# Category: Packing problems

Kissing number
In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sph
The Pursuit of Perfect Packing
The Pursuit of Perfect Packing is a book on packing problems in geometry. It was written by physicists Tomaso Aste and Denis Weaire, and published in 2000 by Institute of Physics Publishing with a sec
Sphere packing
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimension
Parallel task scheduling (also called parallel job scheduling or parallel processing scheduling) is an optimization problem in computer science and operations research. It is a variant of optimal job
Sphere packing in a cylinder
Sphere packing in a cylinder is a three-dimensional packing problem with the objective of packing a given number of identical spheres inside a cylinder of specified diameter and length. For cylinders
Hoffman's packing puzzle
Hoffman's packing puzzle is an assembly puzzle named after , who described it in 1978. The puzzle consists of 27 identical rectangular cuboids, each of whose edges have three different lengths. Its go
The quadratic knapsack problem (QKP), first introduced in 19th century, is an extension of knapsack problem that allows for quadratic terms in the objective function: Given a set of items, each with a
Apollonian sphere packing
Apollonian sphere packing is the three-dimensional equivalent of the Apollonian gasket. The principle of construction is very similar: with any four spheres that are cotangent to each other, it is the
List of shapes with known packing constant
The packing constant of a geometric body is the largest average density achieved by packing arrangements of congruent copies of the body. For most bodies the value of the packing constant is unknown.
Packaging engineering
Packaging engineering, also package engineering, packaging technology and packaging science, is a broad topic ranging from design conceptualization to product placement. All steps along the manufactur
Ellipsoid packing
In geometry, ellipsoid packing is the problem of arranging identical ellipsoid throughout three-dimensional space to fill the maximum possible fraction of space. The currently densest known packing st
Sphere packing in a sphere
Sphere packing in a sphere is a three-dimensional packing problem with the objective of packing a given number of equal spheres inside a unit sphere. It is the three-dimensional equivalent of the circ
Slothouber–Graatsma puzzle
The Slothouber–Graatsma puzzle is a packing problem that calls for packing six 1 × 2 × 2 blocks and three 1 × 1 × 1 blocks into a 3 × 3 × 3 box. The solution to this puzzle is unique (up to mirror ref
Conway puzzle
Conway's puzzle, or blocks-in-a-box, is a packing problem using rectangular blocks, named after its inventor, mathematician John Conway. It calls for packing thirteen 1 × 2 × 4 blocks, one 2 × 2 × 2 b
Knapsack problem
The knapsack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total we
Tripod packing
In combinatorics, tripod packing is a problem of finding many disjoint tripods in a three-dimensional grid, where a tripod is an infinite polycube, the union of the grid cubes along three positive axi
Packing density
A packing density or packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing. In simplest terms, this is the ratio of the volume of bodies
Tetrahedron packing
In geometry, tetrahedron packing is the problem of arranging identical regular tetrahedra throughout three-dimensional space so as to fill the maximum possible fraction of space. Currently, the best l
Cutting stock problem
In operations research, the cutting-stock problem is the problem of cutting standard-sized pieces of stock material, such as paper rolls or sheet metal, into pieces of specified sizes while minimizing
Kepler conjecture
The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no
Smoothed octagon
The smoothed octagon is a region in the plane found by Karl Reinhardt in 1934 and conjectured by him to have the lowest maximum packing density of the plane of all centrally symmetric convex shapes. I
Close-packing of equal spheres
In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement (or lattice). Carl Friedrich Gauss proved that the highest average density –
Sphere packing in a cube
In geometry, sphere packing in a cube is a three-dimensional sphere packing problem with the objective of packing spheres inside a cube. It is the three-dimensional equivalent of the circle packing in
Square packing in a square
Square packing in a square is a packing problem where the objective is to determine how many squares of side one (unit squares) can be packed into a square of side a. If a is an integer, the answer is
Strip packing problem
The strip packing problem is a 2-dimensional geometric minimization problem. Given a set of axis-aligned rectangles and a strip of bounded width and infinite height, determine an overlapping-free pack
Ulam's packing conjecture
Ulam's packing conjecture, named for Stanislaw Ulam, is a conjecture about the highest possible packing density of identical convex solids in three-dimensional Euclidean space. The conjecture says tha
Polygon partition
In geometry, a partition of a polygon is a set of primitive units (e.g. squares), which do not overlap and whose union equals the polygon. A polygon partition problem is a problem of finding a partiti
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 possi
Rectangle packing
Rectangle packing is a packing problem where the objective is to determine whether a given set of small rectangles can be placed inside a given large polygon, such that no two small rectangles overlap