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.
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.
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 has been unsolved since 2002, and there are many more problems in higher dimensions. Visit here to find out more!