We just learned that every convex polyhedron has a general net, and we saw an example in Figure 7.10 of a nonconvex polyhedron that does not have a net (cutting along edges only). This naturally raises the question: Does every nonconvex polyhedron have a general net? This is yet another unsolved problem. As with Dürer's problem, no counterexample is known, but there is not much evidence that the answer to the general unfolding question is yes. If every polyhedron does have a general net, it would certainly make a stunning theorem!

Because this problem seems difficult, researchers have focused on a special class of nonconvex polyhedra known as orthogonal polyhedra, where “orthogonal” means “at right angles.”

Orthogonal Polyhedra

You can think of orthogonal polyhedra as those you could build out of Lego blocks. An orthogonal polyhedron is a polyhedron where each edge is parallel to one of the axes of a standard right-angled xyz-coordinate system. If all edges are parallel to an axis, then all faces are parallel to a coordinate plane: either xy (horizontal) or xz (vertical, front or back) or yz (vertical, side, left or right). Any pair of faces of an orthogonal polyhedron that share an edge either lie in the same plane (they are coplanar) or meet at right angles to each other, that is, orthogonally.