Hinged dissection

[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.

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).

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.
Dudeney's hinged dissection of a triangle into a square.
Animation of hinged dissection from hexagram to triangle to square
Animation of hinged dissection from hexagram to triangle to square
Hinged square to pentagon
Hinged square to pentagon