Tetrahedra and octahedra together can fill all of space in an arrangement known as the tetrahedral-octahedral honeycomb. One of the best packings of regular dodecahedra is based on the aforementioned face-centered cubic (FCC) lattice. Tetrahedra can achieve a packing of at least 85%. No other Platonic solid can tile space on its own, but some preliminary results are known. Packings of Platonic solids in three dimensions Ĭubes can easily be arranged to fill three-dimensional space completely, the most natural packing being the cubic honeycomb. The 8-dimensional E8 lattice and 24-dimensional Leech lattice have also been proven to be optimal in their respective real dimensional space. With 'simple' sphere packings in three dimensions ('simple' being carefully defined) there are nine possible definable packings. In three dimensions, close-packed structures offer the best lattice packing of spheres, and is believed to be the optimal of all packings. Many other shapes have received attention, including ellipsoids, Platonic and Archimedean solids including tetrahedra, tripods (unions of cubes along three positive axis-parallel rays), and unequal-sphere dimers. The Kepler conjecture postulated an optimal solution for packing spheres hundreds of years before it was proven correct by Thomas Callister Hales. This problem is relevant to a number of scientific disciplines, and has received significant attention. Many of these problems, when the container size is increased in all directions, become equivalent to the problem of packing objects as densely as possible in infinite Euclidean space. 1.3 Packings of Platonic solids in three dimensions.1.2 Sphere packings in higher dimensions.In some variants the overlapping (of objects with each other and/or with the boundary of the container) is allowed but should be minimized. More commonly, the aim is to pack all the objects into as few containers as possible. In some variants, the aim is to find the configuration that packs a single container with the maximal packing density. Usually the packing must be without overlaps between goods and other goods or the container walls. The set may contain different objects with their sizes specified, or a single object of a fixed dimension that can be used repeatedly. A set of objects, some or all of which must be packed into one or more containers.Multiple containers may be given depending on the problem. A container, usually a two- or three-dimensional convex region, possibly of infinite size.Each packing problem has a dual covering problem, which asks how many of the same objects are required to completely cover every region of the container, where objects are allowed to overlap. Many of these problems can be related to real-life packaging, storage and transportation issues. The goal is to either pack a single container as densely as possible or pack all objects using as few containers as possible. Numerical results are also available at problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. A couple of improvements were found (for n=37, 101, 146, 169) and numerical values are given for all packings in the extended range 1 ≤ n ≤ 500 (except for n=393, 411, 421, 453). Furthermore, it describes a deterministic procedure for the optimization of packings that differ from usual regular hexagonal patterns by one or more vacancies, without relying on simulation programs. This article introduces a shorter notation for hexagonal dense packings in rectangular containers. However, their work lacks a precise treatment of the closure of observed vacancies as well as any numerical data of the best arrangements found. How must the box be designed to achieve the maximum packing density? This special problem was investigated by Lubachevsky and Graham in 2003, where they classified record packings for n ≤ 213. It arises if one has to decide how a certain number of circular items should be packed into a rectangular box when no assumption is made on the shape of the box.

Arranging a fixed number n of equal non-overlapping circles in a rectangle with variable aspect ratio is a non-standard packing problem.

Packing problem circles in rectangle