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Abstract: Arrange a collection of identical squares side-by-side to form a connected geometric figure — this is a polyomino. Given a 2D polyomino with holes, can we fold it into a 3D polyominoid with a desirable topology? For instance, can all holes be eliminated, resulting in a surface that deformation retracts to a point, or can we transform it into a cylinder or a sphere? We seek to achieve such transformations purely through rigid folding, without tearing the material or overlapping squares. In this paper, we interweave the study of polyominoes and polyominoids with techniques resembling those from origami and kirigami to introduce a mathematical model for classifying and manipulating these transformations. Finally, we explore potential applications in product and puzzle design: we pitch several toy ideas and provide examples of innovative lamp structures where topology plays an interesting role in shaping the properties and effects of light. 

Classification

The classes of disk, cylindrical, and spherical polyominoids are deformation retractable to a point, circle, and sphere, respectively. 

A non-smooth folding allows the inner-perimeter edges of a hole to be glued with the interior edges or outer-perimeter edges of the polyomino.

Enumeration of Closings

A closing is a folding pattern that closes all holes in a polyomino.

Conjecture: Every nxn square fenestration has exactly 4 closings for all 𝑛 ∈N. By exhaustive search, we have checked that this is true for the first three elements of the sequence.

Patching Examples

Let 𝐴 and 𝐵 be polyominoes, a 1D-patching of 𝐴 and 𝐵 is an identification (or gluing) of a subset of their edges (one-dimensional faces). A 2D-patching of 𝐴 and 𝐵 is an identification (gluing) of a subset of their square two-dimensional faces. We construct a compatible patching by choosing a pair of folding patterns, one on 𝐴 and one on 𝐵 that agree on their intersection (the faces that have been identified).