2.1 Feeling the Force of Powers
A wager that cannot be won
Grab a sheet of foil or paper, the largest you can lay hands on. Crease it in half. Crease it again. And again. How far can you push it?
Have a go. Almost everyone jams up at around 7 creases, however wide or wafer-thin the sheet. Naveen swears no sheet can survive more than 7 folds. Priya suspects something flimsier, a serviette, a page of newsprint, might sneak in a fold or two more.
Here is the jolt, though. Imagine you could keep folding forever, with no jamming at all. How tall would the pile climb? Begin with a sheet 0.001 cm thick and track what each fold does.
Each fold doubles the thickness. So after 1 fold it stands at 0.002 cm, after 2 folds 0.004 cm, after 3 folds 0.008 cm, and onward:
| Fold | Thickness | Fold | Thickness | Fold | Thickness |
|---|---|---|---|---|---|
| 1 | 0.002 cm | 7 | 0.128 cm | 13 | 8.192 cm |
| 2 | 0.004 cm | 8 | 0.256 cm | 14 | 16.384 cm |
| 3 | 0.008 cm | 9 | 0.512 cm | 15 | 32.768 cm |
| 4 | 0.016 cm | 10 | 1.024 cm | 16 | 65.536 cm |
| 5 | 0.032 cm | 11 | 2.048 cm | 17 | \approx 131 cm |
| 6 | 0.064 cm | 12 | 4.096 cm |
(The symbol \approx is read “approximately equal to.”) By the 10th fold the pile barely tops 1 cm; by the 17th it reaches roughly 131 cm, just over four feet. Push on, and the figures sprint away from you:
- By fold 30 the height is about 10.7 km, cruising altitude for a jet, and not far short of the Mariana Trench’s depth.
- By fold 26, about 670 m, overtaking the Burj Khalifa (828 m tall, almost).
- By fold 46, over 7{,}00{,}000 km, enough to vault past the Moon!
That runaway behaviour is multiplicative growth, better known as exponential growth.
Math Talk Extend the doubling table yourself for folds 18 through 45. By fold 20 the height is roughly 10.5 m, by fold 27 about 1.3 km. Spot the neat rhythm: every 10 folds scales the height by about 1024. Test it, from fold 0 to fold 10 the height climbs from 0.001 cm to 1.024 cm, a jump of 1024 = 2 \times 2 \times \dots \times 2 (2 multiplied by itself ten times). The same 1024 resurfaces from fold 10 to 20, and from 20 to 30. Why precisely 1024?
Giving the pattern a name
The thickness after each fold is just 0.001 cm multiplied by 2, again and again. Scribbling all those 2s wears thin fast, so we lean on a shorthand:
0.001 \times 2 \times 2 = 0.001 \times 2^2, \qquad 0.001 \times \underbrace{2 \times 2 \times 2}_{3 \text{ times}} = 0.001 \times 2^3.
After 7 folds the thickness equals 0.001 \times 2^7 = 0.128 cm. The tiny lifted number records how often the 2 is multiplied by itself.