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 astan­dard linkingin 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.