A Review

abel’s hell

“Divergent series are the invention of the devil and it is shameful to base on them any demonstration whatsoever.”

niels abel

Last time, we left off talking about Pi.

We talked about how one can imagine Mother Nature using Pi to all its infinite glory, but for mere mortals like you and I, we have to truncate the number — to its tenth decimal place, hundredth decimal place, or even millionth decimal place.

But however we do it, we still miss infinitely many digits (infinity minus a finite number is still infinity, don’t-cha-know).

This is simply an example of our inability to describe the Universe to its fullest.

For the most part, that’s perfectly fine. But in what circumstance does our inability to capture the true nature of a phenomenon become a problem?

A Hopeless Challenge

Let’s talk oceans.

There are few things more majestic than a ship in water. Captain Ahab knew it. Jack and Rose knew it (well, at least until the end). So should you.

Just imagine it. The sparkling blue waters. The expanse of never-ending ocean. The beautiful wedge-shaped pattern that flows behind every ship, like a streaming cape.

And near the ship: turbulent white-water splashes, jets, and ripples, forming in a million different places with a million different patterns, often too quick and too elaborate for the eye to capture.

The mathematical equations which govern the flow of water in an ocean were known as early back as the 18th century. But for the next century and a bit, scientists were absolutely confounded by the equations.

“What in the world,” they asked together, “Can we do with that?”

The equations, as they stand, are much too complex for a direct analysis. Even today, there is a million dollar prize for anyone who can answer even the most basic questions about these equations.

So with no possibility of exact solutions in sight, the mathematicians and physicists turned to developing methods for approximating solutions.

Asymptotic Approximations

One method, in particular, traces its roots back to the time of Henri Poincare (1854-1912) who developed it in order to solve a rather terrifying problem in celestial mechanics.

The idea is to break up our solution into more manageable chunks. The first chunk describes the system — the ocean, in this case — in some idealized state. To this, we add the second chunk, which includes some kind of perturbation to this initial state. To this, we add a third and even smaller perturbation. And on to infinity.

For example, suppose we were to express the height of the water waves in this fashion,

Here, the greek epsilon, ε represents a small perturbation. It can represent, for example, the Froude number, or the surface tension, or some other small effect.

These sums or series are called asymptotic approximations because they get better and better in the asymptotic limit that ε tends to zero. That’s the idealized state.

The amazing thing with these approximations is that they tend to be very good. So good, in fact, that calculating only one or two chunks provides an excellent approximation to reality in most cases.

This was great. People were happy. We’d at least established some kind of systematic way of approaching these intractable formulae and, with a moderate amount of sweat and tears, they seemed to give excellent results.

The Physicists were happy.

So were the Applied Mathematicians.

The Pure Maths chaps, on the other hand, were incensed.

The Puries and their Partypooper Tendencies

The problem, the Puries were quick to point out, is that these asymptotic approximations are usually divergent.

What that means is that the approximations get better and better as we include more and more terms…but only up to a point. After that point, all hell breaks loose, and the whole thing blows up (to infinity).

Thus in effect, we have taken a system of water waves, perfectly well behaved and all very pretty, and approximated it using an infinite series of terms which, when added up, gives us completely nonsense.

The Pure Mathematicians, in all their need for rules and regulations, just weren’t comfortable toying with a concept that was so ill-defined.

But everybody else was all, “Fuggetaboutit. Take a look at all these purty graphs, yo.”

Should We Worry?

By now you’ve hopefully grappled with issue.

Nature in all its magnificence is woefully complicated.

In many cases, it’s so horrendously complicated that its mathematical description needs to be approximated. One such method of approximation involves using a divergent series.

Why do we use them? We use them because they work so nicely.

So the question becomes, in using these divergent approximations, are we ignoring anything useful? In particular, what is the nature of this divergence? Why does our approximation work so well initially, but behave so badly later on?

What’s the price we pay for dancing with the devil?

On Mother Nature

pi

“God, after all, can effortlessly spit out the infinitely long decimals of Pi. On a whim. While he’s sippin’ a brew and munchin’ some cheetos.

Because that’s how the Almighty rolls.”

Imagine a circle.

That’s right. Take that wonderful goody-two-shoes shape of two dimensional Euclidean geometry. It’s neat, it’s tidy, not as obtuse as a triangle (or god forbid, a square), compact, space efficient, and yes, undeniably sexy.

If Kylie Minogue was a Euclidean shape, she’d be a circle.

Now as we all know, the circle (and its three-dimensional brother, the sphere) is nature’s favourite shape. It occurs in everything, from the shape of soap bubbles in the bath to the wave expansion of sound disturbance at a concert. Everything from teensy atoms to humongous stars and planets are connected with the circle.

You know, in the Medieval Ages, they believed that the compass, with its magical ability to produce the most perfect of shapes — must have been one of God’s divine instruments.

Then again, would God really need a compass if he was God?

Never mind.

The point is, the circle is beautiful. It’s divine. It’s perfect.

On Mathematicians

Now we turn to the mathematicians.

It was known as early as the Egyptians that the ratio of a circle’s circumference to its diameter is constant. After that, more and more accurate approximations to this mysterious number proliferated through the years — 6 decimal places by Archimedes’ time and up to 100 decimal places by Newton’s time.

In 1706, William Jones — probably due to a lack of stationary and ink — began using the symbol Pi to represent the tedious number. Then in 1761, the German mathematician, Johannes Lambert proved Pi is irrational, and in 1882, Ferdinand von Lindemann (also German) proved Pi is transcendental.

Let’s talk about what these two facts imply.

Being irrational, Pi can never be written down as a terminating number, nor does it ever repeat. If you possessed an infinite amount of boredom, time, space, and pencil lead, and wrote out Pi all the way to infinity — you’d never see a pattern.

Being transcendental, Pi can never be constructed with a straight edge ruler and a compass — meaning, given theoretically perfect tools, you would never be able to measure out Pi or in particular, you would never be able to construct a square with the same area as a circle.

You put these two mathematical facts together and there is only one thing to conclude:

Pi is an awful, atrociously foul number.

On the Fundamental Question

And so we come to a fundamental, philosophical divide.

Nature has the ability to make perfect circles every time. God, after all, can effortlessly spit out the infinitely long decimals of Pi. On a whim. Just like that.

Because that’s how He rolls.

To the Almighty, Pi is indeed wonderful, beautiful, and simple.

But for us mere mortals with comparatively little brain space, we usually have to truncate Pi. So when we use calculators or a computers, programmed with only a finite number of decimal places, we’re not using Pi as nature intended it, but the imperfect, imprecise version we’ve constructed for our own evil schemes.

For the most part, this is okay.

Nobody (except for the clinically unhinged) loses sleep over the fact that we’re only using 16 decimal places. The resulting error between mathematical prediction and true phenomenon is so inconsequential, nobody cares.

But you see, all this buildup — it was just a metaphor.

An attempt to explain to you the difference between man, mathematics, and nature.

An attempt to explain that yes, nature is indeed wonderful and perfect, but sometimes, we aren’t always able to capture this magnificence completely. Especially when we impose our own mathematical rules to describe it.

So here’s a question: At what point does the difference between mathematical representation, pragmatic implementation, and true reality have severe consequences.

When it comes to maths, I don’t really consider myself a noob.

Of course, there’s a lot I don’t know. In fact, it’s a common belief that the last mathematician to know ‘everything’ (that is, to at least be familiar with everything) was David Hilbert — and that only really applies to the early 20th century.

Today, there is simply no way for you to be an expert in every field or even, in a substantial number of fields. There’s simply too much to know. You’d be spending the whole span of your life, playing catch-up with what’s been done and what’s being done.

But yeah. I’m not a noob, y’know?

For example, I think it’s safe to say I know my way around integration. Single, double, or triple integration — no problem. Laplace integrals, Fourier integrals, Elliptic integrals — piece of cake. Even more deviant and exotic creatures like Error functions, Fresnel integrals, Airy integrals, and so on and so forth I’ve encountered.

Today, however, I met a new beast. At first glance, I treated with disdain.

After longer, more protracted glance, I’m happy to report I’m still…uh…pretty repulsed by the whole thing.

The formula in question was derived by the Australian mathematician, John Henry Mitchell in 1898. The formula, which can be used to calculate the wave resistance for ships of different forms, was intended to revolutionize the Naval Engineering community.

It didn’t.

The popular theory is that the engineers took one look at the formula, went “Eeewww”, and promptly burnt the article.

People then go on to theorize that Mitchell’s short career (10 years?) in research mathematics was due to his disgust at the reception of this work.

But looking at the formula, can you really blame the community?

Yuck.

Things I Don’t Care to See #138: A quintuplet integration.

P.S. If you’re feeling particularly bad about the poor Aussie, don’t be. His work in the 1898 paper is now seen as one of the most important and revolutionary contributions to hydrodynamic ship resistance.

If only I can be so lucky.

Let’s face it. Some things in life are certainties. This is one of them:

Or so I thought.

“I don’t wanna die.”

See, when your plane is plummeting towards the Atlantic and dozens of oxygen masks are simultaneously popped from their compartments, that’s precisely what you’re supposed to be thinking.

That’s what most sane passengers on board Flight 888 were thinking, anyways, but not me; personally, I was content to stare at the gorgeous woman sitting by my side and wonder — amidst all the screaming — whether, she’d be the type to favour wit and humour over a Calvin-Klein physique.

Mind you, this was just the setup (it turned out to be a false alarm, anyways — “Due to horrendous weather conditions,” said the pilot), but I still wanted to give you an idea of the truly epic life-or-death scope of the story to come.

The real story, however, begins with her.

Even in the limited illumination of the cabin, her hair shone a rich, copper red. This framed not only a perfect curving face, but also two of the bluest eyes I’d ever seen. But this wasn’t all. The earlier theatrics had scared her and despite clear skies and smooth sailing, I could tell she was still on edge. But this was perfect, you see, because it gave me a chance to swoop in and sooth her poor, tortured soul (that’s how it’s supposed to work, anyways).

“Just a little turbulence,” I offered helpfully, “Nothing to be scared of — uh…” I trailed off, motioning expectantly towards her.

“Rhea.”

“Right. Rhea. It’s Jeff, by the way. There’s nothing to be scared of, Rhea,” I assured her again.

She was silent for a moment; then leaning over me, pointed a slender finger out the window. “Nothing?” she said, raising an eyebrow, “Yeah, nothing is what’s stopping us from plummeting, oh I dunno, twenty-thousand feet to the ground in a blazing tomb of shattering metal and death, right? …Jeff?”

So it was going to be harder than I thought.

Ten minutes had passed since my disastrous attempt to heroically intervene, and by now, she was already nervously fingering her way through a Cosmo magazine.

I needed to do something — something drastic — or I was going lose her to another one of those ghastly how-does-your-love-life-stack-up quizzes.

“You’re wrong,” I blurted out, loudly.

She turned and looked at me, bemused.

“About what?”

“What you said earlier. About how nothing is holding us up.”

I ran my fingers through my hair and backtracked. “Look, flying wasn’t invented overnight, right? Da Vinci, Cayley, Lanchester, Joukowski, …” I rattled off a few more impressive-sounding names, “Thousands of the greatest minds conspiring over hundreds of years with the sole purpose of reaching one of man’s greatest dreams…”

“So?”

“So…today — today, aerodynamics, all this –,” I gestured out the window, “All this — is grounded in solid, rigorous mathematics. It’s not nothingness that’s holding us up. It’s math.”

“Math,” she repeated disinterestedly, “I’m not very good at math.”

“You don’t have to be,” I replied softly.

She picked up the Cosmo magazine from her lap and froze for what seemed like ages. Then, seeming to have made a decision, she placed it in the pouch before her and leaned back in her seat, sighing.

“Okay,” she said, nodding, “Let’s talk.”

By now, we’d both finished our creations: three crisp paper airplanes assembled with the enthused precision only a 10-year-old could muster.

She threw hers and laughed as it flopped straight to the ground. Mine careened suicidally into a bulkhead and the stewardess, having witnessed all our shenanigans, scowled in our direction.

As I held up the last plane, I explained. “There are four aerodynamic forces we need to talk about: thrust, drag, lift, and gravity.” I said, gesturing in each of the four directions.

“As air flows over the plane, pressure is exerted on the surface of the plane. If there’s more pressure on the bottom than the top, lift is produced, and the plane stays aloft so long as the force is greater than gravity.”

“But this,” she said, waving at our surroundings, “is not made of paper.”

“Ah ha! Obtaining lift is easy — heck, even a barn door raised at an angle produces lift. The second problem is drag, and so flight is actually the problem of maximizing lift, while minimizing drag.”

“But how?” I asked excitedly, “What kind of aerofoil design do we need, Rhea?”

“Beats me,” she said, amused by my enthusiasm, “But I bet you’ll tell me all about it over dinner.”

It was like a date, really.

But with only two meal choices and really crappy wine.

“So…” Rhea said, prodding her chicken suspiciously, “When the Wright Brothers managed the first ever flight in nineteen-hundred-whatever, they didn’t actually know how the math worked?”

With a fork, I poked my own in-flight meal timidly. Half-expecting the chicken to wake up and start clucking hysterically, I wisely opted for the fruit cup instead.

“The Wright Brothers were at the forefront of experimental aerodynamics,” I corrected, “Using their own wind tunnel tests, they produced different aerofoils until they found one that worked. Most importantly, it had to be curved (or cambered) so air flows faster over the top; this reduces the pressure on the top and so produces more lift.”

“But surprisingly, there were almost no meaningful aerodynamic theories at the time. We had the right equations — but nobody knew what to do with them.”

“But this changed?”

I pushed my meal back with a grimace. “Oh, sure. Theoretical aerodynamics was rampant only a handful of years later. But, uh…I dunno,” I said, teasing her, “You sure you’re ready for this stuff?”

I leaned back and pretended to peer at her dubiously.

“I mean, it’s pretty wild.”

She just rolled her eyes and went back to eating.

The crew had turned off the lights and so most passengers were dozing quietly. Not us, though. We insisted on whispering softly, and with the lone reading light providing the only illumination, it had the eerie effect of making it seem like we were the only two passengers on board Flight 888.

“The three governing equations of fluid motion are known as the Navier Stokes Equations,” I explained while scribbling on napkins, “And together, they simply express three fundamental laws of nature in a mathematical form.”

“This one here,” I said pointing to the first equation, “Says that the total amount of fluid — or, in our case, air particles – has to remain constant. This is just conservation of mass. Stuff can’t disappear.”

“The second one here is Newton’s Second Law: Mass times acceleration, is equal to the total applied force. That’s just conservation of momentum.”

“And the last one here is conservation of energy — the total energy in the system must remain constant.”

She looked disdainfully at the row of napkins strewn before her and pointed a single, perfectly manicured finger at one of them. “What can we do with that?” she asked.

“It’s tough,” I conceded, “These equations are too hard to solve directly. And even for the two-dimensional problem (flow around a cross section of the aerofoil), it’s still too complicated because of the geometry.”

“So what we do is we consider a simpler geometry, say, the airflow around a flat plate; we solve the governing equations for this problem, then we construct a mathematical map that brings us back to the original, more difficult geometry.”

“It’s tough, I know. But the point is, the math is all there. It’s not a mystery anymore, like it was for da Vinci and the Wright Brothers.

And that — believe it or not — was the last time we’d talk about math.

She was relaxed now, and her head was gently propped against a pillow resting on my shoulder. But whether or not this Zen-like state of relaxation was because of newfound understanding (as opposed to sheer mental boredom)…well, I’ll let you decide.

In any case, it worked, and hell, she hadn’t touched that damn Cosmo since we both started talking.

“Are you asleep?” I asked.

She took her time answering. “Not yet”.

“Sooooo…”, I said, stretching out the vowel and grinning sheepishly, “Tell me, what do you do?”

At that suggestion, she whipped her head off the pillow and gave me such a look of pure, seething outrage that…well, I tell ya’, it would have immediately sent lesser mortals fleeing.

“Oooooh, I get it,” she squealed, “Good looking stranger corners you on an airplane for a three hour balls-to-wall math lecture, and then asks what you do. Typical,” she sniffed.

“Wha–?”, I said, feigning mock hurt, “But I got you dinner and everything.”

She gave me a mischievous grin, punched her pillow, and put her head back.

“Admit it Jeff,” she said, “You’ve been using me.”

I pushed the button on my armrest turning off the overhead lights and plunging the two of us into blissful darkness.

“Babe,” I said, “You’ve no idea.”

A great mystery has struck the math department.

It seems that every once in a while — most likely during the nights — someone uses the whiteboard in the common room to scribble various notes of the mathematical sort.

It’s not that the math is brilliant. Or that the scribbles prove some hundred-year old mystery.

“But just look at this writing,” a faculty member exclaims one afternoon, “It’s just so neat. And look at these diagrams. Who’s responsible for this?”

Nobody’s really sure.

The board’s been untouched since. Except for single, small addition: “Who’s is this handwriting?” it now says at the top.

And me? I just sit there. Tickled.