Chiral polyhedra in ordinary space, II

A chiral polyhedron has a geometric symmetry group with two orbits on the flags, such that adjacent flags are in distinct orbits. Part I of the paper described the discrete chiral polyhedra in ordinary Euclidean 3-space with finite skew faces and finite skew vertex-figures; they occur in infinite families and are of types {4,6}, {6,4} and {6,6}. Part II completes the enumeration of all discrete chiral polyhedra in 3-space. There exist several families of chiral polyhedra with infinite, helical faces. In particular, there are no discrete chiral polyhedra with finite faces in addition to those described in Part I.Comment: 48 page

The complete classification of five-dimensional Dirichlet-Voronoi polyhedra of translational lattices

In this paper we report on the full classification of Dirichlet-Voronoi polyhedra and Delaunay subdivisions of five-dimensional translational lattices. We obtain a complete list of $110244$ affine types (L-types) of Delaunay subdivisions and it turns out that they are all combinatorially inequivalent, giving the same number of combinatorial types of Dirichlet-Voronoi polyhedra. Using a refinement of corresponding secondary cones, we obtain $181394$ contraction types. We report on details of our computer assisted enumeration, which we verified by three independent implementations and a topological mass formula check.Comment: 16 page

Stanley's Major Contributions to Ehrhart Theory

This expository paper features a few highlights of Richard Stanley's extensive work in Ehrhart theory, the study of integer-point enumeration in rational polyhedra. We include results from the recent literature building on Stanley's work, as well as several open problems.Comment: 9 pages; to appear in the 70th-birthday volume honoring Richard Stanle

Computing Groebner Fans

This paper presents algorithms for computing the Groebner fan of an arbitrary polynomial ideal. The computation involves enumeration of all reduced Groebner bases of the ideal. Our algorithms are based on a uniform definition of the Groebner fan that applies to both homogeneous and non-homogeneous ideals and a proof that this object is a polyhedral complex. We show that the cells of a Groebner fan can easily be oriented acyclically and with a unique sink, allowing their enumeration by the memory-less reverse search procedure. The significance of this follows from the fact that Groebner fans are not always normal fans of polyhedra in which case reverse search applies automatically. Computational results using our implementation of these algorithms in the software package Gfan are included.Comment: 26 page

Polygonal Complexes and Graphs for Crystallographic Groups

The paper surveys highlights of the ongoing program to classify discrete polyhedral structures in Euclidean 3-space by distinguished transitivity properties of their symmetry groups, focussing in particular on various aspects of the classification of regular polygonal complexes, chiral polyhedra, and more generally, two-orbit polyhedra.Comment: 21 pages; In: Symmetry and Rigidity, (eds. R.Connelly, A.Ivic Weiss and W.Whiteley), Fields Institute Communications, to appea