Packing problems
Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to either pack a single container as densely as possible or pack all objects using as few containers as possible. Many of these problems can be related to real-life packaging, storage and transportation issues. 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. In a bin packing problem, you are given: Usually  the packing must be without overlaps between goods and other goods or the container walls. In some variants, the aim is to find the configuration that packs a single container with the maximal packing density. More commonly, the aim is to pack all the objects into as few containers as possible. In some variants the overlapping (of objects with each other and/or with the boundary of the container) is allowed but should be minimized. Packing in infinite space. 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. This problem is relevant to a number of scientific disciplines, and has received significant attention. The Kepler conjecture postulated an optimal solution for packing spheres hundreds of years before it was proven correct by Thomas Callister Hales. 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. Hexagonal packing of circles. These problems are mathematically distinct from the ideas in the circle packing theorem. The related circle packing problem deals with packing circles, possibly of  different sizes, on a surface, for instance the plane or a sphere. The counterparts of a circle in other dimensions can never be packed with complete efficiency in dimensions larger than one (in a one-dimensional universe, the circle analogue is just two points). That is, there will always be unused space if you are only packing circles. The most efficient way of packing circles, hexagonal packing, produces approximately 91% efficiency. Sphere packings in higher dimensions. In three dimensions, close-packed structures offer the best "lattice" packing of spheres, and is believed to be the optimal of all packings. With  'simple' sphere packings in three dimensions ('simple' being carefully defined) there are nine possible definable packings. The 8-dimensional E8 lattice and 24-dimensional Leech lattice have also been proven to be optimal in their respective real dimensional space. Packings of Platonic solids in three dimensions. Cubes can easily be arranged to fill three-dimensional space completely, the most natural packing being the cubic honeycomb. No other Platonic solid can tile space on its own, but some preliminary results are known. Tetrahedra can achieve a packing of at least 85%. One of the best packings of regular dodecahedra is based  on the aforementioned face-centered cubic (FCC) lattice. Tetrahedra and octahedra together can fill all of space in an arrangement known as the tetrahedral-octahedral honeycomb. Simulations combining local improvement methods with random packings suggest that the lattice packings for icosahedra, dodecahedra, and octahedra are optimal in the broader class of all packings. Packing in 3-dimensional containers. Different cuboids into a cuboid. Determine the minimum number of cuboid containers (bins) that are required to pack a given set of item cuboids. The rectangular cuboids to be packed can be rotated by 90 degrees on each axis. Spheres into a  Euclidean ball. The problem of finding the smallest ball such that disjoint open unit balls may be packed inside it has a simple and complete answer in -dimensional Euclidean space if formula_1, and in an infinite-dimensional Hilbert space with no restrictions. It is worth describing in detail here, to give a flavor of the general problem. In this case, a configuration of pairwise tangent unit balls is available. Place the centers at the vertices formula_2 of a regular formula_3 dimensional simplex with edge 2; this is easily realized starting from an orthonormal basis. A small computation shows that the  distance of each vertex from the barycenter is formula_4. Moreover, any other point of the space necessarily has a larger distance from "at least" one of the vertices. In terms of inclusions of balls, the open unit balls centered at formula_2 are included in a ball of radius formula_6, which is minimal for this configuration. To show that this configuration is optimal, let formula_7 be the centers of disjoint open unit balls contained in a ball of radius centered at a point formula_8. Consider the map from the finite set formula_9 into formula_10 taking formula_11 in the corresponding formula_12 for  each formula_13. Since for all formula_14, formula_15 this map is 1-Lipschitz and by the Kirszbraun theorem it extends to a 1-Lipschitz map that is globally defined; in particular, there exists a point formula_16 such that for all formula_17 one has formula_18, so that also formula_19. This shows that there are disjoint unit open balls in a ball of radius if and only if formula_20. Notice that in an infinite-dimensional Hilbert space this implies that there are infinitely many disjoint open unit balls inside a ball of radius if and only if formula_21. For instance, the unit balls  centered at formula_22, where formula_23 is an orthonormal basis, are disjoint and included in a ball of radius formula_24 centered at the origin. Moreover, for formula_25, the maximum number of disjoint open unit balls inside a ball of radius is formula_26. Spheres in a cuboid. Determine the number of spherical objects of given diameter that can be packed into a cuboid of size formula_27. Identical spheres in a cylinder. Determine the minimum height of a cylinder with given radius that will pack identical spheres of radius . For a small radius the spheres arrange to ordered structures, called columnar structures. Polyhedra  in spheres. Determine the minimum radius that will pack identical, unit volume polyhedra of a given shape. Packing in 2-dimensional containers. Many variants of 2-dimensional packing problems have been studied. See the linked pages for more information. Packing of circles. You are given unit circles, and have to pack them in the smallest possible container. Several kinds of containers have been studied: Packing of squares. You are given unit squares and have to pack them into the smallest possible container, where the container type varies: Related fields. In tiling or tessellation problems, there are to be no gaps , nor overlaps. Many of the puzzles of this type involve packing rectangles or polyominoes into a larger rectangle or other square-like shape. There are significant theorems on tiling rectangles (and cuboids) in rectangles (cuboids) with no gaps or overlaps: The study of polyomino tilings largely concerns two classes of problems: to tile a rectangle with congruent tiles, and to pack one of each "n"-omino into a rectangle. A classic puzzle of the second kind is to arrange all twelve pentominoes into rectangles sized 3×20, 4×15, 5×12 or 6×10. Packing of irregular objects. Packing of irregular objects is a problem  not lending itself well to closed form solutions; however, the applicability to practical environmental science is quite important. For example, irregularly shaped soil particles pack differently as the sizes and shapes vary, leading to important outcomes for plant species to adapt root formations and to allow water movement in the soil. The problem of deciding whether a given set of polygons can fit in a given square container has been shown to be complete for the existential theory of the reals. Many puzzle books as well as mathematical journals contain articles on packing problems.