{{Short description|Geometric partition where pieces are connected by "hinged" points}} {{EngvarB|date=December 2013}} {{Use dmy dates|date=May 2024}} [[File:Hinged dissection 3-4-6-3 loop.gif|thumb|right|250px|Loop animation of hinged dissections from triangle to square, then to hexagon, then back again to triangle. Notice that the chain of pieces can be entirely connected in a ring during the rearrangement from square to hexagon.]]
In geometry, a '''hinged dissection''', also known as a '''swing-hinged dissection''' or '''Dudeney dissection''',<ref name="akiyama">{{cite book | author=Akiyama, Jin | author1-link = Jin Akiyama |author2=Nakamura, Gisaku | title=Discrete and Computational Geometry | chapter=Dudeney Dissection of Polygons | year=2000 | volume=1763 | pages=14–29 | doi=10.1007/978-3-540-46515-7_2|series = Lecture Notes in Computer Science|isbn = 978-3-540-67181-7}}</ref> is a kind of geometric dissection in which all of the pieces are connected into a chain by "hinged" points, such that the rearrangement from one figure to another can be carried out by swinging the chain continuously, without severing any of the connections.<ref name=cornell>{{cite web |url=https://www.math.cornell.edu/~mec/GeometricDissections/2.2%20Hinged%20Dissections.html |title=Hinged Dissections |last1=Pitici |first1=Mircea |date=September 2008 |website=Math Explorers Club |publisher=Cornell University |access-date=19 December 2013}}</ref> Typically, it is assumed that the pieces are allowed to overlap in the folding and unfolding process;<ref name=ORourke44>{{cite arXiv |last=O'Rourke |first=Joseph|author-link=Joseph O'Rourke (professor) |eprint=cs/0304025v1 |title=Computational Geometry Column 44 |year=2003}}</ref> this is sometimes called the "wobbly-hinged" model of hinged dissection.<ref name=opp>{{cite web |url=http://cs.smith.edu/~orourke/TOPP/P47.html |title=Problem 47: Hinged Dissections |date=8 December 2012 |website=The Open Problems Project |publisher=Smith College |access-date=19 December 2013}}</ref>
==History== thumb|upright=1.5|Dudeney's hinged dissection of a triangle into a square. alt=Animation of hinged dissection from hexagram to triangle to square|thumb|Animation of hinged dissection from hexagram to triangle to square The concept of hinged dissections was popularised by the author of mathematical puzzles, Henry Dudeney. He introduced the famous hinged dissection of a square into a triangle (pictured) in his 1907 book The Canterbury Puzzles.<ref name=swing1>Frederickson 2002, p.1</ref> The Wallace–Bolyai–Gerwien theorem, first proven in 1807, states that any two equal-area polygons must have a common dissection. However, the question of whether two such polygons must also share a ''hinged'' dissection remained open until 2007, when Erik Demaine ''et al.'' proved that there must always exist such a hinged dissection, and provided a constructive algorithm to produce them.<ref name=opp /><ref name="exist">{{cite book|title=Proceedings of the twenty-fourth annual symposium on Computational geometry - SCG '08|last1=Abbot|first1=Timothy G.|last2=Abel|first2=Zachary|last3=Charlton|first3=David|last4=Demaine|first4=Erik D.|last5=Demaine|first5=Martin L.|last6=Kominers|first6=Scott D.|year=2008|isbn=9781605580715|pages=110|chapter=Hinged Dissections Exist|arxiv=0712.2094|doi=10.1145/1377676.1377695|s2cid=3264789|author4-link=Erik Demaine|author5-link=Martin Demaine}}</ref><ref name=guardian>{{cite news |last=Bellos |first=Alex |date=30 May 2008 |title=The science of fun |url=https://www.theguardian.com/science/2008/may/31/maths.science |newspaper=The Guardian |access-date=20 December 2013}}</ref> This proof holds even under the assumption that the pieces may not overlap while swinging, and can be generalised to any pair of three-dimensional figures which have a common dissection (see Hilbert's third problem).<ref name=exist /><ref name=tony>{{cite journal |last=Phillips |first=Tony |date=November 2008 |title=Tony Phillips' Take on Math in the Media |url=https://www.ams.org/news/math-in-the-media/mmarc-11-2008-media |journal=Math in the Media |access-date=20 December 2013 }}</ref> In three dimensions, however, the pieces are not guaranteed to swing without overlap.<ref name=ORourke50>{{cite journal |last=O'Rourke |first=Joseph|author-link=Joseph O'Rourke (professor) |date=March 2008 |title=Computational Geometry Column 50 |url=http://cs.smith.edu/~orourke/Papers/50.pdf |journal=ACM SIGACT News |volume=39 |issue=1 |access-date=20 December 2013}}</ref>
==Other hinges== alt=Hinged square to pentagon|thumb|Hinged square to pentagon Other types of "hinges" have been considered in the context of dissections. A '''twist-hinge dissection''' is one which use a three-dimensional "hinge" which is placed on the edges of pieces rather than their vertices, allowing them to be "flipped" three-dimensionally.<ref name=swing6>Frederickson 2002, p.6</ref><ref name=bridges>{{cite conference |url=http://archive.bridgesmathart.org/2007/bridges2007-21.pdf |access-date=20 December 2013 |title=Symmetry and Structure in Twist-Hinged Dissections of Polygonal Rings and Polygonal Anti-Rings |last1=Frederickson |first1=Greg N. |year=2007 |publisher=The Bridges Organization |conference=Bridges 2007}}</ref> As of 2002, the question of whether any two polygons must have a common twist-hinged dissection remains unsolved.<ref name=swing7>Frederickson 2002, p. 7</ref>
==References== {{Reflist|33em}}
==Bibliography== *{{cite book |last=Frederickson |first=Greg N. |date=26 August 2002 |title=Hinged Dissections: Swinging and Twisting |url=https://archive.org/details/hingeddissection0000fred |url-access=registration |publisher=Cambridge University Press |isbn=978-0521811927 |access-date=19 December 2013}}
==External links== *[http://lsusmath.rickmabry.org/rmabry/live3d/hinged-triangle-square.htm An applet demonstrating Dudeney's hinged square-triangle dissection] *[https://web.archive.org/web/20150118004925/http://sylvester.math.nthu.edu.tw/d3/thesis-2003/yang/dissection/hinged.htm A gallery of hinged dissections]
Category:Geometric dissection Category:Recreational mathematics Category:Discrete geometry Category:Euclidean plane geometry