My 2019 Mathematics A To Z: Zeno’s Paradoxes

Today’s A To Z term was nominated by Dina Yagodich, who runs a YouTube channel with a host of mathematics topics. Zeno’s Paradoxes exist in the intersection of mathematics and philosophy. Mathematics majors like to declare that they’re all easy. The Ancient Greeks didn’t understand infinite series or infinitesimals like we do. Now they’re no challenge at all. This reflects a belief that philosophers must be silly people who haven’t noticed that one can, say, exit a room.

This is your classic STEM-attitude of missing the point. We may suppose that Zeno of Elea occasionally exited rooms himself. That is a supposition, though. Zeno, like most philosophers who lived before Socrates, we know from other philosophers making fun of him a century after he died. Or at least trying to explain what they thought he was on about. Modern philosophers are expected to present others’ arguments as well and as strongly as possible. This even — especially — when describing an argument they want to say is the stupidest thing they ever heard. Or, to use the lingo, when they wish to refute it. Ancient philosophers had no such compulsion. They did not mind presenting someone else’s argument sketchily, if they supposed everyone already knew it. Or even badly, if they wanted to make the other philosopher sound ridiculous. Between that and the sparse nature of the record, we have to guess a bit about what Zeno precisely said and what he meant. This is all right. We have some idea of things that might reasonably have bothered Zeno.

And they have bothered philosophers for thousands of years. They are about change. The ones I mean to discuss here are particularly about motion. And there are things we do not understand about change. This essay will not answer what we don’t understand. But it will, I hope, show something about why that’s still an interesting thing to ponder.

Cartoony banner illustration of a coati, a raccoon-like animal, flying a kite in the clear autumn sky. A skywriting plane has written 'MATHEMATIC A TO Z'; the kite, with the letter 'S' on it to make the word 'MATHEMATICS'.
Art by Thomas K Dye, creator of the web comics Projection Edge, Newshounds, Infinity Refugees, and Something Happens. He’s on Twitter as @projectionedge. You can get to read Projection Edge six months early by subscribing to his Patreon.

Zeno’s Paradoxes.

When we capture a moment by photographing it we add lies to what we see. We impose a frame on its contents, discarding what is off-frame. We rip an instant out of its context. And that before considering how we stage photographs, making people smile and stop tilting their heads. We forgive many of these lies. The things excluded from or the moments around the one photographed might not alter what the photograph represents. Making everyone smile can convey the emotional average of the event in a way that no individual moment represents. Arranging people to stand in frame can convey the participation in the way a candid photograph would not.

But there remains the lie that a photograph is “a moment”. It is no such thing. We notice this when the photograph is blurred. It records all the light passing through the lens while the shutter is open. A photograph records an eighth of a second. A thirtieth of a second. A thousandth of a second. But still, some time. There is always the ghost of motion in a picture. If we do not see it, it is because our photograph’s resolution is too coarse. If we could photograph something with infinite fidelity we would see, even in still life, the wobbling of the molecules that make up a thing.

A photograph of a blurry roller coaster passing through a vertical loop.
One of the many loops of Vortex, a roller coaster at Kings Island amusement park from 1987 to 2019. Taken by me the last day of the ride’s operation; this was one of the roller coaster’s runs after 7 pm, the close of the park the last day of the season.

Which implies something fascinating to me. Think of a reel of film. Here I mean old-school pre-digital film, the thing that’s a great strip of pictures, a new one shown 24 times per second. Each frame of film is a photograph, recording some split-second of time. How much time is actually in a film, then? How long, cumulatively, was a camera shutter open during a two-hour film? I use pre-digital, strip-of-film movies for convenience. Digital films offer the same questions, but with different technical points. And I do not want the writing burden of describing both analog and digital film technologies. So I will stick to the long sequence of analog photographs model.

Let me imagine a movie. One of an ordinary everyday event; an actuality, to use the terminology of 1898. A person overtaking a walking tortoise. Look at the strip of film. There are many frames which show the person behind the tortoise. There are many frames showing the person ahead of the tortoise. When are the person and the tortoise at the same spot?

We have to put in some definitions. Fine; do that. Say we mean when the leading edge of the person’s nose overtakes the leading edge of the tortoise’s, as viewed from our camera. Or, since there must be blur, when the center of the blur of the person’s nose overtakes the center of the blur of the tortoise’s nose.

Do we have the frame when that moment happened? I’m sure we have frames from the moments before, and frames from the moments after. But the exact moment? Are you positive? If we zoomed in, would it actually show the person is a millimeter behind the tortoise? That the person is a hundredth of a millimeter ahead? A thousandth of a hair’s width behind? Suppose that our camera is very good. It can take frames representing as small a time as we need. Does it ever capture that precise moment? To the point that we know, no, it’s not the case that the tortoise is one-trillionth the width of a hydrogen atom ahead of the person?

If we can’t show the frame where this overtaking happened, then how do we know it happened? To put it in terms a STEM major will respect, how can we credit a thing we have not observed with happening? … Yes, we can suppose it happened if we suppose continuity in space and time. Then it follows from the intermediate value theorem. But then we are begging the question. We impose the assumption that there is a moment of overtaking. This does not prove that the moment exists.

Fine, then. What if time is not continuous? If there is a smallest moment of time? … If there is, then, we can imagine a frame of film that photographs only that one moment. So let’s look at its footage.

One thing stands out. There’s finally no blur in the picture. There can’t be; there’s no time during which to move. We might not catch the moment that the person overtakes the tortoise. It could “happen” in-between moments. But at least we have a moment to observe at leisure.

So … what is the difference between a picture of the person overtaking the tortoise, and a picture of the person and the tortoise standing still? A movie of the two walking should be different from a movie of the two pretending to be department store mannequins. What, in this frame, is the difference? If there is no observable difference, how does the universe tell whether, next instant, these two should have moved or not?

A mathematical physicist may toss in an answer. Our photograph is only of positions. We should also track momentum. Momentum carries within it the information of how position changes over time. We can’t photograph momentum, not without getting blurs. But analytically? If we interpret a photograph as “really” tracking the positions of a bunch of particles? To the mathematical physicist, momentum is as good a variable as position, and it’s as measurable. We can imagine a hyperspace photograph that gives us an image of positions and momentums. So, STEM types show up the philosophers finally, right?

Hold on. Let’s allow that somehow we get changes in position from the momentum of something. Hold off worrying about how momentum gets into position. Where does a change in momentum come from? In the mathematical physics problems we can do, the change in momentum has a value that depends on position. In the mathematical physics problems we have to deal with, the change in momentum has a value that depends on position and momentum. But that value? Put it in words. That value is the change in momentum. It has the same relationship to acceleration that momentum has to velocity. For want of a real term, I’ll call it acceleration. We need more variables. An even more hyperspatial film camera.

… And does acceleration change? Where does that change come from? That is going to demand another variable, the change-in-acceleration. (The “jerk”, according to people who want to tell you that “jerk” is a commonly used term for the change-in-acceleration, and no one else.) And the change-in-change-in-acceleration. Change-in-change-in-change-in-acceleration. We have to invoke an infinite regression of new variables. We got here because we wanted to suppose it wasn’t possible to divide a span of time infinitely many times. This seems like a lot to build into the universe to distinguish a person walking past a tortoise from a person standing near a tortoise. And then we still must admit not knowing how one variable propagates into another. That a person is wide is not usually enough explanation of how they are growing taller.

Numerical integration can model this kind of system with time divided into discrete chunks. It teaches us some ways that this can make logical sense. It also shows us that our projections will (generally) be wrong. At least unless we do things like have an infinite number of steps of time factor into each projection of the next timestep. Or use the forecast of future timesteps to correct the current one. Maybe use both. These are … not impossible. But being “ … not impossible” is not to say satisfying. (We allow numerical integration to be wrong by quantifying just how wrong it is. We call this an “error”, and have techniques that we can use to keep the error within some tolerated margin.)

So where has the movement happened? The original scene had movement to it. The movie seems to represent that movement. But that movement doesn’t seem to be in any frame of the movie. Where did it come from?

We can have properties that appear in a mass which don’t appear in any component piece. No molecule of a substance has a color, but a big enough mass does. No atom of iron is ferromagnetic, but a chunk might be. No grain of sand is a heap, but enough of them are. The Ancient Greeks knew this; we call it the Sorites paradox, after Eubulides of Miletus. (“Sorites” means “heap”, as in heap of sand. But if you had to bluff through a conversation about ancient Greek philosophers you could probably get away with making up a quote you credit to Sorites.) Could movement be, in the term mathematical physicists use, an intensive property? But intensive properties are obvious to the outside observer of a thing. We are not outside observers to the universe. It’s not clear what it would mean for there to be an outside observer to the universe. Even if there were, what space and time are they observing in? And aren’t their space and their time and their observations vulnerable to the same questions? We’re in danger of insisting on an infinite regression of “universes” just so a person can walk past a tortoise in ours.

We can say where movement comes from when we watch a movie. It is a trick of perception. Our eyes take some time to understand a new image. Our brains insist on forming a continuous whole story even out of disjoint ideas. Our memory fools us into remembering a continuous line of action. That a movie moves is entirely an illusion.

You see the implication here. Surely Zeno was not trying to lead us to understand all motion, in the real world, as an illusion? … Zeno seems to have been trying to support the work of Parmenides of Elea. Parmenides is another pre-Socratic philosopher. So we have about four words that we’re fairly sure he authored, and we’re not positive what order to put them in. Parmenides was arguing about the nature of reality, and what it means for a thing to come into or pass out of existence. He seems to have been arguing something like that there was a true reality that’s necessary and timeless and changeless. And there’s an apparent reality, the thing our senses observe. And in our sensing, we add lies which make things like change seem to happen. (Do not use this to get through your PhD defense in philosophy. I’m not sure I’d use it to get through your Intro to Ancient Greek Philosophy quiz.) That what we perceive as movement is not what is “really” going on is, at least, imaginable. So it is worth asking questions about what we mean for something to move. What difference there is between our intuitive understanding of movement and what logic says should happen.

(I know someone wishes to throw down the word Quantum. Quantum mechanics is a powerful tool for describing how many things behave. It implies limits on what we can simultaneously know about the position and the time of a thing. But there is a difference between “what time is” and “what we can know about a thing’s coordinates in time”. Quantum mechanics speaks more to the latter. There are also people who would like to say Relativity. Relativity, special and general, implies we should look at space and time as a unified set. But this does not change our questions about continuity of time or space, or where to find movement in both.)

And this is why we are likely never to finish pondering Zeno’s Paradoxes. In this essay I’ve only discussed two of them: Achilles and the Tortoise, and The Arrow. There are two other particularly famous ones: the Dichotomy, and the Stadium. The Dichotomy is the one about how to get somewhere, you have to get halfway there. But to get halfway there, you have to get a quarter of the way there. And an eighth of the way there, and so on. The Stadium is the hardest of the four great paradoxes to explain. This is in part because the earliest writings we have about it don’t make clear what Zeno was trying to get at. I can think of something which seems consistent with what’s described, and contrary-to-intuition enough to be interesting. I’m satisfied to ponder that one. But other people may have different ideas of what the paradox should be.

There are a handful of other paradoxes which don’t get so much love, although one of them is another version of the Sorites Paradox. Some of them the Stanford Encyclopedia of Philosophy dubs “paradoxes of plurality”. These ask how many things there could be. It’s hard to judge just what he was getting at with this. We know that one argument had three parts, and only two of them survive. Trying to fill in that gap is a challenge. We want to fill in the argument we would make, projecting from our modern idea of this plurality. It’s not Zeno’s idea, though, and we can’t know how close our projection is.

I don’t have the space to make a thematically coherent essay describing these all, though. The set of paradoxes have demanded thought, even just to come up with a reason to think they don’t demand thought, for thousands of years. We will, perhaps, have to keep trying again to fully understand what it is we don’t understand.

And with that — I find it hard to believe — I am done with the alphabet! All of the Fall 2019 A-to-Z essays should appear at this link. Additionally, the A-to-Z sequences of this and past years should be at this link. Tomorrow and Saturday I hope to bring up some mentions of specific past A-to-Z essays. Next week I hope to share my typical thoughts about what this experience has taught me, and some other writing about this writing.

Thank you, all who’ve been reading, and who’ve offered topics, comments on the material, or questions about things I was hoping readers wouldn’t notice I was shorting. I’ll probably do this again next year, after I’ve had some chance to rest.

Reading the Comics, March 19, 2019: Average Edition

This time around, averages seem important.

Mark Anderson’s Andertoons for the 18th is the Mark Anderson’s Andertoons for the week. This features the kids learning some of the commonest terms in descriptive statistics. And, as Wavehead says, the similarity of names doesn’t help sorting them out. Each is a kind of average. “Mean” usually is the arithmetic mean, or the thing everyone including statisticians calls “average”. “Median” is the middle-most value, the one that half the data is less than and half the data is greater than. “Mode” is the most common value. In “normally distributed” data, these three quantities are all the same. In data gathered from real-world measurements, these are typically pretty close to one another. It’s very easy for real-world quantities to be normally distributed. The exceptions are usually when there are some weird disparities, like a cluster of abnormally high-valued (or low-valued) results. Or if there are very few data points.

On the blackboard the teacher's written Median, Mode, and Mean, with a bunch of numbers from 3 through 15. Wavehead: 'I know they're all subtly different, but I have to say, the alliteration doesn't help.'
Mark Anderson’s Andertoons for the 18th of March, 2019. Essays which discuss topics raised by Andertoons can be found at this link. Also at this link, nearly enough.

The word “mean” derives from the Old French “meien”, that is, “middle, means”. And that itself traces to the Late Latin “medianus”, and the Latin “medius”. That traces back to the Proto-Indo-European “medhyo”, meaning “middle”. That’s probably what you might expect, especially considering that the mean of a set of data is, if the data is not doing anything weird, likely close to the middle of the set. The term appeared in English in the middle 15th century.

The word “median”, meanwhile, follows a completely different path. That one traces to the Middle French “médian”, which traces to the Late Latin “medianus” and Latin “medius” and Proto-Indo-European “medhyo”. This appeared as a mathematical term in the late 19th century; Etymology Online claims 1883, but doesn’t give a manuscript citation.

The word “mode”, meanwhile, follows a completely different path. This one traces to the Old French “mode”, itself from the Latin “modus”, meaning the measure or melody or style. We get from music to common values by way of the “style” meaning. Think of something being done “á la mode”, that is, “in the [ fashionable or popular ] style”. I haven’t dug up a citation about when this word entered the mathematical parlance.

So “mean” and “median” don’t have much chance to do anything but alliterate. “Mode” is coincidence here. I agree, it might be nice if we spread out the words a little more.

Edison, pointing to a checkerboard: 'So Grandpa if you put one cookie on the first square, two on the second, four on the next, then eight, and you keep doubling them until you fill all 64 squares do you know what you'll end up with?' Grandpa: 'A stomachache for a month?'
John Hambrock’s The Brilliant Mind of Edison Lee for the 18th of March, 2019. I’ve been talking about this strip a lot lately, it seems to me. Essays where I do discuss Edison Lee are at this link.

John Hambrock’s The Brilliant Mind of Edison Lee for the 18th has Edison introduce a sequence to his grandfather. Doubling the number of things for each square of a checkerboard is an ancient thought experiment. The notion, with grains of wheat rather than cookies, seems to be first recorded in 1256 in a book by the scholar Ibn Khallikan. One story has it that the inventor of chess requested from the ruler that many grains of wheat as reward for inventing the game.

If we followed Edison Lee’s doubling through all 64 squares we’d have, in total, need for 263-1 or 18,446,744,073,709,551,615 cookies. You can see why the inventor of chess didn’t get that reward, however popular the game was. It stands as a good display of how exponential growth eventually gets to be just that intimidatingly big.

Edison, like many a young nerd, is trying to stagger his grandfather with the enormity of this. I don’t know that it would work. Grandpa ponders eating all that many cookies, since he’s a comical glutton. I’d estimate eating all that many cookies, at the rate of one a second, eight hours a day, to take something like eighteen billion centuries. If I’m wrong? It doesn’t matter. It’s a while. But is that any more staggering than imagining a task that takes a mere ten thousand centuries to finish?

Toby, looking at his homework, and a calculator, and a textbook: '... Wow. Crazy ... Huh. How about that? ... Am I stupid if math *always* has a surprise ending?'
Greg Cravens’s The Buckets for the 19th of March, 2019. It also seems like I discuss The Buckets more these days, as seen at this link.

Greg Cravens’s The Buckets for the 19th sees Toby surprised by his mathematics homework. He’s surprised by how it turned out. I know the feeling. Everyone who does mathematics enough finds that. Surprise is one of the delights of mathematics. I had a great surprise last month, with a triangle theorem. Thomas Hobbes, the philosopher/theologian, entered his frustrating sideline of mathematics when he found the Pythagorean Theorem surprising.

Mathematics is, to an extent, about finding interesting true statements. What makes something interesting? That depends on the person surprised, certainly. A good guideline is probably “something not obvious before you’ve heard it, thatlooks inevitable after you have”. That is, a surprise. Learning mathematics probably has to be steadily surprising, and that’s good, because this kind of surprise is fun.

If it’s always a surprise there might be trouble. If you’re doing similar kinds of problems you should start to see them as pretty similar, and have a fair idea what the answers should be. So, from what Toby has said so far … I wouldn’t call him stupid. At most, just inexperienced.

Caption: Eric had good feelings about his date. It turned out, contrary to what he first thought ... [ Venn Diagram picture of two circles with a good amount of overlap ] ... They actually had quite a lot in common.
Eric the Circle for the 19th of March, 2019, this one by Janka. This and other essays with Eric the Circle, by any artist, should be at this link.

Eric the Circle for the 19th, by Janka, is the Venn Diagram joke for the week. Properly any Venn Diagram with two properties has an overlap like this. We’re supposed to place items in both circles, and in the intersection, to reflect how much overlap there is. Using the sizes of each circle to reflect the sizes of both sets, and the size of the overlap to represent the size of the intersection, is probably inevitable. The shorthand calls on our geometric intuition to convey information, anyway.

She: 'What time is it?' He: 'The clock in the hall says 7:56, the oven clock says 8:02, the DVD clock says 8:07, and the bed table clock says 8:13.' She: 'Which one's right?' He: 'I dunno.' She: 'Let's ee. Four clocks ... carry the one ... divided by four ... ' He: 'Whenever we want to know what time it is we have to do algebra.' She: 'We better hurry, it's 8:05 and five-eighths!'
Tony Murphy’s It’s All About You for the 19th of March, 2019. And the occasional time I discuss something from It’s All About You are here.

Tony Murphy’s It’s All About You for the 19th has a bunch of things going on. The punch line calls “algebra” what’s really a statistics problem, calculating the arithmetic mean of four results. The work done is basic arithmetic. But making work seem like a more onerous task is a good bit of comic exaggeration, and algebra connotes something harder than arithmetic. But Murphy exaggerates with restraint: the characters don’t rate this as calculus.

Then there’s what they’re doing at all. Given four clocks, what’s the correct time? The couple tries averaging them. Why should anyone expect that to work?

There’s reason to suppose this might work. We can suppose all the clocks are close to the correct time. If they weren’t, they would get re-set, or not looked at anymore. A clock is probably more likely to be a little wrong than a lot wrong. You’d let a clock that was two minutes off go about its business, in a way you wouldn’t let a clock that was three hours and 42 minutes off. A clock is probably as likely to show a time two minutes too early as it is two minutes too late. This all suggests that the clock errors are normally distributed, or something like that. So the error of the arithmetic mean of a bunch of clock measurements we can expect to be zero. Or close to zero, anyway.

There’s reasons this might not work. For example, a clock might systematically run late. My mantle clock, for example, usually drifts about a minute slow over the course of the week it takes to wind. Or the clock might be deliberately set wrong: it’s not unusual to set an alarm clock to five or ten or fifteen minutes ahead of the true time, to encourage people to think it’s later than it really is and they should hurry up. Similarly with watches, if their times aren’t set by Internet-connected device. I don’t know whether it’s possible to set a smart watch to be deliberately five minutes fast, or something like that. I’d imagine it should be possible, but also that the people programming watches don’t see why someone might want to set their clock to the wrong time. From January to March 2018, famously, an electrical grid conflict caused certain European clocks to lose around six minutes. The reasons for this are complicated and technical, and anyway The Doctor sorted it out. But that sort of systematic problem, causing all the clocks to be wrong in the same way, will foil this take-the-average scheme.

Murphy’s not thinking of that, not least because this comic’s a rerun from 2009. He was making a joke, going for the funnier-sounding “it’s 8:03 and five-eights” instead of the time implied by the average, 8:04 and a half. That’s all right. It’s a comic strip. Being amusing is what counts.

There were just enough mathematically-themed comic strips this past week for one more post. When that is ready, it should be at this link. I’ll likely post it Tuesday.

Why Stuff Can Orbit, Part 10: Where Time Comes From And How It Changes Things

Why Stuff Can Orbit, featuring a dazed-looking coati (it's a raccoon-like creature from Latin America) and a starry background.
Art courtesy of Thomas K Dye, creator of the web comic Newshounds. He has a Patron for those able to support his work.


And the supplemental reading:

And again my thanks to Thomas K Dye, creator of the web comic Newshounds, for the banner art. He has a Patreon to support his creative habit.

In the last installment I introduced perturbations. These are orbits that are a little off from the circles that make equilibriums. And they introduce something that’s been lurking, unnoticed, in all the work done before. That’s time.

See, how do we know time exists? … Well, we feel it, so, it’s hard for us not to notice time exists. Let me rephrase it then, and put it in contemporary technology terms. Suppose you’re looking at an animated GIF. How do you know it’s started animating? Or that it hasn’t stalled out on some frame?

If the picture changes, then you know. It has to be going. But if it doesn’t change? … Maybe it’s stalled out. Maybe it hasn’t. You don’t know. You know there’s time when you can see change. And that’s one of the little practical insights of physics. You can build an understanding of special relativity by thinking hard about that. Also think about the observation that the speed of light (in vacuum) doesn’t change.

When something physical’s in equilibrium, it isn’t changing. That’s how we found equilibriums to start with. And that means we stop keeping track of time. It’s one more thing to keep track of that doesn’t tell us anything new. Who needs it?

For the planet orbiting a sun, in a perfect circle, or its other little variations, we do still need time. At least some. How far the planet is from the sun doesn’t change, no, but where it is on the orbit will change. We can track where it is by setting some reference point. Where the planet is at the start of our problem. How big is the angle between where the planet is now, the sun (the center of our problem’s universe), and that origin point? That will change over time.

But it’ll change in a boring way. The angle will keep increasing in magnitude at a constant speed. Suppose it takes five time units for the angle to grow from zero degrees to ten degrees. Then it’ll take ten time units for the angle to grow from zero to twenty degrees. It’ll take twenty time units for the angle to grow from zero to forty degrees. Nice to know if you want to know when the planet is going to be at a particular spot, and how long it’ll take to get back to the same spot. At this rate it’ll be eighteen time units before the angle grows to 360 degrees, which looks the same as zero degrees. But it’s not anything interesting happening.

We’ll label this sort of change, where time passes, yeah, but it’s too dull to notice as a “dynamic equilibrium”. There’s change, but it’s so steady and predictable it’s not all that exciting. And I’d set up the circular orbits so that we didn’t even have to notice it. If the radius of the planet’s orbit doesn’t change, then the rate at which its apsidal angle changes, its “angular velocity”, also doesn’t change.

Now, with perturbations, the distance between the planet and the center of the universe will change in time. That was the stuff at the end of the last installment. But also the apsidal angle is going to change. I’ve used ‘r(t)’ to represent the radial distance between the planet and the sun before, and to note that what value it is depends on the time. I need some more symbols.

There’s two popular symbols to use for angles. Both are Greek letters because, I dunno, they’ve always been. (Florian Cajori’s A History of Mathematical Notation doesn’t seem to have anything. And when my default go-to for explaining mathematician’s choices tells me nothing, what can I do? Look at Wikipedia? Sure, but that doesn’t enlighten me either.) One is to use theta, θ. The other is to use phi, φ. Both are good, popular choices, and in three-dimensional problems we’ll often need both. We don’t need both. The orbit of something moving under a central force might be complicated, but it’s going to be in a single plane of movement. The conservation of angular momentum gives us that. It’s not the last thing angular momentum will give us. The orbit might happen not to be in a horizontal plane. But that’s all right. We can tilt our heads until it is.

So I’ll reach deep into the universe of symbols for angles and call on θ for the apsidal angle. θ will change with time, so, ‘θ(t)’ is the angular counterpart to ‘r(t)’.

I’d said before the apsidal angle is the angle made between the planet, the center of the universe, and some reference point. What is my reference point? I dunno. It’s wherever θ(0) is, that is, where the planet is when my time ‘t’ is zero. There’s probably a bootstrapping fallacy here. I’ll cover it up by saying, you know, the reference point doesn’t matter. It’s like the choice of prime meridian. We have to have one, but we can pick whatever one is convenient. So why not pick one that gives us the nice little identity that ‘θ(0) = 0’? If you don’t buy that and insist I pick a reference point first, fine, go ahead. But you know what? The labels on my time axis are arbitrary. There’s no difference in the way physics works whether ‘t’ is ‘0’ or ‘2017’ or ‘21350’. (At least as long as I adjust any time-dependent forces, which there aren’t here.) So we get back to ‘θ(0) = 0’.

For a circular orbit, the dynamic equilibrium case, these are pretty boring, but at least they’re easy to write. They’re:

r(t) = a	\\ \theta(t) = \omega t

Here ‘a’ is the radius of the circular orbit. And ω is a constant number, the angular velocity. It’s how much a bit of time changes the apsidal angle. And this set of equations is pretty dull. You can see why it barely rates a mention.

The perturbed case gets more interesting. We know how ‘r(t)’ looks. We worked that out last time. It’s some function like:

r(t) = a + A cos\left(\sqrt{\frac{k}{m}} t\right) + B sin\left(\sqrt{\frac{k}{m}} t\right)

Here ‘A’ and ‘B’ are some numbers telling us how big the perturbation is, and ‘m’ is the mass of the planet, and ‘k’ is something related to how strong the central force is. And ‘a’ is that radius of the circular orbit, the thing we’re perturbed around.

What about ‘θ(t)’? How’s that look? … We don’t seem to have a lot to go on. We could go back to Newton and all that force equalling the change in momentum over time stuff. We can always do that. It’s tedious, though. We have something better. It’s another gift from the conservation of angular momentum. When we can turn a forces-over-time problem into a conservation-of-something problem we’re usually doing the right thing. The conservation-of-something is typically a lot easier to set up and to track. We’ve used it in the conservation of energy, before, and we’ll use it again. The conservation of ordinary, ‘linear’, momentum helps other problems, though not I’ll grant this one. The conservation of angular momentum will help us here.

So what is angular momentum? … It’s something about ice skaters twirling around and your high school physics teacher sitting on a bar stool spinning a bike wheel. All right. But it’s also a quantity. We can get some idea of it by looking at the formula for calculating linear momentum:

\vec{p} = m\vec{v}

The linear momentum of a thing is its inertia times its velocity. This is if the thing isn’t moving fast enough we have to notice relativity. Also if it isn’t, like, an electric or a magnetic field so we have to notice it’s not precisely a thing. Also if it isn’t a massless particle like a photon because see previous sentence. I’m talking about ordinary things like planets and blocks of wood on springs and stuff. The inertia, ‘m’, is rather happily the same thing as its mass. The velocity is how fast something is travelling and which direction it’s going in.

Angular momentum, meanwhile, we calculate with this radically different-looking formula:

\vec{L} = I\vec{\omega}

Here, again, talking about stuff that isn’t moving so fast we have to notice relativity. That isn’t electric or magnetic fields. That isn’t massless particles. And so on. Here ‘I’ is the “moment of inertia” and \vec{w} is the angular velocity. The angular velocity is a vector that describes for us how fast the spinning is and what direction the axis around which the thing spins is. The moment of inertia describes how easy or hard it is to make the thing spin around each axis. It’s a tensor because real stuff can be easier to spin in some directions than in others. If you’re not sure that’s actually so, try tossing some stuff in the air so it spins in each of the three major directions. You’ll see.

We’re fortunate. For central force problems the moment of inertia is easy to calculate. We don’t need the tensor stuff. And we don’t even need to notice that the angular velocity is a vector. We know what axis the planet’s rotating around; it’s the one pointing out of the plane of motion. We can focus on the size of the angular velocity, the number ‘ω’. See how they’re different, what with one not having an arrow over the symbol. The arrow-less version is easier. For a planet, or other object, with mass ‘m’ that’s orbiting a distance ‘r’ from the sun, the moment of inertia is:

I = mr^2

So we know this number is going to be constant:

L = mr^2\omega

The mass ‘m’ doesn’t change. We’re not doing those kinds of problem. So however ‘r’ changes in time, the angular velocity ‘ω’ has to change with it, so that this product stays constant. The angular velocity is how the apsidal angle ‘θ’ changes over time. So since we know ‘L’ doesn’t change, and ‘m’ doesn’t change, then the way ‘r’ changes must tell us something about how ‘θ’ changes. We’ll get into that next time.

How May 2016 Treated My Mathematics Blog

The start of the Theorem Thursdays project did delay my monthly vanity post. That’s all right. I can be vain on a schedule. But I do like spending time mulling over the WordPress statistics around here and pondering their meaning.

My readership dropped in May as I expected. I posted only fifteen times in May, compared to daily in April and for that matter March. But my readership didn’t halve. It dropped back to about what it was before the Leap Day A To Z project, which I’m not sure how to read. It suggests folks around here were reasonably eager to see more stuff from me but that I maybe didn’t draw in so many new readers. Let me break things around:

Readership Numbers:

For simple page views: I fell short of the important-only-to-me threshold of a thousand page views. There were 981 views, down from 1,500 in April and 1,557 in March. Rated per posting — and I know some people were reading archive material rather than new posts — that’s not a bad trend, though. That’s about 65 page views per posting, compared to 50 in the busier months before.

There were, WordPress says, 627 unique visitors in May. That’s down from April’s 757 and March’s 734. But again per-posting … well, that’s nearly 42 per post in May, compared to 25 in April and 24 in March. I’d be interested in the posting schedule that gets the best readership per post, but it’s probably impossible to work that out.

The big measure of reader engagement, comments, looks catastrophic in May: only 22 comments, down from 55 in April and 84 in March. But that’s an illusion. I learned that linking to an old post using its full URL, which for me starts, creates a backlink that WordPress regards as a comment. If I use a short URL, starting, that creates the backlink but it doesn’t count as a comment. So I was curious how many comments there were which weren’t self-made comments and apparently 22 it is. But I haven’t got any way of figuring out how that compares to previous months, not without doing a lot of boring counting.

But the number of likes were down too. There were 133 of them in May, down from 345 in April and 320 in March. Even per-post that’s a collapse. It averages just under 9 per posting, compare to 11 and a half in April and a bit over 10 in March.

Popular Posts:

The five most popular posts in May? The usual blend: me referring to other stuff, me reading comics, and trapezoids.

Listing Countries:

The countries sending me the greatest number of readers were the United States (564 page views), Canada (88), India (34), and Germany (33). That’s all about in order. Single-reader countries — and I’ll put this in a bullet list because I read that people like those — were:

  • Argentina
  • Burkina Faso
  • Chile
  • China (**)
  • Colombia
  • Czech Republic
  • European Union (*****)
  • Hungary (*)
  • Indonesia
  • Malta
  • Nepal
  • Oman
  • Pakistan
  • Portugal
  • Senegal
  • Serbia (*)
  • Slovakia
  • South Korea (*)
  • Thailand
  • Turkey
  • Uganda

Hungary, Serbia, and South Korea were single-reader countries last month. China’s been a single-reader country the last two months. The European Union is in its sixth straight month as a single-reader country despite the fact that, you know, not a country. Whoever’s doing this is trying to provoke some weird reaction from me.

Counting Readers:

June started with my little blog here at 37,238 page views from an alleged 14,900 unique viewers. There’s reportedly 586 WordPress followers, up from 579 at the start of May and 573 at the start of April. And being a WordPress follower should be easier than it used to be, as I put a little blue “Follow On WordPress” button in the upper right corner of the page. It’s right above the “Follow Blog Via Email” card that I really want to rewrite as e-mail because I’m like that. There’s still apparently only eleven e-mail followers but, well, hi, gang.

WordPress’s “Insight” tab on the statics viewer says my most popular reading day is Wednesday, with 17 percent of page views. I don’t think I’ve posted anything on a Wednesday all May. But since Wednesdays are fourteen percent of the week I suppose that’s just a meaningless bit of static. It also says my most popular hour is 3:00 pm, which gets 17 percent of page views. Yes, I’m suspicious about that seventeen turning up again. But I haven’t got any reason to think that’s meaningless either, what with the hour from 3 to 4 pm not being 17 percent of the day. I have no idea if this is 3 pm my time, or Universal Time, or whatever time zone WordPress Master Command’s server is in. I’d appreciate some clarification on this point.

Search Terms:

Stuff WordPress admits brought people to me? Not so much interesting stuff, but, what the heck. Here’s some:

  • jumble comic before
  • origin is the gateway to your entire gaming universe.
  • historical comic stories
  • disney comic strips
  • tiger bud blake
  • nebus test medical
  • wet cement comic

At least I can feel pretty confident that search found the right place.

Reading the Comics, May 3, 2016: Lots Of Images Edition

After the heavy pace of March and April I figure to take it easy and settle to about a three-a-week schedule around here. That doesn’t mean that Comic Strip Master Command wants things to be too slow for me. And this time they gave me more comics than usual that have expiring URLs. I don’t think I’ve had this many pictures to include in a long while.

Bill Whitehead’s Free Range for the 28th presents an equation-solving nightmare. From my experience, this would be … a great pain, yes. But it wouldn’t be a career-wrecking mess. Typically a problem that’s hard to solve is hard because you have no idea what to do. Given an expression, you’re allowed to do anything that doesn’t change its truth value. And many approaches might look promising without quite resolving to something useful. The real breakthrough is working out what approach should be used. For an astrophysics problem, there are some classes of key decisions to make. One class is what to include and what to omit in the model. Another class is what to approximate — and how — versus what to treat exactly. Another class is what sorts of substitutions and transformations turn the original expression into one that reveals what you want. Those are the hard parts, and those are unlikely to have been forgotten. Applying those may be tedious, and I don’t doubt it would be anguishing to have the finished work wiped out. But it wouldn’t set one back years either. It would just hurt.

Christopher Grady’s Lunar Babboon for the 29th I classify as the “anthropomorphic numerals” joke for this essay. Boy, have we all been there.

'Numbers are boring!' complains the audience. 'Not so. They contain high drama and narrative. Here's an expense account that was turned in to me last week. Can you create a *story* based on these numbers?' 'Once upon a time, a guy was fired for malfeasance ... ' 'If you skip right to the big finish, sure.'
Bill Holbrook’s On The Fastrack for the 29th of April, 2016. Spoiler: there aren’t any numbers in the second panel.

Bill Holbrook’s On The Fastrack for the 29th continues the storyline about Fi giving her STEM talk. She is right, as I see it, in attributing drama and narrative to numbers. This is most easily seen in the sorts of finance and accounting mathematics which the character does. And the inevitable answer to “numbers are boring” (or “mathematics is boring”) is surely to show how they are about people. Even abstract mathematics is about things (some) people find interesting, and that must be about the people too.

'Look, Grandpa! I got 100% on my math test! Do you know what that means? It means that out of ten questions, I got at least half of them correct!' 'It must be that new, new, new math.' 'So many friendly numbers!'
Rick Detorie’s One Big Happy for the 3rd of May, 2016. Ever notice how many shirt pockets Grandpa has? I’m not saying it’s unrealistic, just that it’s more than the average.

Rick Detorie’s One Big Happy for the 16th is a confused-mathematics joke. Grandpa tosses off a New Math joke that’s reasonably age-appropriate too, which is always nice to see in a comic strip. I don’t know how seriously to take Ruthie’s assertion that a 100% means she only got at least half of the questions correct. It could be a cartoonist grumbling about how kids these days never learn anything, the same way ever past generation of cartoonists had complained. But Ruthie is also the sort of perpetually-confused, perpetually-confusing character who would get the implications of a 100% on a test wrong. Or would state them weirdly, since yes, a 100% does imply getting at least half the test’s questions right.

Border Collies, as we know, are highly intelligent. 'Yup, the math confirms it --- we can't get by without people.'
Niklas Eriksson’s Carpe Diem for the 3rd of May, 2016. I’m a little unnerved there seems to be a multiplication x at the end of the square root vinculum on the third line there.

Niklas Eriksson’s Carpe Diem for the 3rd uses the traditional board full of mathematical symbols as signifier of intelligence. There’s some interesting mixes of symbols here. The c2, for example, isn’t wrong for mathematics. But it does evoke Einstein and physics. There’s the curious mix of the symbol π and the approximation 3.14. But then I’m not sure how we would get from any of this to a proposition like “whether we can survive without people”.

'What comes after eleven?' 'I can't do it. I don't have enough fingers to count on!' Tiger hands him a baseball glove. 'Use this.'
Bud Blake’s Tiger for the 3rd of May, 2016. How did Punkinhead get up to eleven?

Bud Blake’s Tiger for the 3rd is a cute little kids-learning-to-count thing. I suppose it doesn’t really need to be here. But Punkinhead looks so cute wearing his tie dangling down onto the floor, the way kids wear their ties these days.

Tony Murphy’s It’s All About You for the 3rd name-drops algebra. I think what the author really wanted here was arithmetic, if the goal is to figure out the right time based on four clocks. They seem to be trying to do a simple arithmetic mean of the time on the four clocks, which is fair if we make some assumptions about how clocks drift away from the correct time. Mostly those assumptions are that the clocks all started right and are equally likely to drift backwards or forwards, and do that drifting at the same rate. If some clocks are more reliable than others, then, their claimed time should get more weight than the others. And something like that must be at work here. The mean of 7:56, 8:02, 8:07, and 8:13, uncorrected, is 8:04 and thirty seconds. That’s not close enough to 8:03 “and five-eighths” unless someone’s been calculating wrong, or supposing that 8:02 is more probably right than 8:13 is.

Feynman Online Physics

Likely everybody in the world has already spotted this before, but what the heck: CalTech and the Feynman Lectures Website have put online an edition of volume one of The Feynman Lectures on Physics. This is an HTML 5 edition, so older web browsers might not be able to read it sensibly.

The Feynman Lectures are generally regarded as one of the best expositions of basic physics; they started as part of an introduction to physics class that spiralled out of control and that got nearly all the freshmen who were trying to take it lost. I know the sense of being lost; when I was taking introductory physics I turned to them on the theory they might help me understand what the instructor was going on about. It didn’t help me.

This isn’t because Feynman wasn’t explaining well what was going on. It’s just that he approached things with a much deeper, much broader perspective than were really needed for me to figure out my problems in — oh, I’m not sure, probably something like how long a block needs to slide down a track or something like that. Here’s a fine example, excerpted from Chapter 5-2, “Time”:

Continue reading “Feynman Online Physics”

Reading the Comics, October 25, 2012

As before, this is going to be the comics other than those run through King Features Syndicate, since I haven’t found a solution I like for presenting their mathematics-themed comic strips for discussion. But there haven’t been many this month that I’ve seen either, so I can stick with strips for today at least. (I’m also a little irked that Comics Kingdom’s archives are being shut down — it’s their right, of course, but I don’t like having so many dead links in my old articles.) But on with the strips I have got.

Continue reading “Reading the Comics, October 25, 2012”

Reading the Comics, October 13, 2012

I suppose it’s been long enough to resume the review of math-themed comic strips. I admit there are weeks I don’t have much chance to write regular articles and then I feel embarrassed that I post only comic strips links, but I do enjoy the comics and the comic strip reviews. This one gets slightly truncated because King Features Syndicate has indeed locked down their Comics Kingdom archives of its strips, making it blasted inconvenient to read and nearly impossible to link to them in any practical, archivable way. They do offer a service,, with their comic strips, but I can hardly expect every reader of mine to pay up over there just for the odd day when Mandrake the Magician mentions something I can build a math problem from. Until I work out an acceptable-to-me resolution, then, I’ll be dropping to and a few oddball strips that the Houston Chronicle carries.

Continue reading “Reading the Comics, October 13, 2012”