Using my A to Z Archives: Differential Equations


I’d like today to share a less-old essay. This one is from the 2019 A-to-Z, and it’s about one of those fundamental topics. Differential equations permeate much of mathematics. Someone might mistake them for being all of advanced mathematics, or at least the kind of mathematics that professionals do. The confusion is reasonable. So I talk a bit here about why they seem to be part of everything.

My All 2020 Mathematics A to Z: Delta


I have Dina Yagodich to thank for my inspiration this week. As will happen with these topics about something fundamental, this proved to be a hard topic to think about. I don’t know of any creative or professional projects Yagodich would like me to mention. I’ll pass them on if I learn of any.

Color cartoon illustration of a coati in a beret and neckerchief, holding up a director's megaphone and looking over the Hollywood hills. The megaphone has the symbols + x (division obelus) and = on it. The Hollywood sign is, instead, the letters MATHEMATICS. In the background are spotlights, with several of them crossing so as to make the letters A and Z; one leg of the spotlights has 'TO' in it, so the art reads out, subtly, 'Mathematics A to Z'.
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.

Delta.

In May 1962 Mercury astronaut Deke Slayton did not orbit the Earth. He had been grounded for (of course) a rare medical condition. Before his grounding he had selected his flight’s callsign and capsule name: Delta 7. His backup, Wally Schirra, who did not fly in Slayton’s place, named his capsule the Sigma 7. Schirra chose sigma for its mathematical and scientific meaning, representing the sum of (in principle) many parts. Slayton said he chose Delta only because he would have been the fourth American into space and Δ is the fourth letter of the Greek alphabet. I believe it, but do notice how D is so prominent a letter in Slayton’s name. And S, Σ, prominent in both Slayton and Schirra’s.

Δ is also a prominent mathematics and engineering symbol. It has several meanings, with several of the most useful ones escaping mathematics and becoming vaguely known things. They blur together, as ideas that are useful and related and not identical will do.

If “Δ” evokes anything mathematical to a person it is “change”. This probably owes to space in the popular imagination. Astronauts talking about the delta-vee needed to return to Earth is some of the most accessible technical talk of Apollo 13, to pick one movie. After that it’s easy to think of pumping the car’s breaks as shedding some delta-vee. It secondarily owes to school, high school algebra classes testing people on their ability to tell how steep a line is. This gets described as the change-in-y over the change-in-x, or the delta-y over delta-x.

Δ prepended to a variable like x or y or v we read as “the change in”. It fits the astronaut and the algebra uses well. The letter Δ by itself means as much as the words “the change in” do. It describes what we’re thinking about, but waits for a noun to complete. We say “the” rather than “a”, I’ve noticed. The change in velocity needed to reach Earth may be one thing. But “the” change in x and y coordinates to find the slope of a line? We can use infinitely many possible changes and get a good result. We must say “the” because we consider one at a time.

Used like this Δ acts like an operator. It means something like “a difference between two values of the variable ” and lets us fill in the blank. How to pick those two values? Sometimes there’s a compelling choice. We often want to study data sampled at some schedule. The Δ then is between one sample’s value and the next. Or between the last sample value and the current one. Which is correct? Ask someone who specializes in difference equations. These are the usually numeric approximations to differential equations. They turn up often in signal processing or in understanding the flows of fluids or the interactions of particles. We like those because computers can solve them.

Δ, as this operator, can even be applied to itself. You read ΔΔ x as “the change in the change in x”. The prose is stilted, but we can understand it. It’s how the change in x has itself changed. We can imagine being interested in this Δ2 x. We can see this as a numerical approximation to the second derivative of x, and this gets us back to differential equations. There are similar results for ΔΔΔ x even if we don’t wish to read it all out.

In principle, Δ x can be any number. In practice, at least for an independent variable, it’s a small number, usually real. Often we’re lured into thinking of it as positive, because a phrase like “x + Δ x” looks like we’re making a number a little bigger than x. When you’re a mathematician or a quality-control tester you remember to consider “what if Δ x is negative”. From testing that learn you wrote your computer code wrong. We’re less likely to assume this positive-ness for the dependent variable. By the time we do enough mathematics to have opinions we’ve seen too many decreasing functions to overlook that Δ y might be negative.

Notice that in that last paragraph I faithfully wrote Δ x and Δ y. Never Δ bare, unless I forgot and cannot find it in copy-editing. I’ve said that Δ means “the change in”; to write it without some variable is like writing √ by itself. We can understand wishing to talk about “the square root of”, as a concept. Still it means something else than √ x does.

We do write Δ by itself. Even professionals do. Written like this we don’t mean “the change in [ something ]”. We instead mean “a number”. In this role the symbol means the same thing as x or y or t might, a way to refer to a number whose value we might not know. We might not care about. The implication is that it’s small, at least if it’s something to add to the independent variable. We use it when we ponder how things would be different if there were a small change in something.

Small but not tiny. Here we step into mathematics as a language, which can be as quirky and ambiguous as English. Because sometimes we use the lower-case δ. And this also means “a small number”. It connotes a smaller number than Δ. Is 0.01 a suitable value for Δ? Or for δ? Maybe. My inclination would be to think of that as Δ, reserving δ for “a small number of value we don’t care to specify”. This may be my quirk. Others might see it different.

We will use this lowercase δ as an operator too, thinking of things like “x + δ x”. As you’d guess, δ x connotes a small change in x. Smaller than would earn the title Δ x. There is no declaring how much smaller. It’s contextual. As with δ bare, my tendency is to think that Δ x might be a specific number but that δ x is “a perturbation”, the general idea of a small number. We can understand many interesting problems as a small change from something we already understand. That small change often earns such a δ operator.

There are smaller changes than δ x. There are infinitesimal differences. This is our attempt to make sense of “a number as close to zero as you can get without being zero”. We forego the Greek letters for this and revert to Roman letters: dx and dy and dt and the other marks of differential calculus. These are difficult numbers to discuss. It took more than a century of mathematicians’ work to find a way our experience with Δ x could inform us about dx. (We do not use ‘d’ alone to mean an even smaller change than δ. Sometimes we will in analysis write d with a space beside it, waiting for a variable to have its differential taken. I feel unsettled when I see it.)

Much of the completion of work we can credit to Augustin Cauchy, who’s credited with about 800 publications. It’s an intimidating record, even before considering its importance. Cauchy is, per Florian Cajori’s History Mathematical Notations, one of the persons we can credit with the use of Δ as symbol for “the change in”. (Section 610.) He’s not the only one. Leonhardt Euler and Johann Bernoulli (section 640) used Δ to represent a finite difference, the difference between two values.

I’m not aware of an explicit statement why Δ got the pick, as opposed to other letters. It’s hard to imagine a reason besides “difference starts with d”. That an etymology seems obvious does not make it so. It does seem to have a more compelling explanation than the use of “m” for the slope of a line, or \frac{\Delta y}{\Delta x} , though.

Slayton’s Mercury flight, performed by Scott Carpenter, did not involve any appreciable changes in orbit, a Δ v. No crewed spacecraft would until Gemini III. The Mercury flight did involve tests in orienting the spacecraft, in Δ θ and Δ φ on the angles of the spacecraft’s direction. These might have been in Slayton’s mind. He eventually flew into space on the Apollo-Soyuz Test Project, when an accident during landing exposed the crew to toxic gases. The investigation discovered a lesion on Slayton’s lung. A tiny thing, ultimately benign, which discovered earlier could have kicked him off the mission and altered his life so.


Thank you all for reading. I’m gathering all my 2020 A-to-Z essays at this link, and have all my A-to-Z essays of any kind at this link. Here is hoping there’s a good week ahead.

How June 2020 Taught Me How Many People Just Read Me For The Comics


As part of stepping back how much I’ve committed to writing, I had figured to not do my full write-ups of monthly readership statistics. Too many of the statistics were too common, month to month; I don’t need to keep trying to tease information out about which South American countries got a single page view any given month. But I’m not quite courageous enough to abandon them altogether, either.

In June I published 13 pieces, which is a pretty common number. A-to-Z months usually have more than that — last year I managed a several-month streak where I published every single day — but I’m deliberately trying not to do that this time. The number of page views dropped, though. There were 1,318 page views in June, from a recorded 929 unique visitors. That’s way below the twelve-month running averages of 2,289.3 views from 1,551.2 visitors. It’s my lowest page view count since June of 2019, when everybody had that mysterious drop in readers. It’s my lowest visitor count since December 2019.

Bar chart of monthly readership and unique visitor count. After several months holding steady around two thousand page views, June saw a drop to about 1,300 page views.
Mostly I’m not including the map of countries that sent me any readers because I hate having to look up whether it’s Mercatur or Mercator in describing the map projection. You’d think the spell checker would help me, but I have managed to misspell Cincinnati(?) enough times that it won’t tell me when I’m doing that wrong, so I can’t trust it on other words now. Also I’m not positive that WordPress is using Mercator rather than some other casually-similar-looking projection and somehow that’s a distinction I treat as important for my alt text?

There were 22 comments given in June, above the average of 15.4, thanks in part to how A-to-Z sequences appeal directly for comments. There were 43 likes, which is down from the running average of 60.1.

In all, a stunning rebuke to cutting back on my comic strip content. Maybe, anyway. Viewed per posting, it’s a less dramatic collapse. Per posting, there were 101.4 views, compared to an average of 129.2. That’s about four-fifths my average, rather than the three-fifths that the raw numbers implied. There were 71.5 unique visitors per posting, compared to an average of 86.8. Again, that’s a one-fifth drop rather than the two-fifths that the raw figures said I had. 3.3 likes per posting, compared to an average of 3.6. That’s barely a drop. And 1.7 comments per posting, compared to an average 1.0.

The most popular pieces … you know, I don’t need to support the popularity of my grooves-on-a-record-album or the count of different trapezoids. Let me list the five most popular pieces published in June, from June. You can almost see the transition from comics to A-to-Z:

I started July having posted 1,480 things here, gathering 107,748 views from a recorded 59,837 unique visitors. So somewhere along the lines I’ve missed visitor #60,000. Sorry, whoever you were.

I’d published 9,771 words in June, at an average 751.6 words per posting. My average post length so far this year has been 672 words. I’m curious how this will change with me writing one big piece a week, and then a bunch of shorter ones around it.

And I’m always happy to have more readers, even as I’m afraid of answering their comments. You can “Follow Nebusresearch” using the button on this page. Or you can add my RSS feed to whatever reader you like. I do announce posts on my sometimes-accessible @Nebusj Twitter account, and I’m trying to post announcements of new A-to-Z essays to my account on the mathematics-themed Mastodon account, @nebusj@mathstodon.xyz. Thanks for reading, however many of you remain.

Using my A to Z Archives: Commutative


I’ve learned a couple tricks to these essays. One is that I can always make them longer, and do. I’m trying not to. Another is in how to pick fruitful subjects. There’s always some chance to this and I’m still being surprised. But if I’ve found anything to be a sure bet, it’s picking the candidate topic that seems to be about the most obvious property. For example, everyone interested in mathematics knows about commutative things. But these are great because they make me focus on some “obvious” bit of mathematics lore. The “commutative” essay, in 2018, was an example of this. I didn’t achieve an insight to match that of “asymptote”, in which I finally understood a thing I had known for decades. But I improved my understanding at least.

Using my A to Z Archives: Cantor’s Middle Third


There are some topics that only the most confident pop-mathematics bloggers can avoid writing. The topics are well-covered already. But they are fascinating, and they are accessible, and that is a powerful combination. One of them is the cardinality of infinitely large sets. That we can say some infinitely large sets are the same size, and others are larger, and have something that seems to make coherent sense.

To date I haven’t written that, not exactly. I have come close, though. One of them is from the End 2016 A-to-Z and its essay on Cantor’s Middle Third. It is a scattering of dust along a line segment. It is a set of points which, altogether, cover no length. But there are as many points in this set as there are in the entire real number line. It’s neat to discover. Please consider it.

My All 2020 Mathematics A to Z: Complex Numbers


Mr Wu, author of the Singapore Maths Tuition blog, suggested complex numbers for a theme. I wrote long ago a bit about what complex numbers are and how to work with them. But that hardly exhausts the subject, and I’m happy revisiting it.

Color cartoon illustration of a coati in a beret and neckerchief, holding up a director's megaphone and looking over the Hollywood hills. The megaphone has the symbols + x (division obelus) and = on it. The Hollywood sign is, instead, the letters MATHEMATICS. In the background are spotlights, with several of them crossing so as to make the letters A and Z; one leg of the spotlights has 'TO' in it, so the art reads out, subtly, 'Mathematics A to Z'.
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.

Complex Numbers.

A throwaway joke somewhere in The Hitchhiker’s Guide To The Galaxy has Marvin The Paranoid Android grumble that he’s invented a square root for minus one. Marvin’s gone and rejiggered all of mathematics while waiting for something better to do. Nobody cares. It reminds us while Douglas Adams established much of a particular generation of nerd humor, he was not himself a nerd. The nerds who read The Hitchhiker’s Guide To The Galaxy obsessively know we already did that, centuries ago. Marvin’s creation was as novel as inventing “one-half”. (It may be that Adams knew, and intended Marvin working so hard on the already known as the joke.)

Anyone who’d read a pop mathematics blog like this likely knows the rough story of complex numbers in Western mathematics. The desire to find roots of polynomials. The discovery of formulas to find roots. Polynomials with numbers whose formulas demanded the square roots of negative numbers. And the discovery that sometimes, if you carried on as if the square root of a negative number made sense, the ugly terms vanished. And you got correct answers in the end. And, eventually, mathematicians relented. These things were unsettling enough to get unflattering names. To call a number “imaginary” may be more pejorative than even “negative”. It hints at the treatment of these numbers as falsework, never to be shown in the end. To call the sum of a “real” number and an “imaginary” “complex” is to warn. An expert might use these numbers only with care and deliberation. But we can count them as numbers.

I mentioned when writing about quaternions how when I learned of complex numbers I wanted to do the same trick again. My suspicion is many mathematicians do. The example of complex numbers teases us with the possibilites of other numbers. If we’ve defined \imath to be “a number that, squared, equals minus one”, what next? Could we define a \sqrt{\imath} ? How about a \log{\imath} ? Maybe something else? An arc-cosine of \imath ?

You can try any of these. They turn out to be redundant. The real numbers and \imath already let you describe any of those new numbers. You might have a flash of imagination: what if there were another number that, squared, equalled minus one, and that wasn’t equal to \imath ? Numbers that look like a + b\imath + c\jmath ? Here, and later on, a and b and c are some real numbers. b\imath means “multiply the real number b by whatever \imath is”, and we trust that this makes sense. There’s a similar setup for c and \jmath . And if you just try that, with a + b\imath + c\jmath , you get some interesting new mathematics. Then you get stuck on what the product of these two different square roots should be.

If you think of that. If all you think of is addition and subtraction and maybe multiplication by a real number? a + b\imath + c\jmath works fine. You only spot trouble if you happen to do multiplication. Granted, multiplication is to us not an exotic operation. Take that as a warning, though, of how trouble could develop. How do we know, say, that complex numbers are fine as long as you don’t try to take the log of the haversine of one of them, or some other obscurity? And that then they produce gibberish? Or worse, produce that most dread construct, a contradiction?

Here I am indebted to an essay that ten minutes ago I would have sworn was in one of the two books I still have out from the university library. I’m embarrassed to learn my error. It was about the philosophy of complex numbers and it gave me fresh perspectives. When the university library reopens for lending I will try to track back through my borrowing and find the original. I suspect, without confirming, that it may have been in Reuben Hersh’s What Is Mathematics, Really?.

The insight is that we can think of complex numbers in several ways. One fruitful way is to match complex numbers with points in a two-dimensional space. It’s common enough to pair, for example, the number 3 + 4\imath with the point at Cartesian coordinates (3, 4) . Mathematicians do this so often it can take a moment to remember that is just a convention. And there is a common matching between points in a Cartesian coordinate system and vectors. Chaining together matches like this can worry. Trust that we believe our matches are sound. Then we notice that adding two complex numbers does the same work as adding ordered coordinate pairs. If we trust that adding coordinate pairs makes sense, then we need to accept that adding complex numbers makes sense. Adding coordinate pairs is the same work as adding real numbers. It’s just a lot of them. So we’re lead to trust that if addition for real numbers works then addition for complex numbers does.

Multiplication looks like a mess. A different perspective helps us. A different way to look at where point are on the plane is to use polar coordinates. That is, the distance a point is from the origin, and the angle between the positive x-axis and the line segment connecting the origin to the point. In this format, multiplying two complex numbers is easy. Let the first complex number have polar coordinates (r_1, \theta_1) . Let the second have polar coordinates (r_2, \theta_2) . Their product, by the rules of complex numbers, is a point with polar coordinates (r_1\cdot r_2, \theta_1 + \theta_2) . These polar coordinates are real numbers again. If we trust addition and multiplication of real numbers, we can trust this for complex numbers.

If we’re confident in adding complex numbers, and confident in multiplying them, then … we’re in quite good shape. If we can add and multiply, we can do polynomials. And everything is polynomials.

We might feel suspicious yet. Going from complex numbers to points in space is calling on our geometric intuitions. That might be fooling ourselves. Can we find a different rationalization? The same result by several different lines of reasoning makes the result more believable. Is there a rationalization for complex numbers that never touches geometry?

We can. One approach is to use the mathematics of matrices. We can match the complex number a + b\imath to the sum of the matrices

a \left[\begin{tabular}{c c} 1 & 0 \\ 0 & 1 \end{tabular}\right] + b \left[\begin{tabular}{c c} 0 & 1 \\ -1 & 0  \end{tabular}\right]

Adding matrices is compelling. It’s the same work as adding ordered pairs of numbers. Multiplying matrices is tedious, though it’s not so bad for matrices this small. And it’s all done with real-number multiplication and addition. If we trust that the real numbers work, we can trust complex numbers do. If we can show that our new structure can be understood as a configuration of the old, we convince ourselves the new structure is meaningful.

The process by which we learn to trust them as numbers, guides us to learning how to trust any new mathematical structure. So here is a new thing that complex numbers can teach us, years after we have learned how to divide them. Do not attempt to divide complex numbers. That’s too much work.

Using my A to Z Archives: Benford’s Law


The one topic in my 2017 series that I picked myself, without a nomination, was the second, on Benford’s Law. It’s one that seems to defy the notion that numbers are independent of human construction. It’s a mathematical principle discovered in the modern day by experimentation. It’s one that likely would not have been found, in the form we know, if electronic computers were abundant and cheap two centuries ago.

For a small point I wanted to mention how (United States) street addresses can serve as a rough proxy for position. So I needed some house numbers. One that was small and one larger. It’s hard for a human to pick random numbers. We tend to pick odd numbers more than even. We tend to shy away from ‘edge’ numbers, taking (say) 1 to be somehow less random than 3 or, better, 7. When I have to pick an arbitrary number then I try to pick even numbers, and try to run toward the edges. I know this makes me no more random than anyone else. But it means at least my numbers look different.

A person important important to me lives on the 400 block of their street. So I picked a 400 number, and changed the last two digits away from their actual address. Then I needed a larger number. If 400 is a plausible enough ordinary number, how about the 1400 block? So I wrote into my essay the ideas of a house at 418 and another at 1418. I also wanted an even higher street number, and if -18 is a good plausible low number, why not -88 as a good plausible high number? And so I went with that and put it to press.

And, I swear to you, I did not think about it past that.

The trouble is that white supremacists have adopted the number 1-4-8-8 as a dogwhistle. It’s used as reference to the Hitlerian agenda of boundless evil.

When I realized this I thought about what to do. One tempting option was to leave it as is. As a set of digits this is as good as any other. A symbol has power if it is taken to represent that thing; why give one more inch or jot to evil people? And anyone who knows me would know better than to think —

And there’s the rub. Anyone who knows me would know my longing for a just and decent world. More than seven billion people do not know me, and never will. How much time do I demand they spend studying my politics to know that I did not make this completely arbitrary choice to deniably signal cruelty? One point of a dogwhistle is to make something that the perpetrators understand, and the targets understand. And skeptical onlookers will think a meaningless or coincidental choice. This because the overt action of the dogwhistle is something that looks arbitrary or insignificant. That it is about something that seems trivial is important. It lets the aggressors paint their targets as paranoid and thus ridiculous, finding dire patterns in randomness. Who am I to make other people study me to know whether I intended something or whether I thought “I know someone who lives on the 400 block of their street. What’s a number that also has a 4, but is not too much bigger”?

This decided me. Were the number something relevant to the essay, I could justify keeping it in. If I were writing about James IV’s ascension to the throne of Scotland I couldn’t skip naming the year it happened. But for this? And so my imaginary houses moved to 419 and 1419 and 1489.

Mathematicians like to present the field as a universal thing, free of the human culture and concerns and thoughts that create it. It’s not, and can’t be. This example turned up, with thematic unity, in an essay about a thing that turns up studying things that seem to be independent of human culture. It’s a lesson I shall remember.

Using my A to Z Archives: Bijection


I’d like to bring back to attention another piece from my 2015 A-to-Z. Bijection is another term from analysis, which you end up doing a lot of as a mathematics major, and start to slowly understand in grad school. I’m almost to the point of understanding the basics myself these days.

The essay also shows how my style has changed since I started. That essay’s rather technical, giving more of the sort of definition you could use to see whether something was a bijection. A bunch of things, particularly a string of topics whose precise definition was heaps of technical terms, got me writing more about the context and history and culture of terms. There’s Mathworld if someone needs to know the four conditions a concept must satisfy. But there is something to say for essays that lay out in clear language what a thing is and how to know whether you have one. I may do some more like that.

My All 2020 Mathematics A to Z: Butterfly Effect


It’s a fun topic today, one suggested by Jacob Siehler, who I think is one of the people I met through Mathstodon. Mathstodon is a mathematics-themed instance of Mastodon, an open-source microblogging system. You can read its public messages here.

Color cartoon illustration of a coati in a beret and neckerchief, holding up a director's megaphone and looking over the Hollywood hills. The megaphone has the symbols + x (division obelus) and = on it. The Hollywood sign is, instead, the letters MATHEMATICS. In the background are spotlights, with several of them crossing so as to make the letters A and Z; one leg of the spotlights has 'TO' in it, so the art reads out, subtly, 'Mathematics A to Z'.
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.

Butterfly Effect.

I take the short walk from my home to the Red Cedar River, and I pour a cup of water in. What happens next? To the water, anyway. Me, I think about walking all the way back home with this empty cup.

Let me have some simplifying assumptions. Pretend the cup of water remains somehow identifiable. That it doesn’t evaporate or dissolve into the riverbed. That it isn’t scooped up by a city or factory, drunk by an animal, or absorbed into a plant’s roots. That it doesn’t meet any interesting ions that turn it into other chemicals. It just goes as the river flows dictate. The Red Cedar River merges into the Grand River. This then moves west, emptying into Lake Michigan. Water from that eventually passes the Straits of Mackinac into Lake Huron. Through the St Clair River it goes to Lake Saint Clair, the Detroit River, Lake Erie, the Niagara River, the Niagara Falls, and Lake Ontario. Then into the Saint Lawrence River, then the Gulf of Saint Lawrence, before joining finally the North Atlantic.

Photograph of a small, tree-lined riverbed from a wooden bridge over it.
To the right: East Lansing and the Michigan State University campus. To the left, in a sense: the Atlantic Ocean.

If I pour in a second cup of water, somewhere else on the Red Cedar River, it has a similar journey. The details are different, but the course does not change. Grand River to Lake Michigan to three more Great Lakes to the Saint Lawrence to the North Atlantic Ocean. If I wish to know when my water passes the Mackinac Bridge I have a difficult problem. If I just wish to know what its future is, the problem is easy.

So now you understand dynamical systems. There’s some details to learn before you get a job, yes. But this is a perspective that explains what people in the field do, and why that. Dynamical systems are, largely, physics problems. They are about collections of things that interact according to some known potential energy. They may interact with each other. They may interact with the environment. We expect that where these things are changes in time. These changes are determined by the potential energies; there’s nothing random in it. Start a system from the same point twice and it will do the exact same thing twice.

We can describe the system as a set of coordinates. For a normal physics system the coordinates are the positions and momentums of everything that can move. If the potential energy’s rule changes with time, we probably have to include the time and the energy of the system as more coordinates. This collection of coordinates, describing the system at any moment, is a point. The point is somewhere inside phase space, which is an abstract idea, yes. But the geometry we know from the space we walk around in tells us things about phase space, too.

Imagine tracking my cup of water through its journey in the Red Cedar River. It draws out a thread, running from somewhere near my house into the Grand River and Lake Michigan and on. This great thin thread that I finally lose interest in when it flows into the Atlantic Ocean.

Dynamical systems drops in phase space act much the same. As the system changes in time, the coordinates of its parts change, or we expect them to. So “the point representing the system” moves. Where it moves depends on the potentials around it, the same way my cup of water moves according to the flow around it. “The point representing the system” traces out a thread, called a trajectory. The whole history of the system is somewhere on that thread.

Harvey, sulking: 'What a horrible year! How did it come to this?' Penny: 'I blame chaos theory.' Harvey: 'If it's chaos theory, I know EXACTLY who to blame! Some stupid butterfly flapped its wings and here we are.'
Stephen Beals’s Adult Children for the 21st of June, 2020. There were at least two “chaos theory”/“butterfly effect” comic strips in my feed just last week, and I wasn’t even looking. You can find comics essays where I talk about Adult Children at this link, and comic strips in general here.

Phase space, like a map, has regions. For my cup of water there’s a region that represents “is in Lake Michigan”. There’s another that represents “is going over Niagara Falls”. There’s one that represents “is stuck in Sandusky Bay a while”. When we study dynamical systems we are often interested in what these regions are, and what the boundaries between them are. Then a glance at where the point representing a system is tells us what it is doing. If the system represents a satellite orbiting a planet, we can tell whether it’s in a stable orbit, about to crash into a moon, or about to escape to interplanetary space. If the system represents weather, we can say it’s calm or stormy. If the system is a rigid pendulum — a favorite system to study, because we can draw its phase space on the blackboard — we can say whether the pendulum rocks back and forth or spins wildly.

Come back to my second cup of water, the one with a different history. It has a different thread from the first. So, too, a dynamical system started from a different point traces out a different trajectory. To find a trajectory is, normally, to solve differential equations. This is often useful to do. But from the dynamical systems perspective we’re usually interested in other issues.

For example: when I pour my cup of water in, does it stay together? The cup of water started all quite close together. But the different drops of water inside the cup? They’ve all had their own slightly different trajectories. So if I went with a bucket, one second later, trying to scoop it all up, likely I’d succeed. A minute later? … Possibly. An hour later? A day later?

By then I can’t gather it back up, practically speaking, because the water’s gotten all spread out across the Grand River. Possibly Lake Michigan. If I knew the flow of the river perfectly and knew well enough where I dropped the water in? I could predict where each goes, and catch each molecule of water right before it falls over Niagara. This is tedious but, after all, if you start from different spots — as the first and the last drop of my cup do — you expect to, eventually, go different places. They all end up in the North Atlantic anyway.

Photograph of Niagara Falls, showing the American Falls and the Bridal Veil, with a faint rainbow visible to the left of image and a boat sailing to the right.
Me, screaming to the pilot of the boat at center-right: “There’s my water drop! No, to the left! The left — your other left!”

Except … well, there is the Chicago Sanitary and Ship Canal. It connects the Chicago River to the Des Plaines River. The result is that some of Lake Michigan drains to the Ohio River, and from there the Mississippi River, and the Gulf of Mexico. There are also some canals in Ohio which connect Lake Erie to the Ohio River. I don’t know offhand of ones in Indiana or Wisconsin bringing Great Lakes water to the Mississippi. I assume there are, though.

Then, too, there is the Erie Canal, and the other canals of the New York State Canal System. These link the Niagara River and Lake Erie and Lake Ontario to the Hudson River. The Pennsylvania Canal System, too, links Lake Erie to the Delaware River. The Delaware and the Hudson may bring my water to the mid-Atlantic. I don’t know the canal systems of Ontario well enough to say whether some water goes to Hudson Bay; I’d grant that’s possible, though.

Think of my poor cups of water, now. I had been sure their fate was the North Atlantic. But if they happen to be in the right spot? They visit my old home off the Jersey Shore. Or they flow through Louisiana and warmer weather. What is their fate?

I will have butterflies in here soon.

Imagine two adjacent drops of water, one about to be pulled into the Chicago River and one with Lake Huron in its future. There is almost no difference in their current states. Their destinies are wildly separate, though. It’s surprising that so small a difference matters. Thinking through the surprise, it’s fair that this can happen, even for a deterministic system. It happens that there is a border, separating those bound for the Gulf and those for the North Atlantic, between these drops.

But how did those water drops get there? Where were they an hour before? … Somewhere else, yes. But still, on opposite sides of the border between “Gulf of Mexico water” and “North Atlantic water”. A day before, the drops were somewhere else yet, and the border was still between them. This separation goes back to, even, if the two drops came from my cup of water. Within the Red Cedar River is a border between a destiny of flowing past Quebec and of flowing past Saint Louis. And between flowing past Quebec and flowing past Syracuse. Between Syracuse and Philadelphia.

How far apart are those borders in the Red Cedar River? If you’ll go along with my assumptions, smaller than my cup of water. Not that I have the cup in a special location. The borders between all these fates are, probably, a complicated spaghetti-tangle. Anywhere along the river would be as fortunate. But what happens if the borders are separated by a space smaller than a drop? Well, a “drop” is a vague size. What if the borders are separated by a width smaller than a water molecule? There’s surely no subtleties in defining the “size” of a molecule.

That these borders are so close does not make the system random. It is still deterministic. Put a drop of water on this side of the border and it will go to this fate. But how do we know which side of the line the drop is on? If I toss this new cup out to the left rather than the right, does that matter? If my pinky twitches during the toss? If I am breathing in rather than out? What if a change too small to measure puts the drop on the other side?

And here we have the butterfly effect. It is about how a difference too small to observe has an effect too large to ignore. It is not about a system being random. It is about how we cannot know the system well enough for its predictability to tell us anything.

The term comes from the modern study of chaotic systems. One of the first topics in which the chaos was noticed, numerically, was weather simulations. The difference between a number’s representation in the computer’s memory and its rounded-off printout was noticeable. Edward Lorenz posed it aptly in 1963, saying that “one flap of a sea gull’s wings would be enough to alter the course of the weather forever”. Over the next few years this changed to a butterfly. In 1972 Philip Merrilees titled a talk Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas? My impression is that these days the butterflies may be anywhere, and they alter hurricanes.

Comic strip Chaos Butterfly Man: 'Bitten by a radioactive chaos butterfly, Mike Mason gained the powers of a chaos butterfly!' [ Outside a bank ] Mason flaps his arm at an escaping robber. Robber: 'Ha-ha! What harm can THAT do me?' [ Nine days later ] Robber: 'Where did this thunderstorm finally come fr ... YOW!' (He's struck by lightning.)
RubenBolling’s Super-Fun-Pak Comix for the 23rd of May, 2020. Ruben Bolling uses Chaos Butterfly a good bit in both Super-Fun-Pak Comix and in the main strip, Tom the Dancing Bug. I have fewer essays exploring these Chaos Butterfly strips than you might imagine from that because I ran out of different things to say about the joke. Bolling’s is a great strip, though, and I recommend you consider it.

That we settle on butterflies as agents of chaos we can likely credit to their image. They seem to be innocent things so slight they barely exist. Hummingbirds probably move with too much obvious determination to fit the role. The Big Bad Wolf huffing and puffing would realistically be almost as nothing as a butterfly. But he has the power of myth to make him seem mightier than the storms. There are other happy accidents supporting butterflies, though. Edward Lorenz’s 1960s weather model makes trajectories that, plotted, create two great ellipsoids. The figures look like butterflies, all different but part of the same family. And there is Ray Bradbury’s classic short story, A Sound Of Thunder. If you don’t remember 7th grade English class, in the story time-travelling idiots change history, putting a fascist with terrible spelling in charge of a dystopian world, by stepping on a butterfly.

The butterfly then is metonymy for all the things too small to notice. Butterflies, sea gulls, turning the ceiling fan on in the wrong direction, prying open the living room window so there’s now a cross-breeze. They can matter, we learn.

Using my A to Z Archives: Abacus


I’d like to use today’s publication slot to highlight another older post. This one, less old; it’s from the 2019 sequence.

Abacus led off a series of essays that, generally, I quite liked. Normally right after publication I feel like I just barely had enough words to be presentable and maybe they make any sense. This essay started out with some talk about how to use an abacus. But then I realized the real question was why use an abacus, and that opened the essay in good ways. There’s a writing lesson for me in there somewhere, and I won’t learn it well enough.

Using my A to Z Archives: Ansatz


I want to cut down on the amount of hard new writing I’m doing, at least until I feel at ease with the All 2020 A-to-Z and maybe anything in 2020 at all. But I also want to not disappear into the void except for a weekly appearance. This may be strange since some of the blogs I like best publish once a month or even more rarely, but I don’t have their confidence. Please bear with me as I bring some older posts to attention, then.

The first I want to highlight is the first A-to-Z post I ever did, a million years ago. The summer 2015 sequence started with “ansatz”, and it was the start of my learning a lot. The most important thing was learning that a term I’d picked up in grad school was, at least, idiosyncratic. I don’t know how small a mathematical dialect it is even has the word. If I knew what of other dialect words — not terms that are only used by a specific field, but that are only used by a particular group of people independent of their field — I’d share them. (If you know any please share! Language is fascinating.)

That discovery’s important. One thing I have learned in the past half-decade is to better appreciate the culture and the history that go into mathematics. Catching a word shows some of how human a subject mathematics is.


And to beat the drum once more, I’m eager to hear of topics starting with the letters D, E, and F that I might write about this year. (Or even revisit from an earlier essay sequence.) Thank you.

I’m looking for D, E, and F topics for the All 2020 A-to-Z


It does seem like I only just began the All 2020 Mathematics A-to-Z and already I need some fresh topics. This is because I really want to get to having articles done a week before publication, and while I’m not there yet, I can imagine getting there.

I’d like your nominations for subjects. Please comment here with some topic, name starting ‘D’, ‘E’, or ‘F’, that I might try writing about. This can include people, too. And also please let me know how to credit you for a topic suggestion, including any projects you’re working on that could do with attention.

These are the topics I’ve covered in past essays. I’m open to revisiting ones that I think I have new ideas about. Thanks all for your ideas.

Topics I’ve already covered, starting with the letter D, are:


Topics I’ve already covered, starting with the letter E, are:


Topics I’ve already covered, starting with the letter F, are:


Thank you for reading, and for your thoughts.

My All 2020 Mathematics A to Z: Michael Atiyah


To start this year’s great glossary project Mr Wu, author of the MathTuition88.com blog, had a great suggestion: The Atiyah-Singer Index Theorem. It’s an important and spectacular piece of work. I’ll explain why I’m not doing that in a few sentences.

Mr Wu pointed out that a biography of Michael Atiyah, one of the authors of this theorem, might be worth doing. GoldenOj endorsed the biography idea, and the more I thought it over the more I liked it. I’m not able to do a true biography, something that goes to primary sources and finds a convincing story of a life. But I can sketch out a bit, exploring his work and why it’s of note.

Color cartoon illustration of a coati in a beret and neckerchief, holding up a director's megaphone and looking over the Hollywood hills. The megaphone has the symbols + x (division obelus) and = on it. The Hollywood sign is, instead, the letters MATHEMATICS. In the background are spotlights, with several of them crossing so as to make the letters A and Z; one leg of the spotlights has 'TO' in it, so the art reads out, subtly, 'Mathematics A to Z'.
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.

Michael Atiyah.

Theodore Frankel’s The Geometry of Physics: An Introduction is a wonderful book. It’s 686 pages, including the index. It all explores how our modern understanding of physics is our modern understanding of geometry. On page 465 it offers this:

The Atiyah-Singer index theorem must be considered a high point of geometrical analysis of the twentieth century, but is far too complicated to be considered in this book.

I know when I’m licked. Let me attempt to look at one of the people behind this theorem instead.

The Riemann Hypothesis is about where to find the roots of a particular infinite series. It’s been out there, waiting for a solution, for a century and a half. There are many interesting results which we would know to be true if the Riemann Hypothesis is true. In 2018, Michael Atiyah declared that he had a proof. And, more, an amazing proof, a short proof. Albeit one that depended on a great deal of background work and careful definitions. The mathematical community was skeptical. It still is. But it did not dismiss outright the idea that he had a solution. It was plausible that Atiyah might solve one of the greatest problems of mathematics in something that fits on a few PowerPoint slides.

So think of a person who commands such respect.

His proof of the Riemann Hypothesis, as best I understand, is not generally accepted. For example, it includes the fine structure constant. This comes from physics. It describes how strongly electrons and photons interact. The most compelling (to us) consequence of the Riemann Hypothesis is in how prime numbers are distributed among the integers. It’s hard to think how photons and prime numbers could relate. But, then, if humans had done all of mathematics without noticing geometry, we would know there is something interesting about π. Differential equations, if nothing else, would turn up this number. We happened to discover π in the real world first too. If it were not familiar for so long, would we think there should be any commonality between differential equations and circles?

I do not mean to say Atiyah is right and his critics wrong. I’m no judge of the matter at all. What is interesting is that one could imagine a link between a pure number-theory matter like the Riemann hypothesis and a physical matter like the fine structure constant. It’s not surprising that mathematicians should be interested in physics, or vice-versa. Atiyah’s work was particularly important. Much of his work, from the late 70s through the 80s, was in gauge theory. This subject lies under much of modern quantum mechanics. It’s born of the recognition of symmetries, group operations that you can do on a field, such as the electromagnetic field.

In a sequence of papers Atiyah, with other authors, sorted out particular cases of how magnetic monopoles and instantons behave. Magnetic monopoles may sound familiar, even though no one has ever seen one. These are magnetic points, an isolated north or a south pole without its opposite partner. We can understand well how they would act without worrying about whether they exist. Instantons are more esoteric; I don’t remember encountering the term before starting my reading for this essay. I believe I did, encountering the technique as a way to describe the transitions between one quantum state and another. Perhaps the name failed to stick. I can see where there are few examples you could give an undergraduate physics major. And it turns out that monopoles appear as solutions to some problems involving instantons.

This was, for Atiyah, later work. It arose, in part, from bringing the tools of index theory to nonlinear partial differential equations. This index theory is the thing that got us the Atiyah-Singer Index Theorem too complicated to explain in 686 pages. Index theory, here, studies questions like “what can we know about a differential equation without solving it?” Solving a differential equation would tell us almost everything we’d like to know, yes. But it’s also quite hard. Index theory can tell us useful things like: is there a solution? Is there more than one? How many? And it does this through topological invariants. A topological invariant is a trait like, for example, the number of holes that go through a solid object. These things are indifferent to operations like moving the object, or rotating it, or reflecting it. In the language of group theory, they are invariant under a symmetry.

It’s startling to think a question like “is there a solution to this differential equation” has connections to what we know about shapes. This shows some of the power of recasting problems as geometry questions. From the late 50s through the mid-70s, Atiyah was a key person working in a topic that is about shapes. We know it as K-theory. The “K” from the German Klasse, here. It’s about groups, in the abstract-algebra sense; the things in the groups are themselves classes of isomorphisms. Michael Atiyah and Friedrich Hirzebruch defined this sort of group for a topological space in 1959. And this gave definition to topological K-theory. This is again abstract stuff. Frankel’s book doesn’t even mention it. It explores what we can know about shapes from the tangents to the shapes.

And it leads into cobordism, also called bordism. This is about what you can know about shapes which could be represented as cross-sections of a higher-dimension shape. The iconic, and delightfully named, shape here is the pair of pants. In three dimensions this shape is a simple cartoon of what it’s named. On the one end, it’s a circle. On the other end, it’s two circles. In between, it’s a continuous surface. Imagine the cross-sections, how on separate layers the two circles are closer together. How their shapes distort from a real circle. In one cross-section they come together. They appear as two circles joined at a point. In another, they’re a two-looped figure. In another, a smoother circle. Knowing that Atiyah came from these questions may make his future work seem more motivated.

But how does one come to think of the mathematics of imaginary pants? Many ways. Atiyah’s path came from his first research specialty, which was algebraic geometry. This was his work through much of the 1950s. Algebraic geometry is about the kinds of geometric problems you get from studying algebra problems. Algebra here means the abstract stuff, although it does touch on the algebra from high school. You might, for example, do work on the roots of a polynomial, or a comfortable enough equation like x^2 + y^2 = 1 . Atiyah had started — as an undergraduate — working on projective geometries. This is what one curve looks like projected onto a different surface. This moved into elliptic curves and into particular kinds of transformations on surfaces. And algebraic geometry has proved important in number theory. You might remember that the Wiles-Taylor proof of Fermat’s Last Theorem depended on elliptic curves. Some work on the Riemann hypothesis is built on algebraic topology.

(I would like to trace things farther back. But the public record of Atiyah’s work doesn’t offer hints. I can find amusing notes like his father asserting he knew he’d be a mathematician. He was quite good at changing local currency into foreign currency, making a profit on the deal.)

It’s possible to imagine this clear line in Atiyah’s career, and why his last works might have been on the Riemann hypothesis. That’s too pat an assertion. The more interesting thing is that Atiyah had several recognizable phases and did iconic work in each of them. There is a cliche that mathematicians do their best work before they are 40 years old. And, it happens, Atiyah did earn a Fields Medal, given to mathematicians for the work done before they are 40 years old. But I believe this cliche represents a misreading of biographies. I suspect that first-rate work is done when a well-prepared mind looks fresh at a new problem. A mathematician is likely to have these traits line up early in the career. Grad school demands the deep focus on a particular problem. Getting out of grad school lets one bring this deep knowledge to fresh questions.

It is easy, in a career, to keep studying problems one has already had great success in, for good reason and with good results. It tends not to keep producing revolutionary results. Atiyah was able — by chance or design I can’t tell — to several times venture into a new field. The new field was one that his earlier work prepared him for, yes. But it posed new questions about novel topics. And this creative, well-trained mind focusing on new questions produced great work. And this is one way to be credible when one announces a proof of the Riemann hypothesis.


Here is something I could not find a clear way to fit into this essay. Atiyah recorded some comments about his life for the Web of Stories site. These are biographical and do not get into his mathematics at all. Much of it is about his life as child of British and Lebanese parents and how that affected his schooling. One that stood out to me was about his peers at Manchester Grammar School, several of whom he rated as better students than he was. Being a good student is not tightly related to being a successful academic. Particularly as so much of a career depends on chance, on opportunities happening to be open when one is ready to take them. It would be remarkable if there wre three people of greater talent than Atiyah who happened to be in the same school at the same time. It’s not unthinkable, though, and we may wonder what we can do to give people the chance to do what they are good in. (I admit this assumes that one finds doing what one is good in particularly satisfying or fulfilling.) In looking at any remarkable talent it’s fair to ask how much of their exceptional nature is that they had a chance to excel.

Reading the Comics, June 7, 2020: Hiatus Edition


I think of myself as not a prescriptivist blogger. Here and on my humor blog I do what I feel like, and if that seems to work, I do more of it if I can. If I do enough of it, I try to think of a title, give up and use the first four words that kind of fit, and then ask Thomas K Dye for header art. If it doesn’t work, I drop it without mention. Apart from appealing for A-to-Z topics I don’t usually declare what I intend to do.

This feels different. One of the first things I fell into here, and the oldest hook in my blogging, is Reading the Comics. It’s mostly fun. But it is also work. 2020 is not a year when I am capable of expanding my writing work without bounds. Something has to yield, and my employers would rather it not be my day job. So, at least through the completion of the All 2020 Mathematics A-to-Z, I’ll just be reading the comics. Not Reading the Comics for posting here.

And this is likely a good time for a hiatus. There is much that’s fun about Reading the Comics. First is the comic strips, a lifelong love. Second is that they solve the problem of what to blog about. During the golden age of Atlantic City, there was a Boardwalk performer whose gimmick was to drag a trap along the seabed, haul it up, and identify every bit of sea life caught up in that. My schtick is of a similar thrill, with less harm required of the sea life.

But I have felt bored by this the last several months. Boredom is not a bad thing, of course. And if you are to be a writer, you must be able to write something competent and fresh about a topic you are tired of. Admitting that: I do not have one more sentence in me about kids not buying into the story problem. Or observing that yes, that is a blackboard full of mathematics symbols. Or that lotteries exist and if you play them infinitely many times strange conclusions seem to follow. An exercise that is tiring can be good; an exercise that is painful is not. I will put the painful away and see what I feel like later.

For the time being I figure to write only the A-to-Z essays. And, since I have them, to post references back to old A-to-Z essays. These recaps seemed to be received well enough last year. So why not repeat something that was fine when it was just one of many things?

And after all, the A-to-Z theme is still at heart hauling up buckets of sea life and naming everything in it. It’s just something that I can write farther ahead of deadline, but will not.

Thanks all for reading.

The Boardwalk performer would, if stumped, make up stuff. What patron was going to care if they went away ill-informed? It was a show. The performer just needed a confident air.

How May 2020 Treated My Mathematics Blog


I don’t know why my regular review of my past month’s readership keeps creeping later and later in the month. I understand why it does so on my humor blog: there’s stuff that basically squats on the Sunday, Tuesday, Thursday, and Saturday slots. And a thing has to be written after the 1st of the month. So it can get squeezed along. But my mathematics blog has always been more free-form. I think the trouble is that this is always, in principle, an easy post to write, so it’s always easy enough to push off a little longer, and let harder stuff take my attention. It’s always a mystery how my compulsive need to put things in order will clash with my desire to procrastinate my way out of life.

Still, to May. It was another heck of a month for us all. In it, I published only 13 posts, after a couple of 15-post months in a row. Since the frequency of posting is the one variable I am sure is within my control that affects my readership, how did getting a little more laconic affect my readership?

Bar chart of monthly readership for two years, five months, of work. The last three months have been fairly uniform at about two thousand readers each. October 2019 was abnormally high, around 8500 readers.
You have no idea how upset I am that I didn’t aimlessly hit ‘reload’ just eleven more times in May.

It’s hard to tell, thanks to the October 2019 spike. But my readership crept up a little. There were 1,989 pages viewed in May. This is below the 12-month running average of 2,205.3, but the twelve-month average still includes that October with 8,667 views. There were 1,407 unique visitors, below but still close to the running average of 1,494.0 unique visitors. There were only 35 likes given, below the average of 60.8. But there were 18 comments, above the running average of 14.9. Of course, the twelve-month running average includes December 2019 when nobody left any comments here.

Taking the averages per posting gives me figures that look a little more popular. 153.0 visitors per posting, above the twelve-month running average of 124.6. 108.2 unique visitors per posting, above the average 83.8. Only 2.7 likes per posting, below the 3.7 average. But 1.4 comments per posting, above the 1.0 average.

Where did all these page views come from? Here’s the roster.

Country Readers
United States 1,140
India 128
United Kingdom 109
Canada 96
Australia 45
Philippines 41
Singapore 41
China 22
Turkey 22
Germany 21
Italy 17
Netherlands 17
Austria 14
United Arab Emirates 14
Brazil 13
Sweden 13
Finland 11
Denmark 10
France 10
Japan 10
Malaysia 10
Israel 9
Croatia 8
New Zealand 8
South Africa 8
Colombia 7
El Salvador 6
Hong Kong SAR China 6
Hungary 6
Indonesia 6
Norway 6
Poland 6
Taiwan 6
Egypt 5
Greece 5
Pakistan 5
Romania 5
Belgium 4
Qatar 4
Russia 4
Slovakia 4
Spain 4
Albania 3
Chile 3
Jamaica 3
Jordan 3
Mexico 3
Portugal 3
Serbia 3
Switzerland 3
Thailand 3
Ukraine 3
Argentina 2
Cayman Islands 2
Czech Republic 2
Laos 2
Myanmar (Burma) 2
Palestinian Territories 2
South Korea 2
Vietnam 2
Bahrain 1 (*)
Bangladesh 1
Brunei 1
Bulgaria 1
Cyprus 1
Georgia 1
Guyana 1
Honduras 1
Iraq 1
Ireland 1
Kazakhstan 1
Luxembourg 1
Mauritius 1
Nepal 1
Peru 1
Puerto Rico 1
Zimbabwe 1

This is 77 countries or country-like things all told. There’d been 73 in April and 78 in March. 17 of these were single-view countries. There were 12 of those in April and 30 in March. Only Bahrain has been a single-view country for two months in a row, now.

Mercator-style map of the world with the United States in darkest pink. Most of the Americas, Europe, Asia, Australia, and New Zealand are in a fairly uniform light pink. Almost nothing in Africa is pink; nor are the Baltic states, the Arabian peninsula, Iran, or the adjacent former Soviet republics.
Someday I’ll have a reader in Greenland and I won’t know what to say to them.

All these people looked at, including the home page, 278 posts here. That’s comparable to the 265 of April and 255 of March. 153 pages got more than one view, comparable to the 134 of April and 145 of March. 33 got at least ten views, which is right in line with April’s 36 and March’s 35. The most views were given to some of the usual suspects:

The most popular thing posted in May? That was a tie, actually. One piece was Reading the Comics, May 9, 2020: Knowing the Angles Edition, the usual sort of thing. The other was Reading the Comics, May 2, 2020: What Is The Cosine Of Six Edition, a piece I had meant to follow up on. This is because it so happens that the cosine of six is a number we can, in principle, write out exactly. I had meant to write a post that went through the geometric reasoning that gets you there, but I kept not making time. But, for the short answer, here’s the cosine of six degrees.

First, this will be much easier if we (alas) use the Golden Ratio, φ. That’s a famous number and just about 1.61803. The cosine of six degrees is, to be exact,

\cos(36^\circ) = \left(\frac{1}{2} \cdot \phi\right)\cdot\left(\frac{1}{2}  \sqrt{3}\right) + \sqrt{1 - \frac{1}{4} \phi^2} \cdot \left(\frac{1}{2} \right)

… which you recognize right away reduces to …

\cos(36^\circ) = \frac{1}{4}\sqrt{3} \phi + \frac{1}{4}\sqrt{3 - \phi}

This is a number pretty close to 0.99452, and you can get as many decimal digits as you like. You just have to go through working out decimal digits, ultimately, of \sqrt{5} . I include the first line because if you look closely at it, you’ll get a hint of how to find the cosine of six degrees. It’s the parts of an angle-subtraction formula for cosine.


WordPress estimates me as having published 7,442 words in May. That’s an average of a slender 496.13 words per posting. My average post for the year has fallen to 656 words; at the start of May it had been 691. To the start of June I’ve published 41,978 words here. I don’t know if that counts picture captions and alt text, and have not the faintest idea how it counts LaTeX symbols.

As of the start of June I’ve published 1,467 things, which drew 106,429 views from a recorded 58,907 unique visitors.

I’m hoping next week to start publishing the All 2020 Mathematics A-to-Z. Posts from it should be tagged so as to appear at this link. If you want to be sure you don’t miss any of them, I’m quite flattered. The RSS feed for all my essays as this link, and if you lack an RSS reader, you’re mistaken! You can get a free account at Dreamwidth or Livejournal and use their Reading page to read any RSS feed you like. Or you can click the “Follow Nebusresearch” button on this page, and add it to your WordPress reader.

For a short while there my Twitter account of @Nebusj was working. It’s gone back to where it will just accept WordPress’s automated announcements of posts here, though. I can’t do anything with it. I do have an account on the mathematics-themed Mastodon instance, @nebusj@mathstodon.xyz, and occasionally manage to even just hang out chatting there. It’s hard to get a place in a new social media environment. You need a hook, and you need a playful bit of business anyone can do with you, which both serve to give you an identity. Then you need someone who’s already established to vouch for you as being okay. The A-to-Z is a pretty good hook but the rest is a bit hard. I’m in there trying, though.

Thanks always for reading, however you do it.


Also, because I will someday need this again: to write the ^\circ symbol in WordPress LaTeX, you need the symbol string ^\circ and do not ask me why it’s not, like, \deg (or better, \degree) instead.

Reading the Comics, June 6, 2020: Wrapping Up The Week Edition


Let’s see if I can’t close out the first week of June’s comics. I’d rather have published this either Tuesday or Thursday, but I didn’t have the time to write my statistics post for May, not yet. I’ll get there.

One of Gary Larson’s The Far Side reprints for the 4th is one I don’t remember seeing before. The thing to notice is the patient has a huge right brain and a tiny left one. The joke is about the supposed division between left-brained and right-brained people. There are areas of specialization in the brain, so that the damage or destruction of part can take away specific abilities. The popular imagination has latched onto the idea that people can be dominated by specialties of the either side of the brain. I’m not well-versed in neurology. I will hazard the guess that neurologists see “left-brain” and “right-brain” as amusing stuff not to be taken seriously. (My understanding is the division of people into “type A” and “type B” personalities is also entirely bunk unsupported by any psychological research.)

Psychiatrist talking to a patient whose head is enormously tall on the right and shorter than normal on the left: 'You're a right-brained sort of person, Mr Sommersby. Very creative, artistic, etc ... Unfortunately, I think I also see why you're having trouble figuring out your gas mileage.'
Gary Larson’s The Far Side reprint for the 4th of June, 2020. Essays that showcase something inspired by The Far Side I’ve gathered at this link.

Samson’s Dark Side of the Horse for the 5th is wordplay. It builds on the use of “problem” to mean both “something to overcome” and “something we study”. The mathematics puzzle book is a fanciful creation. The name Lucien Kastner is a Monty Python reference. (I thank the commenters for spotting that.)

Horace, walking and reflecting: 'My childhood wasn't easy. There were all these problems.' Flashback to a childhood Christmas and young Horace delighted to open the book: '1000 Math Problems to Enjoy, by Prof Lucien Kastner.'
Samson’s Dark Side of the Horse for the 5th of June, 2020. This and other essays based on Dark Side of the Horse are at this link.

Dan Collins’s Looks Good on Paper for the 5th is some wordplay on the term “Möbius Strip”, here applied to a particular profession.

A woman on stage is seen from the knees down. Title: 'Mobius Stripper'. Man in the audience thinking: 'I can't tell if she's taking her clothes off or putting them on!'
Dan Collins’s Looks Good on Paper for the 5th of June, 2020. The full, I think, exploration of Looks Good on Paper doing Möbius Strip jokes are gathered at this link.

Bud Blake’s Tiger rerun for the 6th has Tiger complaining about his arithmetic homework. And does it in pretty nice form, really, doing some arithmetic along the way. It does imply that he’s starting his homework at 1 pm, though, so I guess it’s a weekend afternoon. It seems like rather a lot of homework for that age. Maybe he’s been slacking off on daily work and trying to make up for it.

Tiger: 'I've got two plus four hours of homework. I won't be finished until ten minus three o'clock. Or maybe even six plus one and a half o'clock.' Punkinhead: 'What subject?' Tiger: 'Arithmetic, stupid!'
Bud Blake’s Tiger for the 6th of June, 2020. Essays showing off Tiger should all appear at this link.

John McPherson’s Close To Home for the 6th has a cheat sheet skywritten. It’s for a geometry exam. Any subject would do, but geometry lets cues be written out in very little space. The formulas are disappointingly off, though. We typically use ‘r’ to mean the radius of a circle or sphere, but then would use C for its circumference. That would be c = 2\pi r . The area of a circle, represented with A, would be \pi r^2 . I’m not sure what ‘Vol.C’ would mean, although ‘Volume of a cylinder’ would make sense … if the next line didn’t start “Vol.Cyl”. The volume of a circular cylinder is \pi r^2 h , where r is the radius and h the height. For a non-circular cylinder, it’s the area of a cross-section times the height. So that last line may be right, if it extends out of frame.

Kid in school, staring out the window. A cloud skywrites: 'C = pi * r^2', 'Vol C = pi r^2', 'vol. cyl = pi r ... ' Caption: 'With a bit of help from his uncle's skywriting business, Dale was able to pass the geometry final.'
John McPherson’s Close To Home for the 6th of June, 2020. Essays that feature something explored by Close to Home should be at this link.

Granted, though, a cheat sheet does not necessarily make literal sense. It needs to prompt one to remember what one needs. Notes that are incomplete, or even misleading, may be all that one needs.


And this wraps up the comics. This and other Reading the Comics posts are gathered at this link. Next week, I’ll get the All 2020 A-to-Z under way. Thanks once again for all your reading.

Reading the Comics, June 3, 2020: Subjective Opinions Edition


Thanks for being here for the last week before my All-2020 Mathematics A to Z starts. By the time this posts I should have decided on the A-topic, but I’m still up for B or C topics, if you’d be so kind as to suggest things.

Bob Weber Jr’s Slylock Fox for the 1st of June sees Reeky Rat busted for speeding on the grounds of his average speed. It does make the case that Reeky Rat must have travelled faster than 20 miles per hour at some point. There’s no information about when he did it, just the proof that there must have been some time when he drove faster than the speed limit. One can find loopholes in the reasoning, but, it’s a daily comic strip panel for kids. It would be unfair to demand things like proof there’s no shorter route from the diner and that the speed limit was 20 miles per hour the whole way.

Ted Shearer’s Quincy for the 1st originally ran the 7th of April, 1981. Quincy and his friend ponder this being the computer age, and whether they can let computers handle mathematics.

Jef Mallett’s Frazz for the 2nd has the characters talk about how mathematics offers answers that are just right or wrong. Something without “subjective grading”. It enjoys that reputation. But it’s not so, and that’s obvious when you imagine grading. How would you grade an answer that has the right approach, but makes a small careless error? Or how would you grade an approach that doesn’t work, but that plausibly could?

Kid: 'I hate essay assignments. They're so open to subjective grading. It would be a lot simpler if answers would just be right or wrong.' Frazz: 'You're in luck. I understand there's a math test coming up.' Kid: 'What's the lucky part?'
Jef Mallett’s Frazz for the 2nd of June, 2020. Other essays featuring something discussed in Frazz appear at this link.

And how do you know that the approach wouldn’t work? Even in non-graded mathematics, we have subjectivity. Much of mathematics is a search for convincing arguments about some question. What we hope to be convinced of is that there is a sound logical argument making the same conclusions. Whether the argument is convincing is necessarily subjective.

Yes, in principle, we could create a full deductive argument. It will take forever to justify every step from some axiom or definition or rule of inference. And even then, how do we know a particular step is justified? It’s because we think we understand what the step does, and how it conforms to one (or more) rule. That’s again a judgement call.

(The grading of essays is also less subjective than you might think if you haven’t been a grader. The difference between an essay worth 83 points and one worth 85 points may be trivial, yes. But you will rarely see an essay that reads as an A-grade one day and a C-grade the next. This is not to say that essay grading is not subject to biases. Some of these are innocent, such as the way the grader’s mood will affect the grade. Or how the first several papers, or the last couple, will be less consistently graded than the ones done in the middle of the project. Some are pernicious, such as under-rating the work done by ethnic minority students. But these biases affect the way one would grade, say, the partial credit for an imperfectly done algebra problem too.)

Mark Anderson’s Andertoons for the 3rd is the Mark Anderson’s Andertoons for the week. I could also swear that I’ve featured it here before. I can’t find it, if I have discussed this strip before. I may not have. Wavehead’s observing the difference between zero as an additive identity and its role in multiplication.

On the blackboard are written 7 + 0 = 7, 7 - 0 = 7, and 7 x 0 = 0. Wavehead: 'So the takeaway ehre is, if I'm the number 7, avoid multiplication at all costs.'
Mark Anderson’s Andertoons for the 3rd of June, 2020. When I have an essay that features something mentioned in Andertoons the essay’s put up at this link.

Ryan Pagelow’s Buni for the 3rd fits into the anthropomorphic-numerals category of joke. It’s really more of a representation of the year as the four horsemen of the Apocalypse.

Dan Collins’s Looks Good on Paper for the 3rd has a cook grilling a “Möbius Strip Steak”. It’s a good joke for putting on a mathematics instructor’s door.

Doug Savage’s Savage Chickens for the 3rd has, as part of animal facts, the assertion that “llamas have basic math skills”. I don’t know of any specific research on llama mathematics skills. But animals do have mathematics skills. Often counting. Some amount of reasoning. Social animals often have an understanding of transitivity, as well, especially if the social groups have a pecking order.


And this wraps up half of the past week’s mathematically-themed comic strips. I hope to have the rest in a Reading the Comics post at this link in a few days. Thanks for reading.

Reading the Comics, May 29, 2020: Slipping Into Summer More Edition


This is the slightly belated close of last week’s topics suggested by Comic Strip Master Command. For the week we’ve had, I am doing very well.

Werner Wejp-Olsen’s Inspector Danger’s Crime Quiz for the 25th of May sees another mathematician killed, and “identifying” his killer in a dying utterance. Inspector Danger has followed killer mathematicians several times before: the 9th of July, 2012, for instance. Or the 4th of July, 2016, for a case so similar that it’s almost a Slylock Fox six-differences puzzle. Apparently realtors and marine biologists are out for mathematicians’ blood. I’m not surprised by the realtors, but hey, marine biology, what’s the deal? The same gimmick got used the 15th of May, 2017, too. (And in fairness to the late Wejp-Olsen, who could possibly care that similar names are being used in small puzzles used years apart? It only stands out because I’m picking out things that no reasonable person would notice.)

Monty, to his robot pal: 'During a plague, Sir Isaac Newton invented calculus! Shakespeare wrote Lear and Macbeth!' (Two panels of Monty thinking hard and struggling to compose anything.) Monty: 'Maybe I'm more like Charles Darwin. I think he wrote 'On the Origin of Species' when everything was pretty normal.'
Jim Meddick’s Monty for the 25th of May, 2020. Essays with some mention of topics from Monty are at this link.

Jim Meddick’s Monty for the 25th has the title character inspired by the legend of genius work done during plague years. A great disruption in life is a great time to build new habits, and if Covid-19 has given you the excuse to break bad old habits, or develop good new ones, great! Congratulations! If it has not, though? That’s great too. You’re surviving the most stressful months of the 21st century, I hope, not taking a holiday.

Anyway, the legend mentioned here includes Newton inventing Calculus while in hiding from the plague. The actual history is more complicated, and ambiguous. (You will not go wrong supposing that the actual history of a thing is more complicated and ambiguous than you imagine.) The Renaissance Mathematicus describes, with greater authority and specificity than I could, what Newton’s work was more like. And some of how we have this legend. This is not to say that the 1660s were not astounding times for Newton, nor to deny that he worked with a rare genius. It’s more that we are lying to imagine that Newton looked around, saw London was even more a deathtrap than usual, and decided to go off to the country and toss out a new and unique understanding of the infinitesimal and the continuum.

Classroom. The teacher has drawn a geometric ray on the blackboard. Student: 'So that goes on forever? Should we warn people in the hallway?!'
Mark Anderson’s Andertoons rerun for the 27th of May, 2020. It ran at least as recently as the 3rd of August, 2017. and I noticed it then. This, that, and other essays featuring Andertoons can be found at this link.

Mark Anderson’s Andertoons for the 27th is the Mark Anderson’s Andertoons for the week. One of the students — not Wavehead — worries that a geometric ray, going on forever, could endanger people. There’s some neat business going on here. Geometry, like much mathematics, works on abstractions that we take to be universally true. But it also seems to have a great correspondence to ordinary real-world stuff. We wouldn’t study it if it didn’t. So how does that idealization interact with the reality? If the ray represented by those marks on the board goes on to do something, do we have to take care in how it’s used?

Olivia Jaimes’s Nancy for the 29th is set in a (virtual) arithmetic class. It builds on the conflation between “nothing” and “zero”.


And that wraps up my week in comic strips. I keep all my Reading the Comics posts at this link. I am also hoping to start my All 2020 Mathematics A-to-Z shortly, and am open for nominations for topics for the first couple letters. Thank you for reading.

Reading the Comics, May 25, 2020: Slipping into Summer Edition


Comic Strip Master Command wanted to give me a break as I ready for the All 2020 A-to-Z. I appreciate the gesture, especially given the real-world events of the past week. I get to spend this week mostly just listing appearances, even if they don’t inspire deeper thought.

Gordon Bess’s vintage Redeye for the 24th has one of his Cartoon Indians being lousy at counting. Talking about his failures at arithmetic, with how he doesn’t count six shots off well. There’s a modest number of things that people are, typically, able to perceive at once. Six can be done, although it’s easy for a momentary loss of focus to throw you off. This especially for things that have to be processed in sequence, rather than perceived all together.

Wulff and Morgenthaler’s WuMo for the 24th shows a parent struggling with mathematics, billed as part of “the terrible result of homeschooling your kids”. It’s a cameo appearance. It’d be the same if Mom were struggling with history or English. This is just quick for the comic strip reader to understand.

Andrés J. Colmenares’s Wawawiwa for the 25th sets several plants in a classroom. They’re doing arithmetic. This, too, could be any course; it just happens to be mathematics.

Sam Hurt’s Eyebeam for the 25th is built on cosmology. The subject is a blend of mathematics, observation, and metaphysics. The blackboard full of mathematical symbols gets used as shorthand for describing the whole field, not unfairly. The symbols as expressed don’t come together to mean anything. I don’t feel confident saying they don’t mean anything, though.


This is enough for today. I keep all my Reading the Comics posts at this link, and should have another one later this week. And I am trying to get my All 2020 Mathematics A-to-Z ready, with nominations open for the first several letters of the alphabet already. Thank you for reading.

In Our Time podcast repeats episode on Zeno’s Paradoxes


It seems like barely yesterday I was giving people a tip about this podcast. In Our Time, a BBC panel-discussion programme about topics of general interest, this week repeated an episode about Zeno’s Paradoxes. It originally ran in 2016.

The panel this time is two philosophers and a mathematician, which is probably about the correct blend to get the topic down. The mathematician here is Marcus du Sautoy, with the University of Oxford, who’s a renowned mathematics popularizer in his own right. That said I think he falls into a trap that we STEM types often have in talking about Zeno, that of thinking the problem is merely “how can we talk about an infinity of something”. Or “how can we talk about an infinitesimal of something”. Mathematicians have got what seem to be a pretty good hold on how to do these calculations. But that we can provide a logically coherent way to talk about, say, how a line can be composed of points with no length does not tell us where the length of a line comes from. Still, du Sautoy knows rather a few things that I don’t. (The philosophers are Barbara Sattler, with the University of St Andrews, and James Warren, with the University of Cambridge. I know nothing further of either of them.)

The episode also discusses the Quantum Zeno Effect. This is physics, not mathematics, but it’s unsettling nonetheless. The time-evolution of certain systems can be stopped, or accelerated, by frequent measurements of the system. This is not something Zeno would have been pondering. But it is a challenge to our intuition about how change ought to work.

I’ve written some of my own thoughts about some of Zeno’s paradoxes, as well as on the Sorites paradox, which is discussed along the way in this episode. And the episode has prompted new thoughts in me, particularly about what it might mean to do infinitely many things. And what a “thing” might be. This is probably a topic Zeno was hoping listeners would ponder.