Thursday, August 23, 2012

How to Imagine a Billion

Carp on a tarp, who would've thought a book about how we know things could be so research-intensive to write?  The problem I'm running into now is that every time I'm like, "Wait a minute, I should look this up," I end up going on a wiki walk where I kinda-sorta get my question answered but find way more interesting things along the way.  So, OK, now I know how ejection seats work, and I have a reasonable guess as to where the USS Eisenhower was in the spring of 2000, but I'm also finding out all kinds of things I don't really need like stuff about the new robot we put on Mars and how plots of various movies could have been solved in minutes (or would have been solved if the main characters did nothing).


I was talking about space & stuff with a friend, and we were going on and on about how so many people have a really poor grasp of just how astronomically large astronomical distances are.  At one point, it was alleged that humans can't even conceive of a million, let alone the billions and trillions required to understand space.  I thought about that for a while, and as it turns out, yes I can so too imagine a billion - and so can you!  Here, let me teach you how.

First, a teaching tool.  Did you ever use these little yellow blocks to count when you were in gradeschool?  I did, at least in the fifth and sixth grades (I went to different schools every year before that, so I don't know how far back they use these).

Wait, you mean tiny yellow cubes a centimeter on a side are chokeable?

As you can see on the page, they also come in rods of ten, sheets of a hundred, and (not pictured) blocks of a thousand.  It's pretty easy, one is one, ten in one direction is ten, ten in two directions is a hundred, and ten in three directions is a thousand.  Then we just need to go in a fourth direction for ten thousand, right?

There are actually ten thousand here, but the
camera can't show you the fourth dimension.

Well, spatial dimensions are actually tough for humans to suss out once you get past three of 'em - Hell, even that third one is pretty tough!  But those cubes of a thousand are only ten centimeters on a side, you could hold one in your hand pretty easily, and so we can work with them from here on out.  Incidentally, these are also where almost all my understanding of the metric system comes from - one of those, made of water, is a gram, and I just build up from there (running 5K races in my adolescence got me up to and past kilometers, but it's still mostly built on those tiny yellow blocks).

So take that big honkin' 10x10x10 cube up there, and imagine ten of them - ahh, shit.  Cardinality.  Turns out, humans also have trouble imagining more than seven of something (that's why seven-digit phone numbers are so easy for us to remember - or they were, back when we had to remember phone numbers).  But you can get around cardinality easily enough with lug nuts and clocks.  Like, if you're having trouble picturing twelve of something (even the numbers on a clock), just picture four, and rotate them thirty degrees and picture a new set of four there, then another thirty degrees, and you're good (another thirty degrees and you've got ninety, which puts you back at the original position because these are Platonic Lug Nuts which cannot be differentiated from each other).

When I need ten, I picture five lug nuts, then flip 'em over for another set of five.  Technically (read:  if you're an asshole), I'm not imagining "ten" per se so much as I'm imagining "five twice," which is distinct but nobody gives a shit.  So you take those ten and put 'em in a line, and bam, that's ten thousand little yellow cubes.  Spin that line ninety degrees and fill it out, and you've got a hundred thousand little yellow cubes in a square meter that's ten centimeters high.  Easy enough, right?  Now pile up ten of those sheets, and kra-prao!  You've got a thousand thousands, otherwise known as a million.

Now, if you're an asshole, you'll note that we've simply extrapolated by repeating a pattern and we're not actually imagining each yellow cube as a distinct entity all on its own.  But if you're smart, you'll realize that doesn't matter, because we're talking about numbers and scale here, not individuality or distinctness or any of that happy horse shit.  The point is not to bypass the limits of your brain, that obviously can't be done (because if it could, then that wasn't actually a limit), but rather to take something you can easily manipulate with your mind and figure out how much a million of it would be in terms you can also easily grasp.  Translation, if you will.  It's a stretch, not transcendence, is what I'm saying here.  Though, I mean, you might feel a little transcendent, holding a million tiny yellow cubes in your mind's eye.  That's perfectly fine.

So, to recap, we've got little yellow cubes a centimeter on a side, arranged in a bigger cube a meter on a side, so it's 100x100x100 units, or a million.  You could fit that on a pallet (not to be confused with a palette or your palate), so imagine that, because it's easy and the materials are ready to hand.  Then just pile those pallets up ten wide, ten high, and ten deep, and you've got a thousand millions:  a Gin-You-Wine, Bone-A-Fide, Carl Sagan Bee for Billion.  You could stick it in a warehouse, though at ten meters tall, that's actually a Hell of a warehouse.  At least eleven meters tall, in fact, if the pallets are ten centimeters thick.

But let's break it down some, just to make it a little easier.  Let's put two on a floor, since two meters isn't that much to ask of a floor, and make our building five stories tall with rooms, oh, twelve meters on a side (for walking room, you see).  So we've got reasonably spacious rooms with two layers of a hundred pallets, each holding a million tiny yellow cubes.  There's your billion:  a pretty small office building, five stories high, chock-the-fuck-full of a billion tiny yellow cubes.

There's a lot of ways you could go for a trillion, though I like to stick with the 10x10x10 pattern.  That makes for some rather awkward civil engineering when you've got four blocks that are all filled in with tiny office buildings packed full of tiny yellow cubes, but fuck it, it's my imagination and I'll flex it any way I want to!  Even if it means putting nine more buildings inside a perimeter of sixteen other buildings and making them all fifty stories high but with absurdly small square footage.  Again:  raw numbers, and scale; not realism.

Anyway, enjoy your newfound power to imagine billions upon billions!


Steve Bowen said...

Yes, but does it matter if the cubes aren't yellow?

D said...

Well, yeah. The blue ones are only worth three fifths of a cube, so you need like two thirds more to wind up with the same number. That's just common sense.