Figure 1. Convex v. Non-Convex
The figure on the left is a convex polyhedron, while the figure on the right is non-convex. What makes the cube on the left convex is that any two points can be connected by a line segment within the interior of the polyhedron. Looking at the figure on the right, we can see that two points can be connected on its exterior making the polyhedron non-convex.
Let’s quickly define polyhedron. A polyhedron is a three-dimensional figure made up of flat polygonal faces joined by straight edges. Polyhedra can be classified by the number of faces they have. For instance, a 4-sided polyhedron is called a tetrahedron, 5-sided is a pentahedron, 8-sided is an octahedron, and so on.
What makes a polyhedron rigid, you ask?
A polyhedron is rigid if you cannot bend it into another configuration and it remain the same shape. Bending a polyhedron into a congruent configuration would mean that as you move the figure the faces do not change and the edges preserve their length. With rigid polyhedra, however, trying to bend it would result in a tear or break. Take a cube, for example.
Figure 2. A Rigid Polyhedron (Cube)
If you had this cube in your hands, then you could easily find that bending or deforming it could not be done.
This brings us to the Cauchy Rigidity Theorem (1813), in which Cauchy states that any convex polyhedron is rigid. No convex polyhedron can be bent or deformed.
So, what about non-convex polyhedra? It turns out that there exists non-convex polyhedra that can in fact be bent into another configuration and remain the same shape. This is called flexibility. You have a polyhedron (which must be non-convex), you bend it, and you have a polyhedron that is congruent to the one you started with but arranged in a different way. A simple definition of flexibility is continuously moveable. A flexible polyhedron can be moved in a continuous motion.
Raoul Bricard, a French mathematician, constructed the first flexible polyhedron that had self-intersections in 1897. A self-intersection means that two or more edges cross each other. This construction is called the Bricard octahedron and consists of six vertices, twelve edges, and eight triangular faces (hence the name).
Figure 3. Bricard’s Flexible Octahedron With Self-Intersections
Fast forward about 8 decades later when in 1977 an American mathematician by the name of Robert Connelly constructs the first flexible polyhedron that did not have self-intersections. Connelly actually disproved Leonhard Euler, who argued in what is known as the Rigidity Conjecture (1766) that all polyhedra are rigid.
The best and simplest construction of a flexible polyhedron was built by German mathematician, Klaus Steffen. This construction consists of 9 vertices, 21 edges, and 14 triangular faces.
Figure 4. Steffen’s Flexible Polyhedron Without Self-Intersections
I have made Steffen’s polyhedron and you can as well! Follow this template along with the following instructions so that you may physically feel the flexibility. This model is meant to be made out of paper. Note that even if you made Steffan’s flexible polyhedron out of metal plates with the edges as hinges that it would remain flexible.
Now that we have covered rigid and flexible polyhedra, we can discuss the bellows conjecture. Does the volume inside of a flexible polyhedron vary while we bend it? Ijad Sabitov proved in 1995 that there does not exist a construction of a flexible polyhedron that has variable volume as you move it. A flexible polyhedron has the same volume regardless of its state of flex. In other words, a flexible polyhedron would make a terrible bellow.
I have been told that “the bellows conjecture” is quite a misleading name.
Written By: Brittany Braswell
References:
Figure 1. liam.flookes.com/cs/geo/
Figure 2. By Tomruen at en.wikipedia [Public domain], via Wikimedia Commons
Figure 3. en.wikipedia.org/wiki/Bricard_octahedron
Figure 4. By Unknown at en.wikipedia [Public domain], via Wikimedia Commons
Ars Geometrica 