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| {{about|geometric packing problems|numerical packing problems|Knapsack problem}}
| | My name is Teena from Alta studying Business and Management. I did my schooling, secured 73% and hope to find someone with same interests in Rock collecting.<br><br>Here is my homepage; [http://perbiz.co.kr/xe/?document_srl=2187511 Hosgator Reviews] |
| {{Puzzles |Topics}}
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| '''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.
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| In a packing problem, you are given:
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| * 'containers' (usually a single two- or three-dimensional convex region, or an infinite space)
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| * A set of 'objects' some or all of which must be packed into one or more containers. The set may contain different objects with their sizes specified, or a single object of a fixed dimension that can be used repeatedly.
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| 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 density. More commonly, the aim is to pack all the objects into as few containers as possible.<ref>{{cite journal|authors= Lodi, A., Martello, S., Monaci, M.|title = Two-dimensional packing problems: A survey| journal = European Journal of Operational Research|year = 2002|publisher = Elsevier}}</ref> In some variants the overlapping (of objects with each other and/or with the boundary of the container) is allowed but should be minimized.
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| {{Covering-Packing_Problem_Pairs}}
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| ==Packing infinite space==
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| 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 [[sphere packing|packing spheres]] hundreds of years before it was proven correct by [[Thomas Callister Hales]]. Many other shapes have received attention, including ellipsoids,<ref>{{Cite doi|10.1103/PhysRevLett.92.255506}}</ref> Platonic and Archimedean solids<ref name="Torquato"/> including [[tetrahedron packing|tetrahedra]],<ref>{{cite doi|10.1038/nature08641}}</ref><ref>{{cite doi| 10.1007/s00454-010-9273-0 }}</ref> and unequal-sphere dimers.<ref>{{cite doi|10.1088/0953-8984/23/19/194103}}</ref>
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| ===Hexagonal packing of circles===
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| [[File:Circle packing (hexagonal).svg|thumb|right|The hexagonal packing of circles on a 2-dimensional Euclidean plane.]]
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| 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.
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| The [[N-sphere|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, [[Circle packing|hexagonal packing]], produces approximately 91% efficiency.<ref>http://mathworld.wolfram.com/CirclePacking.html</ref>
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| ===Sphere packings in higher dimensions===
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| {{Main|Sphere packing}}
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| In three dimensions, the [[face-centered cubic]] lattice offers the best ''lattice'' packing of spheres, and is believed to be the optimal of all packings. The 8-dimensional [[E8 lattice]] and 24-dimensional [[Leech lattice]] are also believed to be optimal.
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| ===Packings of Platonic solids in three dimensions===
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| 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. [[Tetrahedron packing|Tetrahedra]] can achieve a packing of at least 85%. One of the best packings of regular [[dodecahedron|dodecahedra]] is based on the aforementioned face-centered cubic (FCC) lattice.
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| Tetrahedra and octahedra together can fill all of space in an arrangement known as the [[tetrahedral-octahedral honeycomb]].
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| {| class="wikitable"
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| |-
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| ! Solid
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| ! Optimal density of a lattice packing
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| |-
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| | icosahedra
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| | 0.836357...<ref name="Betke">Betke, U. & Henk, M. Densest lattice packings of 3-polytopes. ''Comput. Geom.'' '''16''', 157–186 (2000)</ref>
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| |-
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| | dodecahedra
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| | (5+sqrt(5))/8=0.904508...<ref name="Betke"/>
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| |-
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| | octahedra
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| | 18/19 = 0.947368...<ref>Minkowski, H. Dichteste gitterfo¨rmige Lagerung kongruenter Ko¨rper. ''Nachr. Akad. Wiss. Go¨ttingen Math. Phys. KI. II'' 311–355 (1904).</ref>
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| |}
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| 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.<ref name="Torquato">{{cite doi|10.1038/nature08239}}</ref>
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| ==Packing in 3-dimensional containers==
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| ===Spheres into a Euclidean ball===
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| The problem of finding the smallest ball such that <math>k</math> disjoint open unit balls may be packed inside it has a simple and complete answer in <math>n</math>-dimensional Euclidean space if <math>\scriptstyle k\leq n+1</math>, 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 <math>k</math> pairwise tangent unit balls is available. Place the centers at the vertices <math>a_1,..,a_k</math> of a regular <math>\scriptstyle(k-1)</math> 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 <math>\scriptstyle\sqrt{2\big(1-\frac{1}{k} \big)}</math>. Moreover, any other point of the space necessarily has a larger distance from ''at least'' one of the <math>\scriptstyle k</math> vertices. In terms of inclusions of balls, the <math>\scriptstyle k</math> open unit balls centered at <math>\scriptstyle a_1,..,a_k</math> are included in a ball of radius <math>\scriptstyle r_k:=1+\sqrt{2\big(1-\frac{1}{k}\big)}</math>, which is minimal for this configuration.
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| To show that this configuration is optimal, let <math>\scriptstyle x_1,...,x_k</math> be the centers of <math>\scriptstyle k</math> disjoint open unit balls contained in a ball of radius <math>\scriptstyle r</math> centered at a point <math>\scriptstyle x_0</math>. Consider the map from the finite set <math>\scriptstyle\{x_1,..x_k\}</math> into <math>\scriptstyle\{a_1,..a_k\}</math> taking <math>\scriptstyle x_j</math> in the corresponding <math>\scriptstyle a_j</math> for each <math>\scriptstyle 1\leq j\leq k</math>. Since for all <math>\scriptstyle 1\leq i<j\leq k</math>, <math>\scriptstyle \|a_i-a_j\|=2\leq\|x_i-x_j\|</math> 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 <math>\scriptstyle a_0</math> such that for all <math>\scriptstyle1\leq j\leq k</math> one has <math>\scriptstyle\|a_0-a_j\|\leq\|x_0-x_j\|</math>, so that also <math>\scriptstyle r_k\leq1+\|a_0-a_j\|\leq 1+\|x_0-x_j\|\leq r</math>. This shows that there are <math>\scriptstyle k</math> disjoint unit open balls in a ball of radius <math>\scriptstyle r</math> if and only if <math>\scriptstyle r\geq r_k</math>. Notice that in an infinite dimensional Hilbert space this implies that there are infinitely many disjoint open unit balls inside a ball of radius <math>\scriptstyle r</math> if and only if <math>\scriptstyle r\geq 1+\sqrt{2}</math>. For instance, the unit balls centered at <math>\scriptstyle\sqrt{2}e_j</math>, where <math>\scriptstyle\{e_j\}_j</math> is an orthonormal basis, are disjoint and included in a ball of radius <math>\scriptstyle 1+\sqrt{2}</math> centered at the origin. Moreover, for <math>\scriptstyle r<1+\sqrt{2}</math>, the maximum number of disjoint open unit balls inside a ball of radius r is <math>\scriptstyle\big\lfloor \frac{2}{2-(r-1)^2}\big\rfloor</math>.
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| ===Spheres in a cuboid===
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| Determine the number of [[sphere|spherical]] objects of given diameter ''d'' can be packed into a [[cuboid]] of size ''a'' × ''b'' × ''c''.
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| ==Packing in 2-dimensional containers==
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| ===Packing circles===
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| {{main|Circle packing}}
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| ====Circles in circle====
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| [[Image:Disk pack10.svg|thumb|120px|right|The optimal packing of 10 circles in a circle]]
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| {{Main|Circle packing in a circle}}
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| Pack ''n'' unit circles into the smallest possible [[Circle (geometry)|circle]]. This is closely related to spreading points in a unit circle with the objective of finding the greatest minimal separation, d<sub>n</sub>, between points.
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| Optimal solutions have been proven for ''n''≤13, and ''n''=19.
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| ====Circles in square====
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| [[Image:Circles packed in square 15.svg|thumb|120px|right|The optimal packing of 15 circles in a square]]
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| {{Main|Circle packing in a square}}
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| Pack ''n'' unit circles into the smallest possible [[Square (geometry)|square]]. This is closely related to spreading points in a unit square with the objective of finding the greatest minimal separation, d<sub>n</sub>, between points.<ref name=upg>{{Cite book|title=Unsolved Problems in Geometry |last=Croft |first=Hallard T. |authorlink= |coauthors= Falconer, Kenneth J.; Guy, Richard K.|year=1991 |publisher=Springer-Verlag |location=New York |isbn=0-387-97506-3 |pages=108–110}}</ref> To convert between these two formulations of the problem, the square side for unit circles will be L=2+2/d<sub>n</sub>.
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| Optimal solutions have been proven for ''n''≤30.<ref>{{Cite web|url=http://hydra.nat.uni-magdeburg.de/packing/csq/csq.html|title=The best known packings of equal circles in a square |author=Eckard Specht |date=20 May 2010 |work= |publisher= |accessdate=25 May 2010}}</ref>
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| ====Circles in isosceles right triangle====
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| [[File:6 cirkloj en 45 45 90 triangulo.png|thumb|120px|right|The optimal packing of 6 circles in a right isosceles triangle]]
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| {{Main|Circle packing in an isosceles right triangle}}
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| Pack ''n'' unit circles into the smallest possible [[Special right triangles#45–45–90 triangle|isosceles right triangle]]. Good estimates are known for ''n''<300.<ref>{{cite web|url=http://hydra.nat.uni-magdeburg.de/packing/crt/crt.html|title=The best known packings of equal circles in an isosceles right triangle|first=Eckard|last=Specht|date=11 March 2011|accessdate=1 May 2011}}</ref>
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| ====Circles in equilateral triangle====
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| [[Image:5 cirkloj en 60 60 60 triangulo v1.png|thumb|120px|right|The optimal packing of 5 circles in an equilateral triangle]]
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| {{Main|Circle packing in an equilateral triangle}}
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| Pack ''n'' unit circles into the smallest possible [[equilateral triangle]]. Optimal solutions are known for ''n''<13, and conjectures are available for ''n''<28.<ref>{{cite doi|10.1016/0012-365X(95)90139-C}}</ref>
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| ===Packing squares===
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| ====Squares in square====
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| [[Image:10 kvadratoj en kvadrato.svg|thumb|120px|right|The optimal packing of 10 squares in a square]]
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| {{Main|Square packing in a square}}
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| Pack ''n'' unit squares into the smallest possible [[Square (geometry)|square]].
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| Optimal solutions have been proven for ''n''=1-10, 14-16, 23-25, 34-36, 62-64, 79-81, 98-100, and any square integer.<ref name="survey">Erich Friedman, [http://www.combinatorics.org/Surveys/ds7.html "Packing unit squares in squares: a survey and new results"], ''The Electronic Journal of Combinatorics'' '''DS7''' (2005).</ref>
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| The wasted space is asymptotically [[Big O notation|O]](''a''<sup>7/11</sup>).<ref>P. Erdős and R. L. Graham, [http://www.math.ucsd.edu/~sbutler/ron/75_06_squares.pdf "On packing squares with equal squares"], ''Journal of Combinatorial Theory, Series A'' '''19''' (1975), pp. 119–123.</ref>
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| ====Squares in circle====
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| Pack ''n'' squares in the smallest possible circle.
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| Minimum solutions:<ref>http://www2.stetson.edu/~efriedma/squincir/</ref>
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| {| class="wikitable"
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| |-
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| ! Number of squares
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| ! Circle radius
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| | 1
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| | 0.707...
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| |-
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| | 2
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| | 1.118...
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| |-
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| | 3
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| | 1.288...
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| |-
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| | 4
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| | 1.414...
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| |-
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| | 5
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| | 1.581...
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| |-
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| | 6
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| | 1.688...
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| |-
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| | 7
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| | 1.802...
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| |-
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| | 8
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| | 1.978...
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| |-
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| | 9
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| | 2.077...
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| |-
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| | 10
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| | 2.121...
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| |-
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| | 11
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| | 2.214...
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| |-
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| |12
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| | 2.236...
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| |}
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| ===Packing rectangles===
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| ====Identical rectangles in a rectangle====
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| The problem of packing multiple instances of a single rectangle of size (''l'',''w''), allowing for 90° rotation, in a bigger rectangle of size (''L'',''W'') has some applications such as loading of boxes on pallets and, specifically, [[woodpulp]] stowage.
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| For example, it is possible to pack 147 rectangles of size (137,95) in a rectangle of size (1600,1230).<ref>E G Birgin, R D Lobato, R Morabito, "An effective recursive partitioning approach for the packing of identical rectangles in a rectangle", ''Journal of the Operational Research Society'', 2010, '''61''', pp. 306-320.</ref>
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| ====Different rectangles in a rectangle====
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| The problem of packing multiple rectangles of varying widths and heights in an enclosing rectangle of minimum area (but with no boundaries on the enclosing rectangle's width or height) has an important application in combining images into a single larger image. A web page that loads a single larger image often renders faster in the browser than the same page loading multiple small images, due to the overhead involved in requesting each image from the web server.
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| An example of a fast algorithm that packs rectangles of varying widths and heights into an enclosing rectangle of minimum area is [http://www.codeproject.com/KB/web-image/rectanglepacker.aspx here].
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| ==Related fields==
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| 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.
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| There are significant theorems on tiling rectangles (and cuboids) in rectangles (cuboids) with no gaps or overlaps:
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| :An ''a'' × ''b'' rectangle can be packed with 1 × ''n'' strips iff ''n'' divides ''a'' or ''n'' divides ''b''.<ref name="Gems2">{{cite book | title = Mathematical Gems II | last1 = Honsberger | first1 = Ross | year = 1976 | publisher = [[The Mathematical Association of America]] | isbn = 0-88385-302-7 | page = 67 }}</ref><ref name="Klarner">{{cite journal | title = Uniformly coloured stained glass windows | journal = Proceedings of the London Mathematical Society |series = 3 | issue = 23 | pages = 613–628 | last1 = Klarner | first1 = D.A. | last2 = Hautus | first2 = M.L.J | authorlink1 = David A. Klarner | year = 1971 }}</ref>
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| :[[de Bruijn's theorem]]: A box can be packed with a [[harmonic brick]] ''a'' × ''a b'' × ''a b c'' if the box has dimensions ''a p'' × ''a b q'' × ''a b c r'' for some [[natural number]]s ''p'', ''q'', ''r'' (i.e., the box is a multiple of the brick.)<ref name="Gems2"/>
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| 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.
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| A classic puzzle of the second kind is to arrange all twelve [[pentomino]]es into rectangles sized 3×20, 4×15, 5×12 or 6×10.
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| ==Packing of irregular objects==
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| 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.<ref>C.Michael Hogan. 2010. [http://www.eoearth.org/article/Abiotic_factor?topic=49461 ''Abiotic factor''. Encyclopedia of Earth. eds Emily Monosson and C. Cleveland. National Council for Science and the Environment]. Washington DC</ref>
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| ==See also==
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| * [[Set packing]]
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| * [[Bin packing problem]]
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| * [[Slothouber-Graatsma puzzle]]
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| * [[Conway puzzle]]
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| * [[Tetris]]
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| * [[Covering problem]]
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| * [[Knapsack problem]]
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| * [[Tetrahedron packing]]
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| * [[Cutting stock problem]]
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| * [[Kissing number problem]]
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| * [[Close-packing of equal spheres]]
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| * [[Random close pack]]
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| ==Notes==
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| <references/>
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| ==References==
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| * {{mathworld|urlname=KlarnersTheorem|title=Klarner's Theorem}}
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| * {{mathworld|urlname=deBruijnsTheorem|title=de Bruijn's Theorem}}
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| ==External links==
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| Many puzzle books as well as mathematical journals contain articles on packing problems.
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| * [http://mathworld.wolfram.com/Packing.html Links to various MathWorld articles on packing]
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| * [http://mathworld.wolfram.com/SquarePacking.html MathWorld notes on packing squares.]
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| * [http://www.stetson.edu/~efriedma/packing.html Erich's Packing Center]
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| * [http://www.packomania.com/ www.packomania.com] A site with tables, graphs, calculators, references, etc.
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| * [http://demonstrations.wolfram.com/BoxPacking/ "Box Packing"] by [[Ed Pegg, Jr.]], the [[Wolfram Demonstrations Project]], 2007.
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| * [http://hydra.nat.uni-magdeburg.de/packing/cci/cci.html#overview Best known packings of equal circles in a circle, up to 1100]
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| {{Use dmy dates|date=September 2010}}
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| * [http://apmonitor.com/me575/index.php/Main/CircleChallenge Circle packing challenge problem in Python]
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| {{Packing problem}}
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| {{DEFAULTSORT:Packing Problem}}
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| [[Category:Packing problem]]
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