Arnold's rouble problem

The napkin folding problem is a problem in geometry and the mathematics of paper folding that explores whether folding a square or a rectangular napkin can increase its perimeter. The problem is known under several names, including the Margulis napkin problem, suggesting it is due to Grigory Margulis, and the Arnold's rouble problem referring to Vladimir Arnold and the folding of a Russian ruble bank note. Some versions of the problem were solved by Robert J. Lang, Svetlana Krat, Alexey S. Tarasov, and Ivan Yaschenko. One form of the problem remains open.

There are several way to define the notion of folding, giving different interpretations. By convention, the napkin is always a unit square.

Considering the folding as a reflection along a line that reflects all the layers of the napkin, the perimeter is always non-increasing, thus never exceeding 4.

By considering more general foldings that possibly reflects only a single layer of the napkin (in this case, each folding is a reflection of a connected component of folded napkin on one side of a straight line), it still open if a sequence of these foldings can increase the perimeter. In other words, it still unknown if exists a solution that can be folded using some combination of mountain folds, valley folds, reverse folds, and/or sink folds (with all folds in the latter two cases being formed along a single line). Also unknown, of course, is whether such a fold would be possible using the more-restrictive pureland origami.

One can ask for a realizable construction within the constraints of rigid origami where the napkin is never stretched whilst being folded. In 2004 A. Tarasov showed that such constructions can indeed be obtained. This can be considered a complete solution to the original problem.

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