Hello, fractal!

There is a wonderful back-door approach to this problem . . .

A string is wound symmetrically around a circular rod.

The string goes exactly 4 times around the rod.

The circumference of the rod is 4 cm and its length is 12 cm.

Find the length of the string.

"Unroll" the cylinderical rod, and we have a 4-by-12 rectangle.

Draw four of them in a row. Code:

* - - - - - * - - - - - * - - - - - * - - - - - *
| | | | * |
| | | * |
| | | * | |
12 | | * | | 12
| | * | | |
| * | | |
| * | | | |
* - - - - - * - - - - - * - - - - - * - - - - - *
4 4 4 4

Draw the diagonal from one corner to the opposite corner.

This is the path of string as it spirals around the cylinder four times.

Using Pythagorus, we can find the length of that hypotenuse.