The Core

I find myself talking a lot to people about exponential growth lately whether talking about Covid (as in my last post), or AI or energy transformation or whatever.

I'm struck by how hard it is to think in terms of exponential processes without visual aids. I've lately taken to presenting this "story" to people to help them grasp how foreign it is to think in non-linear terms.

Here goes:

The world's oceans are suddenly completely drained. I have a method to refill it. The good news is that I can double seawater production every day indefinitely. The bad news is that my starting volume is so low that it's going to take 100 years to fill. At what point is the ocean half full?"

Most people get this with a moment's thought, but many do not. They hedge, say they would need a spreadsheet, need to do some calcs. When I say, "It's doubling every day," they then realize how easy it is and get that it will be half full 99 years and 364 days after I start.

But here's where it starts to blow their minds.

How long does it take to get 1% full? Roughly speaking, that will take 99 years and 358.5 days. The other 99% will come in the last 6.5 days.

How long until I have my first complete, "working" water molecule? This surprised me. It turns out it will take me 99 years and 110 days to build my first complete water molecule. After this slow and arduous process, it will take me just one more day to get to three molecules and just 155 days to get the 4.7E+46 molecules I need to fill the oceans.

If I could start my process with a single molecule, I would have been done in five months, not 100 years.

What if I do have that first molecule and I can increase by 10% per day. Set aside that what it means to add 10% of a molecule in the first day. So instead of doubling, I only have a 10% increase, but I start way ahead of the game with a whole molecule.

It turns out that with that tremendous head start, if I have one molecule and add 10% per day to the ocean, I fill it in just under 1128 days or just a bit more than three years and a month.

So in other words, if my rate is only one tenth of what it was in the first scenario (100% growth each day), it takes me only an extra 2.5 years to fill the oceans once I have my first molecule.

I'm still trying to come up with the best presentation, but it seems to help people understand the effects of large exponents and small base numbers vs larger base numbers and small exponents.

For the first choice, the person earns a grand total of $122,500

For the doubling scheme, the person earns a total of $5,629,499,534,213.11.

A large lake has water lilies growing on it. On the first day, there is one water lily. Each day, the number of water lilies doubles. After 30 days, the water lilies cover half the lake. How long before they also cover the other half of the lake, so the whole lake is full?