How many folding steps do you need to create a $1 \, \text{m}$ measure from an ideal DIN A0 sheet? This sounds easy, but consider that this paper has edge lengths of $\sqrt[4]{2} \, \text{m}$ and $\frac{1}{\sqrt[4]{2}} \, \text{m}$ respectively and I want a (theoretically) exact $1 \, \text{m}$ measure. That's the whole question – but to make sure everybody understands how the construction should be done, I will show the allowed moves in an example:
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In general all moves that don't leave a degree of freedom are allowed. More specifically:
- Folding one corner onto another, for example:
(Unfolding doesn't count as additional step.) - Folding one edge onto another (or itself), for example:
- A generalization of the first listed move: Folding a known point onto another known point, for example:
- Folding around an edge inside the figure, for example:
- Folding a corner onto an edge. To make sure this has no degree of freedom left, the kink can be fixed by going through a known point, for example:
- Folding through two known points, for example:
If somebody comes up with another move that doesn't leave a degree of freedom it is of course also allowed.
In the end you should contruct two points which are exactyl $1 \, \text{m}$ apart. The answer with the least number of moves will be accepted. But to make sure people understand your answer please include a drawing or a photo of how the paper is folded. Also provide a calculation that proves that the edge is really $1 \, \text{m}$. If you prefer to fold a real paper while solving this task you don't need a DIN A0 paper, but instead you can use a DIN A4 paper ($\frac{\sqrt[4]{2}}{4} \, \text{m} \cdot \frac{1}{4 \sqrt[4]{2}} \, \text{m}$) and find a $25 \, \text{cm}$ measure.
Yes, I'm aware of the questions "Create a 3 inch measurement" and "Create a 1 inch measurement". But this question is not so straightforward, because you have to deal with irrational egde lengths to contruct an integer length.