Polygons of equal area can be cut up into identical sets of smaller pieces. Amazing, right?
Before we look at a hinged dissection, let's first consider the much simpler case of dissecting two polygons with equal area, where we want to cut them up smaller pieces which allows us to rearrange one polygon into the other. We present the main ideas, along with a quick sketch of the proof:
With these ideas, we can prove that it is possible to dissect 2 polygons of equal area:
Proof of dissection: The goal is to dissect a polygon of area into a square of side length . Then by claim 1, any 2 polygons of equal area can be dissected into identical pieces.
Suppose we have a polygon of area . We triangulate it into smaller pieces (Claim 2), and dissect each of them into a rectangle (Claim 3), which we then dissect into individual squares square (Claim 4). We combine these squares iteratively (Claim 5) to obtain a single square of side length . Hence we are done.
Now that we understand how dissections amongst polygons work, let's consider having hinged dissections, in which the vertices of the pieces are connected to each other in the same way, which would restrict the possible movement of the pieces. For example, while the image in claim 3 immediately allows for a hinged dissection as indicated by the arrows, the images in claim 4 and 5 do not. We present the main ideas, and the proof of these claims is available in the paper.
Given a hinged figure, define the incidence graph which is constructed by placing a vertex at each piece and hinge, and 2 vertices are connected by an edge if one vertex represents a piece and the other represents a hinge on the piece.
Claim 6. Given a hinged figure, if the incidence graph is a tree , then for any partition into subtrees and , are able to move rooted subtrees using subdivisions and shift to any part of to obtain a new incidence graph .
Claim 7. Given any hinged figure , there is a subdivision such that all valid configurations of are reachable by a continuous non-self-intersecting motion.
With these ideas, we can prove that hinged dissection of 2 polygons of equal area is possible:
Proof of hinged dissection: Start with any valid dissection of the polygons and which have identical pieces. Hinge the vertices of to form a tree, and do the same for . By claim 6, there is a common subdivision of hinged dissections. Using claim 7, there is a further subdivision that results in continuous, non-self-intersecting motion.
To extend to polygons for dissection, we can simply overlay the dissections and move the numerous pieces around, as an extension of claim 1.
Unfortunately for hinged dissections, the main challenge is in figuring out a combination technique that works. This can be done using further machinery that refines the idea in claim 6 to more general structures. Complete details are available in the paper.
Interested in geometry? Hinged dissections are pretty advanced, but our Outside the Box Geometry course has some fun problems that will get your creativity flowing.