Related papers: Dense packings of hard circular arcs
The self-assembly of hard polyhedral particles confined to a flat interface is studied using Monte Carlo simulations. The particles are pinned to the interface by restricting their movement in the direction perpendicular to it while…
It has been widely realized that Monte Carlo methods (approximation via a sample ensemble) may fail in large scale systems. This work offers some theoretical insight into this phenomenon in the context of the particle filter. We demonstrate…
The thesis concentrates on two problems in discrete geometry, whose solutions are obtained by analytic, probabilistic and combinatoric tools. The first chapter deals with the strong polarization problem. This states that for any sequence…
Three-dimensional discrete numerical simulation is used to investigate the properties of close-packed frictionless granular assemblies as a function of particle polydispersity and shape. Unlike some experimental results, simulations show…
We systematically investigate the self-assembly of anisotropic cone-shaped particles decorated by ring-like attractive patches. We demonstrate that the self-assembled clusters, which arise due to the conical particle's anisotropic shape…
We determine putative optimal packings of regular spherical polygons via optimization on smooth manifolds. For several cases, we establish maximality by extending the Lov\'asz theta number to Cayley graphs on the special orthogonal group…
The determination of the densest packings of regular tetrahedra (one of the five Platonic solids) is attracting great attention as evidenced by the rapid pace at which packing records are being broken and the fascinating packing structures…
A random packing of hard particles represents a fundamental model for granular matter. Despite its importance, analytical modeling of random packings remains difficult due to the existence of strong correlations which preclude the…
Let $\Delta$ be the optimal packing density of $\mathbb R^n$ by unit balls. We show the optimal packing density using two sizes of balls approaches $\Delta + (1 - \Delta) \Delta$ as the ratio of the radii tends to infinity. More generally,…
This thesis is a study of large sets of unit vectors in $\cx^n$ such that the absolute value of their standard inner products takes on only a small number of values. We begin with bounds: what is the maximal size of a set of lines with only…
The tangled closure of a collection of subsets of a topological space is the largest subset in which each member of the collection is dense. This operation models a logical `tangle modality' connective, of significance in finite model…
We use a Monte Carlo code to generate synthetic near-IR reflection nebulae that resemble those (normally associated with a bipolar outflow cavity) seen towards massive young stellar objects (YSOs). The 2D axi-symmetric calculations use an…
A collection of spherical particles can be packed tightly together into an amorphous packing known as "random close packing" (RCP). This structure is of interest as a model for the arrangement of molecules in simple liquids and glasses, as…
If a collection of identical particles is poured into a container, different shapes will fill to different densities. But what is the shape that fills a container as close as possible to a pre-specified, desired density? We demonstrate a…
This is the eighth and final paper in a series giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is…
The purpose of this work is to perform a statistical analysis of the location of compact groups in the Universe from observational and semi-analytical points of view. We used the velocity-filtered compact group sample extracted from the Two…
We address the question of which convex shapes, when packed as densely as possible under certain restrictions, fill the least space and leave the most empty space. In each different dimension and under each different set of restrictions,…
In this paper the excluded volume of binary similar hyperparticles with small size difference in D-dimensional Euclidean spaces R2, R3, and R4 is studied using two different statistical geometry approaches. These geometric approaches,…
Although the concept of random close packing with an almost universal packing fraction of ~ 0.64 for hard spheres was introduced more than half a century ago, there are still ongoing debates. The main difficulty in searching the densest…
In many singular metric spaces, the regularity of a shortest-length curve is unknown. Algebraic varieties, or more generally sets defined by finitely many polynomial or real analytic equalities or inequalities, all locally partition into…