this standard domain, we can almost certainly show the same thing is true about f on the other domain. This is considered a kind of mapping, or as a composition of functions. It’s a composition because you can see it as taking your original number called x, seeing what one function does with it — in this case, multiplying it by 32 and adding 16 — and then seeing what a second function — the one named f — does with that.

It’s easy to go the other way around. If we know what g is on the domain [ -1, 1 ] then we can define a related function f on the domain [ -16, 48 ]. We can say $f: x \mapsto g\left(\frac{1}{32}\cdot\left(x - 16\right)\right)$. And again, if we can show something’s true on this standard domain, we can almost certainly show the same thing’s true on the other domain.

I gave examples, mappings, that are simple. They’re linear. They don’t have to be. We just have to match everything in one interval to everything in another. For example, we can match the domain (1, ∞) — all the numbers bigger than 1 — to the domain (0, 1). Let’s again call f the function with domain (1, ∞). Then we can say g is the function with domain (0, 1) and defined by the rule $g: x \mapsto f\left(\frac{1}{x}\right)$. That’s a nonlinear mapping.

Linear mappings are easier to deal with than nonlinear mappings. Usually, mathematically, if something is divided into “linear” and “nonlinear” the linear version is easier. Sometimes a nonlinear mapping is the best one to use to match a function on some convenient domain to a function on some other one. The hard part is often a matter of showing that something true for one function on a common domain like (0, 1) will also be true for the other domain, (a, b).

However, showing that the truth holds can often be done without knowing much about your specific function. You maybe need to know what kind of function it is, that it’s continuous or bounded or something like that. But the actual specific rule? Not so important. You can prove that the truth holds ahead of time. Or find an analysis textbook or paper or something where someone else has proven that. So while a domain might be any interval, often in practice you don’t need to work with more than a couple nice familiar ones.

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