What is your flat plane?

(Still mostly wrong)

First up, I’d like to acknowledge everyone who has followed me from Quora. Thank you!

In my previous post, I declared my conviction—the only good way to do science is to be Mostly Wrong. And science is all we have! As nobody has disagreed (yet), we’re free to dig deeper.

Specifically, I’m going to embrace boredom. We’ll cover a topic so boring that professional mathematicians shy away from it. Let’s not even mention a particular area of data science considered so mind-numbing that even database designers would prefer to lightly anaesthetise themselves with beer and watch rugby, rather than contemplate the sheer tedium of getting this one thing right. Or so it seems. Are you ready to be bored, then?

My kids hate me!

When my children were a lot younger, they’d come to me and complain “I’m bored”. I’d say:

“You’re so lucky. If you were an animal, the only options you’d have would be finding food, running away from something that wants to eat you, sleep, or finding a mate. Humans have the luxury of boredom. Relish every second!”

This is, of course, not only wrong, but infuriating. I was a bad parent. My kids still get angry when they think of this. My older daughter mentioned it just last week. With rage (although she later enjoyed previewing this post). And ‘boring’ remains a problem—especially in the Internet Age where, if your cat pictures aren’t more engaging than other people’s cat pictures, you tend to lose out.

So, this post is boring. Here’s the kicker. If you don’t embrace this boredom—relish every second—then you’re in for a rude shock. You may think life is all about food, sex, sleep, entertainment and (increasingly) avoiding persecution by white male US Republicans. But actually, boring flat planes are far more important. And therefore, due to your inattention, you’re doomed. Pretty near the end of this long, boring post, you’ll even find out why. Zombies may be involved. Shall we begin?

Before we even begin—a boring introduction

Good science starts with problems. Here’s a problem then. We’ve established that there are no absolute truths on which we can build science. So how do we build anything?

We iterate. We make choices and apply the tail-eating process of science I’ve already described. We test our models in reality, toss out those that fail, and try again. And eventually we work out theories that—at least for now—fit with joined-up thinking and work usefully in the real world.

For this we need to start somewhere. Nail something down, at least for now. The picture at the start of the post is of such a standard. A failed standard. That sphere was painstakingly manufactured with really advanced technology, but never even adopted! It’s still pretty awesome. You can read in detail about the Avogadro project at NIST, but here’s my take. It draws us into the severely austere domain of metrologists—scientists who measure things. Obsessively.

The International System of Units (SI) was only established in 1960—a coherent, in-a-sense-arbitrary, science-based set of seven choices. It defines the second, the metre, the kilogram, the ampere, the kelvin, the mole and the candela—and then everything else, every measurement you might possibly wish to make, stems from these. But each of these units has a legacy.

Take the kilogram. In 1799, the Kilogramme des Archives was forged from pure platinum. Using the tech of the day, it equalled the mass of precisely one litre of water at 4 ^oC,¹ where water is at its most dense. One hundred years later, this was considered inadequate, so Johnson, Matthey & Co. of London manufactured a precisely machined cylinder of platinum-iridium—the International Prototype of the Kilogram (IPK, also known to its friends as 𝔎).²

Although just a golfball-sized object, 𝔎 was important. If you weigh anything around the world—at least, if you want to weigh anything reliably—then you need a reference standard. But where does the reference standard come from? Aha! There’s always a traceable chain of calibration—a chain of people comparing weights—that ultimately comes back to 𝔎. Or used to.

Are we there yet?

No. The IPK had a problem. Has a problem. Forty-four copies were made, with most sent around the world to various nations—to maintain the chain of calibration. More than a little pain is involved in re-checking other national copies of the IPK against the original, so this was only done three times: originally in 1889, and subsequently in 1948, and 1989.³

But after comparison, something was off. This is not a small thing.

The IPK is not just kept somewhere on a metrologist’s shelf. It’s too important. It’s stored in an environmentally monitored vault in the basement of the Pavillon de Breteuil in Paris. The enclosing safe can only be accessed if you have all three keys, which are kept apart. The IPK itself is stored under three nested bell jars, and infrequently handled. It is arguably held more securely than the gold in Fort Knox.

And yet, the weight now differs from those of the copies. Some of this is attributable to cleaning—but there’s still a difference. The IPK has lost about 50 μg over the course of the last century. Or its copies have all gained weight. Now 50 millionths of a gram may not seem like much—despite being enough botulinum toxin to kill 250 people by inhalation. But for precision measurements, that’s a problem that affects every measurement not just of weight but also of the amount of matter (moles), current flow (amperes) and luminosity (candelas). Why? Because these other three standards are all interconnected—and depend on the kilogram. Our somewhat arbitrary choices of standards are joined at the hip. And everything derives from these.

We can agonise about precisely why the IPK is secretively taking tirzepatide. Or we can fix it. Several possible ways were contemplated. One was to make a new, physical kilogram. That’s the picture at the start of this post. A single, uniform crystal of 99.9995% pure silicon-28, carefully machined into a near-perfect sphere. How ‘near-perfect’? Well, if you were to blow it up to the size of the Earth, then the difference between the tallest mountain and the deepest ocean trench would be about 4 metres.

But this wasn’t good enough!

Eventually, metrologists tossed these $3.2 million silicon spheres⁴ in favour of something called a Kibble balance, which defines the kilogram using instead the speed of light, a transition frequency of caesium-133, and Planck’s constant. Boring enough?⁵

We can learn a few things from all of this, though. Firstly, that metrologists do indeed pretty much define the term ‘obsessive’. Second, that some people might reasonably consider them pretty boring. And third, that very, very, very tiny differences sometimes matter. Simple choices ramify. Which brings us to …

Finally, a flat plane

Let’s say you’re not into measuring mass to better than 50 parts in a billion. All you want to do is build something—anything—precisely. You’re not a metrologist, you’re an engineer. You might be engineering precision instruments (perhaps for a metrologist—you can’t escape them); or seals for engines; or silicon chips so that people can watch cat pictures on their cellphones. But you want your engineering to work out right. Where do you start?

There’s a pretty good argument that you—in fact everyone—should start with flatness. If you don’t have flatness, then your can’t machine things properly. Add the tiniest tilt to a silicon wafer—put the smallest speck of dirt under it, for example—and the integrated circuit you build on it will be deformed into uselessness. Engineer the surfaces of your seals wrong, and your devices will leak. People won’t want the things you make.

Because it’s so important, we have found really, really precise ways of measuring flatness. In 1665, Robert Hooke published the very first science best-seller: Micrographia. Go ahead! Transiently put aside your cat pictures and read a glorious online copy, courtesy of the Royal Society. Okay, pop-sci publiſhing has advanced ſomewhat ſince then, but the pictures are astonishing.

Not only did he there invent the word ‘cell’—and provide innumerable people a glimpse into the truly minute—Hooke also noticed something strange. When two thin, flat sheets of glass are put together, coloured rings appear. And as you vary the distance between the sheets, so the colours change. Isaac Newton picked this observation up and ran with it a year later, so today we talk about “Newton’s rings”. And Lo and Behold, we have an industrial way of measuring flatness. You can measure flatness to within 0.3 millionths of a metre using contact interferometry: place a slightly convex reference flat on the surface to be assessed, shine a monochromatic sodium lamp on it, and look at the Newton’s rings. (There are multiple other methods too; silicon chip makers use a bevy of them—and some can measure flatness to a third of a billionth of a metre).

That’s very nice, but how do we make that flat plane? When early engineering really took off, engineers found a trick. If you rub two surfaces together, they will tend to conform to one another, but won’t necessarily turn out flat. If however you use three surfaces, they ultimately have no choice. You’ll get flat planes—if you scrape off the areas that are a bit more raised when your rub all three together in sequence. This is the essence of the Whitworth three plates method. That deserves bold emphasis, too.

As we discovered with the kilogram, we need some sort of reference, wherever we turn. If you’re an engineer, you eventually trace everything back to a flat plane. Which comes inexorably back again to metrology. It’s truly remarkable what engineers have achieved. In the 1830’s, Whitworth’s method allowed them to attain flatness down to “a millionth of an inch”. That’s better than you can measure with contact interferometry. Using just skill and hand scraping!

The above image is from an entire, superb book on the Foundations of Mechanical Accuracy (PDF). For a shorter description, try here.

So what’s your flat plane, then?

Consider this. Whatever your passions in life are, I’m pretty sure you haven’t considered the possibility that they might come back to some choice of a flat plane. Or, uhh, metrology. Isn’t metrology boring? But ultimately, whatever you do, it starts with some reference points. All that matters then is where you take things from there—how faithful you are.

We know from my last post that it’s impossible to obtain some sort of absolute ‘true’ reference point. You might join the ‘other half’ of people—non-scientists—who have managed to delude themselves in one of the two main ways already discussed: choosing to believe in some sort of knowable, ultimate Platonic truth (which we know is impossible), or deluding yourself into believing that ‘everything is relative’ and infinitely plastic—reality is an illusion manufactured by the mind.

And yes, you can do this. You can make such a choice. But there are consequences. For me, the distressing thing is the ease with which this happens. I call this “the easiness of the unassumed choice”. You don’t even need to acknowledge that you have chosen an unreasonable path. It’s implicit—you have already made your call.

But in so doing, you’ve also tacitly abandoned the possibility of a reasoned discussion about pretty much everything, starting with basic measurements and moving outwards. Your choice of a flat plane impinges on every other decision you will make, just as scientists choosing reference points have to live with them. But a good scientist will continually re-evaluate their choices.

Simple choices have complex ramifications. In future posts, I’ll explore many of these. For now, I have to keep the two promises I made above. In honouring the first, I’ll show how even mathematicians aren’t immune.

A boring flat plane in mathematics

Admit it. You thought I was going to contrast Euclidean and non-Euclidean geometry, didn’t you? All of the above as a neat segue into General Relativity, or something. Naah. The Escher Circle Limit IV ‘Heaven and Hell’ image above (based on the Poincaré disc) was merely another homage to Escher. The first was of course the picture at the start, which I’m fairly sure is intended to mimic his “Hand with Reflecting Sphere”. Or metrologists are even more boring than you thought.

But a flat plane is also a metaphor. We don’t need to be literal. The oh-so-nearly-perfect spherical kilogram at the start of my post is effectively a ‘flat plane’. Choice of a particular religious idea might be your specific ‘flat plane’ (with some large and unfortunate valleys and hills that obstinately refuse to be exalted or made low). Political choices are ultimately ‘flat planes’. Economists who choose, say, trickle-down economics or Black-Scholes are choosing their own rather bumpy ‘flat plane’. But what about ‘pure’ disciplines like mathematics? Perhaps as an antidote to having some actual fun, let’s explore a flat plane in mathematics—one that may even touch malevolently on geometry.

I’m not a mathematician, but the impression I get is that even most mathematicians find ZFC boring. That is to say, Zermelo-Fraenkel set theory together with the Axiom of Choice (AoC).

At this point, if you are a mathematician and find ZFC to be your catnip, porn or indeed cat video of choice, please feel free to let everyone know in the Comments section below.

But—after that transient excitement—back to the ennui of ZFC. For the vast majority of us who aren’t interested, ZFC is an axiomatic system developed at the start of the twentieth century that resolves the paradoxes inherent in earlier theories of sets, or Wikipedia is a liar.

Okay, I confess. I find ZFC fascinating for one reason. Yeah, right, it provides a consistent set theory. But the catnip is the “C” part.

Will I lose my audience if I digress once more? One last time. And the Banach Tarski paradox is actually rather popular—chances are you’ve heard of it. Not as popular as cat videos, but often held up as an illustration of how counter-intuitive mathematics is, or perhaps how crazy reality is.⁶ What it says is simple, and completely not intuitive: Given a solid ball in 3D space, using standard maths you can chop it up into a finite number of pieces, and then reassemble these pieces to make up two solid balls, each identical to the original. The maths is demonstrably correct. The sort of maths nominally used by physicists.

What it actually tells us, of course, is that ZFC is not a very good way to model reality. And it turns out that the problem is indeed with “C”. If you toss the Axiom of Choice, then Banach Tarski evaporates.

The unfortunate thing here is that abandoning the AoC is pretty painful for pretty much every mathematician. Surely, a tiny subset of mathematicians have done this for ages, but even their few good friends regard them as a bit strange. Somewhere between a sect, and a cult.

What many mathematicians won’t even bother to mention is (a) their dependence on ZFC—the easiness of the unassumed choice; and (b) that rejecting the AoC is weaker. Generally, weak assumptions are more desirable than strong ones, at least, in maths. But here, the apparent power and crutch-like necessity for the AoC overwhelms every other consideration. Truly, a ‘flat plane’ of mathematics. Just, perhaps, not a very good one for some purposes—such as constructing a believable model of reality.

Doomed

So why are we doomed, then? Where’s the terminal melodrama?

I’m sure you’ve worked this out by now. If you’re not a mathematician, furious about my slighting of ZFC. Or a practitioner of religion, about to start your own personal jihad against Dr Jo for in some way impugning your unstated religious choice. Or a Black-Scholes cultist.

We tend to take the foundations of everything we hold dear so unseriously that we don’t even think about them. The easiness of the unassumed choice. Until someone questions these ‘foundations’, of course—when we leap reflexively to their defence. On the rare occasions when we bother to think about foundations, we seldom question, and rarely understand. We often tacitly choose really bad ‘flat planes’ without asking about origins, precisely because delving deeper seems, well, boring.

And then suddenly, when faced with reality—especially changing reality—we stuff it up. It’s 2025. You only need look around you, and see the resurgence of all sorts of zombies we thought we’d buried, to work out that somewhere we screwed up. Perhaps the mistake we made was not shooting at the head? Or are we ourselves to blame?

Fascism is rampant; we are in the midst of a great killing of species that the World hasn’t seen since the Cretaceous-Paleogene extinction event; we can’t even handle a small-ish plague⁷ (30 million deaths) well; people are abandoning reasoned behaviour in favour of cult-like obsessions. How does this all arise?

I suggest this is our human condition. I ask again “What’s your flat plane assumption that perhaps didn’t work out all that well?” What are mine? Perhaps we can explore some of these?

Of course, none of what I’ve said is actually boring. I rather like metrology. Love it even. But I fear that I’m in a minority. Don’t discuss traceable calibration when the competition is cat videos.

A final promise kept

Finally, I promised not to mention a topic so boring that even database designers refuse to contemplate it. So I won’t. But this will be the topic of my next post:

On Being Normal

My 2c, Dr Jo.

1. Technically, water is most dense at 3.984°C ± 0.003°C.

2. You can read the historical details here: http://charm.physics.ucsb.edu/courses/ph21_05/kilogrampaper.pdf Of interest is that when comparing the Kilogramme des Archives with the IPK, you must adjust for air pressure, as their densities differ.

3. For the nit-picking details of how those comparative measurements were made, look here: https://pmc.ncbi.nlm.nih.gov/articles/PMC6664201/ It’s looong, and—dare I say it—boring.

4. All of this work wasn’t however in vain. It did help to define the mole: exactly 602,214,0 76,000,000,000,000,000 atoms. Not one atom more or less.

5. Of course the Kibble balance isn’t boring at all. It’s fascinating: https://en.wikipedia.org/wiki/Kibble_balance. And no, Bryan Kibble had nothing to do with pet food :) He did however play the clarinet, measure the gyromagnetic ratio of the proton, and invent that darn balance.

6. Look around on the Web, if you dare. There is some really crazy stuff written on Banach-Tarski. For obvious reason’s I’m not going to reference it directly—but it’s not hard to find.

7. The last time I mentioned a ‘plague’, someone took me vigorously to task for using the word ‘plague’ about a viral infection. They insisted that it only applies to Yersinia pestis. It’s a metaphor, OK?