Non-connected toric Hilbert schemes

We construct small (50 and 26 points, respectively) point sets in dimension 5 whose graphs of triangulations are not connected. These examples improve our construction in J. Amer. Math. Soc., 13:3 (2000), 611--637 not only in size, but also in that their toric Hilbert schemes are not connected either, a question left open in that article. Additionally, the point sets can easily be put into convex position, providing examples of 5-dimensional polytopes with non-connected graph of triangulations.Comment: 18 pages, 2 figures. Except for Remark 2.6 (see below) changes w.r.t. version 2 are mostly minor editings suggested by an anonimous referee of "Mathematische Annalen". The paper has been accepted in that journal. Most of the contents of Remark 2.6 have been deleted, since there was a flaw in the argumen

On Monotone Sequences of Directed Flips, Triangulations of Polyhedra, and Structural Properties of a Directed Flip Graph

This paper studied the geometric and combinatorial aspects of the classical Lawson's flip algorithm in 1972. Let A be a finite set of points in R2, omega be a height function which lifts the vertices of A into R3. Every flip in triangulations of A can be associated with a direction. We first established a relatively obvious relation between monotone sequences of directed flips between triangulations of A and triangulations of the lifted point set of A in R3. We then studied the structural properties of a directed flip graph (a poset) on the set of all triangulations of A. We proved several general properties of this poset which clearly explain when Lawson's algorithm works and why it may fail in general. We further characterised the triangulations which cause failure of Lawson's algorithm, and showed that they must contain redundant interior vertices which are not removable by directed flips. A special case if this result in 3d has been shown by B.Joe in 1989. As an application, we described a simple algorithm to triangulate a special class of 3d non-convex polyhedra. We proved sufficient conditions for the termination of this algorithm and show that it runs in O(n3) time.Comment: 40 pages, 35 figure

Many projectively unique polytopes

We construct an infinite family of 4-polytopes whose realization spaces have dimension smaller or equal to 96. This in particular settles a problem going back to Legendre and Steinitz: whether and how the dimension of the realization space of a polytope is determined/bounded by its f-vector. From this, we derive an infinite family of combinatorially distinct 69-dimensional polytopes whose realization is unique up to projective transformation. This answers a problem posed by Perles and Shephard in the sixties. Moreover, our methods naturally lead to several interesting classes of projectively unique polytopes, among them projectively unique polytopes inscribed to the sphere. The proofs rely on a novel construction technique for polytopes based on solving Cauchy problems for discrete conjugate nets in S^d, a new Alexandrov--van Heijenoort Theorem for manifolds with boundary and a generalization of Lawrence's extension technique for point configurations.Comment: 44 pages, 18 figures; to appear in Invent. mat

On monotone sequences of directed flips, triangulations of polyhedra, and structural properties of a directed flip graph

This paper studied the geometric and combinatorial aspects of the classical Lawson's flip algorithm  [21, 22]. Let A be a finite point set in R^2 and ω : A → R be a height function which lifts the vertices of A into R^3. Every flip in triangulations of A can be assigned a direction [6, Definition 6.1.1]. A sequence of directed flips is monotone if all its flips follow the same direction. We first established a relatively obvious relation between monotone sequences of directed flips on triangulations of A and triangulations of the lifted point set A^ω in R^3. We then studied the structural properties of a directed flip graph (a poset) on the set of all triangulations of A. We proved several general properties of this poset which clearly explain when Lawson's algorithm works and why it may fail in general. We further characterised the triangulations which cause failure of Lawson's algorithm, and showed that they must contain redundant interior vertices which are not removable by directed flips. A special case of this result in 3d has been shown in [19]. As an application, we described a simple algorithm to triangulate a special class of 3d non-convex polyhedra without using additional vertices. We prove sufficient conditions for the termination of this algorithm, and show it runs in O(n^3) time, where $n$ is the number of input vertices

Triangulations With Very Few Geometric Bistellar Neighbors

We are interested in a notion of elementary change between triangulations of a point configuration, the so-called bistellar flips, introduced by Gel'fand, Kapranov and Zelevinski. We construct sequences of triangulations of point configurations in dimension 3 with n 2 +2n+2 vertices and only 4n \Gamma 3 geometric bistellar flips (for every even integer n), and of point configurations in dimension 4 with arbitrarily many vertices and a bounded number of flips. This drastically improves previous examples and seems to be evidence against the conjecture that any two triangulations of a point configuration can be joined by a sequence of flips. Introduction Given a finite point configuration A in the Euclidean space R d of dimension d we call triangulations of A all the geometrically realized simplicial complexes which cover the convex hull of A and which have their sets of vertices contained in A. In this paper we are interested in a notion of vicinity or elementary change between tri..

On the behavior of spherical and non-spherical grain assemblies, its modeling and numerical simulation

This thesis deals with the numerical modeling and simulation of granular media with large populations of non-spherical particles. Granular media are highly pervasive in nature and play an important role in technology. They are present in fields as diverse as civil engineering, food processing, and the pharmaceutical industry. For the physicist, they raise many challenging questions. They can behave like solids, as well as liquids or even gases and at times as none of these. Indeed, phenomena like granular segregation, arching effects or pattern formation are specific to granular media, hence often they are considered as a fourth state of matter. Around the turn of the century, the increasing availability of large computers made it possible to start investigating granular matter by using numerical modeling and simulation. Most numerical models were originally designed to handle spherical particles. However, making it possible to process non-spherical particles has turned out to be of utmost importance. Indeed, it is such grains that one finds in nature and many important phenomena cannot be reproduced just using spherical grains. This is the motivation for the research of the present thesis. Subjects in several fields are involved. The geometrical modeling of the particles and the simulation methods require discrete geometry results. A wide range of particle shapes is proposed. Those shapes, spheropolyhedra, are Minkowski sums of polyhedra and spheres and can be seen as smoothed polyhedra. Next, a contact detection algorithm is proposed that uses triangulations. This algorithm is a generalization of a method already available for spheres. It turns out that this algorithm relies on a positive answer to an open problem of computational geometry, the connectivity of the flip-graph of all triangulations. In this thesis it has been shown that the flip-graph of regular triangulations that share a same vertex set is connected. The modeling of contacts requires physics. Again the contact model we propose is based on the existing molecular dynamics model for contacts between spheres. Those models turn out to be easily generalizable to smoothed polyhedra, which further motivates this choice of particle shape. The implementation of those methods requires computer science. An implementation of this simulation methods for granular media composed of non-spherical particles was carried out based on the existing C++ code by J.-A. Ferrez that originally handled spherical particles. The resulting simulation code was used to gain insight into the behavior of granular matter. Three experiments are presented that have been numerically carried out with our models. The first of these experiments deals with the flowability (i. e. the ability to flow) of powders. The flowability of bidisperse bead assemblies was found to depend only on their mass-average diameters. Next, an experiment of vibrating rods inside a cylindrical container shows that under appropriate conditions they will order vertically. Finally, experiments investigating the shape segregation of sheres and spherotetrahedra are perfomed. Unexpectedly they are found to mix