## 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.

# 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.

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.

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.

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.

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.

## Reading the Comics, December 30, 2015: Seeing Out The Year Edition

There’s just enough comic strips with mathematical themes that I feel comfortable doing a last Reading the Comics post for 2015. And as maybe fits that slow week between Christmas and New Year’s, there’s not a lot of deep stuff to write about. But there is a Jumble puzzle.

Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips gives us someone so wrapped up in measuring data as to not notice the obvious. The obvious, though, isn’t always right. This is why statistics is a deep and useful field. It’s why measurement is a powerful tool. Careful measurement and statistical tools give us ways to not fool ourselves. But it takes a lot of sampling, a lot of study, to give those tools power. It can be easy to get lost in the problems of gathering data. Plus numbers have this hypnotic power over human minds. I understand Lard’s problem.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 27th of December messes with a kid’s head about the way we know 1 + 1 equals 2. The classic Principia Mathematica construction builds it out of pure logic. We come up with an idea that we call “one”, and another that we call “plus one”, and an idea we call “two”. If we don’t do anything weird with “equals”, then it follows that “one plus one equals two” must be true. But does the logic mean anything to the real world? Or might we be setting up a game with no relation to anything observable? The punchy way I learned this question was “one cup of popcorn added to one cup of water doesn’t give you two cups of soggy popcorn”. So why should the logical rules that say “one plus one equals two” tell us anything we might want to know about how many apples one has?

David L Hoyt and Jeff Knurek’s Jumble for the 28th of December features a mathematics teacher. That’s enough to include here. (You might have an easier time getting the third and fourth words if you reason what the surprise-answer word must be. You can use that to reverse-engineer what letters have to be in the circles.)

Richard Thompson’s Richard’s Poor Almanac for the 28th of December repeats the Platonic Fir Christmas Tree joke. It’s in color this time. Does the color add to the perfection of the tree, or take away from it? I don’t know how to judge.

Hilary Price’s Rhymes With Orange for the 29th of December gives its panel over to Rina Piccolo. Price often has guest-cartoonist weeks, which is a generous use of her space. Piccolo already has one and a sixth strips — she’s one of the Six Chix cartoonists, and also draws the charming Tina’s Groove — but what the heck. Anyway, this is a comic strip about the butterfly effect. That’s the strangeness by which a deterministic system can still be unpredictable. This counter-intuitive conclusion dates back to the 1890s, when Henri Poincaré was trying to solve the big planetary mechanics question. That question is: is the solar system stable? Is the Earth going to remain in about its present orbit indefinitely far into the future? Or might the accumulated perturbations from Jupiter and the lesser planets someday pitch it out of the solar system? Or, less likely, into the Sun? And the sad truth is, the best we can say is we can’t tell.

In Brian Anderson’s Dog Eat Doug for the 30th of December, Sophie ponders some deep questions. Most of them are purely philosophical questions and outside my competence. “What are numbers?” is also a philosophical question, but it feels like something a mathematician ought to have a position on. I’m not sure I can offer a good one, though. Numbers seem to be to be these things which we imagine. They have some properties and that obey certain rules when we combine them with other numbers. The most familiar of these numbers and properties correspond with some intuition many animals have about discrete objects. Many times over we’ve expanded the idea of what kinds of things might be numbers without losing the sense of how numbers can interact, somehow. And those expansions have generally been useful. They strangely match things we would like to know about the real world. And we can discover truths about these numbers and these relations that don’t seem to be obviously built into the definitions. It’s almost as if the numbers were real objects with the capacity to surprise and to hold secrets.

Why should that be? The lazy answer is that if we came up with a construct that didn’t tell us anything interesting about the real world, we wouldn’t bother studying it. A truly irrelevant concept would be a couple forgotten papers tucked away in an unread journal. But that is missing the point. It’s like answering “why is there something rather than nothing” with “because if there were nothing we wouldn’t be here to ask the question”. That doesn’t satisfy. Why should it be possible to take some ideas about quantity that ravens, raccoons, and chimpanzees have, then abstract some concepts like “counting” and “addition” and “multiplication” from that, and then modify those concepts, and finally have the modification be anything we can see reflected in the real world? There is a mystery here. I can’t fault Sophie for not having an answer.

## Reading the Comics, June 30, 2015: Fumigating The Theater Edition

One of my favorite ever episodes of The Muppet Show when I was a kid had the premise the Muppet Theater was being fumigated and so they had to put on a show from the train station instead. (It was the Loretta Lynn episode, third season, number eight.) I loved seeing them try to carry on as normal when not a single thing was as it should be. Since then — probably before, too, but I don’t remember that — I’ve loved seeing stuff trying to carry on in adverse circumstances.

Why this is mentioned here is that Sunday night my computer had a nasty freeze and some video card mishaps. I discovered that my early-2011 MacBook Pro might be among those recalled earlier this year for a service glitch. My computer is in for what I hope is a simple, free, and quick repair. But obviously I’m not at my best right now. I might be even longer than usual answering people and goodness knows how the statistics survey of June will go.

Anyway. Rick Kirkman and Jerry Scott’s Baby Blues (June 26) is a joke about motivating kids to do mathematics. And about how you can’t do mathematics over summer vacation.

Ruben Bolling’s Tom The Dancing Bug (June 26) features a return appearance of Chaos Butterfly. Chaos Butterfly does what Chaos Butterfly does best.

Charles Schulz’s Peanuts Begins (June 26; actually just the Peanuts of March 23, 1951) uses arithmetic as a test of smartness. And as an example of something impractical.

Alex Hallatt’s Arctic Circle (June 28) is a riff on the Good Will Hunting premise. That movie’s particular premise — the janitor solves an impossible problem left on the board — is, so far as I know, something that hasn’t happened. But it’s not impossible. Training will help one develop reasoning ability. Training will provide context and definitions and models to work from. But that’s not essential. All that’s essential is the ability to reason. Everyone has that ability; everyone can do mathematics. Someone coming from outside the academy could do first-rate work. However, I’d bet on the person with the advanced degree in mathematics. There is value in training.

But as many note, the Good Will Hunting premise has got a kernel of truth in it. In 1939, George Dantzig, a grad student in mathematics at University of California/Berkeley, came in late to class. He didn’t know that two problems on the board were examples of unproven theorems, and assumed them to be homework. So he did them, though he apologized for taking so long to do them. Before you draw too much inspiration from this, though, remember that Dantzig was a graduate student almost ready to start work on a PhD thesis. And the problems were not thought unsolvable, just conjectures not yet proven. Snopes, as ever, provides some explanation of the legend and some of the variant ways the story is told.

Mac King and Bill King’s Magic In A Minute (June 28) shows off a magic trick that you could recast as a permutations problem. If you’ve been studying group theory, and many of my Mathematics A To Z terms have readied you for group theory, you can prove why this trick works.

Guy Gilchrist’s Nancy (June 28) carries on Baby Blues‘s theme of mathematics during summer vacation being simply undoable.

Piers Baker’s Ollie and Quentin (June 28) is a gambler’s fallacy-themed joke. It was run — on ComicsKingdom, back then — back in December, and I talked some more about it then.

Mike Twohy’s That’s Life (June 28) is about the perils of putting too much attention into mental arithmetic. It’s also about how perilously hypnotic decimals are: if the pitcher had realized “fourteen million over three years” must be “four and two-thirds million per year” he’d surely have been less distracted.

## Reading the Comics, December 27, 2014: Last of the Year Edition?

I’m curious whether this is going to be the final bunch of mathematics-themed comics for the year 2014. Given the feast-or-famine nature of the strips it’s plausible we might not have anything good through to mid-January, but, who knows? Of the comics in this set I think the first Peanuts the most interesting to me, since it’s funny and gets at something big and important, although the Ollie and Quentin is a better laugh.

Mark Leiknes’s Cow and Boy (December 23, rerun) talks about chaos theory, the notion that incredibly small differences in a state can produce enormous differences in a system’s behavior. Chaos theory became a pop-cultural thing in the 1980s, when Edward Lorentz’s work (of twenty years earlier) broke out into public consciousness. In chaos theory the chaos isn’t that the system is unpredictable — if you have perfect knowledge of the system, and the rules by which it interacts, you could make perfect predictions of its future. What matters is that, in non-chaotic systems, a small error will grow only slightly: if you predict the path of a thrown ball, and you have the ball’s mass slightly wrong, you’ll make a proportionately small error on what the path is like. If you predict the orbit of a satellite around a planet, and have the satellite’s starting speed a little wrong, your prediction is proportionately wrong. But in a chaotic system there are at least some starting points where tiny errors in your understanding of the system produce huge differences between your prediction and the actual outcome. Weather looks like it’s such a system, and that’s why it’s plausible that all of us change the weather just by existing, although of course we don’t know whether we’ve made it better or worse, or for whom.

Charles Schulz’s Peanuts (December 23, rerun from December 26, 1967) features Sally trying to divide 25 by 50 and Charlie Brown insisting she can’t do it. Sally’s practical response: “You can if you push it!” I am a bit curious why Sally, who’s normally around six years old, is doing division in school (and over Christmas break), but then the kids are always being assigned Thomas Hardy’s Tess of the d’Urbervilles for a book report and that is hilariously wrong for kids their age to read, so, let’s give that a pass.