Changing the Topology of Polyominoids Through Rigid Origami
John Mason, Erika Roldan, and Skye Rothstein

Proceedings of Bridges 2025: Mathematics and the Arts
Pages 503–506
Short Papers

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.

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