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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!

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

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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.

$$ \pi ( b^2-a^2) = h \cdot l $$

and so the number of layers obtained by such repetitive windings is

$$ n= \frac{b-a}{h}= \frac{l}{\pi(b+a)}.$$

which is nothing but multiples of average diameter comprising the total length $l $.

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