Flexible polyhedra

If you had to construct a wardrobe at home, you remember that while the back is not nailed, it back is put in place, a wardrobe which is a non-closed poly­he­dron with boundary becomes rigid. If you add the front or add another detail, closing the poly­he­dron, it will for sure remain rigid.

Are there closed flex­ible poly­hedra?

No one could answer this ques­tion for a long time. As always happens in science, one should consider an easier case. In the problem of flex­ible poly­hedra one should work in the plane where poly­hedra are replaced by poly­gons.

Are there flex­ible poly­gons? I.e. such that their sides are fixed, but the angles can change so that the polygon shape changes? Anyone can make such a model from wire using a stan­dard linking in the corners.

If one makes a triangle, it will not bend. I.e. the lengths of the sides deter­mine completely the triangle. And so, deter­mine its area: the Heron's formula allows to calcu­late it from the side lengths.

If one makes a wire quad­rangle or a pentagon or a polygon with a bigger number of vertices, any of them will bend. As a conse­quence, there is no Heron's formula, computing the area from the side lengths, for the number of angles greater than three.

Let's return to the space. What is a flex­ible poly­he­dron if it exists? Analog­i­cally to the flat case, the faces (being of one dimen­sion less than the space) should be rigid plates. And dihe­dral angles connecting the faces should be able to change, as if the edge (a face of dimen­sion one) was real­ized as a hinge.

Let's consider regular poly­hedra. If one makes their models with hinges as edges, one can check that they will not bend. It turns out that this is a general fact for convex poly­hedra. A theorem proved by a french math­e­mati­cian Augustin-Louis Cauchy (1789 – 1857) in 1813 states that a convex poly­he­dron with a given set of faces and gluing condi­tions is unique. I.e. a convex poly­he­dron can not be flex­ible.

The first math­e­mat­ical exam­ples of bend­able poly­hedra, of course, non-convex, as well as the clas­si­fi­ca­tion of such objects, were constructed by a belgian engi­neer R. Bricard in 1897. Math­e­mat­ical, because these poly­hedra were not only non-convex, but also self-inter­secting: their faces inter­sected each other. For the point of view of a math­e­mati­cian, this is also a poly­he­dron that can not be real­ized in our three-dimen­sional space. In 1975 an amer­ican math­e­mati­cian R. Connelly found a way to get rid of self-inter­sec­tions and the first real flex­ible poly­hedra appeared. The simplest known today, consisting of 9 vertices, 17 edges and 14 faces, will be now constructed. It was invented in 1978 by a german math­e­mati­cian Claus Steffen.

A net of the Steffen poly­he­dron consists of two similar parts and a «cover». If you remember the shape of the net, but not the lengths of the edges, it's hard to build such a poly­he­dron your­self: an ability to bend is excep­tional for poly­hedra and there is not a lot of them.

When math­e­mati­cians found out that such poly­hedra exist, they asked a ques­tion which is now called «bellows conjec­ture». Why do bellows fan the coals? Why does the internal volume change. And what about flex­ible poly­hedra: will their volume change while bending? Can one build an accor­dion or bellows out of rigid plates and not of leather?

In the end of the XX century the ques­tion was fully answered by a russian math­e­mati­cian I.H. Sabitov. It turns out that there is a Heron type formula for the volume of poly­hedra, so for flex­ible ones. Namely, there exist a poly­no­mial of one vari­able such that it's coef­fi­cients depend only on the edge lengths of the poly­he­dron and the volume is a root of this poly­no­mial. As edges of flex­ible poly­hedra do not change while bending, the poly­no­mial and its roots do not change either. But different roots of this poly­no­mial are some partic­ular numbers situ­ated at some distance one from another. Small bending should provide small changes of volume, so it can't jump from one root of the poly­no­mial into another. Thus, the volume of flex­ible poly­hedra doesn't change while bending!

We consid­ered both the cases of flex­ible poly­hedra in the plane and in the space. But what happens in greater dimen­sions? There are also some flex­ible poly­hedra, but much less. And the ques­tion about the constancy of volume of flex­ible poly­hedra in higher dimen­sions is still open and waiting for its researcher.

Other etudes in “Polyhedra’s outer geometry”