Over on the Reading Penrose blog, Jean Louis Van Belle (and I apologize if I’ve got the name capitalized or punctuated wrong but I couldn’t find the author’s name except in a run-together, uncapitalized form) is trying to understand Roger Penrose’s Road to Reality, about the various laws of physics as we understand them. In the entry for the 6th of December, “Ordinary Differential equations (II)”, he gets to the question “What’s Physics All About?” and comes to what I have to agree is the harsh fact: a lot of physics is about solving differential equations.
Some of them are ordinary differential equations, some of them are partial differential equations, but really, a lot of it is differential equations. Some of it is setting up models for differential equations. Here, though, he looks at a number of ordinary differential equations and how they can be classified. The post is a bit cryptic — he intends the blog to be his working notes while he reads a challenging book — but I think it’s still worth recommending as a quick tour through some of the most common, physics-relevant, kinds of ordinary differential equation.
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?
It’s been long enough for a couple more mathematics-themed comics to gather, so, let me share them with you. The comics easily available to me may be increasing, too, as dailyink.com has indicated they’re looking to make it easier for people who aren’t subscribers to their service to look at the daily strips. I’d be glad to include them back in my roundup of mathematics strips, at least when I see them making mathematics jokes; there’ve been surprisingly few of them lately. Maybe the King Features Syndicate artists know it’s generally too much effort for me to feature them for a joke about how silly word problems are and have been saving us both the trouble.
Frank Page’s Bob the Squirrel began a sequence November 20 with the kid Lauren doing her math homework and Bob the Squirrel, one of multiple imaginary squirrels which I follow on Twitter, helping. It starts with percentages, a concept I admit that other people find harder than I ever did, probably because the “per cent” just made it clear to me at a young age what the whole thing was about. On the 21st Bob claims to have known a squirrel named Algebra, which wouldn’t be the strangest name for a squirrel. “Algebra”, the word, isn’t drawn from anyone’s name; it’s instead drawn from the title of the book Hisab al-jabr w’al-muqabala, Kitab al-Jabr wa-l-Muqabala (“The Compendious Book On Calculation By Completion and Balancing”), written by Muḥammad ibn Mūsā al-Khwārizmī, whose name did give us the word “algorithm”, which is the kind of successful word-generating power that you usually expect only from obscure Swedish towns. Bob closes things off with your standard breaking-the-word-problem sort of joke.
Hi again. I was hesitant to look at this month’s statistics, as I pretty much fell off the face of the earth for a week there, but I didn’t have the chance to do the serious thinking that’s needed for mathematics writing. The result’s almost exactly the dropoff in readership I might have predicted: from 440 views in October down to 308, and from 220 unique visitors down to 158. That’s almost an unchanged number of views per visitor, 2.00 dropping to 1.95, so at least the people still interested in me are sticking around.
The countries sending me the most viewers were as ever the United States, then Austria (hi, Elke, and thank you), the United Kingdom and then Canada. Sending me a single visitor each were Bulgaria, Cyprus, Czech Republic, Ethiopia, France, Jordan, Lebanon, Nepal, New Zealand, Russia, Singapore, Slovenia, Switzerland, and Thailand. This is also a drop in the number of single-viewer countries, although stalwarts Finland and the Netherlands are off the list. Slovenia’s the only country making a repeat appearance from last month, in fact.
The most popular articles the past month were:
- How Many Trapezoids I Can Draw, the attempt to classify how many different sorts of trapezoids there are, something of obvious steady interest to the public.
- The Intersecting Lines, pointing to Elke Stangl’s debate about the center-of-mass problem.
- Florian Cajori: A History Of Mathematical Notations, my discovery that a great book about where mathematics symbols come from is (partially) online.
- Reading the Comics, November 13, 2013, another of those comics roundups, although this one with more infinite monkeys than average.
- Professor Ludwig von Drake Explains Numerical Mathematics, a comic strip I’m surprised to see pop up on the popularity charts, but maybe he’s just a popular Professor.
- Solving The Price Is Right’s “Any Number” Game, a bit of actual numerical experimentation on my part.
And I apologize for not having produced many essays the past couple weeks, and can only fault myself for being more fascinated by some problems in my day job that’ve been taking up time and mental energy and waking me in the middle of the night with stuff I should try. I’ll be back to normal soon, I’m sure. Don’t tell my boss.
Through the MacTutor archive I learn that today’s the birthday of Émile Michel Hyacinthe Lemoine (1840 – 1912), a mathematician I admit I don’t remember hearing of before. His particular mathematical interests were primarily in geometry (though MacTutor notes professionally he became a civil engineer responsible for Paris’s gas supply).
What interests me is that Lemoine looked into the problem of how complicated a proof is, and in just the sort of thing designed to endear him to my heart, he tried to give a concrete measurement of, at least, how involved a geometric construction was. He identified the classic steps in compass-and-straightedge constructions and classified proofs by how many steps these took. MacTutor cites him as showing that the usual solution to the Apollonius problem — construct a circle tangent to three given circles — required over four hundred operations, but that he was able to squeeze that down to 199.
However, well, nobody seems to have been very interested in this classification. That’s probably because the length doesn’t really measure how complicated a proof (or a construction) is: proofs can have a narrative flow, and a proof that involves many steps each of which seems to flow obviously (or which look like the steps in another, already-familiar proof) is going to be easier to read and to understand than one that involves fewer but more obscure steps. This is the sort of thing that challenges attempts to measure how difficult a proof is, even though it’d be interesting to know.
Here’s one of the things that would be served by being able to measure just how long a proof is: a lot of numerical mathematics depends on having sequences of randomly generated numbers, but, showing that you actually have a random sequence of numbers is a deeply hard problem. If you can specify how you get a particular digit … well, they’re not random, then, are they? Unless it’s “use this digit from this randomly generated sequence”. If you could show there’s no way to produce a particular sequence of numbers in any way more efficiently than just reading them off this list of numbers you’d at least have a fair chance at saying this is a truly unpredictable sequence. But, showing that you have found the most efficient algorithm for producing something is … well, it’s difficult to even start measuring that sort of thing, and while Lemoine didn’t produce a very good measure of algorithmic complexity, he did have an idea, and it’s difficult to see how one could get a good measure if one didn’t start with trying not-very-good ones.
I haven’t had much chance to sit and think about this, but that’s no reason to keep my readers away from it. Elke Stangl has been pondering a probability problem regarding three intersecting lines on a plane, a spinoff of a physics problem about finding the center of mass of an object by the method of pinning it up from a couple different points and dropping the plumb line. My first impulse, of turning this into a matrix equation, flopped for what were as soon as I worked out a determinant obvious reasons, but that hardly means I’m stuck just yet.
For this week’s round of comic strips there’s almost a subtler theme than “they mention math in some way”: several have got links to statistical mechanics and the problem of recurrence. I’m not sure what’s gotten into Comic Strip Master Command that they sent out instructions to do comics that I can tie to the astounding interactions of infinities and improbable events, but it makes me wonder if I need to write a few essays about it.
Gene Weingarten, Dan Weingarten, and David Clark’s Barney and Clyde (October 30) summons the classic “infinite monkeys” problem of probability for its punch line. The concept — that if you had something producing strings of letters at random (taken to be monkeys because, I suppose, it’s assumed they would hit every key without knowing what sensibly comes next), it would, given enough time, produce any given result. The idea goes back a long way, and it’s blessed with a compelling mental image even though typewriters are a touch old-fashioned these days.
It seems to have gotten its canonical formulation in Émile Borel’s 1913 article “Statistical Mechanics and Irreversibility”, as you might expect since statistical mechanics brings up the curious problem of entropy. In short: every physical interaction, say, when two gases — let’s say clear air and some pink smoke as 1960s TV shows used to knock characters out — mix, is time-reversible. Look at the interaction of one clear-gas molecule and one pink-gas molecule and you can’t tell whether it’s playing forward or backward. But look at the entire room and it’s obvious whether they’re mixing or unmixing. How can something be time-reversible at every step of every interaction but not in whole?
The idea got a second compelling metaphor with Jorge Luis Borges’s Library of Babel, with a bit more literary class and in many printings fewer monkeys.
I just noticed that over at archive.org they have Volume I of Florian Cajori’s A History Of Mathematical Notations. There’s a fair chance this means nothing to you, but, Dr Cajori did a great deal of work in writing the history of mathematics in the early 20th century, and with a scope and prose style that still leaves me a bit awed. (He also wrote a history of physics; I remember reading the book, originally written in the mid-1920s, with his description of one of the mysteries of the day. With the advantage of decades on my side I knew this to be the Zeeman effect, a way that magnetic fields affect spectral lines.)
Archive.org has several of Cajori’s books, including the histories mentioned, but Mathematical Notations I like as it’s an indispensable reference. It describes, with abundant examples, the origins of all sorts of the ways we write out mathematical ideas, from numerals themselves to the choices of symbols like the + and x signs to how we got to using letters to represent quantities to something called alligation which was apparently practiced in 15th-century Venice.
Unfortunately archive.org hasn’t yet got Volume II, which includes topics like where the $ symbol for United States currency came from — Cajori had some strong opinions about this, suggesting he was tired of tracking down false leads — but it’s a book you can feel confident in leafing through to find something interesting most any time. I think his description of the way historical opinions had changed particularly fascinating, and recommend particularly Paragraph 96 (pages 64 through 68 of the book, and not one enormous block of text), describing “Fanciful hypotheses on the origins of the numeral forms”, many of them based on ideas that the symbols for numbers contain the number of vertices or strokes or some other mnemonic to how big a number is represented. Of those hypothesis formers he says, “Nor did these writers feel that they were indulging simply in pleasing pastimes or merely contributing to mathematical recreations. With perhaps only one exception, they were as convinced of the correctness of their explanations as are circle-squarers of the soundness of their quadratures”.
Dover publishing, of course, reprints the entire book on paper if you want Volumes I and II together. I admit that’s the form I have, and enjoy, since it becomes one of those books you could use to beat off an intruder if need be.
The Maths History feed on Twitter reminded me that the second of November is the birthday of George Boole, one of a handful of people who’s managed to get a critically important computer data type named for him (others, of course, include Arthur von Integer and the Lady Annabelle String). Reminded is the wrong word; actually, I didn’t have any idea when his birthday was, other than that it was in the first half of the 19th century. To that extent I was right (it was 1815).
He’s famous, to the extent anyone in mathematics who isn’t Newton or Leibniz is, for his work in logic. “Boolean algebra” is even almost the default term for the kind of reasoning done on variables that may have either of exactly two possible values, which match neatly to the idea of propositions being either true or false. He’d also publicized how neatly the study of logic and the manipulation of algebraic symbols could parallel one another, which is a familiar enough notion that it takes some imagination to realize that it isn’t obviously so.
Boole also did work on linear differential equations, which are important because differential equations are nearly inevitably the way one describes a system in which the current state of the system affects how it is going to change, and linear differential equations are nearly the only kinds of differential equations that can actually be exactly solved. (There are some nonlinear differential equations that can be solved, but more commonly, we’ll find a linear differential equation that’s close enough to the original. Many nonlinear differential equations can also be approximately solved numerically, but that’s also quite difficult.)
His MacTutor History of Mathematics biography notes that Boole (when young) spent five years trying to teach himself differential and integral calculus — money just didn’t allow for him to attend school or hire a tutor — although given that he was, before the age of fourteen, able to teach himself ancient Greek I can certainly understand his supposition that he just needed the right books and some hard work. Apparently, at age fourteen he translated a poem by Meleager — I assume the poet from the first century BCE, though MacTutor doesn’t specify; there was also a Meleager who was briefly king of Macedon in 279 BCE, and another some decades before that who was a general serving Alexander the Great — so well that when it was published a local schoolmaster argued that a 14-year-old could not possibly have done that translation. He’d also, something I didn’t know until today, married Mary Everest, niece of the fellow whose name is on that tall mountain.
It’s been a month since I last looked over precisely how not-staggeringly-popular I am, so it’s time again.
For October 2013 I had 440 views, down from September’s 2013. These came from 220 distinct viewers, down again from the 237 that September gave me. This does mean there was a slender improvement in views per visitor, from 1.97 up to 2.00. Neither of these are records, although given that I had a poor updating record again this month that’s all tolerable.
The most popular articles from the past month are … well, mostly the comics, and the trapezoids come back again. I’ve clearly got to start categorizing the other kinds of polygons. Or else plunge directly into dynamical systems as that’s the other thing people liked. October 2013′s top hits were:
- Reading the Comics, October 8, 2013
- How Many Trapezoids I Can Draw
- Reading the Comics, September 11, 2013
- From ElKement: On The Relation Of Jurassic Park and Alien Jelly Flowing Through Hyperspace
- Reading the Comics, September 21, 2013
- The Mathematics of a Pricing Game
The country sending me the most readers again was the United States (226 of them), with the United Kingdom coming up second (37). Austria popped into third for, I think, the first time (25 views), followed by Denmark (21) and at long last Canada (18). I hope they still like me in Canada.
Sending just the lone reader each were a bunch of countries: Bermuda, Chile, Colombia, Costa Rica, Finland, Guatemala, Hong Kong, Laos, Lebanon, Malta, Mexico, the Netherlands, Oman, Romania, Saudi Arabia, Slovenia, Sweden, Turkey, and Ukraine. Finland and the Netherlands are repeats from last month, and the Netherlands is going on at least three months like this.