When I was ten, I read with astonishment that with each breath, I was inhaling molecules that were breathed by the mathematician Archimedes over two thousand years ago.

This sort of invocation of chemistry as a magic history-spanning bridge can be traced back to James Jeans, the English scientist and mathematician, who in his 1940 book “The Kinetic Theory of Gases” wrote: “If we assume that the last breath of, say, Julius Caesar has by now become thoroughly scattered through the atmosphere, then the chances are that each of us inhales one molecule of it with every breath we take.” The science writer Sam Kean recently wrote an entire book, “Caesar’s Last Breath”, that takes this proposition as its starting point.

In between Jeans and Kean, other writers making the same point have replaced Caesar by Archimedes or Jesus or da Vinci. I prefer Archimedes, because he was the first of the ancient Greek mathematicians to come to grips with really big numbers and to connect the macroscopic and microscopic realms; in “The Sand Reckoner” he calculated how many grains of sand would fill the universe as the Greeks understood it.

As I write this essay in April 2020, human society has been violently tipped on its side, and the eight billion or so people who share this planet have come to realize how small the world has become epidemiologically. We’ve also become fearfully conscious of the contents of the air we bring into our bodies. Perhaps now is a good time to take a deep and hopefully healthy breath and think a bit about how the content of our lungs connects us to people far away in space and time, situated in a past that, even at a remove of a few months, feels very distant.

Molecules are tiny; Earth is huge; we’re somewhere in between. Our brains didn’t evolve to handle the difference in scale between microscopic events and events of daily life, or between events of daily life and global processes. We can fling around words and phrases like “nanotech” and “trillion dollar deficit”, but few of us really *get*, on a gut level, how small a nanometer is or how big a trillion is.

And yet, neuroanatomical evolution has already developed a wonderful approach to the problem of scale. Consider for instance the human ear; it must process a gamut of frequencies from 20 to 20,000 cycles per second. It does this using an organ called the cochlea, whose thousands of tiny hair cells respond to different frequencies. When the ear picks up a tone, the position of the hair-cell along the cochlea that responds to that particular tone corresponds roughly to the logarithm of that tone’s frequency — which may sound intimidating if you’re rusty with logarithms, but if you’ve ever played a piano you have an intuitive, kinesthetic sense for the logarithms of frequencies. Each time you shift your hand up an octave, you double the frequency of the note. Frequency is an exponential function of the position of your hand, and conversely, the position of your hand is the logarithm of the frequency produced. The cochlea is just like that, except that it’s receiving sound, not producing it. Some have called the cochlea an “inverse piano” to highlight this analogy.^{1}

We can use exponentials and logarithms to try to get a handle on the large and small, but it’s easy to forget the key difference between counting “one, two, three, four, …” and counting “thousand, million, billion, trillion, …”: the former is an *arithmetic* progression (each term is equal to the previous term *plus* something, namely 1) while the latter is a *geometric* progression (each term is equal to the previous term *times* something, namely 1000). In more concrete terms: If we plot the four numbers one, two, three, and four on a number line, we get this:

On the other hand, if we plot the four numbers one thousand, one million, one billion, and one trillion on a number line, we get this:

Were you expecting to see four dots? Well, the “dot” at the left is actually two dots, one for one thousand and one for one million; at this scale the two dots are too close together to be distinguished. Meanwhile, the dot for one trillion is about a mile off to the right.

If you’ve never seen the video “Powers of Ten” or the similar video “Cosmic Zoom”, I suggest you take a break from reading this essay and watch one or both of them. Touring the universe from the largest scales we know about to the smallest is a great way to get a feeling for how the different levels of our universe fit together. The largest structures we know about are roughly 10^{41} times larger than the smallest. We’re somewhere in between, on a cosmic piano that has roughly a hundred and forty octaves.^{2}

When one does calculations that involve big things, small things, and things that are in between, one sometimes finds that the in-between things are close to the midpoint on a logarithmic scale, in the way that middle C is close to the midpoint of a piano keyboard. One example of this phenomenon is the proposition that there are about as many molecules in a teaspoonful of water as there are teaspoonfuls of water in all Earth’s oceans (about 200 sextillion in both cases). A more mind-boggling example is one I learned from Bill Gosper, who computed that a molecule of polyethylene^{3} spanning the observable Universe, suitably folded, would just about fit in NASA’s Hangar 1, one of the largest buildings ever constructed. Another phenomenon along similar lines is the way you can use an oil drop to measure the wavelength of visible light; the drop is much larger than the wavelength of light, but ponds are much larger than droplets, so when you let a droplet spread evenly over the surface of a pond, you can create a layer of oil so thin that the resulting interference patterns let you determine the wavelength of the light.^{4}

The claim about Caesar’s last breath is yet another a story about three length-scales, spanning the logarithmic ladder from molecules to people to planets. How big are these things? The diameter of the planet is about eight million times the height of the average human adult, which in turn is about five billion times the diameter of a molecule of air. With such disparate ratios (eight million versus five billion) it might seem that in the range from single molecule to entire atmosphere we humans are off-center, logarithmically speaking, but that’s because we’re ignoring two important things: gas kinetics and the shape of the atmosphere. Molecules in a gas aren’t packed like oranges at the grocer’s; they’re constantly jostling one another, in an all-against-all molecular melee that results in far fewer molecules per liter than the size of a molecule would suggest. Also, our atmosphere is not a ball of gas but a *hollow* ball, eight thousand miles across from its northernmost point to its southernmost but only ten miles thin; in relative terms, that’s five time thinner than the shell of a chicken’s egg. When you do the math (as Archimedes would have loved to do, given his famous work on the volume and surface area of spheres), you find that the number of molecules of air in a lung is quite close to the number of lungfuls of air on Earth. And this suggests that the number of molecules from Caesar’s last breath in your lungs right now is approximately 1.

The answer is sufficiently close to 1 that it’s probably sensitive to issues ignored in our oversimplified mixing model.^{5}

Jeans’ claim ignores the massive amount of molecular recombination going on in our atmosphere. In the chemical dance of geological and biological processes, oxygen and nitrogen atoms (the primary constituents of air) change partners all the time. It’s conceivable that most of the oxygen molecules in Caesar’s last breath got split long ago, and hence, strictly speaking, no longer exist. Of course, we could rescue Jeans’ claim by replacing molecules by atoms, and then similar calculations would apply.

I myself prefer to go back to the formulation that I read as a child, the ones that talks about Archimedes’ lifetime pulmonary output instead of his dying breath; aside from the fact that it’s less morbid, it’s also much more likely to be true. That extra factor of half a billion, coming from all those breaths, makes the proposition much more certain, even if some of those breaths contained molecular “repeats”, and even if some of the molecules escaped into outer space, or sit sequestered in permafrost, or were cleft by lightning or metabolism.

I suggest Terry Tao’s lecture “The Cosmic Distance Ladder” as a follow-up to “Powers of Ten” and “Cosmic Zoom”. But such pedagogical tools can only go so far to give us a feeling for the power of raising things to powers. An old story from India tells how a grand vizier, having invented the game of chess for the ruler’s enjoyment, asks that his reward be one rice grain for the first square of the board, two grains for the second, four grains for the next, eight grains for the next, and so on, up until the 64th and last square of the board. The king thinks the vizier is letting him off easy and agrees to his terms. Only later, when he starts trying to pay the rice to the vizier, does he discover that he’s made a mistake. How much rice was the vizier asking for? 1+2+4+8+…+2^{63} comes to about 18 quintillion grains of rice, which is a thousand times greater than the amount of rice that is currently grown in the world in a year. In one version of the story, the king, upon realizing that all the rice in his kingdom wouldn’t suffice, nullifies his promise by having the vizier executed.

Even when you think you understand exponential growth, it’s easy to slip up. Here’s an example from The Giant Golden Book of Mathematics, a book I loved as a child and still admire: “An amoeba is placed in an empty jar. After one second, the amoeba splits into two amoebas, each as big as the mother amoeba. After another second, the daughter amoebas split in the same way. As each new generation splits, the number of amoebas and their total bulk doubles each second. In one hour the jar is full. When is it half-full?” It’s tempting to answer “half an hour”, but the correct answer is one second before the hour is up. Actually, an even better answer is “That’s a ridiculous question.” There are 3600 seconds in an hour, and 3600 rounds of doubling would lead to a growth of the initial biomass by a factor of about 10^{1000}. There aren’t enough octaves on the cosmic piano for that. Before the hour is up, the amoebas would fill all of the the known universe.

Those imaginary amoebas teach us something that we forget at our peril: exponentially growing quantities look negligible until they don’t — or look innocuous until it’s too late to do anything about them. Why worry if ten grains of rice become twenty? It’s still less than a handful. Why worry if ten cases of a communicable disease become twenty? More people die each year from falling out of bed.^{6} It’s easy to dismiss things that are growing exponentially when they’re small. Albert Allen Bartlett famously wrote “The greatest shortcoming of the human race is our inability to understand the exponential function.” Let’s hope we as a species can avoid the grand vizier’s fate.

To end on an upbeat (or dare I say “inspiring”?) note, it’s worth remembering that the same solar energy that kindled Archimedes’ brain by way of chemical bonds in the oxygen he breathed also feeds *our* brains. If we focus enough brainpower on the problems we face as a species, it’s possible we’ll be able to come up with ways to cope with the current crisis and stumble our way through to the next crisis, and the next, and the next.

*Thanks to John Baez, BIll Gosper, Sandi Gubin, Hans Havermann, Michael Kleber, Henri Picciotto, Evan Romer, and Simon Plouffe.*

Next month: Confessions of a Conway Groupie.

**ENDNOTES**

#1. The central nervous system must have its own tricks for dealing with the problem of disparate scale; for instance, perceptible levels of loudness, from the barely discernible to the headache-inducing, span many orders of magnitude, as do perceptible levels of illumination. If you know something about how the brain encodes intensity of auditory and visual stimuli, please post in the comments!

#2. Thinking of this piano puts me in mind of a scene from “The 5000 Fingers of Dr. T.“, which as I child I found so disconcerting that I couldn’t watch the movie.

#3. Polyethylene is a chain of hydrogens attached to a carbon backbone of indefinite length, so in principle a polyethylene molecule could be long enough to span the observable universe; this hypothetical molecule, if folded up tightly, would fit inside Hangar 1.

#4. Can anyone provide a good reference for this?

#5. When we’re dealing with quantities much bigger than 1, or much smaller, an order of magnitude or two usually doesn’t have a qualitative effect on the conclusions we can draw, but that’s not the case when quantities are logarithmically close to 1, or 10^{0}. If the expected number of “special” molecules in our lungs at any given time is computed to be around 10^{−2} = .01, then we could say with confidence that most of the time our lungs don’t contain any. On the other hand, if the expected number of special molecules in our lungs at any given time is computed to be around 10^{2} = 100, and we model the number of such molecules in our lungs as a Poisson random variable, theory tells us that the standard deviation is 10, so that the probability that our lungs contain none at all right now — a “ten-sigma event” — is minuscule.

#6. Propagation of a novel disease through a vulnerable population is described pretty well by an exponential function in the early stages of the epidemic, when most of the population is immunologically naive. In later stages of the epidemic, the sigmoid curve predicted by the logistic model provides a better fit.

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