Napkin folding problem
Encyclopedia
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 Robert J. Lang
, Svetlana Krat, Alexey S. Tarasov, and Ivan Yaschenko. One form of the problem remains open.
, giving different interpretations. By convention, the napkin is always a unit square
.
In this case it can be shown that the perimeter is always non-increasing under such foldings, thus never exceeding 4.
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. That is whether a solution 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
.
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.
However his constructions are not rigid origami
because of their use of sink folds and related forms; although no stretching is needed in sink and unsink folds, it is necessary to curve facets and/or sweep one or more creases continuously through the paper in intermediate steps before obtaining a flat result.
In 1998, I. Yaschenko constructed a 3D folding with projection onto a plane which has a bigger perimeter. This indicated to mathematicians that there was probably a flat folded solution to the problem.
The same conclusion was made by Svetlana Krat.
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.
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 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. 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 the total length goes up approximately as N.
's newsgroup
posting in 1996. Despite attention, it received folklore
status and its origin is often referred as "unknown".
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
explores whether folding a square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
or a rectangular
Rectangle
In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...
napkin
Napkin
A napkin, or face towel is a rectangle of cloth used at the table for wiping the mouth while eating. It is usually small and folded...
can increase its perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...
. The problem is known under several names, including the Margulis napkin problem, suggesting it is due to Grigory Margulis
Grigory Margulis
Gregori Aleksandrovich Margulis is a Russian mathematician known for his far-reaching work on lattices in Lie groups, and the introduction of methods from ergodic theory into diophantine approximation. He was awarded a Fields Medal in 1978 and a Wolf Prize in Mathematics in 2005, becoming the...
, and the Arnold's rouble problem referring to Vladimir Arnold
Vladimir Arnold
Vladimir Igorevich Arnold was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable Hamiltonian systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory,...
and the folding of a Russian ruble
Russian ruble
The ruble or rouble is the currency of the Russian Federation and the two partially recognized republics of Abkhazia and South Ossetia. Formerly, the ruble was also the currency of the Russian Empire and the Soviet Union prior to their breakups. Belarus and Transnistria also use currencies with...
. Some versions of the problem were solved by Robert J. Lang
Robert J. Lang
Dr. Robert J. Lang is an American physicist who is also one of the foremost origami artists and theorists in the world. He is known for his complex and elegant designs, most notably of insects and animals. He has long been a student of the mathematics of origami and of using computers to study the...
, Svetlana Krat, Alexey S. Tarasov, and Ivan Yaschenko. One form of the problem remains open.
Formulations
There are several way to define the notion of foldingPaper folding
Paper craft is the collection of art forms employing paper or card as the primary artistic medium for the creation of three-dimensional objects. It is the most widely used material in arts and crafts. It lends itself to a wide range of techniques, it can for instance be folded, cut, glued,...
, giving different interpretations. By convention, the napkin is always a unit square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
.
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.
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. That is whether a solution 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
Pureland origami
Pureland origami is a style of origami invented by the British paperfolder John Smith which is limited to using only mountain and valley folds. The aim of Pureland Origami is to make origami easier for inexperienced folders and those who have impaired motor skills...
.
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
Robert J. Lang
Dr. Robert J. Lang is an American physicist who is also one of the foremost origami artists and theorists in the world. He is known for his complex and elegant designs, most notably of insects and animals. He has long been a student of the mathematics of origami and of using computers to study the...
showed in 1997 that several classical origami
Origami
is the traditional Japanese art of paper folding, which started in the 17th century AD at the latest and was popularized outside Japan in the mid-1900s. It has since then evolved into a modern art form...
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.
However his constructions are not rigid origami
Rigid origami
Rigid origami is a branch of origami which is concerned with folding structures using flat rigid sheets joined by hinges. It is a part of the study of the mathematics of paper folding, it can be considered as a type of mechanical linkage, and has great practical utility. For instance it was used in...
because of their use of sink folds and related forms; although no stretching is needed in sink and unsink folds, it is necessary to curve facets and/or sweep one or more creases continuously through the paper in intermediate steps before obtaining a flat result.
In 1998, I. Yaschenko constructed a 3D folding with projection onto a plane which has a bigger perimeter. This indicated to mathematicians that there was probably a flat folded solution to the problem.
The same conclusion was made by Svetlana Krat.
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.Lang's solutions
Lang devised two different solutions. Both involved sinkingOrigami techniques
The Yoshizawa-Randlett system is a diagramming system used to describe the folds of origami models. Many origami books begin with a description of basic origami techniques which are used to construct the models. There are also a number of standard bases which are commonly used as a first step in...
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
Sea urchin
Sea urchins or urchins are small, spiny, globular animals which, with their close kin, such as sand dollars, constitute the class Echinoidea of the echinoderm phylum. They inhabit all oceans. Their shell, or "test", is round and spiny, typically from across. Common colors include black and dull...
' 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. 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 the total length goes up approximately as N.
History
Arnold states in his book that he formulated the problem in 1956, but the formulation was left intentionally vague. 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 ProppJim Propp
James Gary "Jim" Propp is a professor of mathematics at the University of Massachusetts Lowell.In high school, Propp was one of the national winners of the United States of America Mathematical Olympiad , and an alumnus of the Hampshire College Summer Studies in Mathematics. Propp obtained his A.B....
's newsgroup
Newsgroup
A usenet newsgroup is a repository usually within the Usenet system, for messages posted from many users in different locations. The term may be confusing to some, because it is usually a discussion group. Newsgroups are technically distinct from, but functionally similar to, discussion forums on...
posting in 1996. Despite attention, it received folklore
Folklore
Folklore consists of legends, music, oral history, proverbs, jokes, popular beliefs, fairy tales and customs that are the traditions of a culture, subculture, or group. It is also the set of practices through which those expressive genres are shared. The study of folklore is sometimes called...
status and its origin is often referred as "unknown".
External links
- Erik DemaineErik DemaineErik D. Demaine , is a professor of Computer Science at the Massachusetts Institute of Technology.-Early life:...
and Joseph O'RourkeJoseph O'Rourke (professor)Joseph O'Rourke is the Olin Professor of Computer Science at Smith College and the chair of the Smith computer science department. His main research interest is computational geometry....
, Geometric Folding Algorithms: Linkages, Origami, Polyhedra - Igor PakIgor PakIgor Pak is a professor of mathematics at the University of California, Los Angeles, working in combinatorics and discrete probability...
, Lectures on Discrete and Polyhedral Geometry, Section 40.