You have $n\in\mathbb{N}^*$ sheets of paper with dimensions $a,b\in\mathbb{R}_+^*$ that can be folded as many times as needed.
What is the set of lengths in $\left]0,\sqrt{a^2+b^2}\right]$ one can measure with those $n$ sheets ?
We can assume that a number is measurable iff one side has it as his length.
Here, given two points, we'll define a fold by the action of reuniting those two edges (e.g putting two opposite corners together, that would makes a diagonal fold) and we'll also consider folding along an axis defined by two points
Is defined as a point what is, or has been in previous folds, an edge
Same question with $n\in\mathbb{N}^*$ sheets of papers with dimensions $a_n,b_n\in\mathbb{R}_+^*$ that can be folded as many times as needed.
You can not use the sheet as a compass, only folding is allowed.
Examples (for the first case where $a,b$ are constant) :
n=1 : You can easily do $\frac{a}{2^N},N\in\mathbb{N}$ (folding on the side with length $a$),$\frac{\sqrt{a^2+b^2}}{2^N},N\in\mathbb{N}$ (folding on the diagonal), etc
n=2 : With a second sheet of paper, we can now 'store' a value, which allows us for instance (by putting the two sheets one after another) to easily have values as $\frac{3a}{4}$ etc.
That problem came to me when I was toying around with a small towel in a store. Its dimensions were written on its label. I began wondering all the lengths I would then measure by just folding the towel on itself, and what I would be able to do with more than one towel.