# paper spiral as dense as possible, given thickness and length

Given a piece of paper of thickness or height $h$ and length $l$, how often can you roll the piece around itself, forming a tight roll in the length direction?

I would use the assumption that the first revolution is of radius $a$ and that there is no gaps between the paper when rolled tightly. But I’m not sure what to do with that and if there is a more elegant assumptions to make. Also I don’t really know how spirals work mathematically.

I used to do these rolls as a kid from leafs and now I do them with bottle etiquettes. I would love to see some of your thoughts on this completely arbitrary math problem!

• Area $= \pi R^2 = l\times h, R = n\times h$ where $n$ is the number of wraps. $n = \sqrt {\frac {l}{h\pi}}$ Dec 4, 2017 at 17:36
• You could see this question Dec 5, 2017 at 17:17

When started on a circular former of given start radius $a$ and if the maximum radius reached is $b$, it can be estimated to a good approximation as follows by equating cylinder end/edge surface area of each of the tightened cylinders to the lateral area.
$$\pi ( b^2-a^2) = h \cdot l$$
$$n= \frac{b-a}{h}= \frac{l}{\pi(b+a)}.$$
which is nothing but multiples of average diameter comprising the total length $l$.