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### Grid Vertex-Unfolding Orthogonal Polyhedra [chapter]

Mirela Damian, Robin Flatland, Joseph O'Rourke
2006 Lecture Notes in Computer Science
We show that any orthogonal polyhedra of genus zero has a grid vertex-unfolding. (There are orthogonal polyhedra that cannot be vertex-unfolded, so some type of "gridding" of the faces is necessary.)  ...  A grid unfolding allows additional cuts along grid edges induced by coordinate planes passing through every vertex.  ...  For orthogonal polyhedra, a grid unfolding is a natural median between edge-unfoldings and unrestricted unfoldings.  ...

### Grid Vertex-Unfolding Orthogonal Polyhedra [article]

Mirela Damian, Robin Flatland, Joseph O'Rourke
2006 arXiv   pre-print
(There are orthogonal polyhedra that cannot be vertex-unfolded, so some type of "gridding" of the faces is necessary.)  ...  A vertex-unfolding permits faces in the net to be connected at single vertices, not necessarily along edges. We show that any orthogonal polyhedron of genus zero has a grid vertex-unfolding.  ...  For orthogonal polyhedra, a grid unfolding is a natural median between edge-unfoldings and unrestricted unfoldings.  ...

### Grid Vertex-Unfolding Orthogonal Polyhedra

Mirela Damian, Robin Flatland, Joseph O'Rourke
2007 Discrete & Computational Geometry
We show that any orthogonal polyhedra of genus zero has a grid vertex-unfolding. (There are orthogonal polyhedra that cannot be vertex-unfolded, so some type of "gridding" of the faces is necessary.)  ...  A grid unfolding allows additional cuts along grid edges induced by coordinate planes passing through every vertex.  ...  For orthogonal polyhedra, a grid unfolding is a natural median between edge-unfoldings and unrestricted unfoldings.  ...

### Grid Vertex-Unfolding Orthogonal Polyhedra [chapter]

Mirela Damian, Robin Flatland, Joseph O'Rourke
Twentieth Anniversary Volume:
We show that any orthogonal polyhedra of genus zero has a grid vertex-unfolding. (There are orthogonal polyhedra that cannot be vertex-unfolded, so some type of "gridding" of the faces is necessary.)  ...  A grid unfolding allows additional cuts along grid edges induced by coordinate planes passing through every vertex.  ...  For orthogonal polyhedra, a grid unfolding is a natural median between edge-unfoldings and unrestricted unfoldings.  ...

### Vertex Unfoldings of Orthogonal Polyhedra: Positive, Negative, and Inconclusive Results

Luis Garcia, Andres Gutierrez, Isaac Ruiz, Andrew Winslow
2018 Canadian Conference on Computational Geometry
The (positive) first result is a simple proof that all genus-0 and genus-1 orthogonal polyhedra have grid vertex unfoldings.  ...  We obtain results for three questions regarding vertex unfoldings of orthogonal polyhedra.  ...  Acknowledgments We would like to thank Erik Demaine for discussion of known results related to vertex unfoldings, and anonymous reviewers whose suggestions improved the readability and correctness of the  ...

### Unfolding Orthogonal Terrains [article]

Joseph O'Rourke
2007 arXiv   pre-print
It is shown that every orthogonal terrain, i.e., an orthogonal (right-angled) polyhedron based on a rectangle that meets every vertical line in a segment, has a grid unfolding: its surface may be unfolded  ...  to a single non-overlapping piece by cutting along grid edges defined by coordinate planes through every vertex.  ...  it may be that all orthogonal polyhedra may be grid-unfoldable.  ...

### Improved Algorithms for Grid-Unfolding Orthogonal Polyhedra

Yi-Jun Chang, Hsu-Chun Yen
2017 International journal of computational geometry and applications
For 1-layer orthogonal polyhedra of genus g, we show a grid-unfolding algorithm using only 2(g − 1) additional cuts, affirmatively answering an open problem raised in a recent literature.  ...  Even for orthogonal polyhedra, it is known that edge-unfolding, i.e., cuts are performed only along the edges of a polyhedron, is not sufficient to guarantee a successful unfolding in general.  ...  Can all orthogonal polyhedra be grid-unfolded with sublinear refinement?  ...

### Unfolding Level 1 Menger Polycubes of Arbitrary Size With Help of Outer Faces [chapter]

Lydie Richaume, Eric Andres, Gaëlle Largeteau-Skapin, Rita Zrour
2019 Lecture Notes in Computer Science
It is worth noticing that this grid-unfolding algorithm is deterministic and without refinement.  ...  In this article, we suggest a grid-unfolding of level 1 Menger polycubes of arbitrary size with L holes along the x-axis, M the y-axis and N the z-axis.  ...  In  , edge-unfolding of several classes of one-layered orthogonal polyhedra (one layer of voxels) with cubic holes are considered and it was shown that several special classes of them admit grid unfoldings  ...

### Epsilon-Unfolding Orthogonal Polyhedra

Mirela Damian, Robin Flatland, Joseph O'Rourke
2007 Graphs and Combinatorics
Here we prove, via an algorithm, that every orthogonal polyhedron (one whose faces meet at right angles) of genus zero may be unfolded.  ...  It is a long unsolved problem to determine whether every polyhedron may be unfolded.  ...  This concept has been used to achieve grid vertex unfoldings of orthostacks  , and later, grid vertex unfoldings of all genus-zero orthogonal polyhedra  .  ...

### Unfolding Orthogonal Polyhedra with Quadratic Refinement: The Delta-Unfolding Algorithm [article]

Mirela Damian, Erik Demaine, Robin Flatland
2011 arXiv   pre-print
We show that every orthogonal polyhedron homeomorphic to a sphere can be unfolded without overlap while using only polynomially many (orthogonal) cuts.  ...  More precisely, given an orthogonal polyhedron with n vertices, the algorithm cuts the polyhedron only where it is met by the grid of coordinate planes passing through the vertices, together with Theta  ...  However, our preliminary investigations embolden us to conjecture that a constant refinement of the vertex grid suffices to grid-unfold all orthogonal polyhedra.  ...

### Unfolding Orthogonal Polyhedra with Quadratic Refinement: The Delta-Unfolding Algorithm

Mirela Damian, Erik D. Demaine, Robin Flatland
2012 Graphs and Combinatorics
We show that every orthogonal polyhedron homeomorphic to a sphere can be unfolded without overlap while using only polynomially many (orthogonal) cuts.  ...  More precisely, given an orthogonal polyhedron with n vertices, the algorithm cuts the polyhedron only where it is met by the grid of coordinate planes passing through the vertices, together with Θ(n 2  ...  However, our preliminary investigations embolden us to conjecture that a constant refinement of the vertex grid suffices to grid-unfold all orthogonal polyhedra.  ...

### Unfolding Manhattan Towers

Mirela Damian, Robin Flatland, Joseph O'Rourke
2008 Computational geometry
The algorithm cuts along edges of a 4 × 5 × 1 refinement of the vertex grid.  ...  We provide an algorithm for unfolding the surface of any orthogonal polyhedron that falls into a particular shape class we call Manhattan Towers, to a nonoverlapping planar orthogonal polygon.  ...  the suggestion was made in  to seek k 1 × k 2 × k 3 -refined grid unfoldings, where every face of the vertex grid is further refined into a grid of edges.  ...

### Unfolding Manhattan Towers [article]

Mirela Damian, Robin Flatland, Joseph O'Rourke
2007 arXiv   pre-print
The algorithm cuts along edges of a 4x5x1 refinement of the vertex grid.  ...  We provide an algorithm for unfolding the surface of any orthogonal polyhedron that falls into a particular shape class we call Manhattan Towers, to a nonoverlapping planar orthogonal polygon.  ...  the suggestion was made in [DO04] to seek k 1 × k 2 × k 3 refined grid unfoldings, where every face of the vertex grid is further refined into a grid of edges.  ...

### Epsilon-Unfolding Orthogonal Polyhedra [article]

Mirela Damian, Robin Flatland, Joseph O'Rourke
2006 arXiv   pre-print
Here we prove, via an algorithm, that every orthogonal polyhedron (one whose faces meet at right angles) of genus zero may be unfolded.  ...  It is a long unsolved problem to determine whether every polyhedron may be unfolded.  ...  This concept has been used to achieve grid vertex unfoldings of orthostacks [DIL04] , and later, grid vertex unfoldings of all genus-zero orthogonal polyhedra [DFO06] .  ...

### Grid Vertex-Unfolding Orthostacks [chapter]

Erik D. Demaine, John Iacono, Stefan Langerman
2005 Lecture Notes in Computer Science
It was conjectured that orthostacks could be unfolded using cuts that lie in a plane orthogonal to a coordinate axis and containing a vertex of the orthostack.  ...  We prove the existence of a vertex-unfolding using only such cuts.  ...  We make partial progress on this problem by show- ing that every orthostack can be grid vertex-unfolded, i.e., cut along some of the grid lines and unfolded into a vertex-connected planar piece without  ...
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