## Ununfoldable polyhedra with convex faces (2002)

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- [erikdemaine.org]
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Venue: | COMPUT. GEOM. THEORY APPL |

Citations: | 26 - 11 self |

### BibTeX

@ARTICLE{Bern02ununfoldablepolyhedra,

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

title = {Ununfoldable polyhedra with convex faces},

journal = {COMPUT. GEOM. THEORY APPL},

year = {2002},

volume = {24},

pages = {51--62}

}

### OpenURL

### Abstract

Unfolding a convex polyhedron into a simple planar polygon is a well-studied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along 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 triangular faces may not be unfoldable no matter how they are cut.

### Citations

155 | 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 =-=[18, 23]-=-, 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... |

101 |
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 =-=[18, 23]-=-, 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... |

83 |
Vorlesungen über die Theorie der Polyeder unter Einschluss der Elemente der Topologie
- Steinitz, Rademacher
- 1976
(Show Context)
Citation Context ...phic 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, a famous theorem of Steinitz =-=[13, 16, 25]-=- 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... |

56 | Folding and unfolding in computational geometry
- O’Rourke
- 1998
(Show Context)
Citation Context ... if cuts are allowed to cross faces. Finally, we prove that “open” polyhedra with triangular faces may not be unfoldable no matter how they are cut. 1 Introduction A classic open question in geometry =-=[5, 12, 20, 24]-=- 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 cutting of the polyhedron, and the resultin... |

43 |
Convex Polytopes, Interscience
- Grünbaum
- 1967
(Show Context)
Citation Context ...phic 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, a famous theorem of Steinitz =-=[13, 16, 25]-=- 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... |

38 | Star unfolding of a polytope with applications
- Agarwal, Aronov, et al.
- 1990
(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 [18, 23], which cuts along points with more than one shortest path ... |

37 |
Dual models
- Wenninger
- 1983
(Show Context)
Citation Context ....unc.edu. 1sa random polytope causes overlap with probability approaching 1 as the number of vertices approaches infinity [22]. While unfoldings were originally used to make paper models of polyhedra =-=[7, 27]-=-, unfoldings have important industrial applications. For example, sheet metal bending is an efficient process for manufacturing [14, 26]. In this process, the desired object is approximated by a polyh... |

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. =-=[4]-=- 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... |

32 |
Nonoverlap of the star unfolding
- Aronov, O'Rourke
- 1991
(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 [18, 23], which cuts along points with more than one shortest path ... |

27 |
Convex polytopes with convex nets
- Shephard
- 1975
(Show Context)
Citation Context ... if cuts are allowed to cross faces. Finally, we prove that “open” polyhedra with triangular faces may not be unfoldable no matter how they are cut. 1 Introduction A classic open question in geometry =-=[5, 12, 20, 24]-=- 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 cutting of the polyhedron, and the resultin... |

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 [7, 27], unfoldings have important industrial applications. For example, sheet metal bending is an efficient process for manufacturing =-=[14, 26]-=-. 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... |

15 | Ununfoldable polyhedra
- Bern, Demaine, et al.
- 1999
(Show Context)
Citation Context ... we present an “open” triangulated polyhedron that cannot be unfolded. Finding a “closed” ununfoldable polyhedron is an intriguing open problem. A preliminary version of this work appeared in CCCG’99 =-=[3]-=-. 2 Basics We begin with formal definitions and some basic results about polyhedra, unfoldings, and cuttings. We define a polyhedron to be a connected set of closed planar polygons in 3-space such tha... |

14 | Basic properties of convex polytopes
- Henk, Richter-Gebert, et al.
(Show Context)
Citation Context ...phic 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, a famous theorem of Steinitz =-=[13, 16, 25]-=- 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... |

14 | Manufacturability-Driven Decomposition of Sheet Metal Prooducts
- Wang
- 1997
(Show Context)
Citation Context ...nfoldings were originally used to make paper models of polyhedra [7, 27], unfoldings have important industrial applications. For example, sheet metal bending is an efficient process for manufacturing =-=[14, 26]-=-. 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... |

11 | Enumerating foldings and unfoldings between polygons and polytopes
- Demaine, Demaine, et al.
(Show Context)
Citation Context ...be flattened without overlap. ✷ A common assumption is that an unfolding must be a simple polygon, which implies that a cutting of a closed polyhedron must furthermore be a (connected) tree; see e.g. =-=[8, 9, 22]-=-. Without this restriction, in most cases, a cutting of a closed polyhedron is a tree, but in fact this is not always the case. Figure 2 illustrates a basic construction for separating off a connected... |

10 |
Unfolding 3dimensional convex polytopes: A package for Mathematica 1.2 or 2.0, Mathematica Notebook
- Namiki, Fukuda
- 1993
(Show Context)
Citation Context ... polyhedra is available, and thus heuristics must be used [20]. There are two freely available heuristic programs for constructing edge unfoldings of polyhedra: the Mathematica package UnfoldPolytope =-=[19]-=-, and the Macintosh program HyperGami [15]. There are no reports of these programs failing to find an edge unfolding for a convex polyhedron; HyperGami even finds unfoldings for nonconvex polyhedra. T... |

7 | Examples, counterexamples, and enumeration results for foldings and unfoldings between polygons and polytopes
- Demaine, Demaine, et al.
- 2000
(Show Context)
Citation Context ...be flattened without overlap. ✷ A common assumption is that an unfolding must be a simple polygon, which implies that a cutting of a closed polyhedron must furthermore be a (connected) tree; see e.g. =-=[8, 9, 22]-=-. Without this restriction, in most cases, a cutting of a closed polyhedron is a tree, but in fact this is not always the case. Figure 2 illustrates a basic construction for separating off a connected... |

5 |
Strange unfoldings of convex polytopes
- Fukuda
- 1997
(Show Context)
Citation Context ... if cuts are allowed to cross faces. Finally, we prove that “open” polyhedra with triangular faces may not be unfoldable no matter how they are cut. 1 Introduction A classic open question in geometry =-=[5, 12, 20, 24]-=- 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 cutting of the polyhedron, and the resultin... |

4 |
The Painter’s Manual: A Manual of Measurement of Lines, Areas, and Solids by Means of Compass and Ruler Assembled by Albrecht Dürer for the Use of All Lovers of Art with Appropriate Illustrations Arranged to be
- Dürer
- 1977
(Show Context)
Citation Context ...olygon is called an edge unfolding or net. While the first explicit description of this problem is by Shephard in 1975 [24], it has been implicit since at least the time of Albrecht Dürer, circa 1500 =-=[11]-=-. It is widely conjectured that every convex polyhedron has an edge unfolding. Some recent support for this conjecture is that every triangulated convex polyhedron has a vertex unfolding, in which the... |

4 |
Pleasures of plication
- Hayes
- 1995
(Show Context)
Citation Context ...cs must be used [20]. There are two freely available heuristic programs for constructing edge unfoldings of polyhedra: the Mathematica package UnfoldPolytope [19], and the Macintosh program HyperGami =-=[15]-=-. There are no reports of these programs failing to find an edge unfolding for a convex polyhedron; HyperGami even finds unfoldings for nonconvex polyhedra. There are also several commercial heuristic... |

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

3 |
Unfolding polyhedra. sci.math Usenet article
- Schevon
- 1987
(Show Context)
Citation Context ...nd planar. The class of topologically convex polyhedra includes all convex-faced polyhedra (i.e., polyhedra whose faces are all convex) that are homeomorphic to spheres. Schevon and other researchers =-=[4, 21]-=- have asked whether all such polyhedra can be unfolded without overlap by cutting along edges. In other words, can the conjecture that every convex polyhedron is edge-unfoldable be extended to topolog... |

3 |
Algorithms for Geodesics on Polytopes
- Schevon
- 1989
(Show Context)
Citation Context ...versity of North Carolina, Chapel Hill, NC 27599-3175, USA, email: snoeyink@cs.unc.edu. 1sa random polytope causes overlap with probability approaching 1 as the number of vertices approaches infinity =-=[22]-=-. While unfoldings were originally used to make paper models of polyhedra [7, 27], unfoldings have important industrial applications. For example, sheet metal bending is an efficient process for manuf... |

1 | The discrete geodesic problem - Touch-3d - 1998 |

1 |
Examples, counterexamples, and enumeration results for foldings and unfoldings between polygons and polytopes
- Polyhedra
- 1961
(Show Context)
Citation Context ....unc.edu. 1a random polytope causes overlap with probability approaching 1 as the number of vertices approaches infinity [22]. While unfoldings were originally used to make paper models of polyhedra =-=[7, 27]-=-, unfoldings have important industrial applications. For example, sheet metal bending is an efficient process for manufacturing [14, 26]. In this process, the desired object is approximated by a polyh... |