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Polyhedral Characterization of Reversible Hinged Dissections

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Abstract

We prove that two polygons A and B have a reversible hinged dissection (a chain hinged dissection that reverses inside and outside boundaries when folding between A and B) if and only if A and B are two noncrossing nets of a common polyhedron. Furthermore, monotone reversible hinged dissections (where all hinges rotate in the same direction when changing from A to B) correspond exactly to noncrossing nets of a common convex polyhedron. By envelope/parcel magic, it becomes easy to design many hinged dissections.

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Notes

  1. For simplicity, we assume that the edges of T are drawn using segments along the surface of P, and that vertices of degree 2 can be used in T to draw any polygonal path.

  2. We here refer to the geodesic distance on the surface of P.

  3. Burago and Zalgaller first proved the result for any orientable surface [14] but later noted and fixed a flaw in their construction [15]. At the same time, they generalized their result to a stronger statement about possibly non-orientable surfaces.

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Correspondence to Stefan Langerman.

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Akiyama, J., Demaine, E.D. & Langerman, S. Polyhedral Characterization of Reversible Hinged Dissections. Graphs and Combinatorics 36, 221–229 (2020). https://doi.org/10.1007/s00373-019-02041-2

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