Is "Napkin conjecture" open ? (ORIGAMI) - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T04:29:42Z http://mathoverflow.net/feeds/question/46866 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46866/is-napkin-conjecture-open-origami Is "Napkin conjecture" open ? (ORIGAMI) Jérôme JEAN-CHARLES 2010-11-21T21:40:44Z 2011-12-11T00:25:54Z <p>If false the following conjecture would be a nice counter intuitive fact. </p> <p>Given a square sheet of perimeter $P$ when folding it along Origami moves you end up with some polygonal flat figure with perimeter $P^'$ :<br> <em>Napkin conjecture</em> : You always have $P^' \leq P$.</p> <p>In other words you cannot increase the perimeter using any finite sequence of origami folds. </p> <p><strong>Q1</strong>: Intuition tells us it is true ( how on hell can it increase?). Yet I think I read somewhere that there was some weird folding (called "mountain urchin"??) that strictly increases the perimeter. Is this true?<br> <em>Note1</em> : I am not even sure that the initial sheet's squareness is required. </p> <p>I cannot find any reference on the net, maybe the name has changed, I heard about this 20 years ago. </p> <p>The second question is about generalizing the conjecture.</p> <p><strong>Q2</strong>: With the idea of generalizing the conjecture to continuous folds or bends ( using some average shadow as a perimeter) I stumble on how you can mathematically define bending a sheet, alternatively : how do you say "a sheet is untearable" in maths?<br> <strong>Note2</strong>: It might also be a matter of physics about how much we idealize bending mathematically. </p> http://mathoverflow.net/questions/46866/is-napkin-conjecture-open-origami/46867#46867 Answer by Andrey Rekalo for Is "Napkin conjecture" open ? (ORIGAMI) Andrey Rekalo 2010-11-21T21:56:27Z 2010-11-22T01:52:36Z <p>There is a general version of this question which is known as <em>"the rumpled dollar problem"</em>. It was posed by V.I. Arnold at his seminar in 1956. It appears as the very first problem in <a href="http://books.google.co.uk/books?id=LqvLnu8c3ToC&amp;printsec=frontcover&amp;dq=arnold%2527s+problems&amp;source=bl&amp;ots=l0ULwBNvsc&amp;sig=mLNHb1upSiddEtfrHRjbjvJPfzE&amp;hl=en&amp;ei=6ZPpTIvqLcuaOozXvZEK&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=1&amp;ved=0CBYQ6AEwAA#v=onepage&amp;q&amp;f=false" rel="nofollow"><em>"Arnold's Problems"</em></a>: </p> <blockquote> <p>Is it possible to increase the perimeter of a rectangle by a sequence of foldings and unfoldings? </p> </blockquote> <p>According to the same source (p. 182), </p> <blockquote> <p>Alexei Tarasov has shown that a rectangle admits a realizable folding with arbitrarily large perimeter. A realizable folding means that it could be realized in such a way as if the rectangle were made of infinitely thin but absolutely nontensile paper. Thus, a folding is a map $f:B\to\mathbb R^2$ which is isometric on every polygon of some subdivision of the rectangle $B$. Moreover, the folding $f$ is realizable as a piecewise isometric homotopy which, in turn, can be approximated by some isotopy of space (which corresponds to the impossibility of self-intersection of a paper sheet during the folding process). </p> </blockquote> <p>Have a look at</p> <ul> <li><p>A. Tarasov, <em>Solution of Arnold’s “folded rouble” problem.</em> (in Russian) Chebyshevskii Sb. 5 (2004), 174–187.</p></li> <li><p>I. Yashenko, <em>Make your dollar bigger now!!!</em> Math. Intelligencer 20 (1998), no. 2, 38–40.</p></li> </ul> <hr> <p>A history of the problem is also briefly discussed in Tabachnikov's <a href="http://www.math.psu.edu/tabachni/prints/Arnoldpr.pdf" rel="nofollow">review</a> of "Arnold's Problems":</p> <blockquote> <p>It is interesting that the problem was solved by origami practitioners way before it was posed (at least, in 1797, in the Japanese origami book “Senbazuru Orikata”).</p> </blockquote> http://mathoverflow.net/questions/46866/is-napkin-conjecture-open-origami/46941#46941 Answer by Joseph O'Rourke for Is "Napkin conjecture" open ? (ORIGAMI) Joseph O'Rourke 2010-11-22T12:48:03Z 2011-12-11T00:25:54Z <p>Permit me to supplement Andrey's definitive answer.</p> <p>First, as Gerry Myerson says, this problem is discussed in Robert Lang's <em><a href="http://173-14-177-170-newengland.hfc.comcastbusiness.net/product.asp?ProdCode=1942" rel="nofollow">Origami Design Secrets</a></em>: pp.315-318, under the title "The Margulis Napkin Problem." He credits the problem to Gregori Margulis.</p> <p>Second, the problem is discussed in Igok Pak's book <em><a href="http://www.math.ucla.edu/~pak/book.htm" rel="nofollow">Lectures on Discrete and Polyhedral Geometry</a></em>, p.354, which is available online. You can pretty much guess the proof from the following instructive figure of Igor's:</p> <p><br />&nbsp;&nbsp;&nbsp;<img src="http://people.csail.mit.edu/~orourke/MathOverflow/NapkinFolding.jpg" alt="Napkin"><br /></p> <p>Third, there is another surprising result that is intellectually analogous to increasing the perimeter by folding: The volume enclosed by any convex polyhedron can be increased by bending the surface (retaining intrinsic isometry) to render it nonconvex! See Chapter 39 of Igor's book, p.339ff.</p> http://mathoverflow.net/questions/46866/is-napkin-conjecture-open-origami/61102#61102 Answer by Anton Petrunin for Is "Napkin conjecture" open ? (ORIGAMI) Anton Petrunin 2011-04-08T23:10:05Z 2011-10-24T22:44:56Z <p>In addition to the answers above, here are some remarks from <a href="http://front.math.ucdavis.edu/1004.0545" rel="nofollow">my paper</a>. (Sorry for self-advertisement.)</p> <p><strong>1.</strong> An other solution. It is based on idea of Yashenko. This way you can incresae the perimeter just a bit, but it is done by repeating one fold (which is very simple but not "simple" in the sense below). </p> <p><img src="http://www.math.psu.edu/petrunin/papers/arnold/pics/yasch-hq.png" alt="alt text"></p> <p><strong>2.</strong> It is still not known if you can increase the perimeter by a sequence of natural folds; i.e., folds like this: <img src="http://www.math.psu.edu/petrunin/papers/arnold/pics/otgib-hq.png" alt="alt text"> I just learned that this problem also appears in <a href="http://www.math.ucla.edu/~pak/book.htm" rel="nofollow">Pak's book</a>, Problem 40.16b; it is marked by [$*$] which means that the problem is open.</p>