Ingeometry, a hinged dissection, also known as a swing-hinged dissectionorDudeney 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]
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.[4][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).[6][8] In three dimensions, however, the pieces are not guaranteed to swing without overlap.[9]
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]
