In this paper we are concerned with three lattice problems: the lattice packing problem, the lattice covering problem and the lattice packing-covering problem. One way to find optimal lattices for these problems is to enumerate all finitely many, locally optimal lattices. For the lattice packing problem there are two classical algorithms going back to Minkowski and Voronoi. For the covering and for the packing-covering problem we propose new algorithms. Here we give a brief survey about these approaches. We report on some recent computer based computations where we were able to reproduce and partially extend the known classification of locally optimal lattices. Furthermore we found new record breaking covering and packing-covering lattices. We describe several methods with examples to show that a lattice is a locally optimal solution to one of the three problems.
We describe algorithms which address two classical problems in lattice geometry: the lattice covering and the simultaneous lattice packing-covering problem. Theoretically our algorithms solve the two problems in any fixed dimension d in the sense that they approximate optimal covering lattices and optimal packing-covering lattices within any desired accuracy. Both algorithms involve semidefinite programming and are based on Voronoi's reduction theory for positive definite quadratic forms, which describes all possible Delone triangulations of Z^d. In practice, our implementations reproduce know…
In this paper we discuss various special problems on packing and covering. Among others we survey the problems and results concerning finite arrangements, Minkowskian, saturated, compact, and totally separable packings. We discuss shortest path problems and questions about stability of packings.
The present work surveys problems in $n$-dimensional space with $n$ large. Early development in the study of packing and covering in high dimensions was motivated by the geometry of numbers. Subsequent results, such as the discovery of the Leech lattice and the linear programming bound, which culminated in the recent solution of the sphere packing problem in dimensions 8 and 24, were influenced by coding theory. After mentioning the known results concerning existence of economical packings and coverings we discuss the different methods yielding upper bounds for the density of packing congruent…
A packing by a body $K$ is collection of congruent copies of $K$ (in either Euclidean or hyperbolic space) so that no two copies intersect nontrivially in their interiors. A covering by $K$ is a collection of congruent copies of $K$ such that for every point $p$ in the space there is copy in the collection containing $p$. A completely saturated packing is one in which it is not possible to replace a finite number of bodies of the packing with a larger number and still remain a packing. A completely reduced covering is one in which it is not possible to replace a finite number of bodies of the …
This paper determines the optimal upper bound for the simultaneous packing and covering constants of the two-dimensional centrally symmetric convex domains. It solved a problem opening for more than thirty years.
We introduce and study certain notions which might serve as substitutes for maximum density packings and minimum density coverings. A body is a compact connected set which is the closure of its interior. A packing $\cal P$ with congruent replicas of a body $K$ is $n$-saturated if no $n-1$ members of it can be replaced with $n$ replicas of $K$, and it is completely saturated if it is $n$-saturated for each $n\ge 1$. Similarly, a covering $\cal C$ with congruent replicas of a body $K$ is $n$-reduced if no $n$ members of it can be replaced by $n-1$ replicas of $K$ without uncovering a portion of …
Supergroups of some hyperbolic space groups are classified as a continuation of our former works. Fundamental domains will be integer parts of truncated tetrahedra belonging to families F1 - F4, for a while, by the notation of E. Molnár et al. in $2006$. As an application, optimal congruent hyperball packings and coverings to the truncation base planes with their very good densities are computed. This covering density is better than the conjecture of L.~Fejes~Tóth for balls and horoballs in $1964$.
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 of points N for which such a configuration exists and obtain the bounds 11 <= N <= 55. The disparity of these bounds is symptomatic, we believe, of the fact that PC^2 belongs in a higher complexity class than the standard packing and covering problems.
This paper suveys different variants of the following problem: Given a convex set $K$ and a sequence $\{C_i\}$ of convex bodies in $E^n$, is it possible to pack the sequence of bodies in $K$ or cover $K$ with the bodies? Algorithmic versions of these problems are on-line packing and on-line covering: The bodies of the sequence are given one at a time and the algorithm is to decide on the placement of the arriving body before the next body is revealed; once placed, the body cannot be moved.
We adapt a construction of Klee (1981) to find a packing of unit balls in $\ell_p$ ($1\leq p<\infty$) which is efficient in the sense that enlarging the radius of each ball to any $R>2^{1-1/p}$ covers the whole space. We show that the value $2^{1-1/p}$ is optimal.
It is well known that the lattice packing density and the lattice covering density of a triangle are $\frac{2}{3}$ and $\frac{3}{2}$ respectively. We also know that the lattices that attain these densities both are unique. Let $δ_{L}(K)$ and $\vartheta_{L}(K)$ 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 $δ_{L}(Q)$ and $\vartheta_{L}(Q)$, where $Q$ is an arbitr…
In this note we prove that for any compact subset $S$ of a Busemann surface $({\mathcal S},d)$ (in particular, for any simple polygon with geodesic metric) and any positive number $δ$, the minimum number of closed balls of radius $δ$ with centers at $\mathcal S$ and covering the set $S$ is at most 19 times the maximum number of disjoint closed balls of radius $δ$ centered at points of $S$: $ν(S) \le ρ(S) \le 19ν(S)$, where $ρ(S)$ and $ν(S)$ are the covering and the packing numbers of $S$ by $δ$-balls.
We prove that for every integer $t\ge 1$ there exists a constant $c_t$ such that for every $K_t$-minor-free graph $G$, and every set $S$ of balls in $G$, the minimum size of a set of vertices of $G$ intersecting all the balls of $S$ is at most $c_t$ times the maximum number of vertex-disjoint balls in $S$. This was conjectured by Chepoi, Estellon, and Vaxès in 2007 in the special case of planar graphs and of balls having the same radius.
In $n$-dimensional hyperbolic space $\mathbf{H}^n$ $(n\ge2)$ there are $3$-types of spheres (balls): the sphere, horosphere and hypersphere. If $n=2,3$ we know an universal upper bound of the ball packing densities, where each ball volume is related to the volume of the corresponding Dirichlet-Voronoi (D-V) cell. E.g. in $\mathbf{H}^3$ the densest horoball packing is derived from the $\{3,3,6\}$ Coxeter tiling consisting of ideal regular simplices $T_{reg}^\infty$ with dihedral angles $\fracπ{3}$. The density of this packing is $δ_3^\infty\approx 0.85328$ and this provides a very rough upper b…
For every convex disk $K$ (a convex compact subset of the plane, with non-void interior), the packing density $δ(K)$ and covering density $\vartheta(K)$ form an ordered pair of real numbers, {\em i.e.}, a point in ${\mathbb R}^2$. The set $Ω$ consisting of points assigned this way to all convex disks is the subject of this article. A few known inequalities on $δ(K)$ and $\vartheta(K)$ jointly outline a relatively small convex polygon $P$ that contains $Ω$, while the exact shape of $Ω$ remains a mystery. Here we describe explicitly a leaf-shaped convex region $Λ$ contained in $Ω$ and occupying …
A coincidence site lattice is a sublattice formed by the intersection of a lattice $Γ$ in $\mathbb{R}^d$ with the image of $Γ$ under a linear isometry. Such a linear isometry is referred to as a linear coincidence isometry of $Γ$. Here, we consider the more general case allowing any affine isometry. Consequently, general results on coincidence isometries of shifted copies of lattices, and of crystallographic point packings are obtained. In particular, we discuss the shifted square lattice and the diamond packing in detail.
A geometric study of twin and grain boundaries in crystals and quasicrystals is achieved via coincidence site lattices (CSLs) and coincidence site modules (CSMs), respectively. Recently, coincidences of shifted lattices and multilattices (i.e. finite unions of shifted copies of a lattice) have been investigated. Here, we solve the coincidence problem for a shifted hexagonal lattice. This result allows us to analyze the coincidence isometries of the hexagonal packing by viewing the hexagonal packing as a multilattice.
In [BL] in relation to the unsolved Bang's plank problem (1951) we obtained a lower bound for the sum of relevant measures of cylinders covering a given d-dimensional convex body. In this paper we provide the packing counterpart of these estimates. We also extend bounds to the case of r-fold covering and packing and show a packing analog of Falconer's results ([Fa]).
Motivated by long-standing conjectures on the discretization of classical inequalities in the Geometry of Numbers, we investigate a new set of parameters, which we call \emph{packing minima}, associated to a convex body $K$ and a lattice $Λ$. These numbers interpolate between the successive minima of $K$ and the inverse of the successive minima of the polar body of $K$, and can be understood as packing counterparts to the covering minima of Kannan & Lovász (1988). As our main results, we prove sharp inequalities that relate the volume and the number of lattice points in $K$ to the sequence of …
A linear isometry $R$ of $\mathbb{R}^d$ is called a similarity isometry of a lattice $Γ\subseteq \mathbb{R}^d$ if there exists a positive real number $β$ such that $βRΓ$ is a sublattice of (finite index in) $Γ$. The set $βRΓ$ is referred to as a similar sublattice of $Γ$. A (crystallographic) point packing generated by a lattice $Γ$ is a union of $Γ$ with finitely many shifted copies of $Γ$. In this study, the notion of similarity isometries is extended to point packings. We provide a characterization for the similarity isometries of point packings and identify the corresponding similar subpac…
Based on Minkowski's work on critical lattices of 3-dimensional convex bodies we present an efficient algorithm for computing the density of a densest lattice packing of an arbitrary 3-polytope. As an application we calculate densest lattice packings of all regular and Archimedean polytopes.
In this note we give a simple proof of the classical fact that the hexagonal lattice gives the highest density circle packing among all lattices in $R^2$. With the benefit of hindsight, we show that the problem can be restricted to the important class of well-rounded lattices, on which the density function takes a particularly simple form. Our proof emphasizes the role of well-rounded lattices for discrete optimization problems.
In this paper we prove an asymptotic lower bound for the sphere packing density in dimensions divisible by four. This asymptotic lower bound improves on previous asymptotic bounds by a constant factor and improves not just lower bounds for the sphere packing density, but also for the lattice sphere packing density and, in fact, the Hurwitz lattice sphere packing density.
It is shown that there is no quasi-sphere packing of the lattice grid Z^{d+1} or a co-compact hyperbolic lattice of H^{d+1} or the 3-regular tree \times Z, in R^d, for all d. A similar result is proved for some other graphs too. Rather than using a direct geometrical approach, the main tools we are using are from non-linear potential theory.
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.
We algorithmically characterize the maximal contact number problem for finite congruent lattice sphere packings in $\mathbb{R}^d$ and show that in $\mathbb{R}^3$ this problem is equivalent to determining the maximal coordination of a monatomic crystal.
We show that for every lattice packing of $n$-dimensional spheres there exists an $(n/\log_2(n))$-dimensional affine plane which does not meet any of the spheres in their interior, provided $n$ is large enough. Such an affine plane is called a free plane and our result improves on former bounds.
G.F. Voronoi (1868-1908) wrote two memoirs in which he describes two reduction theories for lattices, well-suited for sphere packing and covering problems. In his first memoir a characterization of locally most economic packings is given, but a corresponding result for coverings has been missing. In this paper we bridge the two classical memoirs. By looking at the covering problem from a different perspective, we discover the missing analogue. Instead of trying to find lattices giving economical coverings we consider lattices giving, at least locally, very uneconomical ones. We classify local …
After the investigation of the congruent and non-congruent hyperball packings related to doubly truncated Coxeter orthoscheme tilings \cite{SzJ1}, we consider the corresponding covering problems. In \cite{MSSz} the authors gave a partial classification of supergroups of some hyperbolic space groups whose fundamental domains will be integer parts of truncated tetrahedra, and determined the optimal congruent hyperball packing and covering configurations belonging to some of these classes. In this paper we compliment these results with the investigation of the non-congruent covering cases, and th…
A parametrization of integral Descartes configurations (and effectively Apollonian disk packings) by pairs of two-dimensional integral vectors is presented. The vectors, called here tangency spinors defined for pairs of tangent disks, are spinors associated to the Clifford algebra for three-dimensional Minkowski space. A version with Pauli spinors is given. The construction provides a novel interpretation to the known Diophantine equation parametrizing integral Apollonian packings.
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 Tóth, 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 also not completely saturated.
In this article, we will show the existence of lattice packings in a sparse family of dimensions. This construction will be a generalisation of Venkatesh's lattice packing result. In our construction, we replace the appearance of the cyclotomic number field with a division algebra over the rational field. For this, we develop an analogue of Siegel's mean value theorem over lattices that have a prescribed set of symmetries given by a finite non-commutative group inside the multiplicative subgroup of a division algebra. This approach improves the best known lattice packing bounds in many dimensi…
We give a new proof of the Minkowski-Hlawka bound on the existence of dense lattices. The proof is based on an elementary method for constructing dense lattices which is almost effective.
In this paper we construct a new family of lattice packings for superballs in three dimensions (unit balls for the $l^p_3$ norm) with $p \in (1, 1.58]$. We conjecture that the family also exists for $p \in (1.58, \log_2 3 = 1.5849625\ldots]$. Like in the densest lattice packing of regular octahedra, each superball in our family of lattice packings has $14$ neighbors.
Several conditions are given when a packing of equal disks in a torus is locally maximally dense, where the torus is defined as the quotient of the plane by a two-dimensional lattice. Conjectures are presented that claim that the density of any strictly jammed packings, whose graph does not consist of all triangles and the torus lattice is the standard triangular lattice, is at most $\frac{n}{n+1}\fracπ{\sqrt{12}}$, where $n$ is the number of packing disks. Several classes of collectively jammed packings are presented where the conjecture holds.
We reformulate the covering and quantizer problems as the determination of the ground states of interacting particles in $\mathbb{R}^d$ that generally involve single-body, two-body, three-body, and higher-body interactions. This is done by linking the covering and quantizer problems to certain optimization problems involving the "void" nearest-neighbor functions that arise in the theory of random media and statistical mechanics. These reformulations, which again exemplifies the deep interplay between geometry and physics, allow one now to employ theoretical and numerical optimization technique…
One of the basic problems in discrete geometry is to determine the most efficient packing of congruent replicas of a given convex set $K$ in the plane or in space. The most commonly used measure of efficiency is density. Several types of the problem arise depending on the type of isometries allowed for the packing: packing by translates, lattice packing, translates and point reflections, or all isometries. Due to its connections with number theory, crystallography, etc., lattice packing has been studied most extensively. In two dimensions the theory is fairly well developed, and there are seve…
Any self-similar directed graph iterated function system with probabilities, defined on m-dimensional Euclidean space, determines a unique list of self-similar Borel probability measures whose supports are the components of the attractor. Using an application of the Renewal Theorem we obtain an explicit calculable value for the power law behaviour of the qth packing moments of the self-similar measures at scale r as r tends to 0 in the non-lattice case, with a corresponding limit for the lattice case. We do this (i) for any real q if the strong separation condition (SSC) holds, (ii) for non-ne…
After having investigated the densest packings by congruent hyperballs to the regular prism tilings in the $n$-dimensional hyperbolic space $\mathbb{H}^n$ ($n \in \mathbb{N}, n \ge 3)$ we consider the dual covering problems and determine the least dense hyperball arrangements and their densities.
In this paper we prove that the optimal lattice packing of the Minkowski, Davis, and Chebyshev-Cohn balls is realized with respect to the sublattices of index two of the critical lattices of corresponding balls
We present an efficient Monte Carlo method for the lattice sphere packing problem in d dimensions. We use this method to numerically discover de novo the densest lattice sphere packing in dimensions 9 through 20. Our method goes beyond previous methods not only in exploring higher dimensions but also in shedding light on the statistical mechanics underlying the problem in question. We observe evidence of a phase transition in the thermodynamic limit $d\to\infty$. In the dimensions explored in the present work, the results are consistent with a first-order crystallization transition, but leave …
The famous Kepler conjecture has a less spectacular, two-dimensional equivalent: The theorem of Thue states that the densest circle packing in the Euclidean plane has a hexagonal structure. A common proof uses Voronoi cells and analyzes their area applying Jensen's inequality on convex functions to receive a local estimate which is globally valid. Based on the concept of Voronoi cells, we will introduce a new tessellation into so-called L-triangles which can be related to fundamental parallelograms of lattice circle packings. Therefore a globally disordered circle packing can be reduced to loc…
Two constructions of lattice packings of $ n $-dimensional cross-polytopes ($ \ell_1 $ balls) are described, the density of which exceeds that of any prior construction by a factor of at least $ 2^{\frac{n}{\ln n}(1 + o(1))} $ when $ n \to \infty $. The first family of lattices is explicit and is obtained by applying Construction A to a class of Reed-Solomon codes. The second family has subexponential construction complexity and is based on the notion of Sidon sets in finite Abelian groups. The construction based on Sidon sets also gives the highest known asymptotic density of packing discrete…
We generate and study an ensemble of isostatic jammed hard-sphere lattices. These lattices are obtained by compression of a periodic system with an adaptive unit cell containing a single sphere until the point of mechanical stability. We present detailed numerical data about the densities, pair correlations, force distributions, and structure factors of such lattices. We show that this model retains many of the crucial structural features of the classical hard-sphere model and propose it as a model for the jamming and glass transitions that enables exploration of much higher dimensions than ar…
We introduce a parameter space for periodic point sets, given as unions of $m$ translates of point lattices. In it we investigate the behavior of the sphere packing density function and derive sufficient conditions for local optimality. Using these criteria we prove that perfect, strongly eutactic lattices cannot be locally improved to yield a periodic sphere packing with greater density. This applies in particular to the densest known lattice sphere packings in dimension $d\leq 8$ and $d=24$.
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…
