The below describes the effects of exponential growth and is entirely lifted from something I read on the Internet. You may have a copy of the same article in paper form but the links might add to your understanding.
The principle is simple. You start with one. Double it and you have two. Double again and you have four. These numbers are small but in a very short time if you continue to do the same thing you will have some quite large numbers.
Exponential growth
Take a sheet of paper of the ordinary variety – letter size for the Americans, A4 for the rest of the world – and fold it into half. Fold it a second time, and a third time. It’s about as thick as your finger nail. Continue folding if you can. At 7 folds it is as thick as a notebook. If you would have been able to fold it 10 times, it would be as thick as the width of your hand. Unfortunately, it isn’t possible to do so more than about 12 times (worth a read). Try it for yourself. At seventeen folds it would be taller than your average house. Three more folds and that sheet of paper is a quarter way up the Sears tower. Ten more folds and it has crossed the outer limits of the atmosphere. Another twenty and it has reached the sun from the earth. At sixty folds it has the diameter of the solar system. At 100 folds it has the radius of the universe. “Preposterous!”, you exclaim. That is what I thought till I started calculating the thickness myself. If you do not want to pull out your trusty calculator here is a table that contains what I have described above.
| n | 2**n | km (0.1*10**-6 * 2**n) | Comment |
|---|---|---|---|
| 0 | 1 | 0.1 x 10**-6 | |
| 1 | 2 | 0.2 x 10**-6 | |
| 2 | 4 | 0.4 x 10**-6 | |
| 3 | 8 | 0.8 x 10**-6 | finger nail thickness |
| 4 | 16 | 1.6 x 10**-6 | |
| 5 | 32 | 3.2 x 10**-6 | |
| 6 | 64 | 6.4 x 10**-6 | |
| 7 | 128 | 12.8 x 10**-6 | thickness of a notebook |
| 8 | 256 | 25.6 x 10**-6 | |
| 9 | 512 | 51.2 x 10**-6 | |
| 10 | 1024 | 0.1 x 10**-3 | width of a hand (incl. thumb) |
| 11 | 2048 | 0.2 x 10**-3 | |
| 12 | 4096 | 0.4 x 10**-3 | 0.4m height of a stool |
| 13 | 8192 | 0.8 x 10**-3 | |
| 14 | 16384 | 1.6 x 10**-3 | 1.6m: an average person’s height (yeah, a short guy) |
| 15 | 32768 | 3.3 x 10**-3 | |
| 16 | 65536 | 6.6 x 10**-3 | |
| 17 | 131072 | 13.1 x 10**-3 | 13m height of a two story house |
| 18 | 262144 | 26.2 x 10**-3 | |
| 19 | 524288 | 52.4 x 10**-3 | |
| 20 | 1048576 | 104.9 x 10**-3 | quarter of the Sears tower (440m) |
| … | …. | …. | …. |
| 25 | 33554432 | 3.4 x 10**0 | past the Matterhorn |
| 30 | 1073741824 | 107.4 x 10**0 | outer limits of the atmosphere |
| 35 | 34359738368 | 3.4 x 10**3 | |
| 40 | 1099511627776 | 109.9 x 10**3 | |
| 45 | 35184372088832 | 3.5 x 10**6 | |
| 50 | 1125899906842624 | 112.5 x 10**6 | ~ distance to the sun (95 million miles) |
| 55 | 36028797018963968 | 3.6 x 10**9 | |
| 60 | 1152921504606846976 | 115.3 x 10**9 | size of the solar system? |
| 65 | 36893488147419103232 | 3.7 x 10**12 | one-third of a light year |
| 70 | 1180591620717411303424 | 118.1 x 10**12 | 11 light years |
| 75 | 37778931862957161709568 | 3.8 x 10**15 | 377 light years |
| 80 | 1208925819614629174706176 | 120.9 x 10**15 | 12,000 light years |
| 85 | 38685626227668133590597632 | 3.9 x 10**18 | 4x the diameter of our galaxy |
| 90 | 1237940039285380274899124224 | 123.8 x 10**18 | 12 million light years |
| 95 | 39614081257132168796771975168 | 4.0 x 10**21 | |
| 100 | 1267650600228229401496703205376 | 126.8 x 10**21 | (12 billion light years) approx. radius of the known universe? |
Note:
- A sheet of paper is about 0.1 mm thick. I use the common
80gm/m2 variety. - I have represented the exponentiation operator with
**. - The idea for this article and, indeed, the paper-folding
analogy came from an issue of the Economist. According to
Ozgur Ince it was in the 15 July 1995 issue and was titled
The End of the Line. I cannot link to that article as
the online archive at the Economist only goes back to
1997. - If anyone detects a factual mistake in the table, please
contact me with the correction. It is possible that I have
got some numbers wrong while typing this in.
This table should convince anyone about the rapidity of exponential growth. Yes, the example I have taken does double at every step; usual growth is just a few percentage points but the core idea is the same.
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As I said at the top, that is not my work but it does represent the rapidity of exponential growth. Whether the radius of the Universe is right, or indeed if it means anything to suggest that the Universe has a radius, is open for discussion.
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