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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.

Hippalectryon
  • 7,730
  • 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 01 '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 01 '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 01 '14 at 14:16
  • @achillehui I'm not sure that those axioms can be applied here. For instance, axiom 1 states that for any points $p_1,p_2$ there exists a unique fold that passes through both, and gives a parametric equation for the line. However, I hardly see how by just folding your sheet (you have no other geometric tool available) you will be able to create that fold for random points. Or am I missing something ? -- (Looking at the text I added at the bottom of the question might help you understand how the foldings are made; all the folds should be doable with mere (perfect) hands) – Hippalectryon Sep 01 '14 at 14:54
  • Well, I think you already used the axiom 1 when folding on the diagonal (by taking left upper and right lower corner as the 2 points). – cjackal Dec 22 '14 at 14:17
  • @cjackal I do not need axiom 1 to fold on the diagonal, though. – Hippalectryon Dec 22 '14 at 14:21
  • You must clarify the axioms you can use to fold, like "can fold along diagonals" or "can fold in half". Without some preassumed axioms, you cannot fold anything rigorously, or in other words, mathematically. – cjackal Dec 22 '14 at 14:31
  • @cjackal Is it clear with the edit ? – Hippalectryon Dec 22 '14 at 14:39
  • I have some problem understanding your definition of a fold. If a $\neq$ b, we cannot get diagonal fold by overlapping two opposite points. – cjackal Dec 22 '14 at 14:44
  • @cjackal Sorry for that, I realized that mistake and allowed folds along an axis too. – Hippalectryon Dec 22 '14 at 14:45
  • Because this setting do not satisfy the axiom 2 of Huzita-Hatori, there may be quite different set of measurable lengths, I think. In this setting, folding corresponds to some isometries on $\mathbb{R}^2$, and in my opinion, the study of the group generated by initial isometries are essential. – cjackal Dec 22 '14 at 14:58

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