Dudeney’s Hinged Dissections

How do you turn a triangle into a square? Or a pentagon into a nonagon? If I gave you a pair of scissors, you’d probably eventually be able to find a way to cut up the shape into a bunch of pieces and put them back together in a way that at least resembles a shape with the right number of sides.

In 1807, it was proven in the Wallace–Bolyai–Gerwien theorem any two equal-area polygons must have a common dissection. This means it is certainly possible to cut up a heptagon and turn it into a triangle. A simple example of turning a square into a hexagon is shown below, and more can be found here. If you’ve ever played with Tangrams you might understand how hard it can be to find these dissections.

A square and a hexagon with a common dissection

Now what if I told you that your dissections had to be hinged? Here, hinged means that all of the pieces of the dissection should connect to an adjacent piece by a ‘hinge’, forming a chain-like structure as it transforms into the next shape. See the gif below for an example.

A hinged dissection of a triangle into a square, and then into a hexagon

In 1907 the puzzle-maker Henry Dudeney first introduced the hinged dissection of a triangle into a square. The question of whether two polygons could share a hinged dissection was not solved until 2007, when Erik Demaine was able to prove that such a dissection always exists, and provided an algorithm to find them. His method can even be generalised to 3D figures with a common dissection.

After 2 centuries, you might think that we’ve found out all we can about swinging shapes around. In fact, there’s always more to find out! The question of whether a ‘twist hinge’ dissection exists has been unsolved since 2002, and there are many more problems in higher dimensions. Visit here to find out more!

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