Napkin folding problem
The napkin folding problem in geometry 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. Some versions of the problem were solved by Svetlana Krat, Robert J. Lang, Alexey S. Tarasov and Ivan Yaschenko. One form of the problem remains open.
History of the problem
Arnold states in his book that he formulated the problem in 1956, but the formulation was left intentionally vague.[1][2] He called it 'the rumpled rouble problem', and it was the first of many interesting problems he set at seminars in Moscow over 40 years. In the West, it became known as Margulis napkin problem after Jim Propp's newsgroup posting in 1996.[3] Despite attention, it received folklore status and its origin is often referred as "unknown".[4]
Formulations of the problem
There are several way to define the notion of folding, giving different interpretations. By convention, the napkin is always a unit square.
Folding along a straight line
One can consider sequential folding of all layers along a line. In this case it can be shown that the perimeter is always non-increasing under such foldings, thus never exceeding 4.[1][3]
It is still unknown if there is a solution using a sequence of foldings, such that each is a reflection of a connected component of folded napkin on one side of a straight line.[5] That is whether a solution can be got using pureland origami.
Where only the result matters
One can ask whether there exist a folded planar napkin (without regard as to how it was folded into that shape).
In 1998, I. Yaschenko constructed a 3D folding with projection onto a plane which has a bigger perimeter[4]. This indicated to mathematicians that there was probably a flat folded solution to the problem.
Robert J. Lang showed in 2003 that several classical origami constructions give an easy solution[6]. In fact, Lang showed that the perimeter can be made as large as desired by making the construction more complicated. However his constructions are not rigid origami, the paper needs to be able to stretch slightly in intermediate steps.
The same conclusion was made by Svetlana Krat[7]. Her approach is different, she gives very simple construction of a "rumpling" which increase perimeter and then proves that any "rumpling" can be arbitrary well approximated by a "folding". 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.
Folding without stretching
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 that such constructions can indeed be obtained. This can be considered a complete solution to the original problem.[8]
Solutions
Lang's solutions
Lang devised two different solution.[6] Both involved sinking flaps and so were not rigid origami. The simplest was based on the origami bird base and gave a solution with a perimeter of about 4.12 compared to the original perimeter of 4.
The second solution can be used to make a figure with a perimeter as large as desired. He divides the square into a large number of smaller squares and employs the 'sea urchin' type origami construction known for years in the origami literature. The crease pattern shown is the n = 5 case and can be used to produce a figure with 25 flaps, one for each of the large circles, and sinking is used to thin them. When very thin the 25 arms will give a 25 pointed star with a small center and a perimeter approaching N2/(N-1), in the case of N = 5 this is about 6.25 and it goes up approximately as N.
See also
References
- ^ a b Arnold, Vladimir Igorevich (2005). Arnold's Problems. Berlin: Springer. ISBN 3-540-20748-1.
- ^ Tabachnikov, Sergei (2007). "Book review of "Arnold's problems"" (PDF). Math. Intelligencer. 29 (1): 49–52. doi:10.1007/BF02984760.
- ^ a b "The Margulis Napkin Problem, newsgroup discussion of 1996". Geometry Junkyard.
- ^ a b Yaschenko, I. (1998). "Make your dollar bigger now!". Math. Intelligencer. 20 (2): 38–40.
- ^ Petrunin, Anton (2008). "Arnold's problem on paper folding". Zadachi Sankt-peterburgskoj matematicheskoj olimpiady shkol'nikov po matematike (in Russian). arXiv:1004.0545.
- ^ a b Lang, Robert J. (2003). Origami Design Secrets: Mathematical Methods for an Ancient Art. A K Peters. pp. 315–319.
- ^ S. Krat, Approximation Problems in Length Geometry,Ph.D. thesis, Pennsylvania State University, 2005
- ^ Tarasov, A. S. (2004). "Solution of Arnold's "folded ruble" problem". Chebyshevskii Sbornik (in Russian). 5 (1): 174–187.