Fun With Logarithms 3: Orders of Magnitude

I started hearing the term “Orders of magnitude” long before I knew what it meant. It’s actually a simple and extremely handy convention. 10 is one order of magnitude larger than 1. 100 is one order of magnitude larger than 10. 0.000001 is six orders of magnitude smaller than 1. Simple.

The best thing about orders of magnitude is that they allow you to get an intuitive grasp on large numbers, which are just about as unintuitive as it gets, and with hardly any math or research necessary. Here’s an example.

I’m 1.9 meters tall (6’3″, if you insist). A peppercorn has a diameter of about 2 millimeters. A millimeter is 1,000 microns. Ordinary bacteria have dimensions of around 1 micron. Therefore, a peppercorn is about 1,000 times larger than a bacterium.

A peppercorn is pretty small, but if you look at it closely, you can still make out details: peppercorns have weird little wrinkles and chips and dust on their surfaces. A peppercorn is an easy object to comprehend. To comprehend the size of a bacterium, go the other direction: imagine an object 1,000 times larger than a peppercorn. That comes out to about 2,000 millimeters, or 2 meters. That means I am to a peppercorn as a peppercorn is to a bacterium. Or, if you want to use me as your basis, a bacterium sitting on a peppercorn is like me standing on the summit of a 1,900-meter mountain. That’s close to the same as me standing on top of El Capitan:

(Source and licensing.)

Anyway.  Bacteria are pretty small. But they’re not as small as viruses, which have dimensions measured in tens or hundreds of nanometers. Twenty nanometers is about eight orders of magnitude smaller than me, so a virus standing next to me is like me standing next to Jupiter.

That’s a lot less intuitive. I know mathematically how big Jupiter is, but it doesn’t make any sense to my gut, and it’s important to take the gut into consideration if you really want to get a feeling for anything. Luckily, you can chain orders of magnitude end-to-end.

A 20-nanometer virus particle is 100 times smaller than a 2-micron bacterium. Therefore, if El Capitan represents me, and I represent a bacterium (which is a weird thought), the virus will be 1.9 centimeters across, or about the size of a wine grape. A virus compared to a person is like a grape on the summit of a mountain. That’s stretching the powers of intuition, but it’s still comprehensible.

Let’s have more fun! A football field (American football or the football that most of the world calls football but Americans call soccer, the proper name of which people really like to argue about for some reason) is about 100 meters from end to end. I stood on many such fields in gym class as a child, so I have a pretty good intuitive grasp of their size. 100 meters is a large distance, but not so large you can’t wrap your head around it, so it’s useful for making even harder order-of-magnitude comparisons.

Most atoms are around 200 picometers across. 1 nanometer is 1,000 picometers, so 100 picometers (ignoring the factor of two, which you’re allowed to do in order-of-magnitude math) is ten orders of magnitude, or ten billion times, smaller than a meter.

A human hair is about 100 microns in diameter, and is another good basis for comparison, since 100 microns is about as small as something can get and still be visible to the average naked eye. 100 microns is 4 orders of magnitude smaller than 1 meter. 100 picometers is 10 orders of magnitude smaller than 1 meter. Therefore, 6 orders of magnitude (or a factor of 1 million) separate the diameter of an atom and the diameter of a hair. It just so happens that a 100-meter football field is 6 orders of magnitude larger than a 100-micron-diameter hair. So an atom in a strand of hair would be in the same proportion as the diameter of that hair compared to the length of a football field.

This same kind of sloppy-but-useful math can be applied to understand astronomical distances, too, up to a point. The Earth has a diameter of about 12,000 kilometers (closer to 12,700, actually, but we’re not being that precise). 12,000 kilometers is 10 million times larger than 1.2 meters. The earth, therefore, is about 10 million times larger than me. Something ten million times smaller than me would be 1.9 microns across, which is the size of a bacterium. Therefore, the bacteria sitting on my skin feel just as small as me standing on the Earth. Which is weird and almost poetic. Almost.

But my understanding of 1.9 microns is abstract. (How many times have I said “intuitive” and “abstract” so far? You have my permission to begin a drinking game. Intuitive abstract intuitive abstract intuitive abstract intuitive abstract. Enjoy your evening and try to vomit into the toilet. Intuitive. Abstract.) Let’s use a football field as our basis instead. I am 7 orders of magnitude smaller than the Earth. In order to be 7 orders of magnitude smaller than a football field, an object would have to be 7 – 2 = 5 orders of magnitude smaller than a meter, or 10 microns. Still too small.

El Capitan is about 1,000 meters tall (it’s actually over 2,000, but that’s within the same order of magnitude). That’s 3 orders of magnitude larger than a meter. To be 7 orders of magnitude smaller than El Capitan, an object would have to be 4 orders of magnitude smaller than a meter, or 100 microns, which is the diameter of a hair or a speck of dust. So we are all specks of dust sitting on the mountaintop that is the Earth. (Feel free to punch me for that sentence, if you happen to meet me in the street. Seriously. I’d punch myself right now, except I keep flinching out of the way.)

You can also use this kind of tricky math to build toy models of astronomical systems in your head. For instance: what would the Earth and Moon look like, locked in their orbits, if you saw them from a distance?

Well, the Earth is about 10,000 kilometers across. The moon is about 3,000 kilometers across, or one-third of an Earth diameter. Imagine a really big grape sitting next to a smallish grape, and you’ll have about the right proportions. A coin and the bottom of a drinking glass are also in similar proportions.

The Moon’s semimajor axis is about 300,000 kilometers, so it’s usually separated from the Earth by 30 Earth diameters or 100 Moon diameters. Here’s an experiment you can try at home: set a cup on the floor and line up 98 coins next to it (Not 100, because one and a half coins will be inside the cup and half a coin will be inside the coin representing the moon). Actually, I hate it when people say “Here’s an experiment you can try at home.” I’m gonna be nice and do it for you, although admittedly I’m going to have to switch up and use a coin to represent the Earth, because otherwise, the system would be larger than my floor. But I cheated and did the math and got the proportions right.


Consider that penny. That’s here. That’s home. That’s… No. Sorry. I can’t be sarcastic about Carl Sagan’s Pale Blue Dot speech. Just go watch it. Watch it twice. No smart-assery here: it’s one of the best speeches I’ve heard in my life.


What’s the most interesting place in the universe?

We humans are extremely susceptible to information overload. Yeah, our brains are impressive, but there’s an upper limit on how much they can process before they start to overheat and misbehave. The unfortunate cases of ailments like PTSD and career burnout testify to that. In fact, the brain is remarkably good at filtering out data it deems to be irrelevant. I’ve just looked down at my desk. On my desk is a penny, worth US$0.01. I didn’t put it there recently, so it’s been there for at least a few days. Maybe as long as a week. You know how many times I’ve consciously noticed it since I put it there? Zero. Even though it sits right by my keyboard, where I work every day, my brain just glossed over the existence of a small disc of copper and zinc inexplicably pressed into currency. 

There are all sorts of interesting things all around me, items with thousands or millions of little salient details. I’ve just picked up a ruler from the floor. It’s from Office Depot. It has ’80s-station-wagon-style fake wood down the center. A particular machine in a particular factory in a particular place on Earth made this ruler from a particular load of plastic derived from a particular load of crude oil. If I measured them with high precision, I’m sure I’d find that the tick marks on the ruler have their own unique pattern of variation. Each of the numbers is printed slightly differently, even from identical numerals on the same ruler. I laugh when people say “No two snowflakes are alike,” because if you think about it, no two anythings are alike.

But like I said, it’s hard for us to think about this during our day-to-day lives. Imagine if, rather than just stopping to smell the roses, you stopped to smell them, compare the scents between flowers, check how much pollen each flower had, and count each flower’s petals. You’d still only be covering a tiny fraction of the information you could learn about the roses, but you’d still stall on the sidewalk for a few minutes and get some really strange looks from passerby (trust me on that one…). 

But think about it: each of us only passes through a tiny volume of space in a given day, and we still see all kinds of crazy stuff that we never think about. In the driveway outside, most of the gravel is (I think) some kind of blue granite. Each stone has a different shape. If you studied one of them long enough, you would start to find interesting things about it. Maybe you’d find a cool double-stripe in one of them, or a bumpy spot that looked a little like a face, or a surprisingly sharp edge, or a near-perfect pyramid. And we must remember that these stones, being granite, are little chunks of ancient lava and magma. That is to say, ancient fluid rock that forced its way through other rocks and squeezed out like toothpaste

You might be thinking that this article is kind of loopy and hippy-dippy. Never fear. Because I am borderline obsessive-compulsive, you’d better believe I’m going to quantify the hell out of how awesome and complicated the world is.

Let’s start off with a simple thought experiment. If you took a garden trowel and dug up a cube of dirt 10 centimeters on an edge, you’d end up with 1000 cubic centimeters (one liter) of soil, which, in the area where I live, would contain a few dozen or a few hundred individual stalks of grass, a collection of interesting weeds, organic debris in various stages of decomposition, somewhere between zero and five worms, lots of interesting rocks of different kinds, tough red clay, and, in all likelihood, some extremely unhappy ants. Imagine you wanted to a do a complete analysis of this cube of dirt. The first thing you’d want to do would be to lift it up in one piece and photograph it from all sides. Then you’d want to weigh it and record subjective observations like smell and texture. Then, if you were being really scientific, you could freeze the cube of dirt, embed it in paraffin, and slice it into ten-nanometer slices with a laser microtome. Then you scan each slice with a transmission electron microscope at a resolution of 0.5 nanometers, which would be good enough to show you detail down to the subcellular (almost the molecular) level. This is quite achievable with current technology. Good luck storing all that data, though. At 0.5-nanometer resolution, each individual slice would contain 160,000 terabytes of data (assuming you stored them as uncompressed 32-bit grayscale images, which seems reasonable). That’s at least a whole server farm. Probably several. Possibly several hundred. And that’s only to store the data from a single one of the hundred-million slices you’d end up with. And I’ll remind you that we’re working from a cube of dirt you could easily fit into a breadbox, a grocery bag, a suitcase, or (as a really cruel joke) a cake box. But that ludicrous quantity of data would tell you much of what was going on in the cube of dirt at the time you dug it up. Of course, it wouldn’t tell you things like the temperature distribution or magnetic fields or moisture or anything like that, but you could theoretically work those out, too, or just run the whole cube through a mass spectrometer and get an exact read on its composition. Hope you’ve got another hundred server farms handy.

And, once again, I will remind you that this is for a one-liter cube of dirt. There’s that much interesting stuff happening in 1000 cc’s of soil. And do you know how many cubes this size the Earth alone contains? Upwards of 8.5 trillion trillion. That’s a lot of stuff. Atoms are small, but if you had a diamond containing 8.5 trillion trillion carbon atoms, that diamond would be the size of eight sugar cubes stacked together ( which I can guarantee you I would immediately sell to stop people hunting me down). And don’t forget, that each carbon atom in this diamond represents a cubic hole in your lawn large enough to twist an ankle in, containing more information than you can probably store in an average server farm.

I could go on. I could go on for days. I could try to convey how achingly massive the universe really is, and how intricate many parts of it are, but even my head, accustomed to such bizarre calculations, is already near bursting. 

There’s one more way to begin to grasp how complex the world is. Say you wanted to simulate the whole universe, down to the physics of individual protons, neutrons, and electrons. You could divide the universe into a cubic grid with cells one femtometer (about the radius of a proton) on an edge, and label each cell with the relevant variables (what kinds of particles it contains, their velocities, magnetic fields, et cetera). But in some ways, the universe is like a black-and-white line drawing. Yeah, there are a lot of interesting details, but those details exist only in the shapes of the lines. You can use a nifty mathematical thing called a quadtree to turn that image into pixels without storing each individual pixel of blank white space. The most simpleminded kind of quadtree works like this: take the original image. Divide in half in both directions, so you get four subsquares. If a subsquare contains more than one kind of pixel (meaning it’s not all white or all black), then subdivide it into four smaller squares. Otherwise, store its coordinates and its edge length and write “This square is all white,” which, especially for large squares, saves a lot of memory. Eventually, you’ll subdivide until you’ve got a shitload of squares that are two pixels by two pixels. Some of these you won’t have to subdivide, but many of them will be on the edges of lines, and you’ll have to subdivide them and store the individual pixel values. But a large portion of the image (meaning: all the parts that contain large swaths of white or black) don’t have to be stored at nearly so high a resolution.

You can imagine dividing the universe in the same way. For simplicity’s sake, let’s just consider the physics of individual subatomic particles: protons, neutrons, and electrons. Let’s pick a cubic volume of the universe that fits within our current cosmological horizon, that is, a cube about 90 billion light-years on an edge that contains every object whose light has had time to reach Earth (except for the spherical bits that stick out on the sides; you could deal with those using a different subdivision scheme, but let’s keep things simple for now). Divide that cube in half along each axis, giving you eight separate cubes. If a cube contains nothing, say so and leave it alone. If a cube contains a single subatomic particle, store that particle’s variables in the cube and leave it alone. But if the cube contains more than one particle, subdivide it into eight again. This is called the octree method, and it’s the basis of the absolutely brilliant Barnes-Hut algorithm, which makes gravitational simulations for large numbers of particles run a hell of a lot faster by not doing ultra-high-precision calculations for low-density regions.

We’re going to have to do a lot of subdividing, though. Even in the vacuum of intergalactic space, you’ll probably have to subdivide down to cubes 20 centimeters on an edge to make sure you only have one particle per cube. That’s pretty big in particle-physics terms, but shockingly tiny in universe terms. Even in the vacuums between stars and between planets, you’ll have to subdivide down to one-centimeter cubes, and often smaller. When you reach the density of water (which you do when you approach pretty much any planet or star or grain of zodiacal dust), you have to subdivide down to cubes 0.1 nanometers on an edge. When you’re dealing with something as dense as lead, the cubes are 0.03 nanometers, smaller than most atoms. In the core of the sun, your cubes get down to 0.01 nanometers. 

But actually, even within atoms, you have to subdivide. Atoms are, as our science teachers kept telling us, mostly empty space. The nucleus of an atom has a density of about 2e17 kilograms per cubic meter. So, although for the most part an atom can be described by cubes on the order of maybe a tenth of a picometer on an edge (depending on how many electrons there are), in the nucleus, the cubes are two femtometers on an edge, just large enough to enclose a single proton.

What’s the point of all this? Good question. The point of all this is that, in human terms, or in terms of information, the most interesting places are those where the cubes are very small. That is to say, places where there’s a lot of matter packed close together doing interesting things. These are the places that astronomers study: nebulae and galaxies and stars and planets. But when you subdivide small enough, you begin to see that, really, the most interesting place in the universe is the nucleus of an atom, because you basically have no choice but to describe all the individual particles to get a decent description of the nucleus, which is just another way of saying the cubes involved are the size of the particles involved (protons and neutrons). 

But actually, nuclei aren’t the most interesting place in the universe. They’re dense, yeah, but they’re surrounded by a lot of empty space.

Neutron stars, however, formed when large enough stars go supernova and their cores collapse inwards, contain the smallest amount of empty space possible, according to our current understanding of physics (if they were any denser, you’d end up with a black hole, which, once again, is pretty much all empty space). Large neutron stars have about the same density as atomic nuclei (and possibly higher near their centers), but they’re the size of cities. Which is to say, there’s stuff happening in every cubic femtometer of a neutron star, just like an atomic nucleus, but unlike an atomic nucleus, its size is measured in kilometers, not femtometers.

We’ll be meeting neutron stars again, because they’re such incredibly extreme objects. But for now, I’ll leave you with the echoes of what I’ve been babbling about: The universe is complicated and fascinating and frequently horrifying and headache-inducing. Technically speaking, neutron stars are the most interesting places, but really, it’s all pretty weird. And I like weird. Weird is fun.