# Folding sheets of paper to measure distances

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.

• There are many lengths. What kind of description do you expect? – Hagen von Eitzen Aug 31 '14 at 21:53
• @HagenvonEitzen The set of all obtainable lengths for each $n$ – Hippalectryon Aug 31 '14 at 21:54
• What do you mean "you cannot use the sheet as a compass"? – achille hui Sep 1 '14 at 13:01
• @achillehui One could take one edge to be the center and draw a circle around it with a point on the borders of the sheet – Hippalectryon Sep 1 '14 at 14:04
• I see, you mean disallow explicit use as a compass. However, if you allow the seven operations specified in Huzita-Hatori axioms for paper folding, it is still possible to get all number constructible by compass and straightedge using pure paper folding. What sort of limitation are you going to impose upon your allowed set of foldings? – achille hui Sep 1 '14 at 14:16