## Ununfoldable Polyhedra (1999)

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Citations: | 15 - 9 self |

### BibTeX

@MISC{Bern99ununfoldablepolyhedra,

author = {Marshall Bern and Erik D. Demaine and David Eppstein and Eric Kuo},

title = {Ununfoldable Polyhedra},

year = {1999}

}

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### Abstract

A well-studied problem is that of unfolding a convex polyhedron into a simple planar polygon. In this paper, we study the limits of unfoldability. We give an example of a polyhedron with convex faces that cannot be unfolded by cutting along its edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that "open" polyhedra with convex faces may not be unfoldable no matter how they are cut.

### Citations

153 |
The discrete geodesic problem
- Mitchell, Mount, et al.
- 1987
(Show Context)
Citation Context ...gs are known. The simplest to describe is the star unfolding [1, 2], which cuts from a generic point on the polyhedron along shortest paths to each of the vertices. The second is the source unfolding =-=[12, 16]-=-, which cuts along points with more than one shortest path to a generic source point. There has been little theoretical work on unfolding nonconvex polyhedra. In what may be the only paper on this sub... |

96 |
On shortest paths in polyhedral spaces
- Sharir, Schorr
- 1986
(Show Context)
Citation Context ...gs are known. The simplest to describe is the star unfolding [1, 2], which cuts from a generic point on the polyhedron along shortest paths to each of the vertices. The second is the source unfolding =-=[12, 16]-=-, which cuts along points with more than one shortest path to a generic source point. There has been little theoretical work on unfolding nonconvex polyhedra. In what may be the only paper on this sub... |

80 |
Vorlesungen über die Theorie der Polyeder unter Einschluss der Elemente der Topologie. Reprint der 1934 Auflage. Grundlehren der Mathematischen Wissenschaften
- Steinitz, Rademacher
- 1976
(Show Context)
Citation Context ...s homeomorphic to disks. The second example is ruled out because there are pairs of faces that share more than one edge, which is impossible for a convex polyhedron 1 . In general, Steinitz’s theorem =-=[7, 10, 18]-=- tells us that a polyhedron is topologically convex precisely if its graph is 3-connected and planar. The class of topologically convex polyhedra includes all convex-faced polyhedra (i.e., polyhedra w... |

54 | Folding and unfolding in computational geometry - O'Rourke - 1998 |

42 |
Convex Polytopes, Interscience
- Grünbaum
- 1967
(Show Context)
Citation Context ...s homeomorphic to disks. The second example is ruled out because there are pairs of faces that share more than one edge, which is impossible for a convex polyhedron 1 . In general, Steinitz’s theorem =-=[7, 10, 18]-=- tells us that a polyhedron is topologically convex precisely if its graph is 3-connected and planar. The class of topologically convex polyhedra includes all convex-faced polyhedra (i.e., polyhedra w... |

37 | C.: Star unfolding of a polytope with applications
- Agarwal, Aronov, et al.
- 1997
(Show Context)
Citation Context ...s known that if we allow cuts across the faces as well as along the edges, then every convex polyhedron has an unfolding. Two such unfoldings are known. The simplest to describe is the star unfolding =-=[1, 2]-=-, which cuts from a generic point on the polyhedron along shortest paths to each of the vertices. The second is the source unfolding [12, 16], which cuts along points with more than one shortest path ... |

35 | Unfolding some classes of orthogonal polyhedra
- Biedl, Demaine, et al.
- 1998
(Show Context)
Citation Context ...points with more than one shortest path to a generic source point. There has been little theoretical work on unfolding nonconvex polyhedra. In what may be the only paper on this subject, Biedl et al. =-=[3]-=- show the positive result that certain classes of orthogonal polyhedra can be unfolded. They show the negative result that not all nonconvex polyhedra have edge unfoldings. Two of their examples are g... |

31 |
Nonoverlap of the star unfolding
- Aronov, O’Rourke
- 1992
(Show Context)
Citation Context ...s known that if we allow cuts across the faces as well as along the edges, then every convex polyhedron has an unfolding. Two such unfoldings are known. The simplest to describe is the star unfolding =-=[1, 2]-=-, which cuts from a generic point on the polyhedron along shortest paths to each of the vertices. The second is the source unfolding [12, 16], which cuts along points with more than one shortest path ... |

31 |
Polyhedron Models
- Wenninger
- 1971
(Show Context)
Citation Context ...t since at least the time of Albrecht Dürer, circa 1500 [14]. It is widely conjectured that the answer to this question is yes. While unfoldings were originally used to make paper models of polyhedra =-=[5, 20]-=-, unfoldings have important industrial applications. For example, sheet metal bending is an efficient process for manufacturing [8, 19]. In this process, the desired object is approximated by a polyhe... |

27 |
Convex polytopes with convex nets
- Shephard
- 1975
(Show Context)
Citation Context ...lded if cuts are allowed to cross faces. Finally, we prove that “open” polyhedra with convex faces may not be unfoldable no matter how they are cut. 1 Introduction A classic open question in geometry =-=[4, 6, 14, 17]-=- is whether every convex polyhedron can be cut along its edges and flattened into the plane without any overlap. Such a collection of cuts is called an edge unfolding of the polyhedron, and the result... |

22 | process planning for sheet metal bending operations
- Gupta, Bourne, et al.
(Show Context)
Citation Context ...nfoldings were originally used to make paper models of polyhedra [5, 20], unfoldings have important industrial applications. For example, sheet metal bending is an efficient process for manufacturing =-=[8, 19]-=-. In this process, the desired object is approximated by a polyhedron, which is unfolded into a collection of polygons. Then these polygons are cut out of a sheet of material, and each piece is folded... |

14 | G.M.: Basic properties of convex polytopes
- Henk, Richter-Gebert, et al.
- 1997
(Show Context)
Citation Context ...s homeomorphic to disks. The second example is ruled out because there are pairs of faces that share more than one edge, which is impossible for a convex polyhedron 1 . In general, Steinitz’s theorem =-=[7, 10, 18]-=- tells us that a polyhedron is topologically convex precisely if its graph is 3-connected and planar. The class of topologically convex polyhedra includes all convex-faced polyhedra (i.e., polyhedra w... |

13 | Manufacturability-driven decomposition of sheet metal products
- Wang
- 1997
(Show Context)
Citation Context ...nfoldings were originally used to make paper models of polyhedra [5, 20], unfoldings have important industrial applications. For example, sheet metal bending is an efficient process for manufacturing =-=[8, 19]-=-. In this process, the desired object is approximated by a polyhedron, which is unfolded into a collection of polygons. Then these polygons are cut out of a sheet of material, and each piece is folded... |

10 |
Unfolding 3dimensional convex polytopes: A package for Mathematica 1.2 or 2.0, Mathematica Notebook
- Namiki, Fukuda
- 1993
(Show Context)
Citation Context ... USA, email: ehkst@mit.edu. Work performed while at Xerox PARC.There are two freely available heuristic programs for constructing edge unfoldings of polyhedra: the Mathematica package UnfoldPolytope =-=[13]-=-, and the Macintosh program HyperGami [9]. Neither program has ever failed, and HyperGami even works for some nonconvex polyhedra. There are also several commercial heuristic programs; an example is T... |

5 |
Strange unfoldings of convex polytopes
- Fukuda
- 1997
(Show Context)
Citation Context ...lded if cuts are allowed to cross faces. Finally, we prove that “open” polyhedra with convex faces may not be unfoldable no matter how they are cut. 1 Introduction A classic open question in geometry =-=[4, 6, 14, 17]-=- is whether every convex polyhedron can be cut along its edges and flattened into the plane without any overlap. Such a collection of cuts is called an edge unfolding of the polyhedron, and the result... |

4 |
Pleasures of plication
- Hayes
- 1995
(Show Context)
Citation Context ... while at Xerox PARC.There are two freely available heuristic programs for constructing edge unfoldings of polyhedra: the Mathematica package UnfoldPolytope [13], and the Macintosh program HyperGami =-=[9]-=-. Neither program has ever failed, and HyperGami even works for some nonconvex polyhedra. There are also several commercial heuristic programs; an example is Touch-3D [11], which supports nonconvex po... |

4 | Vorlesungen "uber die Theorie der Polyeder - Steinitz, Rademacher - 1934 |

3 |
Unfolding polyhedra. sci.math Usenet article
- Schevon
- 1987
(Show Context)
Citation Context ...convex polyhedra includes all convex-faced polyhedra (i.e., polyhedra whose faces are all convex) that are homeomorphic to spheres. (This will be proved formally later.) Schevon and other researchers =-=[3, 15]-=- have asked whether all such polyhedra can be unfolded by cutting along edges. In other words, can the conjecture that every convex polyhedron is edge unfoldable be extended to topologically convex po... |

1 | Pleasures of plication. American Scientist, November--December - Hayes - 1995 |