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Unfolding Genus-2 Orthogonal Polyhedra with Linear Refinement

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Abstract

We show that every orthogonal polyhedron of genus \(g \le 2\) can be unfolded without overlap while using only a linear number of orthogonal cuts (parallel to the polyhedron edges). This is the first result on unfolding general orthogonal polyhedra beyond genus-0. Our unfolding algorithm relies on the existence of at most 2 special leaves in what we call the “unfolding tree” (which ties back to the genus), so unfolding polyhedra of genus 3 and beyond requires new techniques.

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Notes

  1. The \(\pm y\) faces are given the awkward names “forward” and “rearward” to avoid confusion with other uses of “front” and “back” introduced later.

References

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Acknowledgements

We thank all the participants of the 31st Bellairs Winter Workshop on Computational Geometry for a fruitful and collaborative environment. In particular, we thank Sebastian Morr for important discussions related to Theorem 1, and to the stitching of unfolding strips at the root node.

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Authors and Affiliations

  1. Department of Computer Science, Villanova University, Villanova, PA, 19085, USA

    Mirela Damian

  2. Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, 32 Vassar St., Cambridge, MA, 02139, USA

    Erik Demaine

  3. Department of Computer Science, Siena College, Loudonville, NY, 12211, USA

    Robin Flatland

  4. Department of Computer Science, Smith College, Northampton, MA, 01063, USA

    Joseph O’Rourke

Authors
  1. Mirela Damian
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  2. Erik Demaine
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  3. Robin Flatland
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  4. Joseph O’Rourke
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Correspondence to Mirela Damian.

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Damian, M., Demaine, E., Flatland, R. et al. Unfolding Genus-2 Orthogonal Polyhedra with Linear Refinement. Graphs and Combinatorics 33, 1357–1379 (2017). https://doi.org/10.1007/s00373-017-1849-5

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  • DOI: https://doi.org/10.1007/s00373-017-1849-5

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