Chocolate, Mint And Notation

“But calculus, Mr Moire. Why do they insist we learn calculus? You said that Newton and Leibniz started it but why did they do it?”

“Scoop me a double-dip chocolate-mint gelato, Jeremy, and I’ll tell you about an infamous quarrel. You named Newton first. I expect most Europeans would name Leibniz first.”

“Here’s your gelato. What does geography have to do with it?”

“Thanks. Mmm, love this combination. Part of the geography thing is international history, part of it is personality and part of it is convenience. England and continental Europe have a history of rivalry in everything from the arts to trade to outright warfare. Each naturally tends to favor its own residents and institutions. Some people say that the British Royal Society was founded to compete with the French philosophical clubs. Maybe England’s king appointed Newton as the society’s President to upgrade the rivalry. Dicey choice. From what I’ve read, Newton’s didn’t hate everybody, he just didn’t like anybody. But somehow he ran that group effectively despite his tendency to go full‑tilt against anyone who disagreed with his views.”

“Leibniz did the same thing on the European side?”

“No, quite the opposite. There was no pre‑existing group for him to head up and he didn’t start one. Instead, he served as a sort of Information Central while working as diplomat and counselor for a series of rulers of various countries. He carried on a lively correspondence with pretty much everyone doing science or philosophy. He kept the world up to date and in the process inserted his own ideas and proposals into the conversation. Unlike Newton, Leibniz was a friendly soul, constantly looking for compromise. Their separate calculus notations are a great example.”

“Huh? Didn’t everyone use the same letters and stuff?”

“The letters, yeah mostly, but the stuff part was a long time coming. What’s calculus about?”

“All I’ve seen so far is proofs and recipes for integrating different function types. Nothing about what it’s about.”

Newton approximates arc ABCDEF.

<sigh> “That’s because you’re being taught by a mathematician. Calculus is about change and how to handle it mathematically. That was a hot topic back in the 1600s and it’s still central to Physics. Newton’s momentum‑acceleration‑force perspective led him to visualize things flowing with time. His Laws of Motion made it easy to calculate straight‑line flows but what to do about curves? His solution was to break the curve into tiny segments he called fluxions. He considered each fluxion to be a microscopic straight line that existed for an infinitesimal time interval. A fluxion’s length was its time interval multiplied by the velocity along it. His algebraic shorthand for ‘per time‘ was to put a dot over whatever letter he was using for distance. Velocity along x was . Acceleration is velocity change per time so he wrote that with a double dot like . His version of calculus amounted to summing fluxion lengths across the total travel time.”

“But that only does time stuff. What about how, say, potential energy adds up across a distance?”

“Excellent question. Newton’s notation wasn’t up to that challenge, but Leibniz developed something better.”

“He copied what Newton was doing and generalized it somehow.”

“Uh, no. Newton claimed Leibniz had done that but Leibniz swore he’d been working entirely independently. Two lines of evidence. First, Newton was notoriously secretive about his work. He held onto his planetary orbit calculations for years before Halley convinced him to publish. Second, Leibniz and other European thinkers came to the problem with a different strategy. Descartes invented Cartesian coordinates a half‑century before. That invention naturally led the Europeans to plot anything against anything. Newton’s fluxions combined tiny amounts of distance and time; Leibniz and company split the two dimensions, one increment along each component. Leibniz tried out a dozen different notations for the increment. After much discussion he finally settled on a simple d. The increment along x is dx, but x could be anything quantitative. dy/dx quantifies y‘s change with x.”

“Ah, the increments are the differentials we see in class. But those all come from limit processes.”

“Leibniz’ d symbol and its powerful multi‑dimensional extensions carry that implication. More poetry.”

~~ Rich Olcott

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

Math Poetry

Eddie serves a good pizza. I amble over to the gelato stand for a chaser. “Evening, Jeremy. You’re looking a little distraught.”

“I am, Mr Moire. Just don’t ask me to quantify it! Math is getting me down. Why do they shove so much of it at us? You don’t put much math into your posts and they make sense mostly.”

“Thanks for the mostly. … Do you enjoy poetry?”

“Once I read some poems I liked. Except in English class. They spend too much time classifying genre and rhyme scheme instead of just looking at what the poet wrote. All that gets in the way.”

“Interesting. What is it that you like about poetry?”

“Mmm, part of it is how it can imply things without really saying them, part of it is how compact a really good one is. I like when they cram the maximum impact into the fewest possible words — take out one word and the whole thing falls apart. That’s awesome when it works.”

“Well, how does it work?”

“Oh, there’s lots of techniques. Metaphor’s a biggie — making one thing stand for something else. Word choice, too — an unexpected word or one with several meanings. Sometimes it’s a challenge finding the word that has just the right rhythm and message.”

“Ah, you write, too. When you compose something, do you use English or Navajo?”

“Whichever fits my thought better. Each language is better at some things, worse at others. A couple of times I’ve used both together even though only rez kids would understand the mix.”

“Makes sense. You realize, of course, that we’ve got a metaphor going here.”

“We do? What standing for what?”

“Science and Poetry. I’ve often said that Physics is poetry with numbers. Math is as much a language as English and Navajo. It has its own written and spoken forms just like they do and people do poetry with it. Like them, it’s precise in some domains and completely unable to handle others. Leaning math is like learning a very old language that’s had time to acquire new words and concepts. No wonder learning it is a struggle.”

“Poetry in math? That’s a stretch, Mr Moire.”

“Prettiest example I can think of quickly is rhyming between the circular and hyperbolic trigonometric systems. The circular system’s based on the sine and cosine. The tangent and such are all built from them.”

“We had those in class — I’ll remember ‘opposite over hypotenuse‘ forever and I got confused by all the formulas — but why do you call them circular and what’s ‘hyperbolic‘ about?”

“Here, let me use Ole Reliable to show you some pictures. I’m sure you recognize the wavy sine and cosine graphs in the circular system. The hyperbolic system is also based on two functions, ‘hyperbolic sine‘ and ‘hyperbolic cosine,’ known in the trade as ‘sinh‘ and ‘cosh.’ They don’t look very similar to the other set, do they?”

“Sure don’t.”

“But for every circular function and formula there’s a hyperbolic partner. Now watch what happens when we combine a sine and cosine. I’ll do it two ways, a simple sum and the Pythagorean sum.”


“Remember his a2+b2=c2? The orange curve comes from that, see in the legend underneath?”

“Oh, like a right triangle’s hypotenuse. But the orange curve is just a flat straight line.”

“True, as we’ve known since Euler’s day. Are you familiar with polar coordinates?”

“A little. There’s a center, one coordinate is distance from the center, and the other coordinate is the angle you’ve rotated something, right?”

“Good enough. Here’s what the same two combinations look like in polar coordinates..”

“Wow. Two circles. I never would have guessed that.”

“Mm-hm. Check the orange circle, the one that was just a level straight line on the simple graph. It’s centered on the origin. That tells us the sum of the squares is invariant, doesn’t change with the angle.”

“Do the hyperbolic thingies make hyperbolas when you add them that way?”

“Not really, just up-curving lines. The plots for their differences are interesting though. For these guys the Pythagorean difference is invariant. Einstein’s relativity is based on that property.”

“Pretty, like you say.”

~~ Rich Olcott

Now And Then

“Alright, I suppose there’s no going down below the Universe’s Year Zero, but what about the other direction? Do you physics guys have a handle on Time’s Top?”

“That’d be Cosmology, Mr Feder. We physicists avoid theorizing about stuff we can’t check against data. Well, except for string theory. The far past leaves clues that astronomers like Cathleen can gather. Sad to say, though, we barely have a handle on Now.”

Cathleen grins. Al and Mr Feder go, “Whaaat?”

“No, really. One of Einstein’s insights was that two observers randomly and independently flying through space won’t be able to agree on whether two external events occurred simultaneously. They can’t even agree on what time it is now.”

“Oh, yeah, I know about that. I’ve read about how the GPS system needs to make corrections to account for what relativity does to the satellite timings.”

“You’re right, Al, but that’s a different issue. Some of that relativistic correction has to do with space compression because of Earth’s mass. The simultaneity problem is strictly about rapid motion and geometry.”

“Wait — geometry?”

“Relativistic geometry, which is a bit different from the kind that Descartes built.”

“Whoa, Sy, slow down there. Descartes was the ‘I think therefore I am‘ guy, right? What’s that got to do with geometry?”

“I guess I got a little ahead of myself there, didn’t I? OK. Yeah, Al, same Descartes. Grew up Catholic in France, was a professional mercenary soldier in the Thirty Years War, wound up fighting first on the Catholic French side and later on fought on the Protestant Dutch side but cross‑over was common, both directions. He realized he was in an ostensibly religious war that was really about who ruled over whom. That may have had something to do with him becoming a professional philosopher who rejected all religious dogmas in favor of what he could learn solely from logic and his own senses. That’s where his famous mantra came from — he started by proving to himself that he existed.”

“Logic led to geometry, I suppose.”

“Indeed, but a new kind, one that required a few innovations that Descartes developed. On the one hand, mathematicians traditionally expressed algebraic problems in words and some of them were doozies, like saying ‘the zenzizenzizenzic‘ where we’d just say x8. We got that simple but <ahem> powerful notation from Descartes. On the geometry side, he’d ditch all the confusing line-ending markers in a diagram like this one. Instead, he’d label the whole line representing a known quantity with a front-of-the-alphabet letter like a or b or c. A line representing an unknown quantity would get its label from the alphabet-trailers like x, y and z. Then he used the same character conventions and his new power notation to write and manipulate algebraic expressions. Those notational inventions were foundational for his bridge between algebraic and geometrical problems. Draw your problem with lines and curves, transform it to algebraic equations, solve that problem exactly, transform it back to geometry and you’re done. Or vice-versa.”

The mesolabe instrument (in red).

“That goes back to Descartes, huh?”

“Mm-hm. His big innovation, though, arose from a borrow from an early Greek gadget called a mesolabe. He proposed an idealized version that would let someone break a line into exact fractions or compare a length against a unit length. That broke the rules of classical Geometry but setting his mesolabe’s Y‑angle to 90° prompted him to name points by their distance along the x– and y‑axes. That’s the nub of the Cartesian coordinate system — a rectangular grid of numbered straight lines that go on forever. Graph paper, right? Wrap the grid around the Earth and you’ve got latitudes and longitudes. Add more numbered grid lines perpendicular to either grid and you’ve got z‑axis coordinates. Three coordinates let you name any point in space. Newton and all the physicists who came after him until the dawn of the 20th Century assumed Descartes’ nice, stable coordinate system.”

“20th Century — that’s when Einstein came on the scene. He broke that system?”

“Sure did. You’ve heard about bent space?”

“Who hasn’t?”

“Well, fasten your seat belts, it’s going to be a fun ride.”

~~ Rich Olcott

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.”


“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.”


~~ Rich Olcott