Yeah, Morse theory is really cool! There's a way to adapt this sort of proof to show that every polyhedral sphere (like the polyhedra considered in the article

Yeah, Morse theory is really cool! There's a way to adapt this sort of proof to show that every polyhedral sphere (like the polyhedra considered in the article), when thought of as residing in 3D space, bound a polyhedral ball. For this, you flood all of space and look at how the cross sections of the polyhedron vary with time. (Hatcher has some notes on 3-manifolds that outline a proof sketch.)