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We consider the number of configurations of a surface in two dimensions that has a prescribed length and encloses a prescribed perimeter with respect to a baseline. An approximate analytical treatment in a semi--continuum compares…
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…
A Hamiltonian cycle of a graph is a closed path that visits every vertex once and only once. It serves as a model of a compact polymer on a lattice. I study the number of Hamiltonian cycles, or equivalently the entropy of a compact polymer,…
Discrete conformal mappings based on circle packing, vertex scaling, and related structures has had significant activity since Thurston proposed circle packing as a way to approximate conformal maps in the 1980s. The first convergence…
In this article, we investigate the convergence behavior of two classes of gathering protocols with fixed circulant topologies using tools from dynamical systems. Given a fixed number of mobile entities moving in the Euclidean plane, we…
Sphere packing problems have a rich history in both mathematics and physics; yet, relatively few analytical analyses of sphere packings exist, and answers to seemingly simple questions are unknown. Here, we present an analytical method for…
We use numerical simulation to investigate and analyze the way that rigid disks and spheres arrange themselves when compressed next to incommensurate substrates. For disks, a movable set is pressed into a jammed state against an ordered…
The optimal circle coverage problem aims to find a configuration of circles that maximizes the covered area within a given region. Although theoretical optimal solutions exist for simple cases, the problem's NP-hard characteristic makes the…
Hard spheres are ubiquitous in condensed matter: they have been used as models for liquids, crystals, colloidal systems, granular systems, and powders. Packings of hard spheres are of even wider interest, as they are related to important…
Porous and heterogeneous materials are found in many applications from composites, membranes, chemical reactors, and other engineered materials to biological matter and natural subsurface structures. In this work we propose an integrated…
We study the response of a two-dimensional hexagonal packing of rigid, frictionless spherical grains due to a vertically downward point force on a single grain at the top layer. We use a statistical approach, where each configuration of the…
Granular packings of non-convex or elongated particles can form free-standing structures like walls or arches. For some particle shapes, such as staples, the rigidity arises from interlocking of pairs of particles, but the origins of…
This review paper is devoted to the problems of sphere packings in 4 dimensions. The main goal is to find reasonable approaches for solutions to problems related to densest sphere packings in 4-dimensional Euclidean space. We consider two…
We consider three-dimensional clusters of identical bubbles packed around a central bubble and calculate their energy and optimal shape. We obtain the surface area and bubble pressures to improve on existing growth laws for…
Consider a general circle packing $\mathcal{P}$ in the complex plane $\mathbb{C}$ invariant under a Kleinian group $\Gamma$. When $\Gamma$ is convex-cocompact or its critical exponent is greater than 1, we obtain an effective…
For the binary discs packed in two dimensions, the packing fraction of disc assembly becomes lower than that of the monodisperse system when the size ratio is close to unity. We show that the suppressed packing fraction is caused by an…
The structural properties of packed soft-core particles provide a platform to understand the cross-pollinated physical concepts in solid-state- and soft-matter physics. Confined on spherical surface, the traditional differential geometry…
Understanding fluid flow through porous media with complex geometries is essential for improving the design and operation of packed-bed reactors. Most existing studies focus on spherical packings, having as a consequence that accurate…
Theoretical background is provided towards the mathematical foundation of the minimum enclosing ball problem. This problem concerns the determination of the unique spherical surface of smallest radius enclosing a given bounded set in the…
Discs form a compact packing of the plane if they are interior disjoint and the graph which connects the center of mutually tangent discs is triangulated. There is only one compact packing by discs all of the same size, called hexagonal…