Amazing volumes of polyhedra

Is it possible to construct both not a convex and a convex poly­hedra from the same set of faces? You'd defi­nitely say Yes. One of the exam­ples is presented in the picture.

A poly­he­dron is called convex if it is placed in a single semi-space with respect to any plane containing one of it's faces.

A poly­he­dron is not convex if there exists a plane containing one of it's faces that cuts it into two pieces.

Let's assume that we managed to construct a convex and a non-convex poly­hedra from the same set of faces. Which of them has the greater volume?

It turns out that we can choose faces, so that the volume of the non-convex poly­he­dron will be greater that the volume of the convex one having the same faces. In this film we tell about the best example known so far.

Consider two trian­gles (the accu­rate length of their edges will be given in the end of the film) that will be faces of our poly­hedra. They will be the faces of both poly­hedra. The one shown on the left is convex, the one shown on the right is not.

Both constructed poly­hedra are octa­hedra (though, not regular) i.e. they have 6 vertices and 8 faces.

What is an informal meaning of volume, in partic­ular of a poly­he­dron? It's the amount of liquid that one can pour into it. Let’s cut off the vertices and start to already filled whereas the nonconvex one — not yet. But maybe the speed of water was different: to measure the amount of water prop­erly, let's pour out the water from each poly­he­dron to iden­tical is higher than in the left one. This means that the volume of the nonconvex poly­he­dron is really greater than the volume of the convex one.

If you calcu­late precisely you will get that the ratio of the volume of nonconvex poly­he­dron to that of the convex is 1,163.

In our problem it's more correct to consider the ratio of volumes than it's differ­ence as it doesn't depend on the size of initial trian­gles used to construct poly­hedra.

The volume of the constructed non-convex poly­he­dron is more then 16% greater than those of the convex one. You can make this poly­hedra your­self using faces with the given edges. If you place the centers of octa­hedra in the origin, their vertices will have the same [coor­di­nates] coor­di­nates as in the film.

We consid­ered here an example constructed by S. N. Mikhalev while he was a Ph.D. student in MSU. It's the best known example so far (it has the greatest ratio of poly­hedra volumes).

However, it is still non known how big can the ratio of volumes of such two poly­hedra be (of a non-convex and a convex one constructed from the same set of faces). And this problem is waiting for it's researcher!

Other etudes in “Polyhedra’s outer geometry”