80s Sunglasses and Recursion

Imagine a thousand pairs of sunglasses, all lined up on a giant tarp. This is an enormous outdoor market where everyone is hawking their wares, and the smell of food truck garlic enters your nostrils. You’re excited to see what else is on offer, but you’re stopped in your tracks to notice that someone has carefully laid out a thousand identical sunglasses.

Each row has fifty pairs of shades, and there are twenty rows total. These are all retro 80s style sunglasses, sorted into different colors by row.

If you wore a different pair every day, it would take you nearly three years to wear them all!

That’s a lot of shades, but it’s something your mind can grasp, even if just barely. A thousand days is something we’ve all experienced, so we know how long that is. And, you can get a sense of how big twenty rows of fifty shades looks and feels.

That’s what a thousand of something is. Let’s feel that for a moment.

Now, if it took that much mental effort to imagine a thousand of something, it must be impossible to imagine a million—much less a billion or trillion, right?

Well, not exactly. The human superpower of recursive thought comes to the rescue.

Recursion is the thing that makes human language notable. We say things like, “I told you that I was going to buy sunglasses.” “I was going to buy sunglasses” is itself a complete thought, but you can talk about that complete thought inside of another thought.

It took longer for these recursive concepts to become formalized into mathematics, but that was the real key—to think of things in terms of other things you could easily symbolize and reference.

Al-Khwarizmi was a Persian mathematician, astronomer, and geographer, among other things. He’s often called “the father of algebra." His most significant contribution was to introduce an algorithm for algebra. Up until then, very specific problems were solved, but now there was a recipe for how to solve ALL similar problems.

Back to recursion and sunglasses. Take that concept of a thousand we just worked so hard on—twenty rows of fifty shades each.

Now, just imagine that instead of a pair of sunglasses, each of those pairs is replaced by a thousand sunglasses—the very thing you’re already visualizing. In other words, just reuse all that hard work; there’s no need to do it all again. You’re already there, so just rubber-stamp the concept of a thousand onto a different place.

Now, we have a million pairs of shades, but how much space does that take up? Let’s envision that tarp that originally had all thousand shades on it. How big is it? That depends on how close we stack the sunglasses, but you can probably fit them all into my living room.

Now, imagine taking that tarp with a thousand shades on it over to a much bigger open air market. There aren’t merchant tables any more; this is just a giant field where people gather to buy and sell insane quantities of stuff. You just line that tarp with a thousand shades on it next to another similar tarp with another thousand shades, and your tarp is in a row of fifty; there are twenty rows total.

Fifty tarps means fifty living-room lengths. My living room is pretty small as those sorts of rooms go—let’s call it fifteen feet square. Fifteen feet times fifty tarps gives you a length of 750 feet, and together with the other twenty rows, this makes a rectangle that is 300 feet wide.

I don’t really have much of an inherent sense of what 300 feet looks like until you remind me that 300 feet is 100 yards, or the length of an American football field (not counting the end zones). So, it’s a giant rectangle the size of a couple of football fields, completely covered with sunglasses.

That’s a million.

Now, you can use the same recursive trick to climb up the ladder of insanely big numbers. I’ve written about large numbers before. I think it’s fun to try to understand them, and it’s always a fun challenge for me to try to grasp something that seems ungraspable at first. It’s also really, really useful since we live in a world dominated by large numbers.

Trillions are tougher, but if you want to understand how many transistors there are in the world, you can get to a place of functional understanding by comparing the number to other things.

A trillion is a thousand billion, and we might be able to grasp that in a comparative manner by remembering that there are about 8 billion humans, so a trillion is more than a hundred times bigger than that.

That’s only really useful if you can grasp 8 billion, of course, which requires a lot of recursion. The other way to go about this is with size, like they way I opened with sunglasses on tarps.

I can give you a number, but let me just bring this piece full circle by telling you that you can fit trillions of transistors on a Triscuit.

A thousand isn’t impossible for a human mind to grasp, and through the power of recursion, that means no numbers are really off limits for us.