Thinking in Spacetime

The Open Mic session in Al’s coffee shop is still going string. The crowd’s still muttering after Jeremy stuck a pin in Big Mike’s “coincidence” balloon when Jim steps up. Jim’s an Astrophysics post‑doc now so we quiet down expectantly. “Nice try, Mike. Here’s another mind expander to play with. <stepping over to the whiteboard> Folks, I give you … a hypotenuse. ‘That’s just a line,’ you say. Ah, yes, but it’s part of some right triangles like … these. Say three different observers are surveying the line from different locations. Alice finds her distance to point A is 300 meters and her distance to point B is 400. Applying Pythagoras’ Theorem, she figures the A–B distance as 500 meters. We good so far?”

A couple of Jeremy’s groupies look doubtful. Maybe‑an‑Art‑Major shyly raises a hand. “The formula they taught us is a2+b2=c2. And aren’t the x and y supposed to go horizontal and vertical?”

“Whoa, nice questions and important points. In a minute I’m going to use c for the speed of light. It’s confusing to use the same letter for two different purposes. Also, we have to pay them extra for double duty. Anyhow, I’m using d for distance here instead of c, OK? To your next point — Alice, Bob and Carl each have their own horizontal and vertical orientations, but the A–B line doesn’t care who’s looking at it. One of our fundamental principles is that the laws of Physics don’t depend on the observer’s frame of reference. In this situation that means that all three observers should measure the same length. The Pythagorean formula works for all of them, so long as we’re working on a flat plane and no-one’s doing relativistic stuff, OK?”

Tentative nods from the audience.

“Right, so much for flat pictures. Let’s up our game by a dimension. Here’s that same A–B line but it’s in a 3D box. <Maybe‑an‑Art‑Major snorts at Jim’s amateur attempt at perspective.> Fortunately, the Pythagoras formula extends quite nicely to three dimensions. It was fun figuring out why.”

Jeremy yells out. “What about time? Time’s a dimension.”

“For sure, but time’s not a length. You can’t add measurements unless they all have the same units.”

“You could fix that by multiplying time by c. Kilometers per second, times seconds, is a length.” His groupies go “Oooo.”

“Thanks for the bridge to spacetime where we have four coordinates — x, y, z and ct. That makes a big difference because now A and B each have both a where and a when — traveling between them is traveling in space and time. Computationally there’s two paths to follow from here. One is to stick with Pythagoras. Think of a 4D hypercube with our A–B line running between opposite vertices. We’re used to calculating area as x×y and volume as x×y×z so no surprise, the hypercube’s hypervolume is x×y×z×(ct). The square of the A–B line’s length would be b2=(ct)2+d2. Pythagoras would be happy with all of that but Einstein wasn’t. That’s where Alice and Bob and Carl come in again.”

“What do they have to do with it?”

“Carl’s sitting steady here on good green Earth, red‑shifted Alice is flying away at high speed and blue‑shifted Bob is flashing toward us. Because of Lorentz contractions and dilations, they all measure different A–B lengths and durations. Each observer would report a different value for b2. That violates the invariance principle. We need a ruggedized metric able to stand up to that sort of punishment. Einstein’s math professor Hermann Minkowski came up with a good one. First, a little nomenclature. Minkowski was OK with using the word ‘point‘ for a location in xyz space but he used ‘event‘ when time was one of the coordinates.”

“Makes sense, I put events on my calendar.”

“Good strategy. Minkowski’s next step quantified the separation between two events by defining a new metric he called the ‘interval.’ Its formula is very similar to Pythagoras’ formula, with one small change: s2=(ct)2–d2. Alice, Bob and Carl see different distances but they all see the same interval.”

Minus? Where did that come from?”

~~ Rich Olcott

The Gelato Model

“Eddie, this ginger gelato’s delicious — not too sweet and just the right amount of ginger bite.”

“Glad you like it, Anne.”

On the way down here, Sy was telling me about how so many things in the Universe run on the same mathematics if you look at them with the right coordinate system. Sy, how do you pick ‘the right coordinate system?”

“The same way you pick the right property to serve as a momentum in Newton’s Equation of Motion — physical intuition. You look for things that fit the system. Sometimes that puts you on the road to understanding, sometimes not. Eddie, you keep track of your gelato sales by flavor. How are they doing?”

“Pistachio’s always a good seller, Sy, but ginger has been coming on strong this year.”

“In motion terns, pistachio’s momentum is constant but ginger is gaining momentum, right?”

“S’what I said.”

“Measured in dollars or trayfuls?”

“In batches. I make it all in-house. I’m proud of that. Dollars, too, of course, but that’s just total for all flavors.”

“Batches all the same size?”

“Some are, some not, depending. If I had a bigger machine I could make more but I do what I can.”

“There you go, Anne, each gelato flavor is like a separate degree of freedom. Eddie’s tracked sales since he started so we can take that date as the origin. Measuring change along any degree in either batches or dollars we have perfectly respectable coordinates although the money view of the system is fuzzier. Velocity is batches per unit time, there’s even a speed limit, and ginger has accelerated. Sound familiar?”

“Sounds like you’re setting up a Physics model.”

“Call it gelato trend physics, but I don’t think I can push the analogy much further. The next step would be to define a useful momentum like Newton did with his Law of Motion.”

F=ma? That’s about acceleration, isn’t it?”

“Probably not in Newton’s mind. Back in his day they were arguing about which was conserved, energy or momentum. It was a sloppy argument because no‑one agreed on crisp definitions. People could use words like ‘quantity of motion‘ to refer to energy or momentum or even something else. Finally Newton defined momentum as ‘mass times velocity‘, but first he had to define ‘mass‘ as ‘quantity of matter‘ to distinguish it from weight which he showed is a force that’s indirectly related to mass.”

“So is it energy or momentum that’s conserved?”

“Both, once you’ve got good definitions of them. But my point is, our car culture has trained us to emphasize acceleration. Newton’s thinking centered on momentum and its changes. In modern terms he defined force as momentum change per unit time. I’m trying to think of a force‑momentum pair for Eddie’s gelato. That’s a problem because I can’t identify an analog for inertia.”

“Inertia? What’s that got to do with my gelato?”

“Not much, and that’s the problem. Inertia is resistance to force. Who can resist gelato? If it weren’t for inertia, the smallest touch would be enough to send an object at high speed off to forever. The Universe would be filled with dust because stars and planets would never get the chance to form. But here we are, which I consider a good thing. Where does inertia come from? Newton changed his mind a couple of times. To this day we only have maybe‑answers to that question.”

“You know we want to know, Sy.”

“Einstein’s favorite guess was Mach’s Principle. There’s about a dozen different versions of the basic idea but they boil down to matter interacting with the combined gravitational and electromagnetic fields generated by the entire rest of the Universe.”

“Wow. Wait, the stars are far away and the galaxies are much, much further away. Their fields would be so faint, how can they have any effect at all?”

“You’re right, Anne, field intensity per star does drop with distance squared. But the number of stars goes up with distance cubed. The two trends multiply together so the force trends grow linearly. It’s a big Universe and size matters.”

“So what about my gelato?”

“We’ll need more research, Eddie. Another scoop of ginger, Anne?”

~~ Rich Olcott

Symmetrical Eavesdropping

“Wait, Sy, you’ve made this explanation way more complicated than it has to be. All I asked about was the horrible whirling I’d gotten myself into. The three angular coordinates part would have done for that, but you dragged in degrees of freedom and deep symmetry and even dropped in that bit about ‘if measurable motion is defined.’ Why bother with all that and how can you have unmeasurable motion?”

“Curiosity caught the cat, didn’t it? Let’s head down to Eddie’s and I’ll treat you to a gelato. Your usual scoop of mint, of course, but I recommend combining it with a scoop of ginger to ease your queasy.”

“You’re a hard man to turn down, Sy. Lead on.”

<walking the hall to the elevators> “Have you ever baked a cake, Anne?”

“Hasn’t everyone? My specialty is Crazy Cake — flour, sugar, oil, vinegar, baking soda and a few other things but no eggs.”

“Sounds interesting. Well, consider the path from fixings to cake. You’ve collected the ingredients. Is it a cake yet?”

“Of course not.”

“Ok, you’ve stirred everything together and poured the batter into the pan. Is it a cake yet?”

“Actually, you sift the dry ingredients into the pan, then add the others separately, but I get your point. No, it’s not cake and it won’t be until it’s baked and I’ve topped it with my secret frosting. Some day, Sy, I’ll bake you one.”

<riding the elevator down to 2> “You’re a hard woman to turn down, Anne. I look forward to it. Anyhow, you see the essential difference between flour’s journey to cakehood and our elevator ride down to Eddie’s.”

“Mmm… OK, it’s the discrete versus continuous thing, isn’t it?”

“You’ve got it. Measuring progress along a discrete degree of freedom can be an iffy proposition.”

“How about just going with the recipe’s step number?”

“I’ll bet you use a spoon instead of a cup to get the right amount of baking soda. Is that a separate step from cup‑measuring the other dry ingredients? Sifting one batch or two? Those’d change the step‑number metric and the step-by-step equivalent of momentum. It’s not a trivial question, because Emmy Noether’s symmetry theorem applies only to continuous coordinates.”

“We’re back to her again? I thought—”

The elevator doors open at the second floor. We walk across to Eddie’s, where the tail‑end of the lunch crowd is dawdling over their pizzas. “Hiya folks. You’re a little late, I already shut my oven down.”

“Hi, Eddie, we’re just here for gelato. What’s your pleasure, Anne?”

“On Sy’s recommendation, Eddie, I’ll try a scoop of ginger along with my scoop of mint. Sy, about that symmetry theorem—”

“The same for me, Eddie.”

“Comin’ up. Just find a table, I’ll bring ’em over.”

We do that and he does that. “Here you go, folks, two gelati both the same, all symmetrical.”

“Eddie, you’ve been eavesdropping again!”

“Who, me? Never! Unless it’s somethin’ interesting. So symmetry ain’t just pretty like snowflakes? It’s got theorems?”

“Absolutely, Eddie. In many ways symmetry appears to be fundamental to how the Universe works. Or we think so, anyway. Here, Anne, have an extra bite of my ginger gelato. For one thing, Eddie, symmetry makes calculations a lot easier. If you know a particular system has the symmetry of a square, for instance, then you can get away with calculating only an eighth of it.”

“You mean a quarter, right, you turn a square four ways.”

“No, eight. It’s done with mirrors. Sy showed me.”

“I’m sure he did, Anne. But Sy, what if it’s not a perfect square? How about if one corner’s pulled out to a kite shape?”

“That’s called a broken symmetry, no surprise. Physicists and engineers handle systems like that with a toolkit of approximations that the mathematicians don’t like. Basically, the idea is to start with some nice neat symmetrical solution then add adjustments, called perturbations, to tweak the solution to something closer to reality. If the kite shape’s not too far away from squareness the adjusted solution can give you some insight onto how the actual thing works.”

“How about if it’s too far?”

“You go looking for a kite‑shaped solution.”

~~ Rich Olcott

Deep Symmetry

“Sy, I can understand mathematicians getting seriously into symmetry. They love patterns and I suppose they’ve even found patterns in the patterns.”

“They have, Anne. There’s a whole field called ‘Group Theory‘ devoted to classifying symmetries and then classifying the classifications. The split between discrete and continuous varieties is just the first step.”

“You say ‘symmetry‘ like it’s a thing rather than a quality.”

“Nice observation. In this context, it is. Something may be symmetrical, that’s a quality. Or it may be subject to a symmetry operation, say a reflection across its midline. Or it may be subject to a whole collection of operations that match the operations of some other object, say a square. In that case we say our object has the symmetry of a square. It turns out that there’s a limited number of discrete symmetries, few enough that they’ve been given names. Squares, for instance, have D4 symmetry. So do four-leaf clovers and the Washington Monument.”

“OK, the ‘4’ must be in there because you can turn it four times and each time it looks the same. What’s the ‘D‘ about?”

Dihedral, two‑sided, like two appearances on either side of a reflection. That’s opposed to ‘C‘ which comes from ‘Cyclic’ like 1‑2‑3‑4‑1‑2‑3‑4. My lawn sprinkler has C4 symmetry, no mirrors, but add one mirror and bang! you’ve got eight mirrors and D4 symmetry.”

“Eight, not just four?”

“Eight. Two mirrors at 90° generate another one 45° between them. That’s the thing with symmetry operations, they combine and multiply. That’s also why there’s a limited number of symmetries. You think you’ve got a new one but when you work out all the relationships it turns out to be an old one looked at from a different angle. Cubes, for instance — who knew they have a three‑fold rotation axis along each body diagonal, but they do.”

“I guess symmetry can make physics calculations simpler because you only have to do one symmetric piece and then spread the results around. But other than that, why do the physicists care?”

“Actually they don’t care much about most of the discrete symmetries but they care a whole lot about the continuous kind. A century ago, a young German mathematician named Emmy Noether proved that within certain restrictions, every continuous symmetry comes along with a conserved quantity. That proof suddenly tied together a bunch of Physics specialties that had grown up separately — cosmology, relativity, thermodynamics, electromagnetism, optics, classical Newtonian mechanics, fluid mechanics, nuclear physics, even string theory—”

“Very large to very small, I get that, but how can one theory have that range? And what’s a conserved quantity?”

“It’s theorem, not theory, and it capped two centuries of theoretical development. Conserved quantities are properties that don’t change while a system evolves from one state to another. Newton’s First Law of Motion was about linear momentum as a conserved quantity. His Second Law, F=ma, connected force with momentum change, letting us understand how a straight‑line system evolves with time. F=ma was our first Equation of Motion. It was a short step from there to rotational motion where we found a second conserved quantity, angular momentum, and an Equation of Motion that had exactly the same form as Newton’s first one, once you converted from linear to angular coordinates.”

“Converting from x-y to radius-angle, I take it.”

“Exactly, Anne, with torque serving as F. That generalization was the first of many as physicists learned how to choose the right generalized coordinates for a given system and an appropriate property to serve as the momentum. The amazing thing was that so many phenomena follow very similar Equations of Motion — at a fundamental level, photons and galaxies obey the same mathematics. Different details but the same form, like a snowflake rotated by 60 degrees.”

“Ooo, lovely, a really deep symmetry!”

“Mm-hm, and that’s where Noether came in. She showed that for a large class of important systems, smooth continuous symmetry along some coordinate necessarily entails a conserved quantity. Space‑shift symmetry implies conservation of momentum, time‑shift symmetry implies conservation of energy, other symmetries lock in a collection of subatomic quantities.”

“Symmetry explains a lot, mm-hm.”

~~ Rich Olcott

Edged Things and Smooth Things

Yeughh, Sy, that whirling, the entire Universe spinning around me in every direction at once.”

“Well, you were at a point of spherical symmetry, Anne.”

“There’s that word ‘symmetryagain. Right side matches left side, what else is there to say?”

“A whole lot, especially after the mathematicians and physicists started playing with the basic notion.”

“Which is?”

“Being able to execute a transformation without making a relevant difference.”

“Relevant?”

“To the context. Swapping the king of spades for the king of hearts would be relevant in some card games but not others, right? If it doesn’t affect the play or the scoring, swapping those two when no‑one’s looking would be a legitimate symmetry operation. Spin a snowflake 60° and it looks the same unless you care exactly where each molecule is. That’s rotational symmetry, but there’s lots of geometric symmetry operations — reflections, inversions, glides, translations—”

“Translation is a symmetry operation?”

“In this connection, ‘translation‘ means movement or swapping between two different places in space. The idea came from crystals. Think of a 3D checkerboard, except the borderlines aren’t necessarily perpendicular. Perfect crystals are like that. Every cube‑ish cell contains essentially the same arrangement of atoms. In principle you could swap the contents of any two cells without making a difference in any of the crystal’s measurable properties. That’d be a translation symmetry operation.”

“Glides make me think of ice skating.”

“The glide operation makes me think of a chess knight’s move — a translation plus a reflection across the translation path. Think of wet footprints crossing a dry floor. That’s one example of combining operations to create additional symmetries. You can execute 48 unique symmetry operations on a cube even without the translation‑related ones. In my grad school’s crystallography class they taught us about point group and wallpaper and space group symmetries. It blew me away — beautiful in both mathematical and artistic senses. You’ve seen M C Escher’s art?”

“Of course, I love it. I pushed into his studio once to watch him work but he spotted me and shouted something Dutch at me. I’ve wondered what he thought when I pushed out of there.”

“His pieces drew heavily on geometric symmetries. So did Baroque art, music and architecture.”

“Music? Oh, yes — they had motifs and whole sections you could swap, and rhythm patterns and tunes you could read forwards and backwards like in a mirror… We’ve come a long way from snowflake symmetry, haven’t we?”

“We’re just getting started. Here’s where the Physics folks generalized the idea. Your unfortunate experience in space is right on the edge of what most people consider as symmetry. Were you impressed with the cube’s 48 operations?”

“I suppose. I haven’t had time to think about it.”

“A sphere has an infinite number. You could pick any of an infinite number of lines through its center. Each is an axis for an infinite number of rotational symmetries. Times two because there’s an inversion point at the center so the rotation could go in either direction. Then each line is embedded in an infinite number of reflection planes.”

“Goodness, no wonder I was dizzy. But it’s still geometry. What was the edge that the physicists went past?”

“The border between step‑at‑a‑time discrete symmetries and continuous ones. Rotate that snowflake 60° and you’ve got a match; anything not a multiple of 60° won’t pair things up. Across the border, some of the most important results in modern Physics depend on continuous symmetries.”

“How can you even have a continuous symmetry?”

“Here, I’ll draw a circle on this square of paper. I can rotate the square by 90, 180 or 270 degrees and everything’s just the way it was. But if the square’s not relevant because we’re only interested in the circle, then I can rotate the paper by any amount I like and it’s a no‑difference transformation, right?”

“Continuous like on an infinite line but it’s wrapped around.”

“Exactly, and your infinite line is another example — any translation along that line, by a mile or a millimeter, is a perfectly good symmetry operation.”

“Ooo, and time, too. I experience time as an infinite line.”

“So does everyone. but most only travel in one direction.”

~~ Rich Olcott

Three Ways To Get Dizzy

<FZzzzzzzzzzzzzzzzzzzzzzzttt!> “Urk … ulp … I need to sit down, quick.”

“Anne? Welcome back, the couch is over there. Goodness, you do look a little green. Can I get you something to drink?”

“A little cool water might help, thanks.”

“Here. Just sit and breathe. That wasn’t your usual fizzing sound when you visit my office. When you’re ready tell me what happened. Must have been an experience, considering some of your other superpower adventures. Where did you ‘push‘ to this time?”

“Well, you know when I push forward I go into the future and when I push backward I go into the past. When I push up or down I get bigger or smaller. You figured out how pushing sideways kicks me to alternate probabilities. And then <shudder> there was that time I found a new direction to push and almost blew up the Earth.”

“Yes, that was a bad one. I’d think you’ve pretty well used up all the directions, though.”

“Not quite. This time I pushed outwards, the same in every direction.”

“Creative. And what happened?”

“Suddenly I was out in deep space, just tumbling in the blackness. There wasn’t an up or down or anything. I couldn’t even tell how big I was. I could see stars way off in the distance or maybe they were galaxies, but they were spinning all crazy. It took me a minute to realize it was me that was spinning, gyrating in several ways at once. It was scary and nauseating but I finally stopped part of it.”

“Floating in space with nothing to kill your angular momentum … how’d you manage to stabilize yourself at all?”

“Using my push superpower, of course. The biggest push resistance is against the past. I pulled pastward from just my shoulders and that stopped my nose‑diving but I was still whirling and cart‑wheeling. I tried to stop that with my feet but that only slowed me down and I was getting dizzy. My white satin had transformed into a spacesuit and I definitely didn’t want to get sick in there so I came home.”

“How’d you do that?”

“Oh, that was simple, I pulled inward. I had to um, zig‑zag? until I got just the right amount.”

“That explains the odd fizzing. I’m glad you got back. Looks like you’re feeling better now.”

“Mostly. Whew! So, Mr Physicist Sy, help me understand it all. <her voice that sounds like molten silver> Please?”

“Well. Um. There’s a couple of ways to go here. I’ll start with degrees of freedom, okay?”

“Whatever you say.”

“Right. You’re used to thinking in straight‑line terms of front/back, left/right and up/down, which makes sense if you’re on a large mostly‑flat surface like on Earth. In mathspeak each of those lines marks an independent degree of freedom because you can move along it without moving along either of the other two.”

“Like in space where I had those three ways to get dizzy.”

“Yup, three rotations at right angles to each other. Boatmen and pilots call them pitch, roll and yaw. Three angular degrees of freedom. Normal space adds three x-y-z straight‑line degrees, but you wouldn’t have been able to move along those unless you brought along a rocket or something. I guess you didn’t, otherwise you could have controlled that spinning.”

“Why would I have carried a rocket when I didn’t know where I was going? Anyhow, my push‑power can drive my straight‑line motion except I didn’t know where I was and that awful spinning had me discombobulated”

“Frankly, I’m glad I don’t know how you feel. Anyhow, if measurable motion is defined along a degree of freedom the measurement is called a coordinate. Simple graphs have an x-coordinate and a y-coordinate. An origin plus almost any three coordinates makes a coordinate system able to locate any point in space. The Cartesian x-y-z system uses three distances or you can have two distances and an angle, that’s cylindrical coordinates, or two angles and one distance and that’s polar coordinates.”

“Three angles?”

“You don’t know where you are.”

<shudder>
 <shudder>

~~ Rich Olcott

The Edge of Pinkness

Susan Kim takes a sip of her mocha latte. eyes me over the rim. “That’s quite a set of patterns you’ve gathered together, Sy, but you’ve left out a few important ones.”

“Patterns?”

A log-linear plot

“Regularities we’ve discovered in Nature. You’ve written about linear and exponential growth, the Logistic Curve that describes density‑limited growth, sine waves that wobble up and down, maybe a couple of others down‑stack, but Chemistry has a couple I haven’t seen featured in your blog.”

“Such as?”

“Log-linear relationships are a biggie. We techies use them a lot to handle phenomena with a wide range. Rather than write 1,000,000,000 or 109, we sometimes just write 9, the base‑10 logarithm. The pH scale for acid concentration is my favorite example. It goes from one mole per liter down to ten micro‑nanomoles per liter. That’s 100 to 10-14. We just drop the minus sign and use numbers between 0 and 14. Fifteen powers of ten. Does Physics have any measurements that cover a range like that?”

“A handful, maybe, in theory. The limitation is in confirming the theory across a billion-fold range or wider. Atomic clocks that are good down to the nanosecond are our standards for precision, but they aren’t set up to count years. Mmmm … the Stefan‑Boltzmann Law that links an object’s electromagnetic radiation curve to its temperature — our measurements cover maybe six or seven powers of ten and that’s considered pretty good.”

“Pikers.” <but I like the way she grins when she says it>

“I took those Chemistry labs long ago. All I remember was acids were colorless and bases were pink. Or maybe the other way around.”

“You’ve got it right for the classic phenolphthalein indicator, but there are dozens of other indicators that have different colors at different acidities. I’ll tell you a secret — phenolphthalein doesn’t kick over right at pH 7, the neutral point. It doesn’t turn pink until the solution’s about ten times less acidic, near pH 8.”

Adapted from this file by Damitr, CC BY-SA 4.0

“So all my titrations were off by a factor of ten?”

“Oh, no, that’s not how it works. I’m going to use round numbers here, and I’ll skip a couple of things like the distinction between concentration and activity. Student lab exercises generally use acid and base concentrations on the order of one molar. For most organic acids, that’d give a starting pH near 1 or 2, way over on the sour side. In your titration you’d add base, drop by drop, until the indicator flips color. At that point you conclude the amounts of acid and base are equivalent, not by weight but by moles. If you know the base concentration you can calculate the acid.”

“That’s about what I recall, right.”

“Now consider that last drop. One drop is about 50 microliters. With a one‑molar base solution, that drop holds 50 nanomoles. OK?”

<I scribble on a paper napkin> “Mm-hm, that looks right.”

“Suppose there’s about 50 milliliters of solution in the flask. Because we’re considering the last drop, the solution in the flask must have become nearly neutral, say pH 6. That means the un‑neutralized acid concentration was 10-6 moles per liter, or one micromolar. Fifty milliliters at one micromolar concentration is, guess what, 50 nanomoles. Your final drop neutralizes the last of the acid sample.”

“So the acid concentration goes to zero?”

“Water’s not that cooperative. Water molecules themselves act like acids and bases. An H2O molecule can snag a hydrogen from another H2O giving an H3O+ and an OH. Doesn’t happen often, but with 55½ moles of water per liter and 6×1023 molecules per mole there’s always a few of those guys hanging around. Neutral water runs 10-7 moles per liter of each, which is why neutral pH is 7. Better yet, the product of H3O+ and OH concentrations is always 10-14 so if you find one you can calculate the other. Take our titration for example. One additional drop adds 50 nanomoles more base. In 50 milliliters of solution that’s roughly 10-6+10-7 molar OH. Call it 1.1×10-6, which implies 0.9×10-8 molar H3O+. Log of that and drop the minus sign, you’re a bit beyond pH 8 which sends phenolphthalein into the pink side. Your titration’s good.”

I eye her over my mug of black mud. “A gratifying indication.”

~~ Rich Olcott

The Hysterical Penguin

“Sy, you said that hysteresis researchers filled in two of Newton’s Physics gaps. OK, I get that he couldn’t do atomic stuff ’cause atoms hadn’t been discovered yet. What’s the other one?”

Proposition XI, Problem VI
from Book I of Newton’s Principia

“Non‑linearity.”

“You’re gonna have to explain that.”

“It’s a math thing. I know you don’t go for equations, so here’s a picture to get you started on how Newton solved problems. Look at all familiar?”

“Whoa, looks like something toward the end of my Geometry class.”

“Exactly. Newton was trained as a geometer and he was good at it. His general strategy was to translate a physical system to a geometrical structure and then work out its properties as a series of geometric proofs. The good news was that he proved a lot of things that started us on the way to quantitative science. The bad news was that his proofs were hard to extend to situations where the geometry wasn’t so easy.”

“That’s easy?”

“For Newton, maybe it was. Who knows? Anyway, the toolkit they gave you in Geometry class was what Newton had to work with — logic, straight lines and some special curves like ellipses and parabolas whose properties had been studied since Euclid, all on a flat plane. Nearly everything depended on finding proportionalities between different distances or areas — this line is twice that one but equal to a third, that sort of thing. Proportionality like that is built into equations like here+(velocity×time)=there. See how distance traveled is proportional to time? The equation plots as a straight line, which is why it’s called a linear equation.”

“So what’s non‑linear look like — all wiggle‑waggle?”

“Not necessarily. Things can vary smoothly along curves that aren’t those classical ones. Newton’s methods are blocked on those but Leibniz’s algebra‑based calculus isn’t. That’s why it won out with people who needed answers. What’s important here is that Newton’s lines can’t describe everything. Mmm… where does a straight line end?”

“Either at a T or never. Same thing for a parabola. Hey, ellipses don’t really end, either.”

“Mm-hm. Newton’s lines either stop abruptly or they continue forever. They don’t grow or peter out exponentially like things in real life do. Suppose something’s velocity changes, for instance.”

“That’s acceleration. I like accelerating.”

“So true, I’ve experienced your driving. But even you don’t accelerate at a constant rate. You go heavy or light or maybe brake, whatever, and our speed goes up or down depending. The only way Newton’s geometry can handle variable acceleration is to break it into mostly‑constant pieces and work one piece at a time. Come to think of it, that may be where he got the idea for his fluxions method for calculus. Fortunately for him, some things like planets and artillery shells move pretty close to what his methods predict. Unfortunately, things like disease epidemics and economies don’t, which is why people are interested in non‑linearity.”

“So what do these hysteresis guys do about it?”

“Mostly algebraic calculus or computer approximations. But there wasn’t just one group of hysteresis guys, there was a bunch of groups, each looking at different phenomena where history makes a difference. Each group had their own method of attack.”

“Like your elephant thing with Anne, lots of notions about entropy.”

Typical hysteresis loop
Red — initial evolution
Blue — subsequent changes

“How’d you find out about that?”

You wrote those posts, Sy, about three years ago.”

“Oh, that’s right. Talk about history. Anyway, it took decades for the ecologists, epidemiologists, civil engineers and several kinds of physicist to realize that they all have systems that behave similarly when driven by a stressor. Starting at some neutral situation, the system evolves in the driver’s direction to some maximum deviation where increased stress has no further effect. When the stress is relieved, the system may stick temporarily at the strained position. When it does evolve away from there, maybe a reverse driver is needed to force a return to the starting situation. In fact, if the forward and reverse drivers are applied repeatedly the system may never get back to the initial unstressed position.”

“Like that iron nail. Not magnetic, then magnetic, then reversed.”

~~ 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

Shortfall

<chirp, chirp> The non-business line again. “Moire here.”

“Hiya, Sy, it’s Eddie. I’m taking orders for tonight’s deliveries. I got some nice-looking artichokes here, how about a garlic and artichoke pizza?”

“No thanks, Eddie, I’ll stick with my usual pepperoni. Wait, you got any ham?”

“Sure.”

“Let’s go with a Hawai’ian.”

“Sy, we’ve had this conversation. You want pineapple on pizza you open a can and dump some on there after I leave the premises and don’t tell me. I got standards!”

“Calm down, Eddie, just yanking your chain. Yeah, do me one of those garlic and artichoke ones. Sounds more classical.”

“That’s better. I got you in the 6:15 wave, OK? Hey, that reminds me. I read your post series about waves and that got me thinking.”

“Nice to know someone reads them.”

“Well, things are real quiet, just me in the kitchen these days so I’m scraping the barrel, you know?”

“Ouch.”

“Gotcha back. Anyhow, that series was all about wiggly waves that repeat regular-like, right? I get that scientists like ’em ’cause they’re easy to calculate with. But that Logic Curve you wrote about goes up and doesn’t come back down again. Does anybody do math with that kind?”

Logistic Curve — blue line,
Associated slope — red line

“Logistic Curve. ‘Logic Curve‘ isn’t a thing. The mathematicians have come up with a plethora of curves and curve families. The physicists have found uses for many of them. The Logistic Curve, for instance, is one of the first tools they take off the shelf for systems that have both lower and upper limits. You’ve seen a lot about how it’s applied to epidemiology. People also use it for ecology, economics, linguistics, chemistry, even agriculture.”

“What do the top and bottom lines have to do with each other?”

“Ah. Sorry I hadn’t made that clear. OK, find a blank page in your order pad. At the top draw a horizontal zig-zag line like a series of 45‑degree triangles touching corners.”

“45 degrees is easy — that’s an 8-slice pizza. Done.”

“You’ve just drawn what’s called a triangle wave, no surprise. OK, now right under that, you’re going to draw another wave that shows the slope of each triangle segment. Where the triangle line goes up you’ve got a positive slope that goes up one unit for every unit across so draw a line at plus‑one, OK?”

“A-ha. Got it.”

“Where the triangle line goes down you’ve got a negative slope, minus‑one.”

“What about where the triangles got points?”

“Just draw a vertical line to connect the slope segments. What’s the completed second line look like?”

“A zig-zag bunch of square boxes. Hey, wait, we made the second line be the slopes for all the pieces, right? Lemme go check the picture in the ‘Curve‘ post. So what you’re saying is … the red line is all the slopes along the blue line … OK, can I say that the red line is how fast stuff is coming at me and the blue line is the backlog?”

“Half-right. For what we’re talking about, ‘slope‘ is whatevers per time‑unit. The blue line shows how much total has come at you so far. Backlog is a little more complicated.”

“I gotta go back and read those posts again. Now I see why they’re saying ‘flattening the curve‘ — they want the blue line to not climb so fast.”

“That’s part of it.. Flattening that red-line curve as much as we can is important. That’s what the masks and social distancing are about. Maybe as many people get sick, total, but if they trickle in instead of flooding in then they don’t overload the system. Here, I’ll send a sketch to your phone.”

“Got it, but there’s lots of lines there.”

“The red line is your completion rate — pizza orders per hour, patients per day, whatever. The red line goes flat because having only one oven limits your throughput. The gray part above it is pizzas per hour you couldn’t bake or patients your hospital couldn’t take that day. The green line is doable business; the black line shows how more capacity would have improved things.”

“Reduce the incoming, raise the capacity or lose the people. Whoa.”

~~ Rich Olcott