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Optimization problem in mathematics
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
Rectangle_packing
Problems which attempt to find the most efficient way to pack objects into containers
pack 147 rectangles of size (137,95) in a rectangle of size (1600,1230). Packing different rectangles in a rectangle: The problem of packing multiple
Packing_problems
Two-dimensional packing problem
onwards. Dense packings of circles in non-square rectangles have also been the subject of investigations. Square packing in a circle Circle packing in a circle
Circle_packing_in_a_square
Two-dimensional packing problem
with half-integer vertex coordinates. Circle packing in a square Squaring the square Rectangle packing Moving sofa problem Brass, Peter; Moser, William;
Square_packing
Problem in computer science
Set packing is a classical NP-complete problem in computational complexity theory and combinatorics, and was one of Karp's 21 NP-complete problems. Suppose
Set_packing
Field of geometry closely arranging circles on a plane
packing in a circle Circle packing in a square Circle packing in a rectangle Circle packing in an equilateral triangle Circle packing in an isosceles right
Circle_packing
On tangency patterns of circles
of a rectangle by a Möbius transformation), in such a way that when the invariant is a rational number, the rectangle has a triangulated packing. His
Circle_packing_theorem
Mathematical and computational problem
from bin packing are used in this problem too. In the guillotine cutting problem, both the items and the "bins" are two-dimensional rectangles rather than
Bin_packing_problem
Unrelated vertices in graphs
only one need be output. This problem is sometimes referred to as "vertex packing". In the maximum-weight independent set problem, the input is an undirected
Independent set (graph theory)
Independent_set_(graph_theory)
Method to solve optimization problems
example, the LP relaxations of the set packing problem, the independent set problem, and the matching problem are packing LPs. The LP relaxations of the set
Linear_programming
Classical problem in combinatorics
intersection of the universe and geometric shapes (e.g., disks, rectangles). Set packing is the problem of selecting the maximum number of sets that are
Set_cover_problem
Set of edges without common vertices
packing Minimum edge cover Maximum matching Minimum vertex cover Maximum independent set Bin covering Bin packing Polygon covering Rectangle packing v t e
Matching_(graph_theory)
Subset of a graph's vertices, including at least one endpoint of every edge
packing Minimum edge cover Maximum matching Minimum vertex cover Maximum independent set Bin covering Bin packing Polygon covering Rectangle packing v t e
Vertex_cover
2D geometric minimization 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
Strip_packing_problem
Subset of a graph's edges
packing Minimum edge cover Maximum matching Minimum vertex cover Maximum independent set Bin covering Bin packing Polygon covering Rectangle packing v t e
Edge_cover
Shape with four equal sides and angles
are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal sides. As with all rectangles, a square's angles
Square
Type of computational problem
problems and usually integer linear programs, whose dual problems are called packing problems. The most prominent examples of covering problems are the set
Covering_problems
Process of producing small rectangular items of fixed dimensions
These are variants of the two-dimensional cutting stock, bin packing and rectangle packing problems, where the cuts are constrained to be guillotine cuts
Guillotine_cutting
Layout of major electronic circuit blocks
floorplanning refers to the problem of packing smaller rectangles with a fixed or unfixed orientation into a larger rectangle.[citation needed] The dimensions
Floorplan_(microelectronics)
Polygon in which all angles are right
of this type are rectangles, and the term axis-aligned rectangle is preferred, although orthogonal rectangle and rectilinear rectangle are in use as well
Rectilinear_polygon
Strongly NP-complete problem in computer science
The NP-hardness of 3-partition was used to prove the NP-hardness rectangle packing, as well as of Tetris and some other puzzles, and some job scheduling
3-partition_problem
Geometric shape formed from six squares
parity does not prevent a packing, and a packing is indeed possible. It is also possible for two sets of pieces to fit a rectangle of size 420, or for the
Hexomino
Operations research problem of packing items into the largest number of bins
of the bin packing problem: in bin covering, the bin sizes are bounded from below and the goal is to maximize their number; in bin packing, the bin sizes
Bin_covering_problem
On integer partitions from monotonic functions
the theorem can be read off from this packing as the heights that the i {\displaystyle i} th vertical rectangle rises above the x {\displaystyle x} -axis
Lambek–Moser_theorem
2023 video game
GitHub as early as 2013, with game becoming playable around 2017. A rectangle packing library, called rectpack2D and created specifically for Hypersomnia
Hypersomnia_(video_game)
Self-similar curve related to golden ratio
starts with a rectangle partitioned into 2 squares. In each step, a square the length of the rectangle's longest side is added to the rectangle. Since the
Golden_spiral
Regular tiling of a two-dimensional space
face-centered cubic and hexagonal close packing are common crystal structures. They are the densest sphere packings in three dimensions. Structurally, they
Hexagonal_tiling
Set of basic shapes which assemble into a polygon
partition of a polygon is a set of primitive units (e.g., triangles, rectangles, etc.), which do not overlap and whose union equals the polygon. A polygon
Polygon_partition
Data structures used in spatial indexing
bounding rectangle in the next higher level of the tree; the "R" in R-tree is for rectangle. Since all objects lie within this bounding rectangle, a query
R-tree
Unsolved geometry question on moving a sofa through a 90° angle
quarter-disks of radius 1 on either side of a 1 by 4 / π {\displaystyle 4/\pi } rectangle from which a half-disk of radius 2 / π {\displaystyle 2/\pi } has been
Moving_sofa_problem
R-tree variant and index for multidimensional objects
should group "similar" data rectangles together, to minimize the area and perimeter of the resulting minimum bounding rectangles (MBRs). Packed Hilbert R-trees
Hilbert_R-tree
Assembly puzzle named after Dean Hoffman
Hoffman's packing puzzle is an assembly puzzle named after Dean G. Hoffman, who described it in 1978. The puzzle consists of 27 identical rectangular
Hoffman's_packing_puzzle
Manufacturing method to avoid waste of materials
rolls 3D nesting – for packing optimization of 3D parts such as boxes, shipping containers, 3D printed parts nesting/packing of freeform 3D objects To
Nesting_(process)
Geometric shape formed from squares
prime rectangles for various polyominoes". Archived from the original on 2007-04-16. Retrieved 2007-05-11. Klarner, D.A.; Göbel, F. (1969). "Packing boxes
Polyomino
Geometric shape formed from seven squares
by 107 (749-square) rectangle. Furthermore, the complete set of free heptominoes can tile three 11-by-23 (253-square) rectangles, each with a one-square
Heptomino
Mathematical problem
cubes of higher dimensions. Square packing in a square Dividing a square into similar rectangles Perfect rectangle Sprague, R. (1939). "Beispiel einer
Squaring_the_square
Problem in combinatorial optimization
to the Bin Packing Problem. It differs from the Bin Packing Problem in that a subset of items can be selected, whereas, in the Bin Packing Problem, all
Knapsack_problem
Set of primitive shapes whose union equals a polygon
union is exactly equal to the target polygon. This is in contrast to a packing problem, in which the units must be disjoint and their union may be smaller
Polygon_covering
United States historic place
The J. C. Rhew Co. Packing Shed was a strawberry packing house in rural northern White County, Arkansas. It was located on the south side of Graham Road
J.C._Rhew_Co._Packing_Shed
Mathematical problem in operations research
are known; however the closely related 3D packing problem has many industrial applications, such as packing objects into shipping containers (see e.g
Cutting_stock_problem
Tiling puzzle
and, furthermore, only one color is used for the outside edge of the rectangle. This puzzle can be extended to tiles with permutations of 4 colors, arranged
Edge-matching_puzzle
Concept in computational geometry
intersects at least one rectangle (hence m ≤ n). Each rectangle is intersected by exactly one line. Since the height of all rectangles is H, it is not possible
Maximum_disjoint_set
Geometric shape formed from two squares
in a countably infinite number of ways. The number of tilings of a 2×n rectangle with dominoes is F n {\displaystyle F_{n}} , the nth Fibonacci number
Domino_(mathematics)
Tiling of euclidean or hyperbolic space of three or more dimensions
In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of
Honeycomb_(geometry)
Graph representing tangency between geometric objects
called penny graphs. Representations as contact graphs of triangles, rectangles, squares, line segments, or circular arcs have also been studied. Chaplick
Contact_graph
Words can be read horizontally and vertically
be transposed to form another valid rectangle. For example, a 4×8 rectangle can also be written as an 8×4 rectangle. Palindromic magic squares, like the
Word_square
Lingerie accessories
ring is circle which looks like an "O". Triangle, rectangle, square, heart, star shape, rectangle with saw tooth shape or even flower shape can be found
Ring,_slide_and_hook
Arithmetic mean is greater than or equal to geometric mean
perimeter of a rectangle with sides of length x1 and x2. Similarly, 4√x1x2 is the perimeter of a square with the same area, x1x2, as that rectangle. Thus for
AM–GM_inequality
Geometric shape formed from ten squares
to prove that the complete set of decominoes cannot be packed into a rectangle, and that not all decominoes can be tiled. The 4,460 decominos without
Decomino
American mathematician (1940–1999)
box-packing. Working with Ronald L. Rivest he found upper bounds on the number of n-ominoes. Klarner's Theorem is the statement that an m by n rectangle can
David_A._Klarner
Sphere tangent to every edge of a polyhedron
distances from its two endpoints to their corresponding circles in this circle packing. Every convex polyhedron has a combinatorially equivalent polyhedron, the
Midsphere
Semiregular tiling of the Euclidean plane
square can be distorted into isosceles trapezoids. In the limit, where the rectangles degenerate into edges, a triangular tiling results, constructed as a snub
Rhombitrihexagonal_tiling
British mathematician (1925–2015)
on the large sieve, on the Heilbronn triangle problem, and on square packing in a square. He was a coauthor of the book Sequences on integer sequences
Klaus_Roth
Solitaire card game
a triangular layout of the tableau, building in ascending sequence and packing in descending order. In the U.S. and Canada, it is so well known that the
Klondike_(solitaire)
Overview of and topical guide to geometry
Equidiagonal quadrilateral Kite (geometry) Orthodiagonal quadrilateral Rhombus Rectangle Square Tangential quadrilateral Trapezoid Isosceles trapezoid Sangaku
Outline_of_geometry
Geometric shape formed from five squares
book The Canterbury Puzzles, published in 1907. The earliest tilings of rectangles with a complete set of pentominoes appeared in the Problemist Fairy Chess
Pentomino
Geometric shape formed from nine squares
nonominoes have holes. Therefore a complete set cannot be packed into a rectangle and not all nonominoes have tilings. Of the 1285 free nonominoes, 960
Nonomino
Define a 2-fat rectangle as an axis-parallel rectangle with an aspect ratio of at most 2. Let R0 be a minimal-area 2-fat rectangle that contains the
Geometric_separator
character detailed the packing method (Cartons, Bandoleers, or Belts / Links) and container type used (M1917 Rifle Ammunition Packing Box, M23 Ammo Crate
List of the United States Army munitions by supply catalog designation
List_of_the_United_States_Army_munitions_by_supply_catalog_designation
Classification of a two-dimensional repetitive pattern
Turkish dish A compact packing of two sizes of circle Another compact packing of two sizes of circle Another compact packing of two sizes of circle 3
Wallpaper_group
Branch of geometry that studies combinatorial properties and constructive methods
the late 19th century. Early topics studied were: the density of circle packings by Thue, projective configurations by Reye and Steinitz, the geometry of
Discrete_geometry
Computational physics simulation algorithm
Donev, Aleksandar; Stillinger, Frank H.; Torquato, Salvatore (2006). "Packing hyperspheres in high-dimensional Euclidean spaces". Physical Review E.
Lubachevsky–Stillinger algorithm
Lubachevsky–Stillinger_algorithm
Geometric shape formed from eight squares
to prove that the complete set of octominoes cannot be packed into a rectangle, and that not all octominoes can be tiled. Golomb, Solomon W. (1994).
Octomino
Triangulation method
For four or more points on the same circle (e.g., the vertices of a rectangle) the Delaunay triangulation is not unique: each of the two possible triangulations
Delaunay_triangulation
Covering by shapes without overlaps or gaps
grid Honeycomb (geometry) List of mathematical art software Packing problem Perfect rectangle Space partitioning The mathematical term for identical shapes
Tessellation
Mathematical model of the physical space
exactly. (Flipping it over is allowed.) Thus, for example, a 2x6 rectangle and a 3x4 rectangle are equal but not congruent, and the letter R is congruent to
Euclidean_geometry
Fabric item designed to carry a child on the body
Korean carrier with a medium to large rectangle of fabric hanging from a very long strap. Traditionally the rectangle is quilted for warmth and wraps around
Baby_sling
Process in materials science
including in 2d, disks, randomly oriented squares and rectangles, aligned squares and rectangles, various other shapes, etc. An important result is the
Random_sequential_adsorption
Circle packing – Field of geometry closely arranging circles on a plane Circle packing in a circle – Two-dimensional packing problem Circle packing in an
List_of_circle_topics
Book on shapes formed from squares
Polyominoes: Puzzles, Patterns, Problems, and Packings is a mathematics book on polyominoes, the shapes formed by connecting some number of unit squares
Polyominoes: Puzzles, Patterns, Problems, and Packings
Polyominoes:_Puzzles,_Patterns,_Problems,_and_Packings
Computer encoding of characters
Braille characters are represented using six dot positions, arranged in a rectangle. Each position may contain a raised dot or not, so Braille can be considered
Six-bit_character_code
Semiregular tiling
a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing (kissing
Truncated_square_tiling
Equation in Fourier analysis
kernel on R 2 {\displaystyle \mathbb {R} ^{2}} is known, and that of a rectangle is determined by taking the periodization. The Poisson summation formula
Poisson_summation_formula
Container with a shape between a square and a round tub
square and a round tub. It resembles an oval but is sometimes closer to a rectangle with rounded corners. These allow the contents to be easily scooped out
Squround
Tactic in association football
Vanderlei Luxemburgo proposed basing the "magic rectangle" on the work of the wing-backs. The rectangle becomes a 3–4–3 on the attack because one of the
Formation (association football)
Formation_(association_football)
Branch of mathematics
such as points, lines and circles. Examples include the study of sphere packings, triangulations, the Kneser-Poulsen conjecture, etc. It shares many methods
Geometry
geometry Ernest Vinberg (1937–2020) J. H. Conway (1937–2020) – sphere packing, recreational geometry Robin Hartshorne (1938–) – geometry, algebraic geometry
List_of_geometers
Stationery item used for flat mail
the sheet sides around a central rectangular area. In this manner, a rectangle-faced enclosure is formed with an arrangement of four flaps on the reverse
Envelope
Type of space-filling polyhedron
gyrobifastigium, with faces made of isosceles right triangles and silver rectangles, is a plesiohedron. The triakis truncated tetrahedron, the prototile of
Plesiohedron
Property of objects which are scaled or mirrored versions of each other
other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, and isosceles triangles are not all
Similarity_(geometry)
Nazi extermination camp in Poland (1942–1943)
chambers marked with a cross. Undressing and hair-cropping area marked with rectangle, with fenced-out "Sluice" into the woods, obstructing the view of the
Belzec_extermination_camp
American chess player, composer, puzzle author and mathematician (1841-1911)
which can be assembled into a 5x13 rectangle. Since the area of the square is 64 units but the area of the rectangle is 65 units, this seems paradoxical
Sam_Loyd
Former almshouse in Marseille, France
the direction of his son, François. The main body of the structure is a rectangle, 112 m by 96 m, composed of four walls in pink and yellow-tinted molasse
Vieille_Charité
Natural number
see Ramsey's theorem. Either 16 or 18 unit squares can be formed into rectangles with perimeter equal to the area; and there are no other natural numbers
17_(number)
(n\,\log \,n)} . Bounding sphere Farthest-first traversal Largest empty rectangle G. T. Toussaint, "Computing largest empty circles with location constraints
Largest_empty_sphere
"larger" than lines or curves, and yet "smaller" than filled circles or rectangles. Effective dimension modifies Hausdorff dimension by requiring that objects
Effective_dimension
Symmetric arrangement of finite sets
where r ≤ n. An n × n Latin rectangle is called a Latin square. If r < n, then it is possible to append n − r rows to an r × n Latin rectangle to form a Latin square
Combinatorial_design
Practice in logistics of unloading directly to customer or other transportation
are generally designed in an "I" configuration, which is an elongated rectangle. The goal in using this shape is to maximize the number of inbound and
Cross-docking
Rational numbers with root 5 added
LCCN 56-10138. Liba, Opher; Ilany, Bat-Sheva (2023). From the Golden Rectangle to the Fibonacci Sequences. Contributions by Nativ, Isaac. Cham: Springer
Golden_field
Smallest 3D projective space
lines form symmetric sub-structures like rows, columns, transversals, or rectangles, as seen in the figure. (There are 20160 such orderings, as seen below
PG(3,2)
Numbers obtained by adding the two previous ones
and (n + 1)-th Fibonacci numbers. To see this, begin with a Fibonacci rectangle of size F n × F n + 1 {\displaystyle F_{n}\times F_{n+1}} and decompose
Fibonacci_sequence
Smallest dimension where a graph can be represented as an intersection graph of boxes
(2001), "Efficient approximation algorithms for tiling and packing problems with rectangles", J. Algorithms, 41 (2): 443–470, doi:10.1006/jagm.2001.1188
Boxicity
Brazilian racing driver (1960–1994)
there was a green stripe under the chin, and there was a blue rounded rectangle near the top. Bruno sported a modified helmet design for the final three
Ayrton_Senna
Capability of a computer graphic to allow whatever is "behind" it to be visible
include: an image that is not rectangular can be filled to the required rectangle using transparent surroundings; the image can even have holes (e.g. be
Transparency_(graphic)
Magnesium end-member of olivine, a nesosilicate mineral
different. M2 site is larger and more regular than M1 as shown in Fig. 1. The packing in forsterite structure is dense. The space group of this structure is
Forsterite
Bitmap image file format family
pixels on the display, because the Image Descriptor can define a smaller rectangle to be rescanned instead of the whole image. Browsers or other displays
GIF
Uniform tiling of the Euclidean plane
a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing (kissing
Truncated_trihexagonal_tiling
Regular tiling of the Euclidean plane
Wikimedia Commons has media related to Order-4 square tiling. Tiling with rectangles Fenestrane Langton's ant OpenStructures, design pattern consisting of
Square_tiling
{\displaystyle X} and geometric shapes such as disks and axis-parallel rectangles. The goal is to select a minimum-size subset C ⊆ R {\displaystyle {\mathcal
Geometric_set_cover_problem
Tiling of the hyperbolic plane
each other. With these choices, the tile has four right angles, like a rectangle, with its sides alternating between segments of hyperbolic lines and arcs
Binary_tiling
RECTANGLE PACKING
RECTANGLE PACKING
RECTANGLE PACKING
RECTANGLE PACKING
Boy/Male
British, English
From the Roe-deer Brook
Girl/Female
Indian
Girl/Female
Indian, Kashmiri
Good Letter
Boy/Male
Indian, Sikh
Love
Girl/Female
Muslim
Clean, Pure
Boy/Male
Australian, Danish, French, German, Italian, Latin
Hammer; Warlike
Girl/Female
Hindu
Boy/Male
Anglo Saxon
Exalts.
Girl/Female
Hindu, Indian, Marathi
Well Conducted; Virtuous
Boy/Male
Australian, Gaelic, Irish, Scottish
From the River Island
RECTANGLE PACKING
RECTANGLE PACKING
RECTANGLE PACKING
RECTANGLE PACKING
RECTANGLE PACKING
v. t.
Hence: To insnare; to entangle.
n.
A four-sided figure having only right angles; a right-angled parallelogram.
imp. & p. p.
of Entangle
a.
Rectangular.
v. t. & i.
To change again, or change back.
n.
A figure which has seven angles; a heptagon.
p. pr. & vb. n.
of Entangle
v. t.
To involve in such complications as to render extrication a bewildering difficulty; hence, metaphorically, to insnare; to perplex; to bewilder; to puzzle; as, to entangle the feet in a net, or in briers.
v. t.
To entangle; to insnare.
v. t.
To twist or interweave in such a manner as not to be easily separated; to make tangled, confused, and intricate; as, to entangle yarn or the hair.
a.
Rectangular.
v. t.
To entangle.
v. t.
To trammel; to entangle.
v. t.
To entangle; to intertwine.
n.
A bearing in the form of an oblong rectangle.
n.
A pentagon.
v. t.
To confuse; to entangle.
v. t.
To make intricate; to entangle.
v. t.
See Entangle.
v. t.
Fig.: To entangle; to hamper.