Hinged dissection explained

In geometry, a hinged dissection, also known as a swing-hinged dissection or Dudeney dissection,[1] 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.[2] Typically, it is assumed that the pieces are allowed to overlap in the folding and unfolding process;[3] this is sometimes called the "wobbly-hinged" model of hinged dissection.[4]

History

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.[5] 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.[6] [7] 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).[8] In three dimensions, however, the pieces are not guaranteed to swing without overlap.[9]

Other hinges

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.[10] [11] As of 2002, the question of whether any two polygons must have a common twist-hinged dissection remains unsolved.[12]

Bibliography

External links

Notes and References

  1. Book: Akiyama, Jin . Jin Akiyama . Nakamura, Gisaku . Discrete and Computational Geometry . Dudeney Dissection of Polygons . 2000 . 1763 . 14–29 . 10.1007/978-3-540-46515-7_2. Lecture Notes in Computer Science. 978-3-540-67181-7.
  2. Web site: Hinged Dissections . Pitici . Mircea . September 2008 . Math Explorers Club . Cornell University . 19 December 2013.
  3. O'Rourke . Joseph. Joseph O'Rourke (professor) . cs/0304025v1 . Computational Geometry Column 44 . 2003.
  4. Web site: Problem 47: Hinged Dissections . 8 December 2012 . The Open Problems Project . Smith College . 19 December 2013.
  5. Frederickson 2002, p.1
  6. Book: Proceedings of the twenty-fourth annual symposium on Computational geometry - SCG '08. Abbot. Timothy G.. Abel. Zachary. Charlton. David. Demaine. Erik D.. Demaine. Martin L.. Kominers. Scott D.. 2008. 9781605580715. 110. Hinged Dissections Exist. 0712.2094. 10.1145/1377676.1377695. 3264789. Erik Demaine. Martin Demaine.
  7. News: Bellos . Alex . 30 May 2008 . The science of fun . The Guardian . 20 December 2013.
  8. Phillips . Tony . November 2008 . Tony Phillips' Take on Math in the Media . Math in the Media . 20 December 2013 .
  9. O'Rourke . Joseph. Joseph O'Rourke (professor) . March 2008 . Computational Geometry Column 50 . ACM SIGACT News . 39 . 1 . 20 December 2013.
  10. Frederickson 2002, p.6
  11. Symmetry and Structure in Twist-Hinged Dissections of Polygonal Rings and Polygonal Anti-Rings . 20 December 2013 . Frederickson . Greg N. . 2007 . . Bridges 2007.
  12. Frederickson 2002, p. 7