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## October 04, 2006

Exponential Growth Cannot Last | Environment Essays Science/Technology |

*[Another repost of a piece from a couple of years ago, once again dealing with the meaning of exponential growth. I hope you'll make it to the punchline at the end.]*

I've written about exponential growth before, but the concept is so essential to understanding the future that awaits us that I want to revisit it.

To say something grows exponentially is to say it grows at a constant percentage rate — for example, 3% per year. Anything that grows in this way *doubles* at a constant rate. You can estimate how long it takes to double by dividing the percentage growth rate into 72. So, for example, something that grows at a rate of 3% per year doubles every 24 years (72/3 = 24).

So, **you can think of exponential growth as growth by doubling** at a constant rate.

Doubling is an extraordinarily powerful process. Some examples (from M. King Hubbert):

1. If you start with a single pair, say Adam and Eve, in just 32 doublings you’d have a population greater than the total population of Earth today. Just 14 doublings later you’d haveone person per square yard over the entire land surface of the planet.2. If someone gives you a single grain of wheat for the first square of a chessboard, 2 for the second, 4 for the third, doubling at each square, by the time you finish the 64 squares of the chessboard you’d have

more than a thousand times the total annual wheat production of Earth.3. If you play the chessboard game with automobiles instead of wheat, by the time you finish the 64 squares you’d have so many automobiles that if you stacked them uniformly

over the entire land surface of the earth, you’d have a layer 1,200 miles deep. (Think of)thatthe next time some economist says world GNP can grow at 3% per yearforever.

**What these examples show is that doubling (or exponential growth) is such a powerful process, that it takes only tens of “generations” of doubling — not hundreds, or thousands, or millions — to completely exhaust the physical environment of the planet. Put another way, in the physical world (as opposed to an idealized mathematical world) exponential growth cannot last for long.**

When any living species is placed in a favorable environment — meaning an environment that doesn’t limit growth because of the lack of some necessity (e.g. food), the presence of a predator, or for some other reason — its population grows exponentially. In Nature, over the long run, limitations in their environments prevent species from multiplying exponentially. Otherwise, the world would long ago have been engulfed.

Why is all this important?

Early in a doubling sequence, the numbers grow slowly. Likewise, until recently in human history, human population grew slowly. Use of energy and material resources by humans also grew slowly, and the resources used were entirely of the renewable variety, except for tiny amounts of coal and metals. Everything else (food, energy, shelter, clothing, etc.) came from animals and plants (renewable), plus a small amount of energy from wind and water (renewable). If humans had continued to rely on renewable resources, that fact would have put a ceiling on population size.

**Starting about two centuries ago, however, a revolution occurred in human life: people starting using non-renewable resources — hydrocarbon fuels and a variety of minerals — in a big way. This use of non-renewables removed the constraints on human population and activity, and exponential growth really kicked in. Not only has population grown exponentially, but human use of coal, oil, gas, iron, copper, tin, lead, zinc, etc. have grown exponentially as well, as has human damage to the environment. It’s the use of non-renewables — hydrocarbon fuels, especially — that has made this growth possible.**

But, inevitably, we’re going to hit the wall, and sooner than we think. Even if we had infinite resources to draw on, exponential growth would soon fill up a finite environment, as we've seen. But that hardly matters, since we *do not* have infinite resources to draw on. **Non-renewables are a one-time gift to humanity. They are finite. We’re burning through them at an exponential pace, and when they’re gone they’re gone forever.**

Now, one of the really startling characteristics of growth by doubling is the following fact: if you consider the sequence of doubled numbers — 1, 2, 4, 8, 16, 32, etc. — *each number in the sequence is greater (by one) than the sum of all the numbers that precede it*.

Why do I call this startling? Consider oil. World oil consumption is now growing at a rate that will double it every 15-20 years. This means, **as long as exponential growth continues, in the next 15-20 years the world will consume more oil than was used in all of human history up to this point**. More than in the entire 19th and 20th centuries combined — *in just 15-20 years* — assuming exponential growth continues. I don't know about you, but I find that startling.

I want to finish with a riddle I posed in the earlier post on exponential growth. I repeat it here because **I’d really like this riddle to stay with you. If it does, you’ll understand exponential growth better than 99.9% of your fellow citizens.**

Suppose you put a small amount of bacteria in a Petri dish. Suppose further that the bacteria population grows exponentially (i.e., by doubling) at a pace that causes it to double each hour. Suppose finally that it takes 100 hours for the bacteria to completely fill the dish, thereby exhausting their supply of nutrients. (It's a *large* Petri dish.)

Question: When is the dish half full?

After 50 hours (half of 100)?

No. Because the population doubles each hour (including the final hour), the dish is half full *just one hour before it’s full*. For the first 99 hours the bacteria have got it made. Then *wham!*

**To make this more vivid and memorable, imagine the following as an animated cartoon. For the first 99 hours the bacteria are just partying and congratulating themselves on how smart and successful they are. It’s party hats and noisemakers, Conga lines and champagne, the bacterial Dow Jones going through the roof. Woo hoo! No limits! After 99 hours, some of the bacteria start to worry, but the rest party on — after all, the dish is only half full. Plenty of room left, plenty of nutrients. The first half lasted 99 hours, and there's another whole half to go! Sure, somebody’s gonna have to figure something out eventually, but meanwhile life is good, and nonstop growth will only make it better!**

**An hour later —**

*the world ends*.**When growth is exponential, limits are sudden.**

Posted by Jonathan at October 4, 2006 04:46 PM

## Comments

Divide into 72? I think it's 69 (ln 2).

Posted by: revere at October 8, 2006 10:12 PM

Rule of 72: http://www.ny.frb.org/education/calc.html

Posted by: Jonathan at October 9, 2006 09:05 AM