相关论文: Densest plane group packings of regular polygons
We have previously explored cylindrical packings of disks and their relation to sphere packings. Here we extend the analytical treatment of disk packings, analysing the rules for phyllotactic indices of related structures and the variation…
A relaxed version of Gummelt's covering rules for the aperiodic decagon is considered, which produces certain random-tiling-type structures. These structures are precisely characterized, along with their relationships to various other…
Solution and analysis of mathematical programming problems may be simplified when these problems are symmetric under appropriate linear transformations. In particular, a knowledge of the symmetries may help reduce the problem dimension, cut…
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…
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, we study the problem of hyperball (hypersphere) packings in $n$-dimensional hyperbolic space ($n \ge 4$). We prove that to each $n$-dimensional congruent saturated hyperball packing, there is an algorithm to obtain a…
In the packing-constrained point covering problem, PC^2, one seeks configurations of points in the plane that cannot all be covered by a packing arrangement of unit disks. We consider in particular the problem of finding the minimum number…
We examine packing of $n$ congruent spheres in a cube when $n$ is close but less than the number of spheres in a regular cubic close-packed (ccp) arrangement of $\lceil p^{3}/2\rceil$ spheres. For this family of packings, the previous…
Obtaining general relations between macroscopic properties of random assemblies, such as density, and the microscopic properties of their constituent particles, such as shape, is a foundational challenge in the study of amorphous materials.…
A polygonal complex in euclidean 3-space is a discrete polyhedron-like structure with finite or infinite polygons as faces and finite graphs as vertex-figures, such that a fixed number r of faces surround each edge. It is said to be regular…
Polytopes are the basic finite data structures for convex sets: they appear as feasible regions in linear optimization, as geometric summaries in algorithms, and as random objects in stochastic geometry. A natural geometric question is…
Determining the minimum density of a covering of $\mathbb{R}^{n}$ by Euclidean unit balls as $n\to\infty$ is a major open problem, with the best known results being the lower bound of $\left(\mathrm{e}^{-3/2}+o(1)\right)n$ by Coxeter, Few…
Nearly orthogonal lattices were formally defined in [4], where their applications to image compression were also discussed. The idea of ``near orthogonality" in $2$-dimensions goes back to the work of Gauss. In this paper, we focus on…
Higher-dimensional orthogonal packing problems have a wide range of practical applications, including packing, cutting, and scheduling. Combining the use of our data structure for characterizing feasible packings with our new classes of…
Suppose one has a collection of disks of various sizes with disjoint interiors, a packing in the plane, and suppose the ratio of the smallest radius divided by the largest radius lies between $1$ and $q$. In his 1964 book Regular Figures…
We describe an algorithm that allows one to find dense packing configurations of a number of congruent disks in arbitrary domains in two or more dimensions. We have applied it to a large class of two dimensional domains such as rectangles,…
This is the sixth in a series of papers 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…
Fitting an unknown number of hyperplanes to data is a fundamental yet challenging problem in machine learning, characterized by its non-convexity, non-differentiability, and unknown model order. Existing approaches often struggle with local…
We describe a new numerical procedure for generating dense packings of disks and spheres inside various geometric shapes. We believe that in some of the smaller cases, these packings are in fact optimal. When applied to the previously…
In \cite{Sz13-1} we defined and described the {\it regular infinite or bounded} $p$-gonal prism tilings in $\SLR$ space. We proved that there exist infinitely many regular infinite $p$-gonal face-to-face prism tilings $\cT^i_p(q)$ and…