We generalize Cauchy’s celebrated theorem on the global rigidity of convex polyhedra in Euclidean 3-space E to the context of circle polyhedra in the 2-sphere S. We prove that any two convex and proper non-unitary c-polyhedra with Möbiuscongruent faces that are consistently oriented are Möbius-congruent. Our result implies the global rigidity of convex inversive distance circle packings in the ...

We give an elementary construction of polyhedra whose links are connected bipartite graphs, which are not necessarily isomorphic pairwise. We show, that the fundamental groups of some of our polyhedra contain surface groups. In particular, we construct polyhedra whose links are generalized m-gons. The polyhedra of this type are interesting because of their universal coverings, which are two-dim...

Motivated by recent experiments on bilayer polyhedra composed of amphiphilic molecules, we study the elastic bending energies of bilayer vesicles forming polyhedral shapes. Allowing for segregation of excess amphiphiles along the ridges of polyhedra, we find that bilayer polyhedra can indeed have lower bending energies than spherical bilayer vesicles. However, our analysis also implies that, co...

An earlier paper describes a program to prove the Kepler conjecture on sphere packings. This paper carries out the second step of that program. A sphere packing leads to a decomposition ofR3 into polyhedra. The polyhedra are divided into two classes. The first class of polyhedra, called quasi-regular tetrahedra, have density at most that of a regular tetrahedron. The polyhedra in the remaining ...

Polyhedra are an important basic structure in computational geometry. One of the most beautiful results concerning polyhedra is Cauchy’s theorem, which states that a convex polyhedron is uniquely defined by its graph, edge lengths and facial angles. (See Section 2 for definitions.) The proof of Cauchy’s theorem (see e.g. [2]) unfortunately is nonconstructive, and the only known algorithm to rec...

It is known that not all simple polyhedra can be tetrahedralized, i.e., decomposed into a set of tetrahedra without adding new vertices (tetrahedralization). We investigate several classes of simple and non-simple polyhedra that admit such decompositions. In particular, we show that certain classes of rectilinear (isothetic) simple polyhedra can always be tetrahedralized in O(n2) time where n i...

We make a case for sub-polyhedral scheduling using (Unit-)TwoVariable-Per-Inequality or (U)TVPI Polyhedra. We empirically show that using general convex polyhedra leads to a scalability problem in a widely used polyhedral scheduler. We propose methods in which polyhedral schedulers can beat the scalability challenge by using sub-polyhedral under-aproximations of the polyhedra resulting from the...

An earlier paper describes a program to prove the Kepler conjecture on sphere packings. This paper carries out the second step of that program. A sphere packing leads to a decomposition of R into polyhedra. The polyhedra are divided into two classes. The first class of polyhedra, called quasi-regular tetrahedra, have density at most that of a regular tetrahedron. The polyhedra in the remaining ...

Relations between graph theory and polyhedra are presented in two contexts. In the first, the symbiotic dependence between 3-connected planar graphs and convex polyhedra is described in detail. In the second, a theory of nonconvex polyhedra is based on a graph-theoretic foundation. This approach eliminates the vagueness and inconsistency that pervade much of the literature dealing with polyhedr...

In this paper we present a new, efficient, outputsensitive method for computing the visibility map of a set of axis-parallel polyhedra (i.e., polyhedra with their faces and edges parallel to the coordinate axes) as seen from a given viewpoint. For non-intersecting polyhedra with n edges in total, the algorithm runs in time O( (n + k) log n), where k is the complexity of the visibility map. The ...