Edge-unfolding polycubes
A polycube is an orthogonal polyhedron formed by gluing unit cubes face to face, the three-dimensional analogue of a polyomino. An edge unfolding cuts the surface along edges only and unfolds it into the plane as a single piece without overlap; here every edge of a cube that lies on the surface may be cut.
Biedl, Demaine, Demaine, Lubiw, O'Rourke, Overmars, Robbins and Whitesides gave a simple box-on-box example showing that, without refining the surface, not every orthogonal polyhedron has an edge unfolding. Grid unfoldings, which may also cut along the intersections with coordinate planes through every vertex, are known for only a few classes: orthotubes, orthostacks of orthogonally convex slabs, and orthogonal terrains (O'Rourke). Cutting along all cube edges refines the surface further still.
Problem. Is there a genus-zero polycube that cannot be edge-unfolded when every cube edge on its surface is available for cutting?
Source: Hart and O'Rourke (2004); Problem 64 of The Open Problems Project (Demaine, Mitchell and O'Rourke).
Imported from topp.openproblem.net.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- CO
- Keywords
- edge unfolding, orthogonal polyhedra, polycubes, The Open Problems Project
References
@inproceedings{biedl1998,
author = {Biedl, Therese and Demaine, Erik D. and Demaine, Martin L. and Lubiw, Anna and O'Rourke, Joseph and Overmars, Mark and Robbins, Steve and Whitesides, Sue},
title = {Unfolding some classes of orthogonal polyhedra},
booktitle = {Proc. 10th Canadian Conference on Computational Geometry},
year = {1998},
pages = {70--71},
url = {https://erikdemaine.org/papers/CCCG98b/paper.ps},
}
@inproceedings{demaine2005cccg04,
author = {Demaine, Erik D. and O'Rourke, Joseph},
title = {Open problems from {CCCG} 2004},
booktitle = {Proc. 17th Canadian Conference on Computational Geometry},
year = {2005},
pages = {303--306},
}
@incollection{orourke2008,
author = {O'Rourke, Joseph},
title = {Unfolding orthogonal polyhedra},
booktitle = {Proc. Snowbird Conference Discrete and Computational Geometry: Twenty Years Later},
editor = {Goodman, J. E. and Pach, J. and Pollack, R.},
publisher = {American Mathematical Society},
year = {2008},
pages = {307--317},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01102, "Edge-unfolding polycubes", clopen.solutions.