geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a hinged dissection, also known as a swing-hinged dissection or Dudeney dissection, is a kind of

geometric dissection Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

in which all of the pieces are connected into a chain by "hinged" points, such that the rearrangement from one

figure Figure may refer to: General *A shape, drawing, depiction, or geometric configuration *Figure (wood), wood appearance *Figure (music), distinguished from musical motif *Noise figure, in telecommunication *Dance figure, an elementary dance pattern ...

to another can be carried out by swinging the chain continuously, without severing any of the connections. Typically, it is assumed that the pieces are allowed to overlap in the folding and unfolding process; this is sometimes called the "wobbly-hinged" model of hinged dissection.

History

The concept of hinged dissections was popularised by the author of

mathematical puzzle Mathematical puzzles make up an integral part of recreational mathematics. They have specific rules, but they do not usually involve competition between two or more players. Instead, to solve such a puzzle, the solver must find a solution that sati ...

Henry Dudeney Henry Ernest Dudeney (10 April 1857 – 23 April 1930) was an English author and mathematician who specialised in logic puzzles and mathematical games. He is known as one of the country's foremost creators of mathematical puzzles. Early life ...

. He introduced the famous hinged dissection of a square into a triangle (pictured) in his 1907 book

The Canterbury Puzzles ''The Canterbury Puzzles and Other Curious Problems'' is a 1907 mathematical puzzle book by Henry Dudeney. The first part of the book features a series of puzzles based on the characters from ''The Canterbury Tales'' by Geoffrey Chaucer. Refere ...

.Frederickson 2002, p.1 The

Wallace–Bolyai–Gerwien theorem In geometry, the Wallace–Bolyai–Gerwien theorem, named after William Wallace, Farkas Bolyai and Paul Gerwien, is a theorem related to dissections of polygons. It answers the question when one polygon can be formed from another by cutting it ...

, 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 Erik D. Demaine (born February 28, 1981) is a professor of computer science at the Massachusetts Institute of Technology and a former child prodigy. Early life and education Demaine was born in Halifax, Nova Scotia, to artist sculptor Marti ...

''et al.'' proved that there must always exist such a hinged dissection, and provided a constructive algorithm to produce them. 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 The third of Hilbert's list of mathematical problems, presented in 1900, was the first to be solved. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely m ...

). In three dimensions, however, the pieces are not guaranteed to swing without overlap.

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.Frederickson 2002, p.6 As of 2002, the question of whether any two polygons must have a common twist-hinged dissection remains unsolved.Frederickson 2002, p. 7

References

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

An applet demonstrating Dudeney's hinged square-triangle dissection
Geometric dissection Recreational mathematics Discrete geometry Euclidean plane geometry