Lattice Sphere Packings Research Articles

On Viazovska's modular form inequalities.

Viazovska proved that the [Formula: see text] lattice sphere packing is the densest sphere packing in [Formula: see text] dimensions. Her proof relies on two inequalities between functions defined in terms of modular and quasimodular forms. We give a direct proof of these inequalities that does not rely on computer calculations.

On the density of cyclotomic lattices constructed from codes

Recently, Venkatesh improved the best known lower bound for lattice sphere packings by a factor [Formula: see text] for infinitely many dimensions [Formula: see text]. Here, we prove an effective version of this result, in the sense that we exhibit, for the same set of dimensions, finite families of lattices containing a lattice reaching this bound. Our construction uses codes over cyclotomic fields, lifted to lattices via Construction A. We also prove a similar result for families of symplectic lattices.

Extreme lattices: symmetries and decorrelation

We study statistical and structural properties of extreme lattices, which are the local minima in the density landscape of lattice sphere packings in d-dimensional Euclidean space . Specifically, we ascertain statistics of the densities and kissing numbers as well as the numbers of distinct symmetries of the packings for dimensions 8 through 13 using the stochastic Voronoi algorithm. The extreme lattices in a fixed dimension of space d () are dominated by typical lattices that have similar packing properties, such as packing densities and kissing numbers, while the best and the worst packers are in the long tails of the distribution of the extreme lattices. We also study the validity of the recently proposed decorrelation principle, which has important implications for sphere packings in general. The degree to which extreme-lattice packings decorrelate as well as how decorrelation is related to the packing density and symmetry of the lattices as the space dimension increases is also investigated. We find that the extreme lattices decorrelate with increasing dimension, while the least symmetric lattices decorrelate faster.

Statistical mechanics of the lattice sphere packing problem

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→∞. In the dimensions explored in the present work, the results are consistent with a first-order crystallization transition but leave open the possibility that a glass transition is manifested in higher dimensions.

Efficient linear programming algorithm to generate the densest lattice sphere packings

Finding the densest sphere packing in d-dimensional Euclidean space R(d) is an outstanding fundamental problem with relevance in many fields, including the ground states of molecular systems, colloidal crystal structures, coding theory, discrete geometry, number theory, and biological systems. Numerically generating the densest sphere packings becomes very challenging in high dimensions due to an exponentially increasing number of possible sphere contacts and sphere configurations, even for the restricted problem of finding the densest lattice sphere packings. In this paper we apply the Torquato-Jiao packing algorithm, which is a method based on solving a sequence of linear programs, to robustly reproduce the densest known lattice sphere packings for dimensions 2 through 19. We show that the TJ algorithm is appreciably more efficient at solving these problems than previously published methods. Indeed, in some dimensions, the former procedure can be as much as three orders of magnitude faster at finding the optimal solutions than earlier ones. We also study the suboptimal local density-maxima solutions (inherent structures or "extreme" lattices) to gain insight about the nature of the topography of the "density" landscape.

Kissing numbers for surfaces

The so-called kissing number for hyperbolic surfaces is the maximum number of homotopically distinct systoles a surface of given genus g can have. These numbers, first studied (and named) by Schmutz Schaller by analogy with lattice sphere packings, are known to grow, as a function of genus, at least like g4/3−ε for any ε > 0. The first goal of this article is to give upper bounds on these numbers; in particular, the growth is shown to be subquadratic. In the second part, a construction of (non-hyperbolic) surfaces with roughly g3/2 systoles is given.

Random perfect lattices and the sphere packing problem

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. Their number, however, grows superexponentially with the dimension, so to get an idea of their properties we propose to study a randomized version of the generating algorithm and to define a random ensemble with an effective temperature in a way reminiscent of a Monte Carlo 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 known 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 the Minkowsky bound φ~2(-(0.84±0.06)d), and a competitor in which their packing fraction decreases superexponentially, 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.

Improved sphere packing lower bounds from Hurwitz lattices

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.

Method for dense packing discovery

The problem of packing a system of particles as densely as possible is foundational in the field of discrete geometry and is a powerful model in the material and biological sciences. As packing problems retreat from the reach of solution by analytic constructions, the importance of an efficient numerical method for conducting de novo (from-scratch) searches for dense packings becomes crucial. In this paper, we use the divide and concur framework to develop a general search method for the solution of periodic constraint problems, and we apply it to the discovery of dense periodic packings. An important feature of the method is the integration of the unit-cell parameters with the other packing variables in the definition of the configuration space. The method we present led to previously reported improvements in the densest-known tetrahedron packing. Here, we use the method to reproduce the densest-known lattice sphere packings and the best-known lattice kissing arrangements in up to 14 and 11 dimensions, respectively, providing numerical evidence for their optimality. For nonspherical particles, we report a dense packing of regular four-dimensional simplices with density ϕ=128/219≈0.5845 and with a similar structure to the densest-known tetrahedron packing.

Perfect, strongly eutactic lattices are periodic extreme

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 ⩽ 8 and d = 24 .

Partition of Euclidean space into polyhedra induced by a small deformation of the densest lattice sphere packing

The number of combinatorially nonequivalent Dirichlet-Voronoi diagrams constructed for the centers of balls in the packing obtained from the densest lattice packing of equal spheres by a small displacement of the spheres is estimated.

From symmetry to particles

The notion of a particle-like state emerging from a symmetry breaking is given five corresponding pictures. We start from a geometrical picture in two dimensions involving a modular curve constructed using 336 triangles. The same number of building blocks is found again, this time as 336 contact points in the ten dimensional space of super string theory in the context of the largest kissing number of lattice sphere packing. The next corresponding representation is an abstract one pertinent to the order of the simple linear Lie group SL(2, n) in seven dimensions (n = 7) which leads to 336 symmetries. Subsequently a tensorial picture is given using the Riemannian tensor of relativity theory but this time in an eight dimensional space (n = 8) for which the number of independent components is again 336. Finally we use a physical string theory related picture in the 12 dimensions of F theory to find 336 moduli space dimensions representing the instanton cells of our theory. It is evident that the five preceding pictures are ten fold interconnected and exchangeable. This additional mental freedom does not only enhance the feeling of understanding, but also facilitates the easy recognition of complex mathematical relations and its connection to the physical concepts.

Gitter und Anwendungen

Respecting the different backgrounds of the participants, most of the talks were aimed to a broad audience, among those nine survey talks invited by the organizers. This concept was very successful and opened many discussions and interdisciplinary interactions. It was also very fruitful for the many young participants. Also most of the non speakers took the opportunity to present their recent work via posters. The theory of lattices has many applications and interactions with various other mathematical and technical disciplines such as information technology, topology, algebraic geometry, representation theory, combinatorics, number theory and modular forms to name only the most prominent ones. One of the most classical problems is the construction of dense lattice sphere packings. The densest lattice sphere packings are only known in dimensions up to 8 and, due to a recent work by H. Cohn and A. Kumar, also in dimension 24. A. Kumar gave a very illuminating talk on the strategy of their proof of this great result which uses certain linear programming techniques based on the fact that the minimal vectors of the Leech lattice form a very good design. Design techniques have also been used by B. Venkov to define the notion of strongly perfect lattices, which realize certain local maxima of the density function and which are now classified up to dimension 12. The theory of modular forms is one well established tool in the investigation of lattices as shown in the nice introduction by Krieg. Bannai applies this theory to bound the strength of designs provided by layers of unimodular lattices. On the other hand, lattices provide one important tool for the construction of modular forms. The last talk by Böcherer showed that in many situations all modular forms are linear combinations of theta-series. Of increasing interest but computationally very difficult is the investigation of thin lattice sphere coverings. Vallentin and Schürmann developed new algorithms to find local optimal coverings and discovered lattice coverings better than the ones previously known. Nguyen presented new reduction algorithms for lattices in small dimensions motivated by practical applications to cryptosystems. For certain applications in information technology not only the density of the lattice but also other properties that minimize the fading error play a role. It turns out that lattices constructed from algebraic number fields yield good codes for Rayleigh fading channels. These ideal lattices are also used to investigate the ring of integers in algebraic number fields as shown in the talk by Schoof and also in the applications of ideal lattices introduced by Bayer-Fluckiger. Strongly related to this is the application to Arakelov theory as illustrated in the talks by Bost and Künnemann. Also Voronoi's classical theory of perfect forms (see Martinet's talk for an introduction) has new fruitful applications in number theory more precisely in the calculation of the homology of GL_n({\bf Z}) as presented by Elbaz-Vincent. Last but not least one should mention the talk by J.-P. Serre on BL-bases and unitary groups in characteristic 2.

Variations around disordered close packing

The most suitable paradigms and tools for investigating the structure of granular packingsare reviewed and discussed in the light of some recent empirical results. I examine the‘typical’ local configurations, their relative occurrences, their correlations, theirorganization and the resulting overall hierarchical structure. The analysis of very largesamples of monosized spheres packed in a disorderly fashion, with packing fractionsρ ranging from 0.58 to 0.64, demonstrates the existence of clearrelations between the geometrical structure and the packing densityρ. It is observed that the local Delaunay and Voronoï volumes have distributions which decayexponentially at large volumes, and have characteristic coefficients proportional toρ. The average number of contacts per sphere grows linearly withρ. The topologicaldensity increases with ρ and it is larger than the corresponding one for lattice sphere packing. The heightsand the shapes of the peaks in the radial distribution function also depend onρ. Moreover, the study of several other quantities, such as the dihedral angle distribution,the fractions of four-, five-and six-membered rings and the shape of the Voronoï cells, showsthat the local organization has characteristic patterns which depend on the packingdensity. All the empirical evidence excludes the possibility of the presence of anycrystalline order, even at the minimal correlation length (one bead diameter).Moreover, icosahedral order and other close packed configurations have no statisticalsignificance, excluding therefore the possibility that geometrical frustration could playany significant role in the formation of such amorphous structures. On the otherhand, the analysis of the local geometrical ‘caging’ discloses that, at densitiesabove 0.601, the system can no longer explore the phase space by means of localmoves only and it becomes trapped either in the basin of attractions of inherentstates, with limiting densities in the range 0.61–0.64, or in the crystalline branch.

Free Planes in Lattice Sphere Packings

Abstract We show that for every lattice packing of n-dimensional spheres there exists an (n/log2(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.

Meditation on an Engraving of Fricke and Klein (The Modular Group and Geometrical Chemistry)

AbstractFor Abstract see ChemInform Abstract in Full Text.

Meditation on an Engraving of Fricke and Klein (The Modular Group and Geometrical Chemistry)

A non-technical account of the links between two-dimensional (2D) hyperbolic and three-dimensional (3D) euclidean symmetric patterns is presented, with a number of examples from both spaces. A simple working hypothesis is used throughout the survey: simple, highly symmetric patterns traced in hyperbolic space lead to chemically relevant structures in euclidean space. The prime examples in the former space are derived from Felix Klein's engraving of the modular group structure within the hyperbolic plane; these include various tilings, networks and trees. Disc packings are also derived. The euclidean examples are relevant to condensed atomic and molecular materials in solid-state chemistry and soft-matter structural science. They include extended nets of relevance to covalent frameworks, simple (lattice) sphere packings, and interpenetrating extended frameworks (related to novel coordination polymers). Limited discussion of the projection process from 2D hyperbolic to 3D euclidean space via mapping onto triply periodic minimal surfaces is presented.

Atlas of quasicrystalline tilings

Tilings are created from root lattices using canonical projection. Like diffraction images, these tilings make the quasicrystalline or crystalline nature of many of these structures clear. These tilings may also be viewed as shadows of lattice sphere packings in n dimensions. The atlas gives many new intriguing quasicrystalline tilings in a systematic way.

Radon transforms and packings

We use some basic results and ideas from the integral geometry to study certain properties of group codes. The properties being studied are generalized weights and spectra of linear block codes over a finite field and their analogues for lattice sphere packings in Euclidean space. No new results are obtained about linear codes, although several short and simple proofs for known results are given. As to the lattices, we introduce a generalization of lattice Θ-functions, prove several identities on these functions, and prove generalizations of Siegel mean value and Minkowski–Hlawka theorems.

Riemann surfaces with longest systole and an improved Vorono� algorithm

In this paper we introduce a new method in order to find the Riemann surface M of a fixed topological type with the longest systole; it is based on a cell decomposition of the Teichmuller space of M. The method also works in the Euclidean case and is similar to the so-called Voronoi algorithm for positive definite quadratic forms, or equivalently, for lattice sphere packings. In particular, we give a new proof of Rogers' theorem.