Napkin folding problem
In geometry, napkin folding problem is a problem asking 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 folding of a Russian ruble. Similarly, there is a great deal of confusion as to whether the problem is completely resolved. Versions of the problem were solved by Robert J. Lang, I. Yaschenko and A.S. Tarasov.
History of the problem
Arnold states in his book that he formulated the problem in 1956, but the formulation was left intentionally vague (see Arnold and Tabachnikov). In the West, it became known as Margulis napkin problem after Jim Propp's newsgroup posting in 1996. Despite attention, it received folklore status and its origin is often referred as "unknown" (see Yaschenko).
Formulations of the problem
There are several way to define the notion of folding, giving different interpretations. We always assume that the napkin is a unit square. First, one can consider sequential folding along a line. In this case it can be shown that the perimeter is always non-increasing under such foldings, thus never exceeding 4 (see Arnold). Second, one can ask whether there exist a folded planar napkin (without regard as to how it was folded into that shape). In this case, Robert J. Lang showed that several classical origami constructions give an easy solution (see Lang). In fact, the perimeter can be made as large as desired. A similar construction was independently found by Yaschenko. Both solutions were criticized by Tarasov who claimed that neither Lang nor Yaschenko check whether a napkin can be actually folded without stretching into these shapes. He showed in 2004 that a related construction can indeed be obtained by a continuous piecewise linear folding map (see Tarasov). There are several other variations of the problem, where the answer remains unknown (see Pak).
See also
References
- I. Yaschenko, "Make your dollar bigger now!", Math. Intelligencer, Vol. 20, No. 2 (1998), 36-40.
- Robert J. Lang, Origami Design Secrets: Mathematical Methods for an Ancient Art, A K Peters, 2003.
- A.S. Tarasov, Solution of Arnold's "folded ruble" problem (in Russian), Chebysh. Sbornik, Vol. 5, No. 1 (2004), 174-187.
- Vladimir Arnold, Arnold's Problems, Springer, 2005.
- Sergei Tabachnikov, Book review of "Arnold's problems", Math. Intelligencer, Vol. 29, No. 1 (2007), 49-52.
- Igor Pak, Lectures on Discrete and Polyhedral Geometry, Section 40.
External links
- The Margulis Napkin Problem, newsgroup discussion of 1996; from the Geometry Junkyard.