Napkin folding 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. It is the first problem listed by Arnold in his book Arnold's Problems, where he calls it the rumpled dollar problem.

Arnold's formulation of the problem was ambiguous, perhaps intentionally, and its solution depends on the types of folds that are allowed. For one restriction of the problem to simple folds made one at a time, the answer is no: no perimeter-increasing fold is possible. For other related restrictions, the answer to the question remains unknown. However, for flat-folded states of an arbitrary crease pattern, without consideration of how the state is reached, or for folding using the restrictions of rigid origami, it is indeed possible to increase the perimeter.

Formulations

There are several ways to define the notion of folding, giving different interpretations. By convention, the napkin is always a unit square, with perimeter 4.

Folding along a straight line

One version of the napkin folding problem allows only a sequence of simple folds, in which all of the layers of the napkin on one side of a line are reflected across the line. For these folds, each step decreases the perimeter. The newly added perimeter along one or more segments of the fold line is more than balanced by a decrease in perimeter along the boundary of the region of the plane covered by both parts of the folded shape, which connects the endpoints of the same line segments. Thus, for folds of this type, the perimeter never exceeds its starting value, 4.

A more general class of foldings allows the reflection across a line of only a single layer of the napkin (a connected component of folded napkin on one side of a straight line), leaving the other components in place. It is still open whether a sequence of these foldings can increase the perimeter. In other words, it is still unknown whether there 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 is whether a perimeter-increasing fold would be possible using the more-restrictive pureland origami, in which a sequence of mountain and valley folds are made one crease at a time. The argument showing that simple folds decrease the perimeter does not apply to this case: although the first fold decreases the perimeter, subsequent folds can increase it again.

Where only the result matters

One can ask whether there exists a folded planar napkin (without regard as to how it was folded into that shape).

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

In 1998, I. Yaschenko constructed a solution that solves a different variant of the problem. Yaschenko's solution is a three-dimensional folding, whose projection onto a plane produces a polygon with a bigger perimeter than the original square.

Svetlana Krat constructs a short map that increases the perimeter; this is a mapping from the original origami square to the plane that is allowed to stretch or distort the paper by decreasing distances, but never by increasing distances. Then, she proves that any short map can be arbitrarily well approximated by folding. The same construction works for an arbitrary rectangle.

Folding without stretching

Rigid origami provides a more constrained version of the problem than the pure existence of a flat-folded state. In rigid origami, what is required is a continuous folding motion throughout which the uncreased panels of origami paper remain flat, without bending or stretching. In 2004, A. Tarasov showed that it is indeed possible to reach a flat-folded state with greater perimeter than the original square, through a sequence of rigid origami motions. This can be considered a complete solution to the original problem.

Solutions

Bird base

Lang devised two different solutions. Both involved sinking flaps and so were not necessarily rigidly foldable. 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.

Sea urchin

Crease pattern for Lang's sea urchin-like solution with n = 5

Lang's 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 described in his 1990 book Origami Sea Life. 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. The general construction produces n2 arms that 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 more generally the total length goes up in approximate proportion to n.

Waterbomb short map

The short map that Krat constructs is very simple: for a square, it uses a waterbomb base to bring the perimeter of the square into an X-shaped tree, with the same perimeter, and maps the interior parts of the square near its corners onto this tree. This leaves a central region within which there is enough material to form a fifth branch of the tree, emanating from its central point, which brings the total perimeter up to 3+2≈4.414.

History

Arnold states in his book Arnold's Problems that he posed the problem in 1956. He called it the "rumpled ruble problem" (or, in the English edition of the book, the "rumpled dollar 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 the 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".

References

  1. 1 2 Arnold, Vladimir I. (2004). "1956-1. The rumpled dollar problem". In Arnold, Vladimir I. (ed.). Arnold's Problems. Berlin: Springer-Verlag. p. 2. doi:10.1007/b138219. ISBN 3-540-20614-0. MR 2078115. See also comments, p. 182.
  2. Pak, Igor (April 20, 2010). Lectures on Discrete and Polyhedral Geometry.
    1. ↑ 40.6 Final remarks, page 358
    2. ↑ Exercise 40.16(b), page 357
    3. ↑ 40.4 Folding a napkin, pages 354–355
  3. 1 2 3 "The Margulis Napkin Problem, newsgroup discussion of 1996". Geometry Junkyard.
  4. 1 2 3 4 5 Yaschenko, I. (1998). "Make your dollar bigger now!!!". The Mathematical Intelligencer. 20 (2): 38–40. doi:10.1007/BF03025296. S2CID 124667472.
  5. ↑ Petrunin, Anton (2008). "Arnold's problem on paper folding". Zadachi Sankt-peterburgskoj Matematicheskoj Olimpiady Shkol'nikov Po Matematike (in Russian). arXiv:1004.0545. Bibcode:2010arXiv1004.0545P.
  6. 1 2 Lang, Robert J. (2003). "9.11. The Margulis Napkin Problem". Origami Design Secrets: Mathematical Methods for an Ancient Art. A K Peters. pp. 315–319. ISBN 9781568811949.
    1. ↑ Page 315 and Figure 9.42, page 316
    2. ↑ Page 318
    3. ↑ Table, page 319
    4. ↑ Page 319
  7. 1 2 Demaine, Erik D.; O'Rourke, Joseph (2007). Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press. p. 239.
  8. 1 2 Krat, Svetlana (2005). "4.3 Approximating Short Maps by PL-isometries and Arnold's "Can You Make Your Dollar Bigger" Problem". Approximation Problems in Length Geometry (Ph.D. thesis). Pennsylvania State University. pp. 94–118. ProQuest 305145549. See especially Section 4.3.4, "A Short Map that Increases the Perimeter of a Rectangle", pp. 117–118.
  9. ↑ Tarasov, A. S. (2004). "Solution of Arnold's "folded ruble" problem". Chebyshevskii Sbornik (in Russian). 5 (1): 174–187. Archived from the original on 2007-08-25.
  10. 1 2 Montroll, John and Robert J. Lang (1990). Origami Sea Life. Dover Publications. pp. 195–201.
  11. ↑ Tabachnikov, Sergei (2007). "Book review of Arnold's Problems" (PDF). The Mathematical Intelligencer. 29 (1): 49–52. doi:10.1007/BF02984760. S2CID 120833539. Archived from the original (PDF) on 2017-07-06.