How big are most things?

Backstory time! When I was a little nerdling, I once picked up a rock and said the immortal sentences, “I like this rock. It’s very medium.” It was true; it wasn’t unusually big or small, and about as smooth as other rocks that I’d see lying around, and a perfectly unremarkable color for a rock. It was extraordinarily ordinary. Or so I thought. I hadn’t exactly tried to guess what an average rock actually looked like.

Let’s try. Consider the size of every rock on Earth, from grains of sand to tectonic plates. First: How many grains of sand are there? I don’t know, but it’s bigger than fifteen. To quote the Bible on the subject, the grains of sand on the seashore cannot be measured or counted.* But it never said anything about estimated! According to the University of (of course) Hawaii, there are approximately 7.5*10^18 grains of sand on Earth’s beaches.
*Hosea 1:10. Yes, I know it’s a figure of speech.

Of course, there’s more sand than just beaches. The Sahara Desert alone covers more area than all earth’s beaches, and is probably deeper too. But since I don’t have a reputable source for the number of grains of sand in the desert, I’ll ignore most of it and pretend the sand on the beaches is the only sand there is. It’s already a big enough number for what I want.

A large sand grain, according to Wikipedia, is about 2 mm. And there are 7.5*10^18 of those on the beaches. So, what’s the average size of a rock?
I don’t know. But we can easily find an upper limit. The actual number will be much smaller than the answer we get here. To the math!

Let’s pretend the Earth is made up of the grains of sand from beaches, plus one giant boulder that makes up the rest of the planet. It’s actually not set up that way, obviously. But if it were, the average size of a rock would be bigger than it is now. In real life, most grains of sand are smaller than that, there are more of them (so they’d be counted more in the average) and there is no rock that gigantic. The diameter of Earth is 12,700 kilometers, or 12,700,000,000 millimeters. (1.27*10^10 mm.) So to find our average length of a rock in the hypothetical case, it’s [2mm*(7.5*10^8 rocks)+(1.27*10^10mm)*1 rock]/(7.5*10^8 + 1 rocks). Plug those numbers into a calculator, and you get an average of a little under two centimeters. About four fifths of an inch, if there are any of you who actually prefer that system of units.

So my “very medium” rock was several times the average size. But remember that the real average is much smaller than the number we calculated. To find how much smaller, I’d have to do some numbers that are more complicated than I want to, and I probably don’t even have the data to do an actual estimate. But I can at least say that it’ll be lots of orders of magnitude smaller. (Why? Two reasons. Scroll up.) That means that my “very medium” rock was way bigger than average. Ouch.

But who cares about rocks? Let’s get more general. Let’s estimate the median size of an anything. There is some size such that half of all objects are smaller and half are larger. Considering the number of things in the universe, I’m surprised that you’ve never wondered about this before and no I’m not.

I’m not going to try to define “object” very precisely. I’ll just say that any specific physical thing is an object, and not worry about the confusion. So an anonymous cubic foot of ocean water in the middle of Earth’s ocean is not a thing, the same water would be a thing (an expanding cloud) if it were in a vacuum, and the same water would be a thing if it were filling a one-cubic-foot box. Of course, any particular molecule in the collection counts no matter what. I don’t care that it’s confusing. The criterion I’m using is: If someone pointed and said “Look at that thing,” would I think they were crazy?

We live in a universe (pretty sure about that) and it’s a universe that’s full of stuff. Some of the stuff is tiny by your standards, like quarks, protons, salt crystals, or really anything smaller than about half an ant. Anything bigger than that, and you probably don’t think of it as a tiny object.
I’m trying to guess at what normal people think, so I could be totally wrong, but I’m pretty happy with my guess of about half an ant. Any bigger and it’s not a speck. The quality of speckness is very important for being tiny. Besides, I’m trying to give you the benefit of the doubt by picking a small estimate.

(If it’s, say, a piano the size of an orange, you might think of it as a tiny piano. But “small for a piano” isn’t the same as “something small.” This is why I don’t react badly every time anyone uses an adjective. Usually it’s assumed that if you say “small” you mean “small for its type.” In this case, I’m specifically saying that the type is everything.)

Other stuff in the universe is gigantic, like a continent or the Sun or the distance between whatever you’re standing on and sea level (don’t look down) or the Death Star. Yes, the Death Star is in this universe. It’s in orbit around Saturn, look it up. My guess is that you probably think of anything larger than a large building as “huge.” Certainly if you can’t see to the end of it you’d consider it huge. And there are obviously lots of things bigger than human visual range.

There are lots of things that are “tiny” and lots of things that are “huge.” If you’re wondering which type there are more of, then stop and think for a bit. All those big things you thought of are made of tiny things. You, being approximately normal sized by human standards, have a few octillion protons and neutrons. Octillion is a big enough number that I might as well have said octodecillion, and you wouldn’t have noticed a difference. The Sun, being approximately normal sized by its standards, has a few octodecillion protons and neutrons. That’s a big enough number that I might as well have said octillion and you wouldn’t have noticed a difference.

With numbers this size, the few big things in the universe are just not statistically significant. The median is dominated by the subatomic stuff. All the electrons and photons and neutrinos are so small that we round down to zero. A proton is less than 10^-15 m. An atom might get as big as 10^-10. All of those sizes are way too small for your gigantic brain to comprehend.

If you pick a random thing, the odds of it being big enough to see are so small that you’d have a better chance of winning the lottery while being struck by lightning on a day divisible by three and surviving without paying any taxes. It’s not going to happen.

So what does this mean?
It means that any time you run across an object in normal life, you can make it sound huge: “This TV screen might not look very big, but it’s quadrillions of times the size of most things!”
Or you can use it in insults: “Your mom is so fat, she would be considered massive even if she weighed eight point two nonillion times less!”
Or you can start talking about wider perspective and then suddenly go the exact opposite direction: “We live on a pale blue dot, hanging in the vast reaches of a universe full of things far off any scale that we can picture. And we’re much bigger than those puny things.”

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