Related papers: Calculating the sequences behind a hexagonal latti…
Circle packings with specified patterns of tangencies form a discrete counterpart of analytic functions. In this paper we study univalent packings (with a combinatorial closed disk as tangent graph) which are embedded in (or fill) a…
We obtain an upper bound to the packing density of regular tetrahedra. The bound is obtained by showing the existence, in any packing of regular tetrahedra, of a set of disjoint spheres centered on tetrahedron edges, so that each sphere is…
Packing problems, which ask how to arrange a collection of objects in space to meet certain criteria, are important in a great many physical and biological systems, where geometrical arrangements at small scales control behaviour at larger…
In this article we explore a symplectic packing problem where the targets and domains are $2n$-dimensional symplectic manifolds. We work in the context where the manifolds have first homology group equal to $\mathbb{Z}^n$, and we require…
We call a packing of hyperspheres in $n$ dimensions an Apollonian sphere packing if the spheres intersect tangentially or not at all; they fill the $n$-dimensional space; and every sphere in the packing is a member of a cluster of $n+2$…
The distance geometry problem asks to find a realization of a given simple edge-weighted graph in a Euclidean space of given dimension K, where the edges are realized as straight segments of lengths equal (or as close as possible) to the…
In an old paper of the author the thinnest five-neighbour packing of translates of a convex disc (different from a parallelogram) was determined. The minimal density was $3/7$, and was attained for a certain packing of triangles. In that…
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…
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…
In many areas of research it is interesting how lattices can be filled with particles that have no nearest neighbors, or they are in limited quantities. Examples may be found in statistical physics, chemistry, materials science, discrete…
There are a finite number of distinct mechanically stable (MS) packings in model granular systems composed of frictionless spherical grains. For typical packing-generation protocols employed in experimental and numerical studies, the…
Rigid particles pack into structures, such as sand dunes on the beach, whose overall stability is determined by the average number of contacts between particles. However, when packing spatially extended objects with flexible shapes,…
Two averaging algorithms are considered which are intended for choosing an optimal plane and an optimal circle approximating a group of points in three-dimensional Euclidean space.
In this note we provide a simple approximation theory motivation for the circular kernel density estimation and further explore the usefulness of the wrapped Cauchy kernel in this context. It is seen that the wrapped Cauchy kernel appears…
Polymer systems in slab geometries are studied on the basis of the recently presented Gaussian Ellipsoid Model [J. Chem. Phys. 114, 7655 (2001)].The potential of the confining walls has an exponential shape. For homogeneous systems in…
We introduce two explicit examples of polynomials orthogonal on the unit circle. Moments and the reflection coefficients are expressed in terms of Jacobi elliptic functions. We find explicit expression for these polynomials in terms of a…
The close packing density of log-normal and bimodal distributed, surface-adsorbed particles or discs in 2D is studied by numerical simulation. For small spread in particle size, the system orders in a polycrystalline structure of hexagonal…
This paper derives the exact cumulative density function of the distance between a randomly located node and any arbitrary reference point inside a regular $\el$-sided polygon. Using this result, we obtain the closed-form probability…
Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. Such packings can be described in terms of the Descartes configurations they contain. It observed…
Molecular simulations of the self-assembly of cone-shaped particles with specific, attractive interactions are performed. Upon cooling from random initial conditions, we find that the cones self assemble into clusters and that clusters…