Slow week around here for mathematically-themed comic strips. These happen. I suspect Comic Strip Master Command is warning me to stop doing two-a-week essays on reacting to comic strips and get back to more original content. Message received. If I can get ahead of some projects Monday and Tuesday we’ll get more going.
Patrick Roberts’s Todd the Dinosaur for the 20th is a typical example of mathematics being something one gets in over one’s head about. Of course it’s fractions. Is there anything in elementary school that’s a clearer example of something with strange-looking rules and processes for some purpose students don’t even know what they are? In middle school and high school we get algebra. In high school there’s trigonometry. In high school and college there’s calculus. In grad school there’s grad school. There’s always something.
Jeff Stahler’s Moderately Confused for the 21st is the usual bad-mathematics-of-politicians joke. It may be a little more on point considering the Future Disgraced Former President it names, but the joke is surely as old as politicians and hits all politicians with the same flimsiness.
John Graziano’s Ripley’s Believe It Or Not for the 22nd names Greek mathematician Pythagoras. That’s close enough to on-point to include here, especially considering what a slow week it’s been. It may not be fair to call Pythagoras a mathematician. My understanding is we don’t know that actually did anything in mathematics, significant or otherwise. His cult attributed any of its individuals’ discoveries to him, and may have busied themselves finding other, unrelated work to credit to their founder. But there’s so much rumor and gossip about Pythagoras that it’s probably not fair to automatically dismiss any claim about him. The beans thing I don’t know about. I would be skeptical of anyone who said they were completely sure.
Vic Lee’s Pardon My Planet for the 23rd is the usual sort of not-understanding-mathematics joke. In this case it’s about percentages, which are good for baffling people who otherwise have a fair grasp on fractions. I wonder if people would be better at percentages if they learned to say “percent” as “out of a hundred” instead. I’m sure everyone who teaches percentages teaches that meaning, but that doesn’t mean the warning communicates.
Samson’s Dark Side Of The Horse for the 25th mentions sudokus, and that’s enough for a slow week like this. I thought Horace was reaching for a calculator in the last panel myself, and was going to say that wouldn’t help any. But then I checked the numbers in the boxes and that made it all better.
Greg Evans’s Luann Againn for the 28th of February — reprinting the strip from the same day in 1989 — uses a bit of arithmetic as generic homework. It’s an interesting change of pace that the mathematics homework is what keeps one from sleep. I don’t blame Luann or Puddles for not being very interested in this, though. Those sorts of complicated-fraction-manipulation problems, at least when I was in middle school, were always slogs of shuffling stuff around. They rarely got to anything we’d like to know.
Jef Mallett’s Frazz for the 1st of March is one of those little revelations that statistics can give one. Myself, I was always haunted by the line in Carl Sagan’s Cosmos about how, in the future, with the Sun ageing and (presumably) swelling in size and heat, the Earth would see one last perfect day. That there would most likely be quite fine days after that didn’t matter, and that different people might disagree on what made a day perfect didn’t matter. Setting out the idea of a “perfect day” and realizing there would someday be a last gave me chills. It still does.
Richard Thompson’s Poor Richard’s Almanac for the 1st and the 2nd of March have appeared here before. But I like the strip so I’ll reuse them too. They’re from the strip’s guide to types of Christmas trees. The Cubist Fur is described as “so asymmetrical it no longer inhabits Euclidean space”. Properly neither do we, but we can’t tell by eye the difference between our space and a Euclidean space. “Non-Euclidean” has picked up connotations of being so bizarre or even horrifying that we can’t hope to understand it. In practice, it means we have to go a little slower and think about, like, what would it look like if we drew a triangle on a ball instead of a sheet of paper. The Platonic Fir, in the 2nd of March strip, looks like a geometry diagram and I doubt that’s coincidental. It’s very hard to avoid thoughts of Platonic Ideals when one does any mathematics with a diagram. We know our drawings aren’t very good triangles or squares or circles especially. And three-dimensional shapes are worse, as see every ellipsoid ever done on a chalkboard. But we know what we mean by them. And then we can get into a good argument about what we mean by saying “this mathematical construct exists”.
Mark Litzler’s Joe Vanilla for the 3rd uses a chalkboard full of mathematics to represent the deep thinking behind a silly little thing. I can’t make any of the symbols out to mean anything specific, but I do like the way it looks. It’s quite well-done in looking like the shorthand that, especially, physicists would use while roughing out a problem. That there are subscripts with forms like “12” and “22” with a bar over them reinforces that. I would, knowing nothing else, expect this to represent some interaction between particles 1 and 2, and 2 with itself, and that the bar means some kind of complement. This doesn’t mean much to me, but with luck, it means enough to the scientist working it out that it could be turned into a coherent paper.
Bill Holbrook’s On The Fastrack is this week about the wedding of the accounting-minded Fi. And she’s having last-minute doubts, which is why the strip of the 3rd brings in irrational and anthropomorphized numerals. π gets called in to serve as emblematic of the irrational numbers. Can’t fault that. I think the only more famously irrational number is the square root of two, and π anthropomorphizes more easily. Well, you can draw an established character’s face onto π. The square root of 2 is, necessarily, at least two disconnected symbols and you don’t want to raise distracting questions about whether the root sign or the 2 gets the face.
That said, it’s a lot easier to prove that the square root of 2 is irrational. Even the Pythagoreans knew it, and a bright child can follow the proof. A really bright child could create a proof of it. To prove that π is irrational is not at all easy; it took mathematicians until the 19th century. And the best proof I know of the fact does it by a roundabout method. We prove that if a number (other than zero) is rational then the tangent of that number must be irrational, and vice-versa. And the tangent of π/4 is 1, so therefore π/4 must be irrational, so therefore π must be irrational. I know you’ll all trust me on that argument, but I wouldn’t want to sell it to a bright child.
Holbrook continues the thread on the 4th, extends the anthropomorphic-mathematics-stuff to call people variables. There’s ways that this is fair. We use a variable for a number whose value we don’t know or don’t care about. A “random variable” is one that could take on any of a set of values. We don’t know which one it does, in any particular case. But we do know — or we can find out — how likely each of the possible values is. We can use this to understand the behavior of systems even if we never actually know what any one of it does. You see how I’m going to defend this metaphor, then, especially if we allow that what people are likely or unlikely to do will depend on context and evolve in time.
Comic Strip Master Command sent another normal-style week for mathematics references. There’s not much that lets me get really chatty or gossippy about mathematics lore. That’s all right. The important thing is: we’ve got Jumble back.
Greg Cravens’s The Buckets for the 25th features a bit of parental nonsense-telling. The rather annoying noise inside a car’s cabin when there’s one window open is the sort of thing fluid mechanics ought to be able to study. I see references claiming this noise to be a Helmholz Resonance. This is a kind of oscillation in the air that comes from wind blowing across the lone hole in a solid object. Wikipedia says it’s even the same phenomenon producing an ocean-roar in a seashell held up to the ear. It’s named for Hermann von Helmholtz, who described it while studying sound and vortices. Helmholz is also renowned for making a clear statement of the conservation of energy — an idea many were working towards, mind — and in thermodynamics and electromagnetism and for that matter how the eye works. Also how fast nerves transmit signals. All that said, I’m not sure that all the unpleasant sound heard and pressure felt from a single opened car window is Helmholz Resonance. Real stuff is complicated and the full story is always more complicated than that. I wouldn’t go farther than saying that Helmholz Resonance is one thing to look at.
Michael Cavna’s Warped for the 25th uses two mathematics-cliché equations as “amazingly successful formulas”. One can quibble with whether Einstein should be counted under mathematics. Pythagoras, at least for the famous theorem named for him, nobody would argue. John Grisham, I don’t know, the joke seems dated to me but we are talking about the comics.
Tony Carrillos’ F Minus for the 28th uses arithmetic as as something no reasonable person can claim is incorrect. I haven’t read the comments, but I am slightly curious whether someone says something snarky about Common Core mathematics — or even the New Math for crying out loud — before or after someone finds a base other than ten that makes the symbols correct.
Cory Thomas’s college-set soap-opera strip Watch Your Head for the 28th name-drops Introduction to Functional Analysis. It won’t surprise you it’s a class nobody would take on impulse. It’s an upper-level undergraduate or a grad-student course, something only mathematics majors would find interesting. But it is very interesting. It’s the reward students have for making it through Real Analysis, the spirit-crushing course about why calculus works. Functional Analysis is about what we can do with functions. We can make them work like numbers. We can define addition and multiplication, we can measure their size, we can create sequences of them. We can treat functions almost as if they were numbers. And while we’re working on things more abstract and more exotic than the ordinary numbers Real Analysis depends on, somehow, Functional Analysis is easier than Real Analysis. It’s a wonder.
Mark Anderson’s Andertoons for the 29th features a student getting worried about the order of arithmetic operations. I appreciate how kids get worried about the feelings of things like that. Although, truly, subtraction doesn’t go “last”; addition and subtraction have the same priority. They share the bottom of the pile, though. Multiplication and division similarly share a priority, above addition-and-subtraction. Many guides to the order of operations say to do addition-and-subtraction in order left to right, but that’s not so. Setting a left-to-right order is okay for deciding where to start. But you could do a string of additions or subtractions in any order and get the same answer, unless the expression is inconsistent.
Justin Boyd’s Invisible Bread for the 30th has maybe my favorite dumb joke of the week. It’s just a kite that’s proven its knowledge of mathematics. I’m a little surprised the kite didn’t call out a funnier number, by which I mean 37, but perhaps … no, that doesn’t work, actually. Of course the kite would be comfortable with higher mathematics.
So, that was a fairly successful month. For June this blog managed a record 1,051 pages viewed. That’s just above April’s high of 1,047, and is a nice rebound from May’s 936. I feel comfortable crediting this mostly to the number of articles I published in the month. Between the Mathematics A To Z and the rush of Reading The Comics posts, and a couple of reblogged or miscellaneous bits, June was my most prolific month: I had 28 articles. If I’d known how busy it was going to be I wouldn’t have skipped the first two Sundays. And i start the month at 25,871 total views.
It’s quite gratifying to get back above 1,000 for more than the obvious reasons. I’ve heard rumors — and I’m not sure where because most of my notes are on my not-yet-returned main computer — that WordPress somehow changed its statistics reporting so that mobile devices aren’t counted. That would explain a sudden drop in both my mathematics and humor blogs, and drops I heard reported from other readership-watching friends. It also implies many more readers out there, which is a happy thought.
Unfortunately because of my computer problems I can’t give reports on things like the number of visitors, or the views per visitor. I can get at WordPress’s old Dashboard statistics page, and that had been showing the number of unique visitors and views per visitor and all that. But on Firefox 3.6.16, and on Safari 5.0.6, this information isn’t displayed. I don’t know if they’ve removed it altogether from the Dashboard Statistics page in the hopes of driving people to their new, awful, statistics page or what. I also can’t find things like the number of likes, because that’s on the New Statistics page, which is inaccessible on browsers this old.
Worse, I can’t find the roster of countries that sent me viewers. I trust that when I get my main computer back, and can look at the horrible new statistics page, I’ll be able to fill that in, but for now — nothing. I’m sorry. I will provide these popular lists when I’m able.
I can say what the most popular posts were in June. As you might expect for a month dominated by the A-To-Z project, the five most popular posts were all Reading The Comics entries:
So last month amongst the talk about the radius of a circle inscribed in a Pythagorean right triangle I mentioned that I had, briefly, floated a conjecture that might have spun off it. It didn’t, though I promised to describe the chain of thought I had while exploring it, on the grounds that the process of coming up with mathematical ideas doesn’t get described much, and certainly doesn’t get described for the sorts of fiddling little things that make up a trifle like this.
The point from which I started was a question about the radius of a circle inscribed in the right triangle with legs of length 5, 12, and 13. This turns out to have a radius of 2, which is interesting because it’s a whole number. It turns out to be simple to show that for a Pythagorean right triangle, that is, a right triangle whose legs are a Pythagorean triple — like (3, 4, 5), or (5, 12, 13), any where the square of the biggest number is the same as what you get adding together the squares of the two smaller numbers — the inscribed circle has a radius that’s a whole number. For example, the circle you could inscribe in a triangle of sides 3, 4, and 5 would have radius 1. The circle inscribed in a triangle of sides 8, 15, and 17 would have radius 3; so does the circle inscribed in a triangle of sides 7, 24, and 25.
Since I now knew that (and in multiple ways: HowardAt58 had his own geometric solution, and you can also do this algebraically) I started to wonder about the converse. If a Pythagorean right triangle’s inscribed circle has a whole number for a radius, can does knowing a circle has a whole number for a radius tell us anything about the triangle it’s inscribed in? This is an easy way to build new conjectures: given that “if A is true, then B must be true”, can it also be that “if B is true, then A must be true”? Only rarely will that be so — it’s neat when it is — but we might be able to patch something up, like, “if B, C, and D are all simultaneously true, then A must be true”, or perhaps, “if B is true, then at least E must be true”, where E resembles A but maybe doesn’t make such a strong claim. Thus are tiny little advances in mathematics created.
For that “About An Inscribed Circle” problem I posted the other day: HowardAt58 worked out one way of proving what the radius of the circle that just fits within the 5-12-13 right triangle has to be, and in a pretty neat geometric fashion. Worth the read. I recommend following his steps along by hand, writing each out, but that reflects that I’m much more likely to follow mathematical reasoning if I write it out, even if I don’t do something past what the original author does. HowardAt58 also includes a little conjecture, inspired by playing around with a couple of Pythagorean triangles (playing around with a couple of examples is a great way to find conjectures), which I at least believe to be true.
Interestingly, his proof isn’t the same geometric proof that I’d realized we could do, so, I’m thinking to include that as another follow-up around here when I can make a couple diagrams that explain it.
Since my last roundup of mathematics-themed comic strips there’s been a modest drizzle of new ones, and I’m not sure that I can find any particular themes to them, except that Zach Weinersmith and the artistic collective behind Eric the Circle apparently like my attention. Well, what the heck; that’s easy enough to give.
Zach Weinersmith’s Saturday Morning Breakfast Cereal (September 29) hopes to be that guy who appears somewhere around the fourth comment of every news article ever that mentions a correlation being found between two quantities. A lot of what’s valuable about science is finding causal links between things, but it’s only in rare and, often, rather artificial circumstances that such links are easy to show. What’s more often necessary is showing that as one quantity changes so does another, which allows one to suspect a link. Then, typically, one would look for a plausible reason they might have anything to do with one another, and look for ways to experiment and prove whether there is or is not.
But just because there is a correlation doesn’t by itself mean that one thing necessarily has anything to do with another. They could be coincidence, for example, or they could be influenced by some other confounding factor. To be worth mention in a decent journal, a correlation is probably going to be strong enough that it’s hard to believe it’s just coincidence, but there could yet be some confounding factor. And even if there is a causal link, in the complicated mess that is reality it can be difficult to discern which way the link flows. This is summarized in deductive logic by saying that correlation does not imply causation, but that uses deductive logic’s definition of “imply”.
In deductive logic to say “this implies that” means it is impossible for “this” to be true and “that” false simultaneously. It is perfectly permissible for both “this” and “that” to be true, and permissible for “this” to be false and “that” false, and — this is the point where Intro to Logic students typically crash — permissible for “this” to be false and “that” true. Colloquially, though, “imply” has a different connotation, something more along the lines of “this” and “that” have to both be false or both be true together. Don’t make that mistake on your logic test.
When a logician says that correlation does not imply causation, she is saying that it is imaginable for the correlation to be true while the causation is false. She is not saying the causation is false; she is just saying that the case is not proved from the fact of a correlation being true. And that’s so; if we just knew two things were correlated we would have to experiment to find whether there is a causal link. But finding a correlation one of the ways to start finding casual links; it’d be obviously daft not to use them as the start of one’s search. Anyway, that guy in about the fourth comment of every news report about a correlation just wants you to know it’s very important he tell you he’s smarter than journalists.
Mikael Wulff and Anders Morgenthaler’s Truth Facts (September 30) — a panel strip that’s often engaging in showing comic charts — gives a guide to what the number of digits you’ve memorized says about you. (For what it’s worth, I peter out at “897932”.) I’m mildly delighted to find that their marker for Isaac Newton is more or less correct: Newton did work out pi to fifteen decimal places, by using his binomial theorem and a calculation of the area within a particular wedge of the circle. (As I make it out Wulff and Morgenthaler put Newton at fourteen decimal points, but they might have read references to Newton working out “fifteen decimal points” as meaning something different to what I do.) Newton’s was not the best calculation of pi in the 1660s when he worked it out — Christoph Grienberger, an Austrian Jesuit astronomer, had calculated 38 decimal places a generation earlier — but I can’t blame Wulff and Morgenthaler for supposing Newton to be a more recognizable name than Grienberger. I imagine if Einstein or Stephen Hawking had done any particularly unique work in calculating the digits of pi they’d have appeared on the chart too.
John Graziano’s Ripley’s Believe It or Not (October 1) — and don’t tell me that attribution doesn’t look weird — shares a story about the followers of the Ancient Greek mathematician, philosopher, and mystic Pythagoras, that they were forbidden to wear wool, eat beans, or pick up things they had dropped. I have heard the beans thing before and I think I’ve heard the wool prohibition before, but I don’t remember hearing about them not being able to pick up things before.
I’m not sure I can believe it, though: Pythagoras was a strange fellow, so far as the historical record is clear. It’s hard to be sure just what is true about him and his followers, though, and what is made up, either out of devoted followers building up the figure they admire or out of critics making fun of a strange fellow with his own little cult. Perhaps it’s so, perhaps it’s not. I would like to see a primary source, and I don’t think any exist.
Seyma Erbas had a post recently that I quite liked. It’s a nearly visual proof of the irrationality of the square root of two. Proving that the square root of two is irrational isn’t by itself a great trick: either that or the proof there are infinitely many prime numbers is probably the simplest interesting proof-by-contradiction someone could do. The Pythagoreans certainly knew of it, and being the Pythagoreans, inspired confusing legends about just what they did about this irrationality.
Anyway, in the reblogged post here, a proof (by contradiction) that the square root of two can’t be rational is done nearly entirely in pictures. The paper which Seyma Erbas cites, Steven J Miller and David Montague’s “Irrationality From The Book”, also includes similar visual proofs of the irrationality of the square roots of three, five, and six, and if the pictures don’t inspire you to higher mathematics they might at least give you ideas for retiling the kitchen. Miller and Montague talk about the generalization problem — making similar diagrams for larger and larger numbers, such as ten — and where their generalization stops working.
Yesterday I came a across a new (new to me, that is) proof of the irrationality of . I found it in the paper “Irrationality From The Book,” by Steven J. Miller, David Montague, which was recently posted to arXiv.org.
Apparently the proof was discovered by Stanley Tennenbaum in the 1950′s but was made widely known by John Conway around 1990. The proof appeared in Conway’s chapter “The Power of Mathematics” of the book Power, which was edited by Alan F. Blackwell, David MacKay (2005).
I haven’t been able to avoid people on my Twitter feed pointing out today’s a Pythagorean Triple, if you write out the month and day as digits and only use the last two digits of the year. There aren’t many such days; if I haven’t missed one there’s only fourteen per century, and we’ve just burned through the tenth of them. But if you want to have a little fun you might try working out whether I’m correct, and when the next one is going to be.
I don’t know of an efficient way of doing this, the sort of thing where you set up a couple of equations and let your favorite version of Mathematica grind away a bit and spit out an array of dates. This seems like the sort of problem best done by working out sets of integers a, b, and c, where , and figure out what sets of those numbers can plausibly even be arranged as dates.
The more mysterious thing to me is that I don’t remember this being so much pointed out when we had the same Pythagorean Triple day in May, and not at all when we were really rich with them back nearly a decade ago. But I wasn’t on Twitter back then; maybe that’s the problem. I also haven’t seen people complaining that it’s a trivial thing not worth pointing out; it may be trivial, but if we aren’t going to enjoy pretty alignments of numbers, what are we supposed to do?