Changing the Topology of Polyominoids Through Rigid Origami

Given a 2D polyomino with holes, can we close all holes by applying rigid 90° origami folds on its interior edges, gluing inner-perimeter edges together to transform it into a 3D polyominoid with different topology? How many folding patterns exist which close all holes? We seek to achieve such transformations without overlapping squares or tearing the material, so there is a bijection on square faces from 2D to 3D and all initial adjacencies are preserved (some may be added where gluings occur). We present enumerative results for closing nxn square holes, and several tools for constructing larger closable polyominoes containing multiple square holes. We also describe several interestingly shaped holes and other folding techniques we have explored along the journey of seeking out a non-closable hole. This talk includes many pretty pictures and is joint work with Erika Roldan and John Mason.