We prove that any polyhedron of genus zero or genus one built out of rectangular faces must be an orthogonal polyhedron, but that there are nonorthogonal polyhedra of genus seven all of whose faces are rectangles. This leads to a resolution of a question posed by Biedl, Lubiw, and Sun [BLS99].
Donoso, Melody and O'Rourke, Joseph, "Nonorthogonal Polyhedra Built from Rectangles" (2002). Computer Science: Faculty Publications, Smith College, Northampton, MA.
Author’s submitted manuscript.