Napkin folding problem

[1] Some versions of the problem were solved by Robert J. Lang, Svetlana Krat, Alexey S. Tarasov, and Ivan Yaschenko.

Robert J. Lang showed in 1997[2] that several classical origami constructions give rise to an easy solution.

[6][7] In fact, Lang showed that the perimeter can be made as large as desired by making the construction more complicated, while still resulting in a flat folded solution.

Although no stretching is needed in sink and unsink folds, it is often (though not always) necessary to curve facets and/or sweep one or more creases continuously through the paper in intermediate steps before obtaining a flat result.

Whether a general rigidly foldable solution exists based on sink folds is an open problem.

In essence she shows that the precise details of the how to do the folds don't matter much if stretching is allowed in intermediate steps.

[9] The crease pattern shown is the n = 5 case and can be used to produce a flat figure with 25 flaps, one for each of the large circles, and sinking is used to thin them.