Related papers: Perfect packing of squares
This article presents uniform random generators of plane partitions according to the size (the number of cubes in the 3D interpretation). Combining a bijection of Pak with the method of Boltzmann sampling, we obtain random samplers that are…
The main goal of this paper is to address the following problem: given a positive integer $n$, find the largest value $S(n)$ such that a square of edge length $S(n)$ in the Euclidean plane can be covered by $n$ unit squares. We investigate…
A \emph{cylinder packing} is a family of congruent infinite circular cylinders with mutually disjoint interiors in $3$-dimensional Euclidean space. The \emph{local density} of a cylinder packing is the ratio between the volume occupied by…
This note concerns the so-called pyjama problem, whether it is possible to cover the plane by finitely many rotations of vertical strips of half-width $\varepsilon$. We first prove that there exist no periodic coverings for…
We study saturated packings produced according to random sequential adsorption (RSA) protocol built of identical rectangles deposited on a flat, continuous plane. An aspect ratio of rectangles is defined as the length-to-width ratio,…
This is the eighth and final paper in a series 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…
We study the well-known two-dimensional strip packing problem. Given is a set of rectangular axis-parallel items and a strip of width $W$ with infinite height. The objective is to find a packing of these items into the strip, which…
This paper characterizes when an $m \times n$ rectangle, where $m$ and $n$ are integers, can be tiled (exactly packed) by squares where each has an integer side length of at least 2. In particular, we prove that tiling is always possible…
By using totally isotropic subspaces in an orthogonal space Omega^{+}(2i,2), several infinite families of packings of 2^k-dimensional subspaces of real 2^i-dimensional space are constructed, some of which are shown to be optimal packings. A…
We examine sequences of dense packings of n congruent non-overlapping disks inside a square which follow specific patterns as n increases along certain values, n = n(1), n(2),... n(k),.... Extending and improving previous work of Nurmela…
We present and analyze PackIt!, a turn-based game consisting of packing rectangles on an $n \times n$ grid. PackIt! can be easily played on paper, either as a competitive two-player game or in \emph{solitaire} fashion. On the $t$-th turn, a…
We consider packings of the plane using discs of radius 1 and r. A packing is compact if every disc D is tangent to a sequence of discs D_1, D_2, ..., D_n such that D_i is tangent to D_{i+1}. We prove that there are only nine values of r…
Given any set of points $S$ in the unit square that contains the origin, does a set of axis aligned rectangles, one for each point in $S$, exist, such that each of them has a point in $S$ as its lower-left corner, they are pairwise interior…
We consider the hexagonal circle packing with radius 1/2 and perturb it by letting the circles move as independent Brownian motions for time t. It is shown that, for large enough t, if \Pi_t is the point process given by the center of the…
The representation of a given quantity with less information is often referred to as `quantization' and it is an important subject in information theory. In this paper, we have considered absolutely continuous probability measures on unit…
We present an algorithm to compute the exact value of the packing measure of self-similar sets satisfying the so called SSC and prove its convergence to the value of the packing measure. We also test the algorithm with examples that show…
Any space-filling packing of spheres can be cut by a plane to obtain a space-filling packing of disks. Here, we deal with space-filling packings generated using inversive geometry leading to exactly self-similar fractal packings. First, we…
Strip packing is a classical packing problem, where the goal is to pack a set of rectangular objects into a strip of a given width, while minimizing the total height of the packing. The problem has multiple applications, e.g. in scheduling…
In 1963, Graham proved that all integers greater than 77 (but not 77 itself) can be partitioned into distinct positive integers whose reciprocals sum to 1. He further conjectured that for any sufficiently large integer, it can be…
A number $N$ is a triangular number if it can be written as $N = t(t + 1)/2$ for some nonnegative integer number $t$. A triangular number $N$ is called square if it is a perfect square, that is, $N = d^2$ for some integer number $d$. Square…