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Discrete Mathematics and Combinatorics Commons™
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Articles 1 - 6 of 6
Full-Text Articles in Discrete Mathematics and Combinatorics
Common Edge-Unzippings For Tetrahedra, Joseph O'Rourke
Common Edge-Unzippings For Tetrahedra, Joseph O'Rourke
Computer Science: Faculty Publications
It is shown that there are examples of distinct polyhedra, each with a Hamiltonian path of edges, which when cut, unfolds the surfaces to a common net. In particular, it is established for infinite classes of triples of tetrahedra.
Continuous Blooming Of Convex Polyhedra, Erik D. Demaine, Martin L. Demaine, Vi Hart, Joan Iacono, Stefan Langerman, Joseph O'Rourke
Continuous Blooming Of Convex Polyhedra, Erik D. Demaine, Martin L. Demaine, Vi Hart, Joan Iacono, Stefan Langerman, Joseph O'Rourke
Computer Science: Faculty Publications
We construct the first two continuous bloomings of all convex polyhedra. First, the source unfolding can be continuously bloomed. Second, any unfolding of a convex polyhedron can be refined (further cut, by a linear number of cuts) to have a continuous blooming.
Conical Existence Of Closed Curves On Convex Polyhedra, Joseph O'Rourke, Costin Vîlcu
Conical Existence Of Closed Curves On Convex Polyhedra, Joseph O'Rourke, Costin Vîlcu
Computer Science: Faculty Publications
Let C be a simple, closed, directed curve on the surface of a convex polyhedron P. We identify several classes of curves C that "live on a cone," in the sense that C and a neighborhood to one side may be isometrically embedded on the surface of a cone Lambda, with the apex a of Lambda enclosed inside (the image of) C; we also prove that each point of C is "visible to" a. In particular, we obtain that these curves have non-self-intersecting developments in the plane. Moreover, the curves we identify that live on cones to both sides support …
Convex Polyhedra Realizing Given Face Areas, Joseph O'Rourke
Convex Polyhedra Realizing Given Face Areas, Joseph O'Rourke
Computer Science: Faculty Publications
Given n ≥ 4 positive real numbers, we prove in this note that they are the face areas of a convex polyhedron if and only if the largest number is not more than the sum of the others.
A Note On Solid Coloring Of Pure Simplicial Complexes, Joseph O'Rourke
A Note On Solid Coloring Of Pure Simplicial Complexes, Joseph O'Rourke
Computer Science: Faculty Publications
We establish a simple generalization of a known result in the plane. The simplices in any pure simplicial complex in Rd may be colored with d+1 colors so that no two simplices that share a (d-1)-facet have the same color. In R2 this says that any planar map all of whose faces are triangles may be 3-colored, and in R3 it says that tetrahedra in a collection may be "solid 4-colored" so that no two glued face-to-face receive the same color.
Some Properties Of Yao Y4 Subgraphs, Joseph O'Rourke
Some Properties Of Yao Y4 Subgraphs, Joseph O'Rourke
Computer Science: Faculty Publications
The Yao graph for k = 4, Y4, is naturally partitioned into four subgraphs, one per quadrant. We show that the subgraphs for one quadrant differ from the subgraphs for two adjacent quadrants in three properties: planarity, connectedness, and whether the directed graphs are spanners.