## The End 2016 Mathematics A To Z: Algebra

So let me start the End 2016 Mathematics A To Z with a word everybody figures they know. As will happen, everybody’s right and everybody’s wrong about that.

## Algebra.

Everybody knows what algebra is. It’s the point where suddenly mathematics involves spelling. Instead of long division we’re on a never-ending search for ‘x’. Years later we pass along gifs of either someone saying “stop asking us to find your ex” or someone who’s circled the letter ‘x’ and written “there it is”. And make jokes about how we got through life without using algebra. And we know it’s the thing mathematicians are always doing.

Mathematicians aren’t always doing that. I expect the average mathematician would say she almost never does that. That’s a bit of a fib. We have a lot of work where we do stuff that would be recognizable as high school algebra. It’s just we don’t really care about that. We’re doing that because it’s how we get the problem we are interested in done. the most recent few pieces in my “Why Stuff can Orbit” series include a bunch of high school algebra-style work. But that was just because it was the easiest way to answer some calculus-inspired questions.

Still, “algebra” is a much-used word. It comes back around the second or third year of a mathematics major’s career. It comes in two forms in undergraduate life. One form is “linear algebra”, which is a great subject. That field’s about how stuff moves. You get to imagine space as this stretchy material. You can stretch it out. You can squash it down. You can stretch it in some directions and squash it in others. You can rotate it. These are simple things to build on. You can spend a whole career building on that. It becomes practical in surprising ways. For example, it’s the field of study behind finding equations that best match some complicated, messy real data.

The second form is “abstract algebra”, which comes in about the same time. This one is alien and baffling for a long while. It doesn’t help that the books all call it Introduction to Algebra or just Algebra and all your friends think you’re slumming. The mathematics major stumbles through confusing definitions and theorems that ought to sound comforting. (“Fermat’s Little Theorem”? That’s a good thing, right?) But the confusion passes, in time. There’s a beautiful subject here, one of my favorites. I’ve talked about it a lot.

We start with something that looks like the loosest cartoon of arithmetic. We get a bunch of things we can add together, and an ‘addition’ operation. This lets us do a lot of stuff that looks like addition modulo numbers. Then we go on to stuff that looks like picking up floor tiles and rotating them. Add in something that we call ‘multiplication’ and we get rings. This is a bit more like normal arithmetic. Add in some other stuff and we get ‘fields’ and other structures. We can keep falling back on arithmetic and on rotating tiles to build our intuition about what we’re doing. This trains mathematicians to look for particular patterns in new, abstract constructs.

Linear algebra is not an abstract-algebra sort of algebra. Sorry about that.

And there’s another kind of algebra that mathematicians talk about. At least once they get into grad school they do. There’s a huge family of these kinds of algebras. The family trait for them is that they share a particular rule about how you can multiply their elements together. I won’t get into that here. There are many kinds of these algebras. One that I keep trying to study on my own and crash hard against is Lie Algebra. That’s named for the Norwegian mathematician Sophus Lie. Pronounce it “lee”, as in “leaning”. You can understand quantum mechanics much better if you’re comfortable with Lie Algebras and so now you know one of my weaknesses. Another kind is the Clifford Algebra. This lets us create something called a “hypercomplex number”. It isn’t much like a complex number. Sorry. Clifford Algebra does lend to a construct called spinors. These help physicists understand the behavior of bosons and fermions. Every bit of matter seems to be either a boson or a fermion. So you see why this is something people might like to understand.

Boolean Algebra is the algebra of this type that a normal person is likely to have heard of. It’s about what we can build using two values and a few operations. Those values by tradition we call True and False, or 1 and 0. The operations we call things like ‘and’ and ‘or’ and ‘not’. It doesn’t sound like much. It gives us computational logic. Isn’t that amazing stuff?

So if someone says “algebra” she might mean any of these. A normal person in a non-academic context probably means high school algebra. A mathematician speaking without further context probably means abstract algebra. If you hear something about “matrices” it’s more likely that she’s speaking of linear algebra. But abstract algebra can’t be ruled out yet. If you hear a word like “eigenvector” or “eigenvalue” or anything else starting “eigen” (or “characteristic”) she’s more probably speaking of abstract algebra. And if there’s someone’s name before the word “algebra” then she’s probably speaking of the last of these. This is not a perfect guide. But it is the sort of context mathematicians expect other mathematicians notice.

• #### John Friedrich 2:13 am on Thursday, 3 November, 2016 Permalink | Reply

The cruelest trick that happened to me was when a grad school professor labeled the Galois Theory class “Algebra”. Until then, the lowest score I’d ever gotten in a math class was a B. After that, I decided to enter the work force and abandon my attempts at a master’s degree.

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• #### Joseph Nebus 3:32 pm on Friday, 4 November, 2016 Permalink | Reply

Well, it’s true enough that it’s part of algebra. But I’d feel uncomfortable plunging right into that without the prerequisites being really clear. I’m not sure I’ve even run into a nice clear pop-culture explanation of Galois Theory past some notes about how there’s two roots to a quadratic equation and see how they mirror each other.

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## Reading the Comics, August 27, 2016: Calm Before The Term Edition

Here in the United States schools are just lurching back into the mode where they have students come in and do stuff all day. Perhaps this is why it was a routine week. Comic Strip Master Command wants to save up a bunch of story problems for us. But here’s what the last seven days sent into my attention.

Jeff Harris’s Shortcuts educational feature for the 21st is about algebra. It’s got a fair enough blend of historical trivia and definitions and examples and jokes. I don’t remember running across the “number cruncher” joke before.

Mark Anderson’s Andertoons for the 23rd is your typical student-in-lecture joke. But I do sympathize with students not understanding when a symbol gets used for different meanings. It throws everyone. But sometimes the things important to note clearly in one section are different from the needs in another section. No amount of warning will clear things up for everybody, but we try anyway.

Tom Thaves’s Frank and Ernest for the 23rd tells a joke about collapsing wave functions, which is why you never see this comic in a newspaper but always see it on a physics teacher’s door. This is properly physics, specifically quantum mechanics. But it has mathematical import. The most practical model of quantum mechanics describes what state a system is in by something called a wave function. And we can turn this wave function into a probability distribution, which describes how likely the system is to be in each of its possible states. “Collapsing” the wave function is a somewhat mysterious and controversial practice. It comes about because if we know nothing about a system then it may have one of many possible values. If we observe, say, the position of something though, then we have one possible value. The wave functions before and after the observation are different. We call it collapsing, reflecting how a universe of possibilities collapsed into a mere fact. But it’s hard to find an explanation for what that is that’s philosophically and physically satisfying. This problem leads us to Schrödinger’s Cat, and to other challenges to our sense of how the world could make sense. So, if you want to make your mark here’s a good problem for you. It’s not going to be easy.

John Allison’s Bad Machinery for the 24th tosses off a panel full of mathematics symbols as proof of hard thinking. In other routine references John Deering’s Strange Brew for the 26th is just some talk about how hard fractions are.

While it’s outside the proper bounds of mathematics talk, Tom Toles’s Randolph Itch, 2 am for the 23rd is a delight. My favorite strip of this bunch. Should go on the syllabus.

## Reading the Comics, August 12, 2016: Skipping Saturday Edition

I have no idea how many or how few comic strips on Saturday included some mathematical content. I was away most of the day. We made a quick trip to the Michigan’s Adventure amusement park and then to play pinball in a kind-of competitive league. The park turned out to have every person in the world there. If I didn’t wave to you from the queue on Shivering Timbers I apologize but it hasn’t got the greatest lines of sight. The pinball stuff took longer than I expected too and, long story short, we got back home about 4:15 am. So I’m behind on my comics and here’s what I did get to.

Tak Bui’s PC and Pixel for the 8th depicts the classic horror of the cleaning people wiping away an enormous amount of hard work. It’s a primal fear among mathematicians at least. Boards with a space blocked off with the “DO NOT ERASE” warning are common. At this point, though, at least, the work is probably savable. You can almost always reconstruct work, and a few smeared lines like this are not bad at all.

The work appears to be quantum mechanics work. The tell is in the upper right corner. There’s a line defining E (energy) as equal to something including $\imath \hbar \frac{\partial}{\partial t}\phi(r, t)$. This appears in the time-dependent Schrödinger Equation. It describes how probability waveforms look when the potential energies involved may change in time. These equations are interesting and impossible to solve exactly. We have to resort to approximations, including numerical approximations, all the time. So that’s why the computer lab would be working on this.

Mark Anderson’s Andertoons! Where would I be without them? Besides short on content. The strip for the 10th depicts a pollster saying to “put the margin of error at 50%”, guaranteeing the results are right. If you follow elections polls you do see the results come with a margin of error, usually of about three percent. But every sampling technique carries with it a margin of error. The point of a sample is to learn something about the whole without testing everything in it, after all. And probability describes how likely it is the quantity measured by a sample will be far from the quantity the whole would have. The logic behind this is independent of the thing being sampled. It depends on what the whole is like. It depends on how the sampling is done. It doesn’t matter whether you’re sampling voter preferences or whether there are the right number of peanuts in a bag of squirrel food.

So a sample’s measurement will almost never be exactly the same as the whole population’s. That’s just requesting too much of luck. The margin of error represents how far it is likely we’re off. If we’ve sampled the voting population fairly — the hardest part — then it’s quite reasonable the actual vote tally would be, say, one percent different from our poll. It’s implausible that the actual votes would be ninety percent different. The margin of error is roughly the biggest plausible difference we would expect to see.

Except. Sometimes we do, even with the best sampling methods possible, get a freak case. Rarely noticed beside the margin of error is the confidence level. This is what the probability is that the actual population value is within the sampling error of the sample’s value. We don’t pay much attention to this because we don’t do statistical-sampling on a daily basis. The most normal people do is read election polling results. And most election polls settle for a confidence level of about 95 percent. That is, 95 percent of the time the actual voting preference will be within the three or so percentage points of the survey. The 95 percent confidence level is popular maybe because it feels like a nice round number. It’ll be off only about one time out of twenty. It also makes a nice balance between a margin of error that doesn’t seem too large and that doesn’t need too many people to be surveyed. As often with statistics the common standard is an imperfectly-logical blend of good work and ease of use.

For the 11th Mark Anderson gives me less to talk about, but a cute bit of wordplay. I’ll take it.

Anthony Blades’s Bewley for the 12th is a rerun. It’s at least the third time this strip has turned up since I started writing these Reading The Comics posts. For the record it ran also the 27th of April, 2015 and on the 24th of May, 2013. It also suggests mathematicians have a particular tell. Try this out next time you do word problem poker and let me know how it works for you.

Julie Larson’s The Dinette Set for the 12th I would have sworn I’d seen here before. I don’t find it in my archives, though. We are meant to just giggle at Larson’s characters who bring their penny-wise pound-foolishness to everything. But there is a decent practical mathematics problem here. (This is why I thought it had run here before.) How far is it worth going out of one’s way for cheaper gas? How much cheaper? It’s simple algebra and I’d bet many simple Javascript calculator tools. The comic strip originally ran the 4th of October, 2005. Possibly it’s been rerun since.

Bill Amend’s FoxTrot Classics for the 12th is a bunch of gags about a mathematics fighting game. I think Amend might be on to something here. I assume mathematics-education contest games have evolved from what I went to elementary school on. That was a Commodore PET with a game where every time you got a multiplication problem right your rocket got closer to the ASCII Moon. But the game would probably quickly turn into people figuring how to multiply the other person’s function by zero. I know a game exploit when I see it.

The most obscure reference is in the third panel one. Jason speaks of “a z = 0 transform”. This would seem to be some kind of z-transform, a thing from digital signals processing. You can represent the amplification, or noise-removal, or averaging, or other processing of a string of digits as a polynomial. Of course you can. Everything is polynomials. (OK, sometimes you must use something that looks like a polynomial but includes stuff like the variable z raised to a negative power. Don’t let that throw you. You treat it like a polynomial still.) So I get what Jason is going for here; he’s processing Peter’s function down to zero.

That said, let me warn you that I don’t do digital signal processing. I just taught a course in it. (It’s a great way to learn a subject.) But I don’t think a “z = 0 transform” is anything. Maybe Amend encountered it as an instructor’s or friend’s idiosyncratic usage. (Amend was a physics student in college, and shows his comfort with mathematics-major talk often. He by the way isn’t even the only syndicated cartoonist with a physics degree. Bud Grace of The Piranha Club was also a physics major.) I suppose he figured “z = 0 transform” would read clearly to the non-mathematician and be interpretable to the mathematician. He’s right about that.

## A Leap Day 2016 Mathematics A To Z: Yukawa Potential

Yeah, ‘Y’ is a lousy letter in the Mathematics Glossary. I have a half-dozen mathematics books on the shelf by my computer. Some is semi-popular stuff like Richard Courant and Herbert Robbins’s What Is Mathematics? (the Ian Stewart revision). Some is fairly technical stuff, by which I mean Hidetoshi Nishimori’s Statistical Physics of Spin Glasses and Information Processing. There’s just no ‘Y’ terms in any of them worth anything. But I can rope something into the field. For example …

## Yukawa Potential

When you as a physics undergraduate first take mechanics it’s mostly about very simple objects doing things according to one rule. The objects are usually these indivisible chunks. They’re either perfectly solid or they’re points, too tiny to have a surface area or volume that might mess things up. We draw them as circles or as blocks because they’re too hard to see on the paper or board otherwise. We spend a little time describing how they fall in a room. This lends itself to demonstrations in which the instructor drops a rubber ball. Then we go on to a mass on a spring hanging from the ceiling. Then to a mass on a spring hanging to another mass.

Then we go onto two things sliding on a surface and colliding, which would really lend itself to bouncing pool balls against one another. Instead we use smaller solid balls. Sometimes those “Newton’s Cradle” things with the five balls that dangle from wires and just barely touch each other. They give a good reason to start talking about vectors. I mean positional vectors, the ones that say “stuff moving this much in this direction”. Normal vectors, that is. Then we get into stars and planets and moons attracting each other by gravity. And then we get into the stuff that really needs calculus. The earlier stuff is helped by it, yes. It’s just by this point we can’t do without.

The “things colliding” and “balls dropped in a room” are the odd cases in this. Most of the interesting stuff in an introduction to mechanics course is about things attracting, or repelling, other things. And, particularly, they’re particles that interact by “central forces”. Their attraction or repulsion is along the line that connects the two particles. (Impossible for a force to do otherwise? Just wait until Intro to Mechanics II, when magnetism gets in the game. After that, somewhere in a fluid dynamics course, you’ll see how a vortex interacts with another vortex.) The potential energies for these all vary with distance between the points.

Yeah, they also depend on the mass, or charge, or some kind of strength-constant for the points. They also depend on some universal constant for the strength of the interacting force. But those are, well, constant. If you move the particles closer together or farther apart the potential changes just by how much you moved them, nothing else.

Particles hooked together by a spring have a potential that looks like $\frac{1}{2}k r^2$. Here ‘r’ is how far the particles are from each other. ‘k’ is the spring constant; it’s just how strong the spring is. The one-half makes some other stuff neater. It doesn’t do anything much for us here. A particle attracted by another gravitationally has a potential that looks like $-G M \frac{1}{r}$. Again ‘r’ is how far the particles are from each other. ‘G’ is the gravitational constant of the universe. ‘M’ is the mass of the other particle. (The particle’s own mass doesn’t enter into it.) The electric potential looks like the gravitational potential but we have different symbols for stuff besides the $\frac{1}{r}$ bit.

The spring potential and the gravitational/electric potential have an interesting property. You can have “closed orbits” with a pair of them. You can set a particle orbiting another and, with time, get back to exactly the original positions and velocities. (Three or more particles you’re not guaranteed of anything.) The curious thing is this doesn’t always happen for potentials that look like “something or other times r to a power”. In fact, it never happens, except for the spring potential, the gravitational/electric potential, and — peculiarly — for the potential $k r^7$. ‘k’ doesn’t mean anything there, and we don’t put a one-seventh or anything out front for convenience, because nobody knows anything that needs anything like that, ever. We can have stable orbits, ones that stay within a minimum and a maximum radius, for a potential $k r^n$ whenever n is larger than -2, at least. And that’s it, for potentials that are nothing but r-to-a-power.

Ah, but does the potential have to be r-to-a-power? And here we see Dr Hideki Yukawa’s potential energy. Like these springs and gravitational/electric potentials, it varies only with the distance between particles. Its strength isn’t just the radius to a power, though. It uses a more complicated expression:

$-K \frac{e^{-br}}{r}$

Here ‘K’ is a scaling constant for the strength of the whole force. It’s the kind of thing we have ‘G M’ for in the gravitational potential, or ‘k’ in the spring potential. The ‘b’ is a second kind of scaling. And that a kind of range. A range of what? It’ll help to look at this potential rewritten a little. It’s the same as $-\left(K \frac{1}{r}\right) \cdot \left(e^{-br}\right)$. That’s the gravitational/electric potential, times e-br. That’s a number that will be very large as r is small, but will drop to zero surprisingly quickly as r gets larger. How quickly will depend on b. The larger a number b is, the faster this drops to zero. The smaller a number b is, the slower this drops to zero. And if b is equal to zero, then e-br is equal to 1, and we have the gravitational/electric potential all over again.

Yukawa introduced this potential to physics in the 1930s. He was trying to model the forces which keep an atom’s nucleus together. It represents the potential we expect from particles that attract one another by exchanging some particles with a rest mass. This rest mass is hidden within that number ‘b’ there. If the rest mass is zero, the particles are exchanging something like light, and that’s just what we expect for the electric potential. For the gravitational potential … um. It’s complicated. It’s one of the reasons why we expect that gravitons, if they exist, have zero rest mass. But we don’t know that gravitons exist. We have a lot of trouble making theoretical gravitons and quantum mechanics work together. I’d rather be skeptical of the things until we need them.

Still, the Yukawa potential is an interesting mathematical creature even if we ignore its important role in modern physics. When I took my Introduction to Mechanics final one of the exam problems was deriving the equivalent of Kepler’s Laws of Motion for the Yukawa Potential. I thought then it was a brilliant problem. I still do. It struck me while writing this that I don’t remember whether it allows for closed orbits, except when b is zero. I’m a bit afraid to try to work out whether it does, lest I learn that I can’t follow the reasoning for that anymore. That would be a terrible thing to learn.

• #### elkement (Elke Stangl) 1:31 pm on Wednesday, 27 April, 2016 Permalink | Reply

That’s an interesting one!! Re closed orbits: I just remember that there are only two potentials that will make sure that every bound orbit is closed: A quadratic (Hooke’s Law, a spring) and a gravitational 1/r potential. Other potentials can have closed orbits, but it depends on initial conditions.
Proofs usually make use of all the constants – energy, angular momentum – to be subsituted in the equations of motion (or the constants emerge from applying Langrange’s formalism) and angular momentum gives rise to an effective ‘add-on’ potential. Then different substitutions are applied that better fit the geometry of the problem, like using 1/r rather than r and angles or polar coordinates … and the statement about closed orbits should be a consequence of calculating the change in angle for moving from maximum to minimum radius.
The procecure felt a bit like so-called early quantum mechanics, where theorems about integer changes in angular momentum were ‘tacked on’ classical theory … and all worked out nicely (and only) with harmonic or 1/r potentials.

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• #### Joseph Nebus 7:01 pm on Friday, 29 April, 2016 Permalink | Reply

Hm. On reading my copy of Davis’s Classical Mechanics — my old textbook on this — I see he says the kr7 potential allows for closed orbits, but doesn’t say one thing or another about whether every orbit with that potential is closed.

But the section has got that tone like you describe, about early quantum mechanics and other proofs like this, of being ad hoc. Describing where an equilibrium might be is fine. The added talk about what makes it stable? … I suppose that’s more obvious when you’ve got some experience in similar problems, but I remember as a freshman finding it baffling why this should be a calculation. And then the part about apsidal angles, to say whether the orbits are closed, seems to come from a particularly deep field of nowhere.

This does remind me that I’ve got a book I mean to read, partly for education, partly for recreation, that is about introducing the most potent tools of mechanics while studying the simplest orbiting-bodies problems.

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• #### elkement (Elke Stangl) 2:08 pm on Tuesday, 3 May, 2016 Permalink | Reply

I searched for a reference now – this is the theorem I meant and its proof (translated to English from French): https://arxiv.org/pdf/0704.2396v1.pdf
Quote: “In 1873, Joseph Louis Francois Bertrand (1822-1900) published a short but important paper in which he proved that there are two central fields only for which all bounded orbits are closed, namely, the isotropic harmonic oscillator law and Newton’s universal gravitation law”

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• #### Joseph Nebus 3:50 pm on Wednesday, 4 May, 2016 Permalink | Reply

Ooh, thank you. This is interesting. And remarkable for being so compact, too! Who knew there’d be results that interesting with barely five pages of work?

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• #### Joseph Nebus 3:00 pm on Saturday, 16 April, 2016 Permalink | Reply Tags: density, packing ( 6 ), quantum mechanics, set theory ( 9 )

My long streak of posting something every day will end. There’s just no keeping up mathematics content like this indefinitely, not with my stamina. But it’s not over just quite yet. I wanted to share some stuff that people had brought to my attention and that’s just too interesting to pass up.

The first comes from … I’m not really sure. I lost my note about wherever it did come from. It’s from the Continuous Everywhere But Differentiable Nowhere blog. It’s about teaching the Crossed Chord Theorem. It’s one I had forgotten about, if I heard it in the first place. The result is one of those small, neat things and it’s fun to work through how and why it might be true.

Next comes from a comment by Gerry on a rather old article, “What’s The Worst Way To Pack?” Gerry located a conversation on MathOverflow.net that’s about finding low-density packings of discs, circles, on the plane. As these sorts of discussions go, it gets into some questions about just what we mean by packings, and whether Wikipedia has typos. This is normal for discovering new mathematics. We have to spend time pinning down just what we mean to talk about. Then we can maybe figure out what we’re saying.

And the last I picked up from Elke Stangl, of what’s now known as the elkemental Force blog. She had pointed me first to lecture notes from Dr Scott Aaronson which try to explain quantum mechanics from starting principles. Normally, almost invariably, they’re taught in historical sequence. Aaronson here skips all the history to look at what mathematical structures make quantum mechanics make sense. It’s not for casual readers, I’m afraid. It assumes you’re comfortable with things like linear transformations and p-norms. But if you are, then it’s a great overview. I figure to read it over several more times myself.

Those notes are from a class in Quantum Computing. I haven’t had nearly the time to read them all. But the second lecture in the series is on Set Theory. That’s not quite friendly to a lay audience, but it is friendlier, at least.

## Reading the Comics, April 5, 2016: April 5, 2016 Edition

I’ve mentioned I like to have five or six comic strips for a Reading The Comics entry. On the 5th, it happens, I got a set of five all at once. Perhaps some are marginal for mathematics content but since when does that stop me? Especially when there’s the fun of a single-day Reading The Comics post to consider. So here goes:

Mark Anderson’s Andertoons is a student-resisting-the-problem joke. And it’s about long division. I can’t blame the student for resisting. Long division’s hard to learn. It’s probably the first bit of arithmetic in which you just have to make an educated guess for an answer and face possibly being wrong. And this is a problem that’ll have a remainder in it. I think I remember early on in long division finding a remainder left over feeling like an accusation. Surely if I’d done it right, the divisor would go into the original number a whole number of times, right? No, but you have to warm up to being comfortable with that.

Ted Key’s Hazel rerun the 5th of April, 2016. Were the Pac-Man ghosts ever called space demons? It seems like they might’ve been described that way in some boring official manual that nobody ever read. I always heard them as “ghosts” anyway.

Ted Key’s Hazel feels less charmingly out-of-date when you remember these are reruns. Ted Key — who created Peabody’s Improbable History as well as the sitcom based on this comic panel — retired in 1993. So Hazel’s attempt to create a less abstract version of the mathematics problem for Harold is probably relatively time-appropriate. And recasting a problem as something less abstract is often a good way to find a solution. It’s all right to do side work as a way to get the work you want to do.

John McNamee’s Pie Comic is a joke about the uselessness of mathematics. Tch. I wonder if the problem here isn’t the abstractness of a word like “hypotenuse”. I grant the word doesn’t evoke anything besides “hypotenuse”. But one irony is that hypotenuses are extremely useful things. We can use them to calculate how far away things are, without the trouble of going out to the spot. We can imagine post-apocalyptic warlords wanting to know how far things are, so as to better aim the trebuchets.

Percy Crosby’s Skippy is a rerun from 1928, of course. It’s also only marginally on point here. The mention of arithmetic is irrelevant to the joke. But it’s a fine joke and I wanted people to read it. Longtime readers know I’m a Skippy fan. (Saturday’s strip follows up on this. It’s worth reading too.)

Bill Griffith’s Zippy the Pinhead for the 5th of April, 2016. Not what Neil DeGrasse Tyson is like when he’s not been getting enough sleep.

Bill Griffith’s Zippy the Pinhead has picked up some quantum mechanics talk. At least he’s throwing around the sorts of things we see in pop science and, er, pop mathematical talk about the mathematics of cutting-edge physics. I’m not aware of any current models of everything which suppose there to be fourteen, or seventeen, dimensions of space. But high-dimension spaces are common points of speculation. Most of those dimensions appear to be arranged in ways we don’t see in the everyday world, but which leave behind mathematical traces. The crack about God not playing dice with the universe is famously attributed to Albert Einstein. Einstein was not comfortable with the non-deterministic nature of quantum mechanics, that there is this essential randomness to this model of the world.

## Reading the Comics, January 27, 2015: Rabbit In A Trapezoid Edition

So the reason I fell behind on this Reading the Comics post is that I spent more time than I should have dithering about which ones to include. I hope it’s not disillusioning to learn that I have no clearly defined rules about what comics to include and what to leave out. It depends on how clearly mathematical in content the comic strip is; but it also depends on how much stuff I have gathered. If there’s a slow week, I start getting more generous about what I might include. And last week gave me a string of comics that I could argue my way into including, but few that obviously belonged. So I had a lot of time dithering.

To make it up to you, at the end of the post I should have our pet rabbit tucked within a trapezoid of his own construction. If that doesn’t make everything better I don’t know what will.

Mark Pett’s Mr Lowe for the 22nd of January (a rerun from the 19th of January, 2001) is really a standardized-test-question joke. But it brings up a debate about cultural biases in standardized tests that I don’t remember hearing lately. I may just be moving in the wrong circles. I remember self-assured rich white males explaining how it’s absurd to think cultural bias could affect test results since, after all, they’re standardized tests. I’ve sulked some around these parts about how I don’t buy mathematics’ self-promoted image of being culturally neutral either. A mathematical truth may be universal, but that we care about this truth is not. Anyway, Pett uses a mathematics word problem to tell the joke. That was probably the easiest way to put a cultural bias into a panel that

T Lewis and Michael Fry’s Over The Hedge for the 25th of January uses a bit of calculus to represent “a lot of hard thinking”. Hammy the Squirrel particularly is thinking of the Fundamental Theorem of Calculus. This particular part is the one that says the derivative of the integral of a function is the original function. It’s part of how integration and differentiation link together. And it shows part of calculus’s great appeal. It has those beautiful long-s integral signs that make this part of mathematics look like artwork.

Leigh Rubin’s Rubes for the 25th of January is a panel showing “Schrödinger’s Job Application”. It’s referring to Schrödinger’s famous thought experiment, meant to show there are things we don’t understand about quantum mechanics. It sets up a way that a quantum phenomenon can be set up to have distinct results in the everyday environment. The mathematics suggests that a cat, poisoned or not by toxic gas released or not by the decay of one atom, would be both alive and dead until some outside observer checks and settles the matter. How can this be? For that matter, how can the cat not be a qualified judge to whether it’s alive? Well, there are things we don’t understand about quantum mechanics.

Roy Schneider’s The Humble Stumble for the 26th of January (a rerun from the 30th of January, 2007) uses a bit of mathematics to mark Tommy, there, as a frighteningly brilliant weirdo. The equation is right, although trivial. The force it takes to keep something with a mass m moving in a circle of radius R at the linear speed v is $\frac{m v^2}{R}$. The radius of the Moon’s orbit around the Earth is strikingly close to sixty times the Earth’s radius. The Ancient Greeks were able to argue that from some brilliantly considered geometry. Here, RE gets used as a name for “the radius of the Earth”. So the force holding the Moon in its orbit has to be approximately $\frac{m v^2}{60 R_e}$. That’s if we say m is the mass of the Moon, and v is its linear speed, and if we suppose the Moon’s orbit is a circle. It nearly is, and this would give us a good approximate answer to how much force holds the Moon in its orbit. It would be only a start, though; the precise movements of the Moon are surprisingly complicated. Newton himself could not fully explain them, even with the calculus and physics tools he invented for the task.

Dave Whamond’s Reality Check for the 26th of January isn’t quite the anthropomorphic-numerals joke for this essay. But we do get personified geometric constructs, which is close, and some silly wordplay. Much as I like the art for Over The Hedge showcasing a squirrel so burdened with thoughts that his head flops over, this might be my favorite of this bunch.

Dave Blazek’s Loose Parts for the 27th of January is a runner-up for the silly jokes trophy this time around.

Cardboard boxes are normally pretty good environments for rabbits, given that they’re places the rabbits can do in and not be seen. We set the box up, but he did all the chewing.

Now I know what you’re thinking: isn’t that actually a trapezoidal prism, underneath a rectangular prism? Yes, I suppose so. The only people who’re going to say so are trying to impress people by saying so, though. And those people won’t be impressed by it. I’m sorry. We gave him the box because rabbits generally like having cardboard boxes to go in and chew apart. He did on his own the pulling-in of the side flaps to make it stand so trapezoidal.

• #### scifihammy 3:32 pm on Monday, 1 February, 2016 Permalink | Reply

I think it’s cute he made his bed the way he liked it :)
We used to have rabbits. They have such personalities :)

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• #### Joseph Nebus 12:42 am on Tuesday, 2 February, 2016 Permalink | Reply

He’s been working hard at becoming more cute lately. I’m surprised he isn’t resting on his cute laurels considering how certain his place in the household is.

I’m new to rabbit-keeping. As a kid I kept guinea pigs, which I liked. But they weren’t nearly so extroverted as our rabbit, and their personality was more one of “gazing out wondering if they were supposed to be invited into this meeting”. It’s a style I like, certainly, but I understand people not seeing the appeal of that.

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• #### scifihammy 10:10 am on Tuesday, 2 February, 2016 Permalink | Reply

Aw :) We also had a guinea pig, to keep the first rabbit company. He Loved his food and would squeak loudly when he thought it was dinner time. Very different to the rabbits, who could only try the Jedi mind trick on you! :)

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• #### Joseph Nebus 11:40 pm on Tuesday, 2 February, 2016 Permalink | Reply

Oh, yes, the squeaking. Guinea pigs have a knack for that. Our rabbit sneezes sometime, and I swear one time I heard him bark, but those are rare events.

Guinea pigs also have that popcorning habit. Rabbits will jump up sometimes too, although our rabbit’s reached the point in life where he would rather not do something quite that time-consuming if he can help it.

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• #### scifihammy 3:01 pm on Wednesday, 3 February, 2016 Permalink | Reply

haha Our rabbits were pretty active – of course having “wolves” prowling round their enclosure might have had something to do with it! :)

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• #### Joseph Nebus 8:18 pm on Friday, 5 February, 2016 Permalink | Reply

Our pet rabbit’s only real experience with dogs has been with my love’s parents’ dogs, a very elderly, frail pair that were afraid of him. Past that it’s just the occasional glimpse of one while at the vet.

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• #### sheldonk2014 3:48 pm on Tuesday, 2 February, 2016 Permalink | Reply

I got this idea for a probability problem
You take aspirin every day
The bottle holds 250 pills
You only take a half
But every time you want one a whole comes out
What will it take for the halves to start to come out
I think I explained this rite
Sheldon

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• #### Joseph Nebus 11:47 pm on Tuesday, 2 February, 2016 Permalink | Reply

It’s a hard problem to answer, actually. What you need to provide an answer is to know how many halves there are, and how many wholes there are, and how well-mixed they are. If you have, say, 10 half-pills and 90 whole-pills, and you’re equally likely to pick a half or a whole, then the chance of picking a half-pill is ten out of a hundred, or ten percent. (There are 10 half-pills wanted, and there are 90 + 10 or 100 things to pick from.)

However, in a real pill bottle, the half-pills and the whole-pills aren’t going to be equally likely to come out. The entire bottle starts out as whole-pills, after all. Half-pills are added when you’ve taken out a whole pill, cut it in half, and tossed one of the halves back in. So they’re going to start out almost entirely on top, closer to the lid and presumably more likely to be shaken or picked out.

However again — in a jumble of large and small things, that gets shaken up, the small things are likely to drift to the bottom, and the large ones to the top. You’ve seen this when it seems like all the raisins sank to the bottom and the bran to the top of the cereal box; or when all the large peanuts are at the top of the mixed-nuts jar and the crumbly little things at the bottom. Half- and whole-pills aren’t as variable in size as mixed nuts, and the bottle isn’t shaken as thoroughly, but the effect is going to hold.

So I’m not sure the problem can be answered purely by reasoning about it. I don’t think we can count on half-pills being as likely to be pulled out as whole-pills. And without some idea of the relatively likelihood of a half versus a whole there’s not a real way to answer. We can make some assumptions that might seem reasonable. But we can’t rely on those until they’re tested by experiment.

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## Reading the Comics, November 18, 2015: All Caught Up Edition

Yes, I feel a bit bad that I didn’t have anything posted yesterday. I’d had a nice every-other-day streak going for a couple weeks there. But I had honestly expected more mathematically themed comic strips, and there just weren’t enough in my box by the end of the 17th. So I didn’t have anything to schedule for a post the 18th. The 18th came through, though, and now I’ve got enough to talk about. And that before I get to reading today’s comics. So, please, enjoy.

Scott Adams’s Dilbert Classics for the 16th of November (originally published the 21st of September, 1992) features Dilbert discovering Bell’s Theorem. Bell’s Theorem is an important piece of our understanding of quantum mechanics. It’s a theorem that excites people who first hear about it. It implies quantum mechanics can’t explain reality unless it can allow information to be transmitted between interacting particles faster than light. And quantum mechanics does explain reality. The thing is, and the thing that casual readers don’t understand, is that there’s no way to use this to send a signal. Imagine that I took two cards, one an ace and one an eight, seal them in envelopes, and gave them to astronauts. The astronauts each travel to ten light-years away from me in opposite directions. (They took extreme offense at something I said and didn’t like one another anyway.) Then one of them opens her envelope, finding that she’s got the eight. Then instantly, even though they’re twenty light-years apart, she knows the other astronaut has an ace in her envelope. But there is no way the astronauts can use this to send information to one another, which is what people want Bell’s Theorem to tell us. (My example is not legitimate quantum mechanics and do not try to use it to pass your thesis defense. It just shows why Bell’s Theorem does not give us a way to send information we care about faster than light.) The next day Dilbert’s Garbageman, the Smartest Man in the World, mentions Dilbert’s added something to Bell’s Theorem. It’s the same thing everybody figuring they can use quantum entanglement to communicate adds to the idea.

Tom Thaves’ Frank and Ernest for the 16th of November riffs on the idea of a lottery as a “tax on people who are bad at math”. Longtime readers here know that I have mixed feelings about that, and not just because I’m wary of cliché. If the jackpot is high enough, you can reach the point where the expectation value of the prize is positive. That is, you would expect to make money if you played the game under the same conditions often enough. But that chance is still vanishingly small. Even playing a million times would not make it likely you would more earn money than you spent. I’m not dogmatic enough to say what your decision should be, at least if the prize is big enough. (And that’s not considering the value placed on the fun of playing. One may complain that it shouldn’t be any fun to buy a soon-to-be-worthless ticket. But many people do enjoy it and I can’t bring myself to say they’re all wrong about feeling enjoyment.)

And it happens that on the 18th Brant Parker and Johnny Hart’s Wizard of Id Classics (originally run the 20th of November, 1965) did a lottery joke. That one is about a lottery one shouldn’t play, except that the King keeps track of who refuses to buy a ticket. I know when we’re in a genre.

Peter Mann’s The Quixote Syndrome for the 16th of November explores something I had never known but that at least the web seems to think is true. Apparently in 1958 Samuel Beckett knew the 12-year-old André Roussimoff. People of my age cohort have any idea who that is when they hear Roussimoff became pro wrestling star André the Giant. And Beckett drove the kid to school. Mann — taking, I think, a break from his usual adaptations of classic literature — speculates on what they might have talked about. His guess: Beckett attempting to ease one of his fears through careful study and mathematical treatment. The problem is goofily funny. But the treatment is the sort of mathematics everyone understands needing and understands using.

John Deering’s Strange Brew for the 17th of November tells a rounding up joke. Scott Hilburn’s The Argyle Sweater told it back in August. I suspect the joke is just in the air. Most jokes were formed between 1922 and 1978 anyway, and we’re just shuffling around the remains of that fruitful era.

Tony Cochrane’s Agnes for the 18th of November tells a resisting-the-word-problem joke. I admit expecting better from Cochrane. But casting arithmetic problems into word problems is fraught with peril. It isn’t enough to avoid obsolete references. (If we accept trains as obsolete. I’m from the United States Northeast, where subways and even commuter trains are viable things.) The problem also has to ask something the problem-solver can imagine wanting to know. It may not matter whether the question asks how far apart two trains, two cars, or two airplanes are, if the student can’t see their distance as anything but trivia. We may need better practice in writing stories if we’re to write story problems.

• #### ivasallay 7:09 pm on Thursday, 19 November, 2015 Permalink | Reply

“One may complain that it shouldn’t be any fun to buy a soon-to-be-worthless ticket.” You may not want to tell people that, but I think it’s a very good point. My favorite, believe it or not, was the rounding up comic.

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• #### Joseph Nebus 4:18 am on Friday, 20 November, 2015 Permalink | Reply

I certainly believe you about the rounding up comic. It’s one of those kinds of jokes that puts the punch line so close to the setup that you have to go back and notice where thing happened, and that’s reliably disorienting and fun.

I understand the reasoning that a lottery ticket should be a completely irrational purchase and that one shouldn’t get pleasure from buying one. But I’m not sure I can draw a distinction between buying a ticket and spending one or two dollars on any other short-lived consumable item. We don’t regard it as inherently stupid that someone might, say, buy a pack of toy gun blasting caps and throw them on the ground to make a couple bangs. Making it a purchase of a chance of money somehow offends people who don’t share the thrill.

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## Reading the Comics, November 1, 2015: Uncertainty and TV Schedules Edition

Brian Fies’s Mom’s Cancer is a heartbreaking story. It’s compelling reading, but people who are emotionally raw from lost love ones, or who know they’re particularly sensitive to such stories, should consider before reading that the comic is about exactly what the title says.

But it belongs here because in the October 29th and the November 2nd installments are about a curiosity of area, and volume, and hypervolume, and more. That is that our perception of how big a thing is tends to be governed by one dimension, the length or the diameter of the thing. But its area is the square of that, its volume the cube of that, its hypervolume some higher power yet of that. So very slight changes in the diameter produce great changes in the volume. Conversely, though, great changes in volume will look like only slight changes. This can hurt.

Tom Toles’s Randolph Itch, 2 am from the 29th of October is a Roman numerals joke. I include it as comic relief. The clock face in the strip does depict 4 as IV. That’s eccentric but not unknown for clock faces; IIII seems to be more common. There’s not a clear reason why this should be. The explanation I find most nearly convincing is an aesthetic one. Roman numerals are flexible things, and can be arranged for artistic virtue in ways that Arabic numerals make impossible.

The aesthetic argument is that the four-character symbol IIII takes up nearly as much horizontal space as the VIII opposite it. The two-character IV would look distractingly skinny. Now, none of the symbols takes up exactly the same space as their counterpart. X is shorter than II, VII longer than V. But IV-versus-VIII does seem like the biggest discrepancy. Still, Toles’s art shows it wouldn’t look all that weird. And he had to conserve line strokes, so that the clock would read cleanly in newsprint. I imagine he also wanted to avoid using different representations of “4” so close together.

Jon Rosenberg’s Scenes From A Multiverse for the 29th of October is a riff on both quantum mechanics — Schödinger’s Cat in a box — and the uncertainty principle. The uncertainty principle can be expressed as a fascinating mathematical construct. It starts with Ψ, a probability function that has spacetime as its domain, and the complex-valued numbers as its range. By applying a function to this function we can derive yet another function. This function-of-a-function we call an operator, because we’re saying “function” so much it’s starting to sound funny. But this new function, the one we get by applying an operator to Ψ, tells us the probability that the thing described is in this place versus that place. Or that it has this speed rather than that speed. Or this angular momentum — the tendency to keep spinning — versus that angular momentum. And so on.

If we apply an operator — let me call it A — to the function Ψ, we get a new function. What happens if we apply another operator — let me call it B — to this new function? Well, we get a second new function. It’s much the way if we take a number, and multiply it by another number, and then multiply it again by yet another number. Of course we get a new number out of it. What would you expect? This operators-on-functions things looks and acts in many ways like multiplication. We even use symbols that look like multiplication: AΨ is operator A applied to function Ψ, and BAΨ is operator B applied to the function AΨ.

Now here is the thing we don’t expect. What if we applied operator B to Ψ first, and then operator A to the product? That is, what if we worked out ABΨ? If this was ordinary multiplication, then, nothing all that interesting. Changing the order of the real numbers we multiply together doesn’t change what the product is.

Operators are stranger creatures than real numbers are. It can be that BAΨ is not the same function as ABΨ. We say this means the operators A and B do not commute. But it can be that BAΨ is exactly the same function as ABΨ. When this happens we say that A and B do commute.

Whether they do or they don’t commute depends on the operators. When we know what the operators are we can say whether they commute. We don’t have to try them out on some functions and see what happens, although that sometimes is the easiest way to double-check your work. And here is where we get the uncertainty principle from.

The operator that lets us learn the probability of particles’ positions does not commute with the operator that lets us learn the probability of particles’ momentums. We get different answers if we measure a particle’s position and then its velocity than we do if we measure its velocity and then its position. (Velocity is not the same thing as momentum. But they are related. There’s nothing you can say about momentum in this context that you can’t say about velocity.)

The uncertainty principle is a great source for humor, and for science fiction. It seems to allow for all kinds of magic. Its reality is no less amazing, though. For example, it implies that it is impossible for an electron to spiral down into the nucleus of an atom, collapsing atoms the way satellites eventually fall to Earth. Matter can exist, in ways that let us have solid objects and chemistry and biology. This is at least as good as a cat being perhaps boxed.

Jan Eliot’s Stone Soup Classics for the 29th of October is a rerun from 1995. (The strip itself has gone to Sunday-only publication.) It’s a joke about how arithmetic is easy when you have the proper motivation. In 1995 that would include catching TV shows at a particular time. You see, in 1995 it was possible to record and watch TV shows when you wanted, but it required coordinating multiple pieces of electronics. It would often be easier to just watch when the show actually aired. Today we have it much better. You can watch anything you want anytime you want, using any piece of consumer electronics you have within reach, including several current models of microwave ovens and programmable thermostats. This does, sadly, remove one motivation for doing arithmetic. Also, I’m not certain the kids’ TV schedule is actually consistent with what was on TV in 1995.

Oh, heck, why not. Obviously we’re 14 minutes before the hour. Let me move onto the hour for convenience. It’s 744 minutes to the morning cartoons; that’s 12.4 hours. Taking the morning cartoons to start at 8 am, that means it’s currently 14 minutes before 24 minutes before 8 pm. I suspect a rounding error. Let me say they’re coming up on 8 pm. 194 minutes to Jeopardy implies the game show is on at 11 pm. 254 minutes to The Simpsons puts that on at midnight, which is probably true today, though I don’t think it was so in 1995 just yet. 284 minutes to Grace puts that on at 12:30 am.

I suspect that Eliot wanted it to be 978 minutes to the morning cartoons, which would bump Oprah to 4:00, Jeopardy to 7:00, Simpsons and Grace to 8:00 and 8:30, and still let the cartoons begin at 8 am. Or perhaps the kids aren’t that great at arithmetic yet.

Stephen Beals’s Adult Children for the 30th of October tries to build a “math error” out of repeated use of the phrase “I couldn’t care less”. The argument is that the thing one cares least about is unique. But why can’t there be two equally least-cared-about things?

We can consider caring about things as an optimization problem. Optimization problems are about finding the most of something given some constraints. If you want the least of something, multiply the thing you have by minus one and look for the most of that. You may giggle at this. But it’s the sensible thing to do. And many things can be equally high, or low. Take a bundt cake pan, and drizzle a little water in it. The water separates into many small, elliptic puddles. If the cake pan were perfectly formed, and set on a perfectly level counter, then the bottom of each puddle would be at the same minimum height. I grant a real cake pan is not perfect; neither is any counter. But you can imagine such.

Just because you can imagine it, though, must it exist? Think of the “smallest positive number”. The idea is simple. Positive numbers are a set of numbers. Surely there’s some smallest number. Yet there isn’t; name any positive number and we can name a smaller number. Divide it by two, for example. Zero is smaller than any positive number, but it’s not itself a positive number. A minimum might not exist, at least not within the confines where we are to look. It could be there is not something one could not care less about.

So a minimum might or might not exist, and it might or might not be unique. This is why optimization problems are exciting, challenging things.

Niklas Eriksson’s Carpe Diem for the 1st of November, 2015. I’m not sure how accurately the art depicts bedbugs, although I’m also not sure how accurately Eriksson should.

Niklas Eriksson’s Carpe Diem for the 1st of November is about understanding the universe by way of observation and calculation. We do rely on mathematics to tell us things about the universe. Immanuel Kant has a bit of reputation in mathematical physics circles for this observation. (I admit I’ve never seen the original text where Kant observed this, so I may be passing on an urban legend. My love has several thousands of pages of Kant’s writing, but I do not know if any of them touch on natural philosophy.) If all we knew about space was that gravitation falls off as the square of the distance between two things, though, we could infer that space must have three dimensions. Otherwise that relationship would not make geometric sense.

Jeff Harris’s kids-information feature Shortcuts for the 1st of November was about the Harvard Computers. By this we mean the people who did the hard work of numerical computation, back in the days before this could be done by electrical and then electronic computer. Mathematicians relied on people who could do arithmetic in those days. There is the folkloric belief that mathematicians are inherently terrible at arithmetic. (I suspect the truth is people assume mathematicians must be better at arithmetic than they really are.) But here, there’s the mathematics of thinking what needs to be calculated, and there’s the mathematics of doing the calculations.

Their existence tends to be mentioned as a rare bit of human interest in numerical mathematics books, usually in the preface in which the author speaks with amazement of how people who did computing were once called computers. I wonder if books about font and graphic design mention how people who typed used to be called typewriters in their prefaces.

• #### ivasallay 11:13 pm on Wednesday, 4 November, 2015 Permalink | Reply

I wouldn’t have seen any of these without your blog. Thank you for including all of them.
Mom’s Cancer is sad but appears to be slightly improving. I hope for remission.
Adult Children makes an awesome point.

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• #### Joseph Nebus 1:16 am on Friday, 6 November, 2015 Permalink | Reply

I’m glad you enjoy. (I’m assuming enjoy.) Part of what’s fun about doing these, besides that it provokes me to write about stuff I didn’t plan ahead of time to do, is that I get to read a great diversity of comic strips. And sometimes introduce people to comics they had no idea existed.

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• #### sheldonk2014 11:01 am on Monday, 16 November, 2015 Permalink | Reply

Hey Joseph
Thank you for visiting
As always Sheldon

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• #### Joseph Nebus 4:07 am on Tuesday, 17 November, 2015 Permalink | Reply

Quite welcome. Glad to see you again.

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## 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 14, 2015: Back To School Edition, Part II

Today’s mathematical-comics blog should get us up to the present. Or at least to Monday. Yes, I’m aware of which paradox of Zeno this evokes. Be nice.

Scott Adams’s Dilbert Classics for the 11th of September is a rerun from, it looks like, the 18th of July, 1992. Anyway, Dilbert has acquired a supercomputer and figures to calculate π to a lot of decimal places, to finally do something about the areas of circles. The calculation of the digits of pi is done, often on home-brewed supercomputers, to lots of digits. But the calculation of areas, or volumes, or circumferences or whatever, isn’t why. We just don’t need that many digits; forty digits of pi would be plenty for almost any calculation involving measuring things in the real world.

It’s actually a bit mysterious why the digits of pi should be worth bothering with. It’s not yet known if pi is a “normal” number, and that’s of some interest. In a normal number every finite sequence of digits appears, and as often as every other sequence of digits just as long. That is, if you break the digits of pi up into four-digit blocks, you should see “1701” and “2038” and “2468” and “9999” and all, each appearing roughly once per thousand blocks. It’s almost certain that pi is normal, because just about every number is. But it’s not proven. And if there were numerical evidence that pi wasn’t normal that would be mathematically interesting, though I wouldn’t blame everybody not in number theory for saying “huh” and “so what?” before moving on. As it is, calculating digits of pi is a good, challenging task that can be used to prove coding and computing abilities, and it might turn up something interesting. It may as well be that as anything.

Steve Breen and Mike Thompson’s Grand Avenue for the 11th of September is a “motivate the word problem” joke. So is Bill Amend’s FoxTrot Classics for the 14th (a rerun from the same day in 2004). I like Amend’s version better, partly because it gives more realistic problems. I also like that it mixes in a bit of French class. It’s not always mathematics that needs motivation.

J C Duffy’s Lug Nuts for the 11th of September is another pun strip badly timed for Pi Day.

Ruben Bolling’s Tom The Dancing Bug gave us a Super Fun-Pak Comics installment on the 11th. And that included a Chaos Butterfly installment pitting deterministic chaos against Schrödinger’s Cat. The Cat represents one (of many) interpretations of quantum mechanics, the “superposition” interpretation. It’s difficult to explain the idea philosophically, to say what is really going on. The mathematics is straightforward, though. In the most common model of quantum mechanics we describe what is going on by a probability distribution, a function that describes how likely each possible outcome is. Quantum mechanics describes how that distribution changes in time. In the superpositioning we have two, or more, probability distributions that describe very different states, but (in a way) averaged together. The changes of this combined distribution then become our idea of how the system changes in time. It’s just hard to say what it could mean when the superposition implies very different things, like a cat being both wet and dry, being equally true at once.

Julie Larson’s Dinette Set for the 12th of September is about double negatives. It’s also about the doomed attempt to bring logic to the constructions of English. At least in English a double negative — “not unwanted”, say — generally parses to a positive, even if the connotation is that the thing is only a bit wanted. This corresponds to what logicians would say. A logican might use “C” to stand in for some statement that can only be true or false. Then, saying “not not C” — an “is true” gets implicitly added to the end of that — is equivalent to saying “C [is true]”. My love, the philosopher, who knows much more Spanish than I do has pointed out that in Spanish the “not not” construction can intensify the strength of the negation, rather than annulling it. This causes us to wonder if Spanish-speaking logic students have a harder time understanding the “not not C” construction. I don’t know and would welcome insight. (Also I hope I have it right that a “not not” is an intensifier, rather than a softener. But I suppose it doesn’t matter, as long as the Spanish equivalent of “not not wanted” still connotes “unwanted”.)

Dan Collins’s Looks Good On Paper for the 12th of September is a simple early-autumn panorama kind of strip. Mathematics — particularly, geometry — gets used as the type case for elementary school. I suppose as long as diagramming sentences is out of fashion there’s no better easy-to-draw choice.

David L Hoyt and Jeff Knurek’s Jumble for the 14th of September.

David L Hoyt and Jeff Knurek’s Jumble for the 14th of September is an abacus joke. For folks who want to do the Jumble themselves, a hint: the second word is not “Dummy” however appealing an unscramble that looks.

Stephen Beals’s Adult Children for the 14th builds on the idea of what if the universe were made wrong. And that’s expressed as a mathematics error in the building of the universe. The idea of mathematics as a transcendent and even god-touching thing is an old one. I imagine this reflects how a logically deduced fact has some kind of universal truth, that a sound argument is sound regardless of who makes it, or considers it. It’s a heady idea. Mathematics also allows us to say some very specific, and remarkable, things about the infinite. This is another god-touching notion. But we don’t have sound reason to think that universe-making must be mathematical. Mathematics can describe many aspects of the universe eerily well, yes. But is it necessary that a universe be mathematically consistent? The question seems to defy any kind of empirical answer; we must listen to philosophers, who can at least give us a reasoned answer.

Tom Thaves’s Frank and Ernest for the 14th of September depicts cave-Frank and cave-Ernest at the dawn of numbers. It suggests the symbol 1 being a representation of a stick, and 0 as a stone. The 1 as a stick seems at least imaginable; counting off things by representing them as sticks or as stroke marks feels natural. Of course I say that coming from a long heritage of doing just that. 0, as I understand it, seems to derive from making with a dot a place where zero of whatever was to be studied should appear; the dot grew into a loop probably to make it harder to miss.

• #### ivasallay 5:30 pm on Tuesday, 15 September, 2015 Permalink | Reply

I really liked Foxtrot, the Jumble, and Frank and Ernest. I would not have seen any of them if you hadn’t written this post. Thank you!

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• #### Joseph Nebus 12:17 am on Friday, 18 September, 2015 Permalink | Reply

Happy to be of service. Any luck on the Jumble words?

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• #### ivasallay 4:28 pm on Monday, 28 September, 2015 Permalink | Reply

I don’t usually do the Jumbles, but since you recommended it:
WAger
Muddy
AsSure
BoUncE
AWE-SUM

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## Reading the Comics, June 21, 2015: Blatantly Padded Edition, Part 2

I said yesterday I was padding one mathematics-comics post into two for silly reasons. And I was. But there were enough Sunday comics on point that splitting one entry into two has turned out to be legitimate. Nice how that works out sometimes.

Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. (June 19) uses mathematics as something to heap upon a person until they yield to your argument. It’s a fallacious way to argue, but it does work. Even at a mathematical conference the terror produced by a screen full of symbols can chase follow-up questions away. On the 21st, they present mathematics as a more obviously useful thing. Well, mathematics with a bit of physics.

Nate Frakes’s Break Of Day (June 19) is this week’s anthropomorphic algebra joke.

Niklas Eriksson’s Carpe Diem for the 20th of June, 2015.

Niklas Eriksson’s Carpe Diem (June 20) is captioned “Life at the Quantum Level”. And it’s built on the idea that quantum particles could be in multiple places at once. Whether something can be in two places at once depends on coming up with a clear idea about what you mean by “thing” and “places” and for that matter “at once”; when you try to pin the ideas down they prove to be slippery. But the mathematics of quantum mechanics is fascinating. It cries out for treating things we would like to know about, such as positions and momentums and energies of particles, as distributions instead of fixed values. That is, we know how likely it is a particle is in some region of space compared to how likely it is somewhere else. In statistical mechanics we resort to this because we want to study so many particles, or so many interactions, that it’s impractical to keep track of them all. In quantum mechanics we need to resort to this because it appears this is just how the world works.

(It’s even less on point, but Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 21st of June has a bit of riffing on Schrödinger’s Cat.)

Brian and Ron Boychuk’s Chuckle Brothers (June 20) name-drops algebra as the kind of mathematics kids still living with their parents have trouble with. That’s probably required by the desire to make a joking definition of “aftermath”, so that some specific subject has to be named. And it needs parents to still be watching closely over their kids, something that doesn’t quite fit for college-level classes like Intro to Differential Equations. So algebra, geometry, or trigonometry it must be. I am curious whether algebra reads as the funniest of that set of words, or if it just fits better in the space available. ‘Geometry’ is as long a word as ‘algebra’, but it may not have the same connotation of being an impossibly hard class.

Jimmy Hatlo’s Little Iodine for the 18th of April, 1954, and rerun the 18th of June, 2015.

And from the world of vintage comic strips, Jimmy Hatlo’s Little Iodine (June 21, originally run the 18th of April, 1954) reminds us that anybody can do any amount of arithmetic if it’s something they really want to calculate.

Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney (June 21) is another strip using the idea of mathematics — and particularly word problems — to signify great intelligence. I suppose it’s easier to recognize the form of a word problem than it is to recognize a good paper on the humanities if you only have two dozen words to show it in.

Juba’s Viivi and Wagner (June 21) is a timely reminder that while sudokus may be fun logic puzzles, they are ultimately the puzzle you decide to make of them.

• #### ivasallay 1:08 am on Thursday, 25 June, 2015 Permalink | Reply

They’re corny, but I did like Break of Day and Chuckle Brothers a lot.

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• #### Joseph Nebus 3:50 am on Thursday, 25 June, 2015 Permalink | Reply

They are corny, yeah, but fun. I think Break of Day the better of that pair because it’s more character-based.

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• #### BunKaryudo 3:27 pm on Monday, 30 November, 2015 Permalink | Reply

I love that Life at the Quantum Level cartoon. :)

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• #### Joseph Nebus 9:43 pm on Thursday, 3 December, 2015 Permalink | Reply

I’m tickled by it myself.

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• #### BunKaryudo 1:48 pm on Friday, 4 December, 2015 Permalink | Reply

It is funny. :)

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## Reading the Comics, March 15, 2015: Pi Day Edition

I had kind of expected the 14th of March — the Pi Day Of The Century — would produce a flurry of mathematics-themed comics. There were some, although they were fewer and less creatively diverse than I had expected. Anyway, between that, and the regular pace of comics, there’s plenty for me to write about. Recently featured, mostly on Gocomics.com, a little bit on Creators.com, have been:

Brian Anderson’s Dog Eat Doug (March 11) features a cat who claims to be “pondering several quantum equations” to prove something about a parallel universe. It’s an interesting thing to claim because, really, how can the results of an equation prove something about reality? We’re extremely used to the idea that equations can model reality, and that the results of equations predict real things, to the point that it’s easy to forget that there is a difference. A model’s predictions still need some kind of validation, reason to think that these predictions are meaningful and correct when done correctly, and it’s quite hard to think of a meaningful way to validate a predication about “another” universe.

• #### ivasallay 6:23 pm on Monday, 16 March, 2015 Permalink | Reply

I actually liked the out of order joke!
My favorite pi comics were Long Story Short and Working Daze.

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• #### Joseph Nebus 9:09 pm on Tuesday, 17 March, 2015 Permalink | Reply

The phone out of order joke most tickled me. It’s probably because of the quiet delivery.

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## Reading The Comics, November 9, 2014: Finally, A Picture Edition

I knew if I kept going long enough some cartoonist not on Gocomics.com would have to mention mathematics. That finally happened with one from Comics Kingdom, and then one from the slightly freak case of Rick Detorie’s One Big Happy. Detorie’s strip is on Gocomics.com, but a rerun from several years ago. He has a different one that runs on the normal daily pages. This is for sound economic reasons: actual newspapers pay much better than the online groupings of them (considering how cheap Comics Kingdom and Gocomics are for subscribers I’m not surprised) so he doesn’t want his current strips run on Gocomics.com. As for why his current strips do appear on, for example, the fairly good online comics page of AZcentral.com, that’s a good question, and one that deserves a full answer.

Vic Lee’s Pardon My Planet for the 6th of November, 2014.

Vic Lee’s Pardon My Planet (November 9), which broke the streak of Comics Kingdom not making it into these pages, builds around a quote from Einstein I never heard of before but which sounds like the sort of vaguely inspirational message that naturally attaches to famous names. The patient talks about the difficulty of finding something in “the middle of four-dimensional curved space-time”, although properly speaking it could be tricky finding anything within a bounded space, whether it’s curved or not. The generic mathematics problem you’d build from this would be to have some function whose maximum in a region you want to find (if you want the minimum, just multiply your function by minus one and then find the maximum of that), and there’s multiple ways to do that. One obvious way is the mathematical equivalent of getting to the top of a hill by starting from wherever you are and walking the steepest way uphill. Another way is to just amble around, picking your next direction at random, always taking directions that get you higher and usually but not always refusing directions that bring you lower. You can probably see some of the obvious problems with either approach, and this is why finding the spot you want can be harder than it sounds, even if it’s easy to get started looking.

Reuben Bolling’s Super Fun-Pak Comix (November 6), which is technically a rerun since the Super Fun-Pak Comix have been a longrunning feature in his Tom The Dancing Bug pages, is primarily a joke about the Heisenberg Uncertainty Principle, that there is a limit to what information one can know about the universe. This limit can be understood mathematically, though. The wave formulation of quantum mechanics describes everything there is to know about a system in terms of a function, called the state function and normally designated Ψ, the value of which can vary with location and time. Determining the location or the momentum or anything about the system is done by a process called “applying an operator to the state function”. An operator is a function that turns one function into another, which sounds like pretty sophisticated stuff until you learn that, like, “multiply this function by minus one” counts.

In quantum mechanics anything that can be observed has its own operator, normally a bit tricker than just “multiply this function by minus one” (although some are not very much harder!), and applying that operator to the state function is the mathematical representation of making that observation. If you want to observe two distinct things, such as location and momentum, that’s a matter of applying the operator for the first thing to your state function, and then taking the result of that and applying the operator for the second thing to it. And here’s where it gets really interesting: it doesn’t have to, but it can depend what order you do this in, so that you get different results applying the first operator and then the second from what you get applying the second operator and then the first. The operators for location and momentum are such a pair, and the result is that we can’t know to arbitrary precision both at once. But there are pairs of operators for which it doesn’t make a difference. You could, for example, know both the momentum and the electrical charge of Scott Baio simultaneously to as great a precision as your Scott-Baio-momentum-and-electrical-charge-determination needs are, and the mathematics will back you up on that.

Ruben Bolling’s Tom The Dancing Bug (November 6), meanwhile, was a rerun from a few years back when it looked like the Large Hadron Collider might never get to working and the glitches started seeming absurd, as if an enormous project involving thousands of people and millions of parts could ever suffer annoying setbacks because not everything was perfectly right the first time around. There was an amusing notion going around, illustrated by Bolling nicely enough, that perhaps the results of the Large Hadron Collider would be so disastrous somehow that the universe would in a fit of teleological outrage prevent its successful completion. It’s still a funny idea, and a good one for science fiction stories: Isaac Asimov used the idea in a short story dubbed “Thiotimoline and the Space Age”, published 1959, which resulted in the attempts to manipulate a compound which dissolves before it adds water might have accidentally sent hurricanes Carol, Edna, and Diane into New England in 1954 and 1955.

Chip Sansom’s The Born Loser (November 7) gives me a bit of a writing break by just being a pun strip that you can save for next March 14.

Dan Thompson’s Brevity (November 7), out of reruns, is another pun strip, though with giant monsters.

Francesco Marciuliano’s Medium Large (November 7) is about two of the fads of the early 80s, those of turning everything into a breakfast cereal somehow and that of playing with Rubik’s Cubes. Rubik’s Cubes have long been loved by a certain streak of mathematicians because they are a nice tangible representation of group theory — the study of things that can do things that look like addition without necessarily being numbers — that’s more interesting than just picking up a square and rotating it one, two, three, or four quarter-turns. I still think it’s easier to just peel the stickers off (and yet, the die-hard Rubik’s Cube Popularizer can point out there’s good questions about polarity you can represent by working out the rules of how to peel off only some stickers and put them back on without being detected).

Rick Detorie’s One Big Happy for the 9th of November, 2014.

Rick Detorie’s One Big Happy (November 9), and I’m sorry, readers about a month in the future from now, because that link’s almost certainly expired, is another entry in the subject of word problems resisted because the thing used to make the problem seem less abstract has connotations that the student doesn’t like.

Fred Wagner’s Animal Crackers (November 9) is your rare comic that could be used to teach positional notation, although when you actually pay attention you realize it doesn’t actually require that.

Mac and Bill King’s Magic In A Minute (November 9) shows off a mathematically-based slight-of-hand trick, describing a way to make it look like you’re reading your partner-monkey’s mind. This is probably a nice prealgebra problem to work out just why it works. You could also consider this a toe-step into the problem of encoding messages, finding a way to send information about something in a way that the original information can be recovered, although obviously this particular method isn’t terribly secure for more than a quick bit of stage magic.

## Realistic Modeling

“Economic Realism (Wonkish)”, a blog entry by Paul Krugman in The New York Times, discusses a paper, “Chameleons: The Misuse Of Mathematical Models In Finance And Economics”, by Paul Pfleiderer of Stanford University, which surprises me by including a color picture of a chameleon right there on the front page, and in an academic paper at that, and I didn’t know you could have color pictures included just for their visual appeal in academia these days. Anyway, Pfleiderer discusses the difficulty of what they term filtering, making sure that the assumptions one makes to build a model — which are simplifications and abstractions of the real-world thing in which you’re interested — aren’t too far out of line with the way the real thing behaves.

This challenge, which I think of as verification or validation, is important when you deal with pure mathematical or physical models. Some of that will be at the theoretical stage: is it realistic to model a fluid as if it had no viscosity? Unless you’re dealing with superfluid helium or something exotic like that, no, but you can do very good work that isn’t too far off. Or there’s a classic model of the way magnetism forms, known as the Ising model, which in a very special case — a one-dimensional line — is simple enough that a high school student could solve it. (Well, a very smart high school student, one who’s run across an exotic function called the hyperbolic cosine, could do it.) But that model is so simple that it can’t model the phase change, that, if you warm a magnet up past a critical temperature it stops being magnetic. Is the model no good? If you aren’t interested in the phase change, it might be.

And then there is the numerical stage: if you’ve set up a computer program that is supposed to represent fluid flow, does it correctly find solutions? I’ve heard it claimed that the majority of time spent on a numerical project is spent in validating the results, and that isn’t even simply in finding and fixing bugs in the code. Even once the code is doing perfectly what we mean it to do, it must be checked that what we mean it to do is relevant to what we want to know.

Pfleiderer’s is an interesting paper and I think worth the read; despite its financial mathematics focus (and a brief chat about quantum mechanics) it doesn’t require any particularly specialized training. There’s some discussions of particular financial models, but what’s important are the assumptions being made behind those models, and those are intelligible without prior training in the field.

• #### elkement 12:30 pm on Wednesday, 2 April, 2014 Permalink | Reply

Thanks a lot for this pointer! (Mis-)used of physics analogies and related models in finance is something I am very interested in.

As fan of Nassim Taleb’s writings I am totally wary of such models. I would like to learn more about the history of modeling though, and I might read “The Physics of Wall Street” by James Owen Weatherall someday.

My current bias is that Wall Street quants are often former physicists who still indulge in playing with some intricate differential equations as they did in academia. But the consequences – when their calculations finally have an impact on real-world financial products – are more far-reaching.

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• #### Joseph Nebus 8:30 pm on Wednesday, 2 April, 2014 Permalink | Reply

I’ve gotten awfully interested in mathematical finance but fear I’ve got too much basic material to learn to get to the really interesting stuff.

My only real squabble with the Wall Street quants, apart from their destroying the world’s economy, is that their existence has given my father the idea that since I know mathematics, and they do mathematics, I just need to whip up an algorithm and start collecting quarter-million-dollar paychecks. As best I can figure it isn’t quite that straightforward, unfortunately.

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## Reading the Comics, March 1, 2014: Isn’t It One-Half X Squared Plus C? Edition

So the subject line references here a mathematics joke that I never have heard anybody ever tell, and only encounter in lists of mathematics jokes. It goes like this: a couple professors are arguing at lunch about whether normal people actually learn anything about calculus. One of them says he’s so sure normal people learn calculus that even their waiter would be able to answer a basic calc question, and they make a bet on that. He goes back and finds their waiter and says, when she comes with the check he’s going to ask her if she knows what the integral of x is, and she should just say, “why, it’s one-half x squared, of course”. She agrees. He goes back and asks her what the integral of x is, and she says of course it’s one-half x squared, and he wins the bet. As he’s paid off, she says, “But excuse me, professor, isn’t it one-half x squared plus C?”

Let me explain why this is an accurately structured joke construct and must therefore be classified as funny. “The integral of x”, as the question puts it, has not just one correct answer but rather a whole collection of correct answers, which are different from one another only by the addition of a constant whole number, by convention denoted C, and the inclusion of that “plus C” denotes that whole collection. The professor was being sloppy in referring to just a single example from that collection instead of the whole set, as the waiter knew to do. You’ll see why this is relevant to today’s collection of mathematics-themed comics.

Jef Mallet’s Frazz (February 22) points out one of the grand things about mathematics, that if you follow the proper steps in a mathematical problem you get to be right, and to be extraordinarily confident in that rightness. And that’s true, although, at least to me a good part of what’s fun in mathematics is working out what the proper steps are: figuring out what the important parts of something you want to study should be, and what follows from your representation of them, and — particularly if you’re trying to represent a complicated real-world phenomenon with a model — whether you’re representing the things you find interesting in the real-world phenomenon well. So, while following the proper steps gets you an answer that is correct within the limits of whatever it is you’re doing, you still get to work out whether you’re working on the right problem, which is the real fun.

Mark Pett’s Lucky Cow (February 23, rerun) uses that ambiguous place between mathematics and physics to represent extreme smartness. The equation the physicist brings to Neil is the (time-dependent) Schrödinger Equation, describing how probability evolves in time, and the answer is correct. If Neil’s coworkers at Lucky Cow were smarter they’d realize the scam, though: while the equation is impressively scary-looking to people not in the know, a particle physicist would have about as much chance of forgetting this as of forgetting the end of “E equals m c … ”.

Hilary Price’s Rhymes With Orange (February 24) builds on the familiar infinite-monkeys metaphor, but misses an important point. Price is right that yes, an infinite number of monkeys already did create the works of Shakespeare, as a result of evolving into a species that could have a Shakespeare. But the infinite monkeys problem is about selecting letters at random, uniformly: the letter following “th” is as likely to be “q” as it is to be “e”. An evolutionary system, however, encourages the more successful combinations in each generation, and discourages the less successful: after writing “th” Shakespeare would be far more likely to put “e” and never “q”, which makes calculating the probability rather less obvious. And Shakespeare was writing with awareness that the words mean things and they must be strings of words which make reasonable sense in context, which the monkeys on typewriters would not. Shakespeare could have followed the line “to be or not to be” with many things, but one of the possibilities would never be “carport licking hammer worbnoggle mrxl 2038 donkey donkey donkey donkey donkey donkey donkey”. The typewriter monkeys are not so selective.

Dan Thompson’s Brevity (February 26) is a cute joke about a number’s fashion sense.

Mark Pett’s Lucky Cow turns up again (February 28, rerun) for the Rubik’s Cube. The tolerably fun puzzle and astoundingly bad Saturday morning cartoon of the 80s can be used to introduce abstract algebra. When you rotate the nine little cubes on the edge of a Rubik’s cube, you’re doing something which is kind of like addition. Think of what you can do with the top row of cubes: you can leave it alone, unchanged; you can rotate it one quarter-turn clockwise; you can rotate it one quarter-turn counterclockwise; you can rotate it two quarter-turns clockwise; you can rotate it two quarter-turns counterclockwise (which will result in something suspiciously similar to the two quarter-turns clockwise); you can rotate it three quarter-turns clockwise; you can rotate it three quarter-turns counterclockwise.

If you rotate the top row one quarter-turn clockwise, and then another one quarter-turn clockwise, you’ve done something equivalent to two quarter-turns clockwise. If you rotate the top row two quarter-turns clockwise, and then one quarter-turn counterclockwise, you’ve done the same as if you’d just turned it one quarter-turn clockwise and walked away. You’re doing something that looks a lot like addition, without being exactly like it. Something odd happens when you get to four quarter-turns either clockwise or counterclockwise, particularly, but it all follows clear rules that become pretty familiar when you notice how much it’s like saying four hours after 10:00 will be 2:00.

Abstract algebra marks one of the things you have to learn as a mathematics major that really changes the way you start looking at mathematics, as it really stops being about trying to solve equations of any kind. You instead start looking at how structures are put together — rotations are seen a lot, probably because they’re familiar enough you still have some physical intuition, while still having significant new aspects — and, following this trail can get for example to the parts of particle physics where you predict some exotic new subatomic particle has to exist because there’s this structure that makes sense if it does.

Jenny Campbell’s Flo and Friends (March 1) is set off with the sort of abstract question that comes to mind when you aren’t thinking about mathematics: how many five-card combinations are there in a deck of (52) cards? Ruthie offers an answer, although — as the commenters get to disputing — whether she’s right depends on what exactly you mean by a “five-card combination”. Would you say that a hand of “2 of hearts, 3 of hearts, 4 of clubs, Jack of diamonds, Queen of diamonds” is a different one to “3 of hearts, Jack of diamonds, 4 of clubs, Queen of diamonds, 2 of hearts”? If you’re playing a game in which the order of the deal doesn’t matter, you probably wouldn’t; but, what if the order does matter? (I admit I don’t offhand know a card game where you’d get five cards and the order would be important, but I don’t know many card games.)

For that matter, if you accept those two hands as the same, would you accept “2 of clubs, 3 of clubs, 4 of diamonds, Jack of spades, Queen of spades” as a different hand? The suits are different, yes, but they’re not differently structured: you’re still three cards away from a flush, and two away from a straight. Granted there are some games in which one suit is worth more than another, in which case it matters whether you had two diamonds or two spades; but if you got the two-of-clubs hand just after getting the two-of-hearts hand you’d probably be struck by how weird it was you got the same hand twice in a row. You can’t give a correct answer to the question until you’ve thought about exactly what you mean when you say two hands of cards are different.

• #### ivasallay 5:28 am on Sunday, 2 March, 2014 Permalink | Reply

In one of my teacher education classes, the instructor said, “You can teach ANYTHING with a picture book.” Picture books can help us recall prior knowledge and give the instructor something to build upon. I think the same thing can be said about comics. I enjoyed reading all of the comics you put in this post, and I learned something new about mathematics because of several of your explanations. Thank you!

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• #### Joseph Nebus 6:32 pm on Monday, 3 March, 2014 Permalink | Reply

Aw, my, thank you.

I’ve thought about including the comics themselves as graphics in these posts. I’m fairly confident that it would qualify as fair use, but something still has me shy away from that.

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• #### elkement 10:53 am on Wednesday, 12 March, 2014 Permalink | Reply

Of course I love the Schrödinger equation joke. I thought the funny thing was that knowing a complete different equation is not knowing the particular solution. I often think about this when a “formula” is presented in movies – I guess most people think that “formulas” in physics are like Ohm’s law … just plug in some numbers.

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• #### Joseph Nebus 4:28 am on Saturday, 15 March, 2014 Permalink | Reply

You’re right; it is a funny part of the strip that the solution really can’t solve any particular problem that anyone has. I wonder if there is a specific quantum mechanics problem that could fit the comic strip’s need — it’s got to be stated in one or two panels, and answered in one, and the answer has to be something that Neil could plausibly memorize and recite without an error. That kind of suggests either a free particle or a particle in an infinite well.

Really, though, I’m invariably delighted when a cartoonist gets the equations correct. The structure of the joke would be the same if a bunch of gibberish were put up, but I appreciate the craftsmanship that goes into getting right the things that don’t need to be.

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• #### LucyJartz 1:41 pm on Wednesday, 23 July, 2014 Permalink | Reply

I thought it was funny that math geniuses grew up to be waiters who went home thrilled to finally use calculus in “real life”.

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• #### Joseph Nebus 3:41 pm on Wednesday, 23 July, 2014 Permalink | Reply

That could be some of the fun, too!

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• #### LucyJartz 4:57 pm on Wednesday, 23 July, 2014 Permalink | Reply

I needed a fun moment of not hating math to get me through homework. Thanks.

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• #### Joseph Nebus 7:34 pm on Thursday, 24 July, 2014 Permalink | Reply

You’re very welcome, and I might just steal your “a fun moment of not hating math” to be my new tagline. Thank you.

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• #### LucyJartz 8:02 pm on Thursday, 24 July, 2014 Permalink | Reply

As it inspiration, you are welcome to it.

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• #### Joseph Nebus 8:13 pm on Saturday, 26 July, 2014 Permalink | Reply

Thanks again.

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## Reading the Comics, February 21, 2014: Circumferences and Monkeys Edition

And now to finish off the bundle of mathematic comics that I had run out of time for last time around. Once again the infinite monkeys situation comes into play; there’s also more talk about circumferences than average.

Brian and Ron Boychuk’s The Chuckle Brothers (February 13) does a little wordplay on how “circumference” sounds like it could kind of be a knightly name, which I remember seeing in a minor Bugs Bunny cartoon back in the day. “Circumference” the word derives from the Latin, “circum” meaning around and “fero” meaning “to carry”; and to my mind, the really interesting question is why do we have the words “perimeter” and “circumference” when it seems like either one would do? “Circumference” does have the connotation of referring to just the boundary of a circular or roughly circular form, but why should the perimeter of circular things be so exceptional as to usefully have its own distinct term? But English is just like that, I suppose.

Paul Trapp’s Thatababy (February 13) brings back the infinite-monkey metaphor. The infinite monkeys also appear in John Deering’s Strange Brew (February 20), which is probably just a coincidence based on how successfully tossing in lots of monkeys can produce giggling. Or maybe the last time Comic Strip Master Command issued its orders it sent out a directive, “more infinite monkey comics!”

Ruben Bolling’s Tom The Dancing Bug (February 14) delivers some satirical jabs about Biblical textual inerrancy by pointing out where the Bible makes mathematical errors. I tend to think nitpicking the Bible mostly a waste of good time on everyone’s part, although the handful of arithmetic errors are a fair wedge against the idea that the text can’t have any errors and requires no interpretation or even forgiveness, with the Ezra case the stronger one. The 1 Kings one is about the circumference and the diameter for a vessel being given, and those being incompatible, but it isn’t hard to come up with a rationalization that brings them plausibly in line (you have to suppose that the diameter goes from outer wall to outer wall, while the circumference is that of an inner wall, which may be a bit odd but isn’t actually ruled out by the text), which is why I think it’s the weaker.

Bill Whitehead’s Free Range (February 16) uses a blackboard full of mathematics as a generic “this is something really complicated” signifier. The symbols as written don’t make a lot of sense, although I admit it’s common enough while working out a difficult problem to work out weird bundles of partly-written expressions or abuses of notation (like on the middle left of the board, where a bracket around several equations is shown as being less than a bracket around fewer equations), just because ideas are exploding faster than they can be written out sensibly. Hopefully once the point is proven you’re able to go back and rebuild it all in a form which makes sense, either by going into standard notation or by discovering that you have soem new kind of notation that has to be used. It’s very exciting to come up with some new bit of notation, even if it’s only you and a couple people you work with who ever use it. Developing a good way of writing a concept might be the biggest thrill in mathematics, even better than proving something obscure or surprising.

Jonathan Lemon’s Rabbits Against Magic (February 18) uses a blackboard full of mathematics symbols again to give the impression of someone working on something really hard. The first two lines of equations on 8-Ball’s board are the time-dependent Schrödinger Equations, describing how the probability distribution for something evolves in time. The last line is Euler’s formula, the curious and fascinating relationship between pi, the base of the natural logarithm e, imaginary numbers, one, and zero.

Todd Clark’s Lola (February 20) uses the person-on-an-airplane setup for a word problem, in this case, about armrest squabbling. Interesting to me about this is that the commenters get into a squabble about how airplane speeds aren’t measured in miles per hour but rather in nautical miles, although nobody not involved in air traffic control really sees that. What amuses me about this is that what units you use to measure the speed of the plane don’t matter; the kind of work you’d do for a plane-travelling-at-speed problem is exactly the same whatever the units are. For that matter, none of the unique properties of the airplane, such as that it’s travelling through the air rather than on a highway or a train track, matter at all to the problem. The plane could be swapped out and replaced with any other method of travel without affecting the work — except that airplanes are more likely than trains (let’s say) to have an armrest shortage and so the mock question about armrest fights is one in which it matters that it’s on an airplane.

Bill Watterson’s Calvin and Hobbes (February 21) is one of the all-time classics, with Calvin wondering about just how fast his sledding is going, and being interested right up to the point that Hobbes identifies mathematics as the way to know. There’s a lot of mathematics to be seen in finding how fast they’re going downhill. Measuring the size of the hill and how long it takes to go downhill provides the average speed, certainly. Working out how far one drops, as opposed to how far one travels, is a trigonometry problem. Trying the run multiple times, and seeing how the speed varies, introduces statistics. Trying to answer questions like when are they travelling fastest — at a single instant, rather than over the whole run — introduce differential calculus. Integral calculus could be found from trying to tell what the exact distance travelled is. Working out what the shortest or the fastest possible trips introduce the calculus of variations, which leads in remarkably quick steps to optics, statistical mechanics, and even quantum mechanics. It’s pretty heady stuff, but I admit, yeah, it’s math.

## From ElKement: On The Relation Of Jurassic Park and Alien Jelly Flowing Through Hyperspace

I’m frightfully late on following up on this, but ElKement has another entry in the series regarding quantum field theory, this one engagingly titled “On The Relation Of Jurassic Park and Alien Jelly Flowing Through Hyperspace”. The objective is to introduce the concept of phase space, a way of looking at physics problems that marks maybe the biggest thing one really needs to understand if one wants to be not just a physics major (or, for many parts of the field, a mathematics major) and a grad student.

As an undergraduate, it’s easy to get all sorts of problems in which, to pick an example, one models a damped harmonic oscillator. A good example of this is how one models the way a car bounces up and down after it goes over a bump, when the shock absorbers are working. You as a student are given some physical properties — how easily the car bounces, how well the shock absorbers soak up bounces — and how the first bounce went — how far the car bounced upward, how quickly it started going upward — and then work out from that what the motion will be ever after. It’s a bit of calculus and you might do it analytically, working out a complicated formula, or you might do it numerically, letting one of many different computer programs do the work and probably draw a picture showing what happens. That’s shown in class, and then for homework you do a couple problems just like that but with different numbers, and for the exam you get another one yet, and one more might turn up on the final exam.

• #### elkement 9:29 am on Wednesday, 16 October, 2013 Permalink | Reply

Thanks again, Joseph. I guess your example concerned with a car makes more sense than my rather philosophical ramblings so I will reblog your post. From questions I have got on my article I conclude that the introduction of those large number of dimensions was probably not self-explanatory, so I am working on an update to this article with more illustrations related to hyperspace. Since I have not seen Liouville equation popularized (in the same way as ‘chaos theory’ is) I have to prepare some illustrations myself which is going to take a while.

I am also considering to pick a non-physics example as I have noticed that ‘many dimensions’ in the sense of these statistical sense (different states of a single system) got confused with those fancy dimensions related to string theory. I think I need to stress more that this is a space of ‘possibilities’ and not some space which is out there (and we are just those infamouse beetles on an inflating balloon that are not able to feel those dimensions).

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• #### Joseph Nebus 3:02 am on Friday, 18 October, 2013 Permalink | Reply

The car example was actually on my mind because I was thinking about writing a post regarding how I set up a differential equations laboratory project (where the students had to answer a set of problems both analytically and numerically, using a Mathematica-like program called Maple to work it out), and then I realized it was a pretty good setup to introducing phase space. I imagine trying to introduce them has to be done in either pendulums or masses-on-springs for want of other simple systems that are interesting and don’t have too many spatial dimensions.

Also I suspect you’re right in talk about dimensions confusing people because “dimension” is a word with so many meanings. I’m not sure of a good alternate word to use, though, and if I do carry on I might just give in and speak of “phase space dimension” instead. The construction is horrible but at least it makes the context clear enough.

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• #### elkement 9:32 am on Wednesday, 16 October, 2013 Permalink | Reply

Reblogged this on Theory and Practice of Trying to Combine Just Anything and commented:
I am still working on a more self-explanatory update to my previous physics post … trying to explain that multi-dimensional hyperspace is really a space of all potential states a single system might exhibit – a space of possibilities and not those infamous multi-dimensional world that might really be ‘out there’ according to string theorists. In the meantime, please enjoy mathematician Joseph Nebus’ additions to my post which includes a down-to-earth example.

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• #### elkement 9:37 am on Wednesday, 16 October, 2013 Permalink | Reply

(I would really like the typos in my reblog here ;-) )

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• #### Joseph Nebus 3:04 am on Friday, 18 October, 2013 Permalink | Reply

Aw, and I do want to thank you for kindly reblogging me. I think yours might be my first referral.

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• #### Pairodox Farm 3:58 pm on Wednesday, 16 October, 2013 Permalink | Reply

It’s nice to see Elke’s influence spreading far and wide about the internet. D

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• #### Joseph Nebus 3:17 am on Friday, 18 October, 2013 Permalink | Reply

It is, and I’m glad to have found her blog.

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• #### elkement 7:12 am on Friday, 18 October, 2013 Permalink | Reply

Thanks, Dave and Joseph! Flattering to hear people talk about me (in a nice way) :-)

I am also very grateful for those referrals of yours, Joseph. My blog is rather anti-popular – concluding from my numbers (views, followers) when comparing with other bloggers who blog for about the same time. Probably my blog is also not enough ‘niche’ and specialist and too much all over the place (Search term poetry, physics, random thoughts on society and the corporate world…)
Blog visitors constitute probably another hyperspace whose dynamics should be modeled. I guess we might find out that occasional spikes in popularity (Freshly Pressed, going viral) are dictated a chance.

But I believe it is better to have a small number of loyal followers you can have very interesting discussions with instead of 1000 people following and liking your blog because of its established popularity (winner-take-all effect in networks). I suppose if we would investigate the numbers of followers on all blogs in the world, assuming a reasonable time you need to ‘invest’ in reading as a follower… we would find out that most followers of those very popular blogs actually do not really follow because then they would need to spend 24/7 on speed reading.

This was probably quite random a comment. but I am re-reading Nassim Taleb’s books currently – so I am intrigued by anything Black-Swan-like …

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• #### Joseph Nebus 11:35 pm on Tuesday, 22 October, 2013 Permalink | Reply

Ah, well, the blog is always greener … I worry about my own lack of popularity and rather admire the community you’ve got over at your site. I haven’t figured out the knack for drawing people out to making comments, yet. I might start giving out challenges.

As you say, though, a small group of people who’re actually listening is rewarding, and I just hope to get one new reader for every three spambots offering to get me to get rich by blogging.

Oddly I know that I got one of Taleb’s books from the library, as an audio book (I enjoy listening to them while commuting), but I didn’t get far into it. I’m not sure what happened; conceivably it was recalled back to the library.

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## From ElKement: May The Force Field Be With You

I’m derelict in mentioning this but ElKement’s blog, Theory And Practice Of Trying To Combine Just Anything, has published the second part of a non-equation-based description of quantum field theory. This one, titled “May The Force Field Be With You: Primer on Quantum Mechanics and Why We Need Quantum Field Theory”, is about introducing the idea of a field, and a bit of how they can be understood in quantum mechanics terms.

A field, in this context, means some quantity that’s got a defined value for every point in space and time that you’re studying. As ElKement notes, the temperature is probably the most familiar to people. I’d imagine that’s partly because it’s relatively easy to feel the temperature change as one goes about one’s business — after all, gravity is also a field, but almost none of us feel it appreciably change — and because weather maps make the changes of that in space and in time available in attractive pictures.

The thing the field contains can be just about anything. The temperature would be just a plain old number, or as mathematicians would have it a “scalar”. But you can also have fields that describe stuff like the pull of gravity, which is a certain amount of pull and pointing, for us, toward the center of the earth. You can also have fields that describe, for example, how quickly and in what direction the water within a river is flowing. These strengths-and-directions are called “vectors” [1], and a field of vectors offers a lot of interesting mathematics and useful physics. You can also plunge into more exotic mathematical constructs, but you don’t have to. And you don’t need to understand any of this to read ElKement’s more robust introduction to all this.

[1] The independent student newspaper for the New Jersey Institute of Technology is named The Vector, and has as motto “With Magnitude and Direction Since 1924”. I don’t know if other tech schools have newspapers which use a similar joke.

• #### elkement 6:22 am on Thursday, 3 October, 2013 Permalink | Reply

Thanks again for your kind pingback and publicity :-)
I need to get to vectors and tensors in the next post(s) but I am still trying to figure out how to do this without mentioning those terms. Fluid dynamics is often a good starting point, e.g. to introduce, ‘derive’ or better motivate Schrödinger’s equation. On the other hand Feynman used to plunge directly into path integrals – presenting them as a rule along the lines of “This is the way nature works – live with it” – and deriving Schrödinger’s equation later.

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• #### Joseph Nebus 3:20 am on Saturday, 5 October, 2013 Permalink | Reply

I’m not quite sure how I’d do either. I think I could probably explain vectors without having to use mathematical symbolism, since the idea of stuff moving at particular speeds in directions can call on physical intuition. Tensors I don’t know how I’d try to explain in popular terms, partly because I’m not really as proficient in them as I should be. I probably need to think seriously about my own understanding of them.

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• #### elkement 6:26 pm on Monday, 7 October, 2013 Permalink | Reply

I have also always considered easier to imagine the different aspects of a vector – the abstract object and the ‘arrow’ as it lives in a specific base. But how do you really imagine the ‘abstract tensor object’ – in contrast to a ‘matrix’ (with more than 3 dimensions probably…)

I have started to read about general relativity (… will finish after I have finally understood the Higgs…) and it took me quite a while to comprehend that you are not allowed to shift a vector in curved space as you shift the ‘arrow’. Actually it made me think about vectors in a new way…

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• #### Joseph Nebus 2:46 am on Friday, 18 October, 2013 Permalink | Reply

(I’m embarrassed that I lost this comment somehow.)

I can sort of reconstruct the process when I think I started to get vectors as a concept, particularly in thinking of them as not tied to some particular point, or even containing information about a point, but somehow floating freely off that. If I get around to trying to explain vectors I might even be able to make all that explicit again.

Tensors I keep feeling like I’m on the verge of having that intuitive leap to where I have some mental model for how they work but I keep finding I don’t quite do enough work with them that it gets past following the rules and into really understanding the rules.

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## From ElKement: Space Balls, Baywatch, and the Geekiness of Classical Mechanics

Over on Elkement’s blog, Theory and Practice of Trying To Combine Just Anything, is the start of a new series about quantum field theory. Elke Stangl is trying a pretty impressive trick here in trying to describe a pretty advanced field without resorting to the piles of equations that maybe are needed to be precise, but, which also fill the page with piles of equations.

The first entry is about classical mechanics, and contrasting the familiar way that it gets introduced to people —- the whole forceequalsmasstimesacceleration bit — and an alternate description, based on what’s called the Principle of Least Action. This alternate description is as good as the familiar old Newton’s Laws in describing what’s going on, but it also makes a host of powerful new mathematical tools available. So when you get into serious physics work you tend to shift over to that model; and, if you want to start talking Modern Physics, stuff like quantum mechanics, you pretty nearly have to start with that if you want to do anything.

So, since it introduces in clear language a fascinating and important part of physics and mathematics, I’d recommend folks try reading the essay. It’s building up to an explanation of fields, as the modern physicist understands them, too, which is similarly an important topic worth being informed about.

• #### elkement 11:03 am on Thursday, 19 September, 2013 Permalink | Reply

Thanks a lot, Joseph – I am really honored :-) I hope I will be able to meet the expectations raised by your post :-D

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• #### Joseph Nebus 2:45 am on Friday, 20 September, 2013 Permalink | Reply

Well, thank you, and I’m confident in you.

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• #### elkement 11:06 am on Thursday, 19 September, 2013 Permalink | Reply

Reblogged this on Theory and Practice of Trying to Combine Just Anything and commented:
This is self-serving, but I can’t resist reblogging Joseph Nebus’ endorsement of my posts on Quantum Field Theory. Joseph is running a great blog on mathematics, and he manages to explain math in an accessible and entertaining way. I hope I will be able to do the same to theoretical physics!

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