Lattice packings with gap defects are not completely saturated (original) (raw)
2003
We show that a honeycomb circle packing in R 2 with a linear gap defect cannot be completely saturated, no matter how narrow the gap is. The result is motivated by an open problem of G. Fejes Toth, G. Kuperberg, and W. Kuperberg, which asks whether of a honeycomb circle packing with a linear shift defect is completely saturated. We also show that an fcc sphere packing in R 3 with a planar gap defect is not completely saturated.
Further lattice packings in high dimensions
Mathematika, 1982
Barnes and Sloane recently described a “general construction” for lattice packings of equal spheres in Euclidean space. In the present paper we simplify and further generalize their construction, and make it suitable for iteration. As a result we obtain lattice packings in ℝm with density Δ satisfying , as m → ∞ where is the smallest value of k for which the k-th iterated logarithm of m is less than 1. These appear to be the densest lattices that have been explicitly constructed in high-dimensional space. New records are also established in a number of lower dimensions, beginning in dimension 96.
International Journal of Mathematics and Mathematical Sciences, 2005
Given a bounded sequence of integers {d 0 ,d 1 ,d 2 ,...}, 6 ≤ d n ≤ M, there is an associated abstract triangulation created by building up layers of vertices so that vertices on the nth layer have degree d n . This triangulation can be realized via a circle packing which fills either the Euclidean or the hyperbolic plane. We give necessary and sufficient conditions to determine the type of the packing given the defining sequence {d n }.
Random perfect lattices and the sphere packing problem
Physical Review E, 2012
Motivated by the search for best lattice sphere packings in Euclidean spaces of large dimensions we study randomly generated perfect lattices in moderately large dimensions (up to d = 19 included). Perfect lattices are relevant in the solution of the problem of lattice sphere packing, because the best lattice packing is a perfect lattice and because they can be generated easily by an algorithm. Their number however grows super-exponentially with the dimension so to get an idea of their properties we propose to study a randomized version of the algorithm and to define a random ensemble with an effective temperature in a way reminiscent of a Montecarlo simulation. We therefore study the distribution of packing fractions and kissing numbers of these ensembles and show how as the temperature is decreased the best know packers are easily recovered. We find that, even at infinite temperature, the typical perfect lattices are considerably denser than known families (like A d and D d) and we propose two hypotheses between which we cannot distinguish in this paper: one in which they improve Minkowsky's bound φ ∼ 2 −(0.84±0.06)d , and a competitor, in which their packing fraction decreases super-exponentially, namely φ ∼ d −ad but with a very small coefficient a = 0.06 ± 0.04. We also find properties of the random walk which are suggestive of a glassy system already for moderately small dimensions. We also analyze local structure of network of perfect lattices conjecturing that this is a scale-free network in all dimensions with constant scaling exponent 2.6 ± 0.1.
On Lattice Packings and Coverings of Asymmetric Limited-Magnitude Balls
IEEE Transactions on Information Theory, 2021
We construct integer error-correcting codes and covering codes for the limited-magnitude error channel with more than one error. The codes are lattices that pack or cover the space with the appropriate error ball. Some of the constructions attain an asymptotic packing/covering density that is constant. The results are obtained via various methods, including the use of codes in the Hamming metric, modular Bt-sequences, 2-fold Sidon sets, and sets avoiding arithmetic progression.
Locally Optimal 2-Periodic Sphere Packings
Discrete & Computational Geometry, 2019
The sphere packing problem is an old puzzle. We consider packings with m spheres in the unit cell (m-periodic packings). For the case m = 1 (lattice packings), Voronoi proved there are finitely many inequivalent local optima and presented an algorithm to enumerate them, and this computation has been implemented in up to d = 8 dimensions. We generalize Voronoi's method to m > 1 and present a procedure to enumerate all locally optimal 2-periodic sphere packings in any dimension, provided there are finitely many. We implement this computation in d = 3, 4, and 5 and show that no 2-periodic packing surpasses the density of the optimal lattices in these dimensions. A partial enumeration is performed in d = 6.
Honeycomb lattices with defects
Physical Review E, 2016
In this paper we introduce a variant of the honeycomb lattice in which we create defects by randomly exchanging adjacent bonds, producing a random tiling with a distribution of polygon edges. We study the percolation properties on these lattices as a function of the number of exchanged bonds using a novel computational method. We find the site and bond percolation thresholds are consistent with other three-coordinated lattices with the same standard deviation in the degree distribution of the dual; here we can produce a continuum of lattices with a range of standard deviations in the distribution. These lattices should be useful for modeling other properties of random systems as well as percolation.
The Local Optimality of the Double Lattice Packing
Discrete and Computational Geometry, 2016
This paper introduces a technique for proving the local optimality of packing configurations. Applying this technique to a general convex polygon, we prove that the construction of the optimal double lattice packing by Kuperberg and Kuperberg is also locally optimal in the full space of packings.
On the Lattice Packings and Coverings of Convex
2014
It is well known that the lattice packing density and the lattice covering density of a triangle are 2 3 and 3 2 respectively [3]. We also know that the lattices that attain these densities both are unique. Let δL(K) and ϑL(K) denote the lattice packing density and the lattice covering density of K, respectively. In this paper, I study the lattice packings and coverings for a special class of convex disks, which includes all triangles and convex quadrilaterals. In particular, I determine the densities δL(Q) and ϑL(Q), where Q is an arbitrary convex quadrilateral. Furthermore, I also obtain all of lattices that attain these densities. Finally, I show that δL(Q)ϑL(Q) ≥ 1 and 1 δ L (Q) + 1 ϑ L (Q) ≥ 2, for each convex quadrilateral Q.
JAMS 1 Sphere Packing: asymptotic behavior and existence of solution
2003
Abstract. Lattices in n-dimensional Euclidean spaces may be parameterized by the non-compact symmetric space SL(n, R)/SO(n, R). We consider sphere packings determined by lattices and study the density function in the symmetric space, showing that the density function ρ(Ak) decreases to 0 if Ak is a sequence of matrices in SL(n, R) with limk→ ∞ ‖Ak ‖ = ∞. As a consequence, we give a simple prove that the optimal solution for the sphere packing problem is attained. The sphere packing problem is one of the famous open problems in mathematics. In short, it asks about the densest way a set of equal spheres can be packed in space n-dimensional Euclidean space R n, without overlapping one the other. In this context, the density means the proportion between the covered and the uncovered amount of space. It has many variations: one could replace spheres of equal radii by spheres of radii 0 < a ≤ r ≤ b bounded from above and below, replace spheres by a collection of identical (preferably c...
Random Lattices and Random Sphere Packings: Typical Properties
Moscow Mathematical Journal
We review results about the density of typical lattices in R n . They state that such density is of the order of 2 −n . We then obtain similar results for random packings in R n : after taking suitably a fraction ν of a typical random packing σ, the resulting packing τ has density C (ν) 2 −n , with a reasonable C (ν) . We obtain estimates on C (ν) .
The packing of two species of polygons on the square lattice
Journal of Physics A: Mathematical and General, 2004
We decorate the square lattice with two species of polygons under the constraint that every lattice edge is covered by only one polygon and every vertex is visited by both types of polygons. We end up with a 24 vertex model which is known in the literature as the fully packed double loop model (FPL 2 ). In the particular case in which the fugacities of the polygons are the same, the model admits an exact solution. The solution is obtained using coordinate Bethe ansatz and provides a closed expression for the free energy. In particular we find the free energy of the four colorings model and the double Hamiltonian walk and recover the known entropy of the Ice model. When both fugacities are set equal to two the model undergoes an infinite order phase transition.
Finite and Uniform Stability of Sphere Packings
Discrete & Computational Geometry, 1998
The main purpose of this paper is to discuss how firm or steady certain known ball packing are, thinking of them as structures. This is closely related to the property of being locally maximally dense. Among other things we show that many of the usual best-known candidates, for the most dense packings with congruent spherical balls, have the property of being uniformly stable, i.e., for a sufficiently small ε > 0 every finite rearrangement of the balls of this packing, where no ball is moved more than ε, is the identity rearrangement. For example, the lattice packings D d and A d for d ≥ 3 in E d are all uniformly stable. The methods developed here can work for many other packings as well. We also give a construction to show that the densest cubic lattice ball packing in E d for d ≥ 2 is not uniformly stable.
Uniqueness and Symmetry in Problems of Optimally Dense Packings
2003
Part of Hilbert's eighteenth problem is to classify the symmetries of the densest packings of bodies in Euclidean and hyperbolic spaces, for instance the densest packings of balls or simplices. We prove that when such a packing problem has a unique solution up to congruence then the solution must have cocompact symmetry group, and we prove that the densest packing of unit disks in the Euclidean plane is unique up to congruence. We also analyze some densest packings of polygons in the hyperbolic plane.
Optimal substructures in optimal and approximate circle packings
Contributions to Algebra and Geometry, 2005
This paper deals with the densest packing of equal circles in a square problem. Sharp bounds for the density of optimal circle packings have given. Several known optimal and approximate circle packings contain optimal substructures. Based on this observation it is sometimes easy to determine the minimal polynomials of the arrangements.
On the Lattice Packings and Coverings of the Plane with Convex Quadrilaterals
arXiv (Cornell University), 2014
It is well known that the lattice packing density and the lattice covering density of a triangle are 2 3 and 3 2 respectively [3]. We also know that the lattices that attain these densities both are unique. Let δL(K) and ϑL(K) denote the lattice packing density and the lattice covering density of K, respectively. In this paper, I study the lattice packings and coverings for a special class of convex disks, which includes all triangles and convex quadrilaterals. In particular, I determine the densities δL(Q) and ϑL(Q), where Q is an arbitrary convex quadrilateral. Furthermore, I also obtain all of lattices that attain these densities. Finally, I show that δL(Q)ϑL(Q) ≥ 1 and 1 δ L (Q) + 1 ϑ L (Q) ≥ 2, for each convex quadrilateral Q.