Making Things Simpler

“How about a pumpkin spice gelato, Mr Moire?”

“I don’t think so, Jeremy. I’m a traditionalist. A double‑dip of pistachio, please.”

“Coming right up, sir. By the way, I’ve been thinking about the Math poetry you find in the circular and hyperbolic functions. How about what you’d call Physics poetry?”

“Sure. Starting small, Physics has symmetries for rhymes. If you can pivot an experiment or system through some angle and get the same result, that’s rotational symmetry. If you can flip it right‑to‑left that’s parity symmetry. I think of a symmetry as like putting the same sound at the end of each line in rhymed verse. Physicists have identified dozens of symmetries, some extremely abstract and some fundamental to how we understand the Universe. Our quantum theory for electrons in atoms is based on the symmetries of a sphere. Without those symmetries we wouldn’t be able to use Schrodinger’s equation to understand how atoms work.”

“Symmetries as rhymes … okaaayy. What else?”

“You mentioned the importance of word choice in poetry. For the Physics equivalent I’d point to notation. You’ve heard about the battle between Newton and Leibniz about who invented calculus. In the long run the algebraic techniques that Leibniz developed prevailed over Newton’s geometric ones because Leibniz’ way of writing math was far simpler to read, write and manipulate — better word choice. Trying to read Newton’s Principia is painful, in large part because Euler hadn’t yet invented the streamlined algebraic syntax we use today. Newton’s work could have gone faster and deeper if he’d been able to communicate with Euler‑style equations instead of full sentences.”


“Leonhard Euler, though it’s pronounced like ‘oiler‘. Europe’s foremost mathematician of the 18th Century. Much better at math than he was at engineering or court politics — both the Russian and Austrian royal courts supported him but they decided the best place for him was the classroom and his study. But while he was in there he worked like a fiend. There was a period when he produced more mathematics literature than all the rest of Europe. Descartes outright rejected numbers involving ‑1, labeled them ‘imaginary.’ Euler considered ‑1 a constant like any other, gave it the letter i and proceeded to build entire branches of math based upon it. Poor guy’s vision started failing in his early 30s — I’ve often wondered whether he developed efficient notational conventions as a defense so he could see more meaning at a glance.”

“He invented all those weird squiggles in Math and Physics books that aren’t even Roman or Greek letters?”

“Nowhere near all of them, but some important ones he did and he pointed the way for other innovators to follow. A good symbol has a well‑defined meaning, but it carries a load of associations just like words do. They lurk in the back of your mind when you see it. π makes you think of circles and repetitive function like sine waves, right? There’s a fancy capital‑R for ‘the set of all real numbers‘ and a fancy capital‑Z for ‘the set of all integers.’ The first set is infinitely larger than the second one. Each symbol carries implications abut what kind of logic is valid nearby and what to be suspicious of. Depends on context, of course. Little‑c could be either speed‑of‑light or a triangle’s hypotenuse so defining and using notation properly is important. Once you know a symbol’s precise meaning, reading an equation is much like reading a poem whose author used exactly the right words.”

“Those implications help squeeze a lot of meaning into not much space. That’s the compactness I like in a good poem.”

“It’s been said that a good notation can drive as much progress in Physics as a good experiment. I’m not sure that’s true but it certainly helps. Much of my Physics thinking is symbol manipulation. Give me precise and powerful symbols and I can reach precise and powerful conclusions. Einstein turned Physics upside down when he wrote the thirteen symbols his General Relativity Field Equation use. In his incredibly compact notation that string of symbols summarizes sixteen interconnected equations relating mass‑energy’s distribution to distorted spacetime and vice‑versa. Beautiful.”

“Beautiful, maybe, but cryptic.”

~~ Rich Olcott

Something of Interest

“OK, Sy, I get how money is sorta like Physics ‘energy‘ except you can’t create energy but you can create money. And I get how Economics ‘velocity of money‘ and Physics ‘velocity don’t have much to do with each other. Your ‘Money Physics‘ phrase doesn’t make much sense unless you’ve got something with more overlap than that.”

“You’re a tough man, Vinnie. How about the word ‘exponential‘?”

“Means something goes up really fast. What about it?”

“Well, first off that’s not really what it means and that’s one of my personal peeves, thank you very much. Yes, quantities can increase exponentially, but not necessarily rapidly, and they can also decrease exponentially, either fast or slow. It’s a math thing.”

“Alright, I got myself into this. You’re gonna tell me how that works and it probably involves equations.”

“You made the phone call, I’m just sitting here, but you’re good, no equations just arithmetic. Ten times ten’s a hundred, right, and you can write that either 10×10 or 10², OK? The little two is the exponent, tells you how many factors to multiply together.”

“And 10 with a little three makes a thousand and ten with a little … six makes a million. See, it goes up really fast.”

“Depends on what the base number is. I’ve sent a tabulation to your phone…”

Exp’t 10 2 99% 100% 101%
2 100 4 98.01% 100% 102.01%
3 1 000 8 97.03% 100% 103.03%
4 10 000 16 96.06% 100% 104.06%
5 100 000 32 95.10% 100% 105.10%
6 1 000 000 64 94.15% 100% 106.15%
7 10 000 000 128 93.21% 100% 107.21%

“What’s all that?”

“Well, the top-row headers are just numbers I multiplied by themselves according to some exponents, and the first column is the series of exponents I used. Like we said, 10² is a hundred and so on down the second column. Number 2 multiplied by itself according to the same exponents gave me the third column and you see the products don’t grow anywhere near as fast. Do you see how the growth rate depends on the number that’s being multiplied and re‑multiplied?”

“No problem. What about the other columns?”

“Start with the fifth column. What’s 100% of 100%?”

“All of it.”

“And 100% of 100% of 100%?”

“I get it — no change no matter the exponent.”

“Absolutely. Now compare that to the 99% and 101% columns that give you the effect of a 1% growth factor. As you’d expect, very little change in either one, but there’s a lesson in the 99% column. It’s exponential by definition, but the results go down, not up. By the way, both of those are such small factors that the results are practically linear. You need to get beyond 15% factors for visible curvature in the usual graphs.”

“OK, so exponential says some arithmetic factor gets applied again and again. What’s that got to do with Physics or Economics?”

“Ever since Newton, Physics has been the study of change, all different kinds. Gradually we’ve built up a catalog of change patterns. Newton pointed out the simplest one in his first Law of Motion — constant velocity, say in meters per second. Plot cumulative distance moved against time and you get a rising straight line. His Second Law implies another simple pattern, constant acceleration. That’s one where velocity’s line rises linearly but distance goes up as the square of the time traveled. But Newton never tackled another very simple, very common pattern.”

“I thought Newton did everything.”

“Not the case. He was an amazing geometer, but to handle this pattern you need algebraic tools like the ones Liebniz was developing. Newton would rather have dunked his arm in boiling rancid skunk oil than do that. It took another century or so until the Bernoulis and Euler beat that problem into the ground.”

“So what’s the simple pattern?”

“Suppose instead of a quantity increasing by some absolute number of thingies per second, it increases by some constant percentage. That’s uncommon in the kinds of mechanical phenomena that Newton studied but it does happen. Say you’re a baby planet in the middle of a dust cloud. Get 15% bigger, you’re 15% better at attracting even more dust. Biological things do that a lot — the more bugs or bacteria you’ve got, the faster they multiply and that’s usually at a constant percentage-per-time rate. Exponential growth in a nutshell.”

“Planets, bugs, what’s that got to do with Economics?”

“Ever hear of ‘compound interest‘?”

“Low rates on bank accounts, high rates on credit cards, compounded. Gotcha.”

“Inflation does compounding, too.”

~~ Rich Olcott