56,455 research outputs found
Rigid ball-polyhedra in Euclidean 3-space
A ball-polyhedron is the intersection with non-empty interior of finitely
many (closed) unit balls in Euclidean 3-space. One can represent the boundary
of a ball-polyhedron as the union of vertices, edges, and faces defined in a
rather natural way. A ball-polyhedron is called a simple ball-polyhedron if at
every vertex exactly three edges meet. Moreover, a ball-polyhedron is called a
standard ball-polyhedron if its vertex-edge-face structure is a lattice (with
respect to containment). To each edge of a ball-polyhedron one can assign an
inner dihedral angle and say that the given ball-polyhedron is locally rigid
with respect to its inner dihedral angles if the vertex-edge-face structure of
the ball-polyhedron and its inner dihedral angles determine the ball-polyhedron
up to congruence locally. The main result of this paper is a Cauchy-type
rigidity theorem for ball-polyhedra stating that any simple and standard
ball-polyhedron is locally rigid with respect to its inner dihedral angles.Comment: 11 pages, 2 figure
Vertex-Facet Incidences of Unbounded Polyhedra
How much of the combinatorial structure of a pointed polyhedron is contained
in its vertex-facet incidences? Not too much, in general, as we demonstrate by
examples. However, one can tell from the incidence data whether the polyhedron
is bounded. In the case of a polyhedron that is simple and "simplicial," i.e.,
a d-dimensional polyhedron that has d facets through each vertex and d vertices
on each facet, we derive from the structure of the vertex-facet incidence
matrix that the polyhedron is necessarily bounded. In particular, this yields a
characterization of those polyhedra that have circulants as vertex-facet
incidence matrices.Comment: LaTeX2e, 14 pages with 4 figure
Epsilon-Unfolding Orthogonal Polyhedra
An unfolding of a polyhedron is produced by cutting the surface and
flattening to a single, connected, planar piece without overlap (except
possibly at boundary points). It is a long unsolved problem to determine
whether every polyhedron may be unfolded. Here we prove, via an algorithm, that
every orthogonal polyhedron (one whose faces meet at right angles) of genus
zero may be unfolded. Our cuts are not necessarily along edges of the
polyhedron, but they are always parallel to polyhedron edges. For a polyhedron
of n vertices, portions of the unfolding will be rectangular strips which, in
the worst case, may need to be as thin as epsilon = 1/2^{Omega(n)}.Comment: 23 pages, 20 figures, 7 references. Revised version improves language
and figures, updates references, and sharpens the conclusio
Faces, Edges, Vertices of Some Polyhedra
A proof that: for any given polyhedron so shaped that every closed non-self intersecting broken line composed of edges of the polyhedron divides the surface of the polyhedron into precisely two disjoint regions each of which is bounded by the closed broken line, v - e + f = 2, where v is the number of vertices of the polyhedron, e the number of edges and f the number of faces
An infinitesimally nonrigid polyhedron with nonstationary volume in the Lobachevsky 3-space
We give an example of an infinitesimally nonrigid polyhedron in the
Lobachevsky 3-space and construct an infinitesimal flex of that polyhedron such
that the volume of the polyhedron isn't stationary under the flex.Comment: 10 pages, 2 Postscript figure
Volume estimates for equiangular hyperbolic Coxeter polyhedra
An equiangular hyperbolic Coxeter polyhedron is a hyperbolic polyhedron where
all dihedral angles are equal to \pi/n for some fixed integer n at least 2. It
is a consequence of Andreev's theorem that either n=3 and the polyhedron has
all ideal vertices or that n=2. Volume estimates are given for all equiangular
hyperbolic Coxeter polyhedra.Comment: 27 pages, 11 figures; corrected typo in Theorem 2.
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