Tighter bounds on the genus of nonorthogonal polyhedra built from rectangles.
Thérèse Biedl, Timothy M. Chan, Erik D. Demaine, Martin L. Demaine, Paul Nijjar, Ryuhei Uehara, Mingwei Wang · 2002
We prove that there is a polyhedron with genus 6 whose faces are orthogonal polygons (equivalently, rectangles) and yet the angles between some faces are not multiples , so the polyhedron itself is not orthogonal. On the other hand, we prove that any such polyhedron must have genus at least 3. These results improve the bounds of Donoso and O'Rourke [4] that there are nonorthogonal polyhedra with orthogonal faces and genus 7 or larger, and any such polyhedron must have genus at least 2. We also demonstrate nonoverlapping one-piece edge-unfoldings (nets) for the genus-7 and genus-6 polyhedra.