Exponents in the physical world

We know what an exponent is, so why are they still so non-intuitive to encounter in the world? If the jar of lights gave you a jumpscare, you wouldn't be the only one. They can be abrupt and startling.

But phenomena like this do exist in the physical world. They are just relatively rare. A nuclear reaction, sure. A chemist probably sees a few in the form of chemical reactions… Certain types of population growth, for limited windows, but I'd argue those aren't truly exponential; there's always a break, due to physics, competition, ecology. A bankruptcy? Ha. I'm sure there will be a few others. Write in if you think of any.

But considering all that, perhaps it's not been very important for humans to evolve an intuition for exponential phenomena.

I'll tell you about seeing the demo, a few months ago, that inspired the one on the site. I'd say that one was tuned a little longer. (I couldn't see wasting your precious time here, when quicker still gets it across.)

When there were just four balls, I wondered if it might be some kind of prank. Maybe, like the infamous bouncing DVD logo, I thought, there might be fixed, regular patterns that ensure the balls would never collide. Even in the jar, it can take a surprisingly long time for the balls to make their first collision and spawn. By seven balls, I was confident that wasn't the case. By 12 or 14, despite my knowledge otherwise, it had the appearance of an approximately linear process.

By then, I could feel the velocity. By 30 or so, I could begin to feel the acceleration. At this point, I was thinking, “Oh boy! There’s about to be a LOT of marbles appearing quickly from here.”

Finally I could feel the jerk, the acceleration of the acceleration. There were suddenly a lot of balls appearing all at once, just as I knew there would be. But crucially, the demo did not stop there.

Next was an abrupt violence of growth, an explosion. The analogous moment in a “technological singularity” scenario, people in the field sometimes call “foom” or a “foom moment” — more commonly and more properly, FOOM! There’s nothing lowercase about it.

1. 1Four lights. Some kind of prank?
2. 2Seven. Confident it wasn't.
3. 3Twelve or fourteen. It looks almost linear. I can feel the velocity.
4. 4Thirty or so. I can feel the acceleration.
5. 5Then the acceleration of the acceleration, a lot all at once.
6. 6And it doesn't stop there. FOOM!

It’s strange, isn’t it? How could something like that surprise me when I was expecting it? How indeed. I think that moment of discomfort deserves our piqued attention, just now. Because I am not sure our brains are built with the mental software to easily intuit such things.