Chiral polyhedra in ordinary euclidean space E3 are nearly regular polyhedra; their geometric symmetry groups have two orbits on the flags, such that adjacent flags are in distinct orbits. This paper completely enumerates the discrete infinite chiral polyhedra in E3 with finite skew faces and finite skew vertex-figures. There are several families of such polyhedra of types {4, 6}, {6, 4} and {6...

This article defines a pair of combinatorial operations on the combinatorial structure of compact right-angled hyperbolic polyhedra in dimension three called decomposition and edge surgery. It is shown that these operations simplify the combinatorics of such a polyhedron, while keeping it within the class of right-angled objects, until it is a disjoint union of Löbell polyhedra, a class of poly...

In this paper we investigate timed polyhedra, i.e. polyhedra which are finite unions of full dimensional simplices of a special kind. Such polyhedra form the basis of timing analysis and in particular of verification tools based on timed automata. We define a representation scheme for these polyhedra based on their extreme vertices, and show that this compact representation scheme is canonical ...

We introduce a novel method for constructing polyhedra by blending together two or more existing polyhedra, using a dualization method. The method results in a Minkowski sum, analogous to that of overlay tilings [5], which uses a topological dual operation to create tilings from a network (a graph embedded in a plane). This paper extends that method to networks on the surface of the unit sphere...

We present the results of an investigation into the representations of Archimedean polyhedra (those polyhedra containing only one type of vertex gure) as quotients of regular abstract polytopes. Two methods of generating these presentations are discussed, one of which may be applied in a general setting, and another which makes use of a regular polytope with the same automorphism group as the d...

We study various classes of polyhedra that can be clamped using parallel jaw grippers. We show that all n-vertex convex polyhedra can be clamped regardless of the gripper size and present an O(n+k) time algorithm to compute all positions of a polyhedron that allow a valid clamp where k is the number of antipodal pairs of features. We also observe that all terrain polyhedra and orthogonal polyhe...

Let L be a family of lattice-free polyhedra in R containing the splits. Given a polyhedron P in R, we characterize when a valid inequality for P ∩(Z×R) can be obtained with a finite number of disjunctive cuts corresponding to the polyhedra in L. We also characterize the lattice-free polyhedra M such that all the disjunctive cuts corresponding to M can be obtained with a finite number of disjunc...