Piecewise linear embedding of a polyhedron

Can one fold a devel­op­ment of a poly­he­dron into a closed body, whose borders will be not parts of planes but rather pieces of smooth surfaces?

Take a cube with edge length equal to $\pi$. Unfold it into the cruci­form devel­op­ment and plot a graph of the func­tion this body.

While the example we consid­ered answers the ques­tion, it has a big disad­van­tage. It's border contains two pieces that remained form the cube and are plane. Having built this example, a construc­tion that doesn't have this disad­van­tage appeared really quickly.

Take a rectan­gular sheet of paper with the edge ratio $\pi/2$. As any rectan­gular sheet, we can fold it into a trian­gular pyramid. To do that just draw edges connecting the centers of neighbor sides and an edge connecting the middles of the longer sides. Folding along the edges one gets a trian­gular pyramid.

From the same sheet of paper we can get another figure, whose surface will be constructed of smooth parts. Connect the middles of neighbor sides with three cylin­ders — two tangent and one perpen­dic­ular to them. Thus, the border is constructed of pieces of cylin­ders.

It's amazing how young math­e­matics is. It seems that such exam­ples should have existed maybe not along with Archimedes, but long long ago. But in fact, the exam­ples we consid­ered appeared only during the fall 2004.

Other etudes in “Polyhedra’s outer geometry”