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Dimension of a subset of a metric space
the packing dimension is one of a number of concepts that can be used to define the dimension of a subset of a metric space. Packing dimension is in
Packing_dimension
Method of determining fractal dimension
is the correlation dimension. It is possible to define the box dimensions using balls, with either the covering number or the packing number. The covering
Minkowski–Bouligand_dimension
Invariant measure of fractal dimension
dimension that, like Hausdorff dimension, is defined using coverings by balls Fractal dimension Intrinsic dimension Packing dimension MacGregor Campbell, 2013
Hausdorff_dimension
Problems which attempt to find the most efficient way to pack objects into containers
sizes specified, or a single object of a fixed dimension that can be used repeatedly. Usually the packing must be without overlaps between goods and other
Packing_problems
Two-dimensional packing problem
Circle packing in a circle is a two-dimensional packing problem with the objective of packing unit circles into the smallest possible larger circle. If
Circle_packing_in_a_circle
Geometrical structure
of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres
Sphere_packing
3D fractal composed of tangential spheres
Apollonian sphere packing is the three-dimensional equivalent of the Apollonian gasket. The principle of construction is very similar: with any four spheres
Apollonian_sphere_packing
Real-valued number of spatial dimensions
}(X):=\inf\{d\geq 0:C_{H}^{d}(X)=0\}.} Packing dimension Assouad dimension Local connected dimension Degree dimension describes the fractal nature of the
Fractal_dimension
Three-dimensional packing problem
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
Sphere_packing_in_a_sphere
{some\ c.e.} \ s\mathrm {-gale\ succeeds\ on\ } X\}} . The effective packing dimension of X is inf { s : s o m e c . e . s − g a l e s u c c e e d
Effective_dimension
Three-dimensional packing problem
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
Sphere_packing_in_a_cylinder
continuous on the right for all t ≥ 0. Packing dimension is constructed in a very similar way to Hausdorff dimension, except that one "packs" E from inside
Dimension_function
Mathematical theory
This holds for packings in three-dimensional Euclidean space. If the midpoints of the spheres are arranged throughout 3D space, the packing is termed a cluster
Finite_sphere_packing
Two-dimensional packing problem
Square packing is a packing problem where the objective is to determine how many congruent squares can be packed into some larger shape, often a square
Square_packing
Field of geometry closely arranging circles on a plane
two-dimensional Euclidean plane, Joseph Louis Lagrange proved in 1773 that the highest-density lattice packing of circles is the hexagonal packing arrangement
Circle_packing
Fractal composed of tangent circles
mathematics, an Apollonian gasket, Apollonian net, or Apollonian circle packing is a fractal generated by starting with a triple of circles, each tangent
Apollonian_gasket
Least variables needed to represent data
practice, the box-counting dimension and the packing dimension often are identical to the Hausdorff dimension. Let X , d {\textstyle X,d} be a metric space
Intrinsic_dimension
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
Geometric space with five dimensions
A five-dimensional (5D) space is a mathematical or physical space that has five independent dimensions. In physics and geometry, such a space extends
Five-dimensional_space
Mathematical and computational problem
and pallet loading. Other variants are two-dimensional bin packing, three-dimensional bin packing, bin packing with delivery, BPPLIB - a library of surveys
Bin_packing_problem
Topics referred to by the same term
spaces: Complex dimension Hausdorff dimension Inductive dimension Lebesgue covering dimension Packing dimension Isoperimetric dimension Measurements of
Dimension_(disambiguation)
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)
Condition for fractals in math
simplify computation of the packing measure. An equivalent statement of the open set condition is to require that the s-dimensional Hausdorff measure of the
Open_set_condition
Industrial production of food and by-products from animals
The meat-packing industry (also spelled meatpacking industry or meat packing industry) handles the slaughtering, processing, packaging, and distribution
Meat-packing_industry
Packing problem
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
Sphere_packing_in_a_cube
Geometry hypothesis
there any three-dimensional convex body with lower packing density than the sphere? More unsolved problems in mathematics Ulam's packing conjecture, named
Ulam's_packing_conjecture
Property of objects which are scaled or mirrored versions of each other
which is often (but not always) equal to the set's Hausdorff dimension and packing dimension. If the overlaps between the fs(K) are "small", we have the
Similarity_(geometry)
Dense arrangement of congruent spheres in an infinite, regular arrangement
In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement (or lattice). Carl Friedrich
Close-packing of equal spheres
Close-packing_of_equal_spheres
October 2016). "Three Variable Dimension Surfaces". ResearchGate. The Fractal dimension of the apollonian sphere packing Archived 6 May 2016 at the Wayback
List of fractals by Hausdorff dimension
List_of_fractals_by_Hausdorff_dimension
Geometric concept
Equilateral dimension Spherical code Soddy's hexlet Cylinder sphere packing Conway, John H.; Neil J.A. Sloane (1999). Sphere Packings, Lattices and
Kissing_number
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
On tangency patterns of circles
The circle packing theorem (also known as the Koebe–Andreev–Thurston theorem) describes the possible patterns of tangent circles among non-overlapping
Circle_packing_theorem
On lattices and sphere packing in Euclidean space
{\displaystyle n} -dimensional Euclidean space, anisohedral tiling in three-dimensional Euclidean space, and the densest sphere packing in Kepler conjecture
Hilbert's_eighteenth_problem
Concept in three-dimensional geometry
In geometry, tetrahedron packing is the problem of arranging identical regular tetrahedra throughout three-dimensional space so as to fill the maximum
Tetrahedron_packing
Ukrainian mathematician (born 1984)
the sphere-packing problem in dimension 8. Her dimension 8 solution quickly led to collaboration with others, and a solution in dimension 24. Previously
Maryna_Viazovska
In geometry, ellipsoid packing is the problem of arranging identical ellipsoid throughout three-dimensional space to fill the maximum possible fraction
Ellipsoid_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
On packing congruent rectangular bricks (of any dimension) into larger rectangular boxes
Nicolaas Govert de Bruijn proved several results about packing congruent rectangular bricks (of any dimension) into larger rectangular boxes, in such a way that
De_Bruijn's_theorem
2016), "Sphere Packing Solved in Higher Dimensions", Quanta Magazine Viazovska, Maryna (2016). "The sphere packing problem in dimension 8". Annals of Mathematics
List of shapes with known packing constant
List_of_shapes_with_known_packing_constant
Shape containing unit line segments in all directions
radius 1/2 in the Euclidean plane, or a ball of radius 1/2 in three-dimensional space, forms a Kakeya set. Much of the research in this area has studied
Kakeya_set
Math theorem about sphere packing
astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of equally sized
Kepler_conjecture
Well-spaced set of points in a metric space
In the mathematical theory of metric spaces, ε-nets, ε-packings, ε-coverings, uniformly discrete sets, relatively dense sets, and Delone sets (named after
Delone_set
Finnish mathematician
in 1994. Her doctoral dissertation, On the Upper Minkowski Dimension, the Packing Dimension, and Orthogonal Projections, was supervised by Pertti Mattila
Maarit_Järvenpää
Lattice in 8-dimensional space with special properties
2016). "Sphere Packing Solved in Higher Dimensions". Quanta Magazine. Viazovska, Maryna (2017). "The sphere packing problem in dimension 8". Annals of
E8_lattice
Two-dimensional packing problem
Circle packing in a square is a packing problem in recreational mathematics where the aim is to pack n unit circles into the smallest possible square.
Circle_packing_in_a_square
Type of crystal structure
structure) than the packing factors for the face-centered and body-centered cubic lattices. Zincblende structures have higher packing factors than 0.34
Diamond_cubic
Limit on the parameters of a block code
code: it is also known as the sphere-packing bound or the volume bound from an interpretation in terms of packing balls in the Hamming metric into the
Hamming_bound
Type of mathematical set
simplices (for example, points, line segments, triangles, and their n-dimensional counterparts) such that all the faces and intersections of the elements
Simplicial_complex
Artificial intelligence method for mathematical discovery
combinatorics and to the online bin packing problem, where it found new mathematical constructions and new packing heuristics. FunSearch frames a problem
FunSearch
Natural number
form the binary tetrahedral group. The optimal sphere packing problem has been solved in dimension 24, one of the only dimensions where this has been solved
24_(number)
Pricing technique for commercial freight transport
avoid dimensional weight charges by using smaller boxes, by compressing their goods, and by reducing the use of packing materials. Dimensional weight
Dimensional_weight
Graph-theoretic description of polyhedra
minimum-energy state of a two-dimensional spring system and lifting the result into three dimensions, or by using the circle packing theorem. Several extensions
Steinitz's_theorem
American mathematician
19 January 2025. Viazovska, Maryna (1 May 2017). "The sphere packing problem in dimension $8$". Annals of Mathematics. 185 (3): 991–1015. arXiv:1603.04246
Henry_Cohn
conversion to and from equivalent jigsaw puzzles and polyomino packing puzzle. Three-dimensional edge-matching puzzles are not currently under direct U.S.
Three-dimensional edge-matching puzzle
Three-dimensional_edge-matching_puzzle
Shape made from cubes joined together
the Slothouber–Graatsma puzzle, and the Conway puzzle are examples of packing problems based on polycubes. Like polyominoes, polycubes can be enumerated
Polycube
Complex structures in matter physics
or hexagonal close packing (hcp) lattices. Up to some extent amorphous metals and quasicrystals can also be modeled by close packing of spheres. The local
Geometrical_frustration
Ronald L. Graham. Worst-Case Performance Bounds for Simple One-Dimensional Packing Algorithms. SICOMP, Volume 3, Issue 4. 1974. Garey, M. R; Graham
Best-fit_bin_packing
Optimization algorithm
(FF) is an online algorithm for bin packing. Its input is a list of items of different sizes. Its output is a packing - a partition of the items into bins
First-fit_bin_packing
Problem of grouping into triples
common vertex). In case of 2-dimensional matching, we have Y = Z. A 3-dimensional matching is a special case of a set packing: we can interpret each element
3-dimensional_matching
represented by Schläfli symbol {3,3,4,3}, and constructed by a 4-dimensional packing of 16-cell facets, three around every (triangular) face. Its dual
16-cell_honeycomb
Ordered arrangement of atoms, ions, or molecules in a crystalline material
symmetric patterns that repeat along the principal directions of three-dimensional space in matter. The smallest group of particles in a material that constitutes
Crystal_structure
Concept in molecular modelling
the hypercubic lattice packing. It is then preferred to choose a unit cell which corresponds to the dense packing of that dimension. In 4D this is D4 lattice;
Periodic_boundary_conditions
Natural number
are 26 sporadic groups. The 26-dimensional Lorentzian unimodular lattice II25,1 plays a significant role in sphere packing problems and the classification
26_(number)
Mathematical problem in operations research
nesting problem. Not many three-dimensional (3D) applications involving cutting are known; however the closely related 3D packing problem has many industrial
Cutting_stock_problem
Japanese art of paper folding
and technique of folding paper. It also refers to the two- and three-dimensional forms created in the process. The use of the term has been extended in
Origami
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
Generalized sphere of dimension n (mathematics)
{\displaystyle n} -dimensional generalization of the 1 {\displaystyle 1} -dimensional circle and 2 {\displaystyle 2} -dimensional sphere to any non-negative
N-sphere
Problem in combinatorial optimization
multiple-choice multi-dimensional knapsack. The IHS (Increasing Height Shelf) algorithm is optimal for 2D knapsack (packing squares into a two-dimensional unit size
Knapsack_problem
Visual art practice using adhesive tape
tape-based techniques intended for long-term exhibition, including illuminated packing-tape images mounted on acrylic or Plexiglas and presented as light boxes
Tape_art
Shape with four equal sides and angles
subdividing squares into unequal squares. Mathematicians have also studied packing squares as tightly as possible into other shapes. Squares can be constructed
Square
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)
Size of a mathematical ball
ball in an n-dimensional Euclidean space. The volume of a n-ball is the Lebesgue measure of this ball, which generalizes to any dimension the usual volume
Volume_of_an_n-ball
Existence theorem on the lattice packing of hyperspheres
result on the lattice packing of hyperspheres in dimension n > 1. It states that there is a lattice in Euclidean space of dimension n, such that the corresponding
Minkowski–Hlawka_theorem
Mathematical model of the physical space
solid geometry is the determination of packing arrangements, such as the problem of finding the most efficient packing of spheres in n dimensions. This problem
Euclidean_geometry
2024 studio album by Gracie Abrams
"offers glimpses into the singer's interior world", although it "lacks dimension". NME's Hannah Mylrea said that Abrams "embraces her growing pains and
The_Secret_of_Us
Three-dimensional packing problem
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 ×
Conway_puzzle
One of the five 2D Bravais lattices
lattice (sometimes called triangular lattice) is one of the five two-dimensional Bravais lattice types. The symmetry category of the lattice is wallpaper
Hexagonal_lattice
Three-dimensional packing problem
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 (all shapes
Slothouber–Graatsma_puzzle
Several sets of circles associated with Apollonius of Perga
also known as a Leibniz packing or an Apollonian packing. This gasket is a fractal, being self-similar and having a dimension d that is not known exactly
Circles_of_Apollonius
Natural number
simplest parallelotope that is not a zonotope. Seventeen is the highest dimension for paracompact Vineberg polytopes with rank n + 2 {\displaystyle n+2}
17_(number)
Voronoi tessellation where the generating point of each Voronoi cell is also its centroid
depends on the dimension." In two dimensions, the basic cell for the optimal CVT is a regular hexagon as it is proven to be the most dense packing of circles
Centroidal Voronoi tessellation
Centroidal_Voronoi_tessellation
Branch of geometry that studies combinatorial properties and constructive methods
usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, n-dimensional Euclidean space
Discrete_geometry
Number of balls of a given size needed to cover a given space
can be applied to general metric spaces. Two related concepts are the packing number, the number of disjoint balls that fit in a space, and the metric
Covering_number
24-dimensional repeating pattern of points
D.; Radchenko, Danylo; Viazovska, Maryna (2017), "The sphere packing problem in dimension 24", Annals of Mathematics, 185 (3): 1017–1033, arXiv:1603.06518
Leech_lattice
2005 mathematics text
distinguished from sphere packing, which considers higher dimensions (here, everything is two dimensional) and is more focused on packing density than on combinatorial
Introduction to Circle Packing
Introduction_to_Circle_Packing
Packing material
create a three-dimensional plastic wallpaper. Although the idea was a failure, they found that what they made could be used as packing material. Sealed
Bubble_wrap
1988 mathematical book
Sphere Packings, Lattices and Groups is a book about geometry and group theory by John Conway and Neil Sloane, with contributions by other mathematicians
Sphere Packings, Lattices and Groups
Sphere_Packings,_Lattices_and_Groups
mathematics In combinatorics, tripod packing is a problem of finding many disjoint tripods in a three-dimensional grid, where a tripod is an infinite polycube
Tripod_packing
Linear stacking of regular tetrahedra that form helices
ISBN 052120125X. Boerdijk, A.H. (1952). "Some remarks concerning close-packing of equal spheres". Philips Res. Rep. 7: 303–313. Fuller, R.Buckminster
Boerdijk–Coxeter_helix
Marvel Comics fictional character
teenage counterpart, Psi-Lord, who had been raised by Nathaniel in a dimension outside of time. Franklin, as Psi-Lord, founds the team Fantastic Force
Franklin_Richards_(character)
High-multiplicity bin packing is a special case of the bin packing problem, in which the number of different item-sizes is small, while the number of items
High-multiplicity_bin_packing
Geometric system with a finite number of points
geometries are Galois geometries, since any finite projective space of dimension three or greater is isomorphic to a projective space over a finite field
Finite_geometry
Classification of a two-dimensional repetitive pattern
plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns
Wallpaper_group
Puzzles involving the assembly of flat shapes
Tiling puzzles are puzzles involving two-dimensional packing problems in which a number of flat shapes have to be assembled into a larger given shape
Tiling_puzzle
Two-dimensional packing problem
Circle packing in a right isosceles triangle is a packing problem where the objective is to pack n unit circles into the smallest possible isosceles right
Circle packing in an isosceles right triangle
Circle_packing_in_an_isosceles_right_triangle
Law of sediment aggradation
\eta } , over time, t {\displaystyle t} , is equal to one over the grain packing density, ε o {\displaystyle \varepsilon _{o}} , times the negative divergence
Exner_equation
Continuous function that is not absolutely continuous
differentiability is usually given in terms of fractal dimension, with the Hausdorff dimension the most popular choice. This line of research was started
Cantor_function
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
Natural number, composite number
Retrieved 2023-01-18. Baez, John C. (February 2015). "Pentagon-Decagon Packing". AMS Blogs. American Mathematical Society. Retrieved 2023-01-18. Coxeter
14_(number)
2007 American film
the women he targets. The film was originally released theatrically by Dimension Films on April 6, 2007 as part of Grindhouse, a double feature that combined
Death_Proof
PACKING DIMENSION
PACKING DIMENSION
Surname or Lastname
English (chiefly Devon)
English (chiefly Devon) : from a Middle English pet form of the Old English personal name Hocca.Dutch : patronymic from Hock 4.
Girl/Female
Arabic, Muslim
Abstinent; Lacking Mundane Ambitions
Surname or Lastname
English (Lancashire)
English (Lancashire) : habitational name from Hacking in Lancashire, the name of which is of uncertain origin. Early forms appear with the definite article, and the name may represent an Old English term for a fish weir, a derivative of hæcc ‘hatch’, ‘low gate’, or haca ‘hook’.
Surname or Lastname
English
English : variant of Markin.
Boy/Male
Muslim/Islamic
Walking
Surname or Lastname
English
English : patronymic from Parkin.Americanized form of one or more like-sounding Jewish names.
Girl/Female
Gujarati, Indian
Sweet Eyes
Surname or Lastname
English and German
English and German : patronymic from the personal name Paul.
Girl/Female
Tamil
Making
Boy/Male
English
Little rock.
Girl/Female
Hindu
Making
Surname or Lastname
English
English : possibly from Middle English Old French personal name Pic (see Pike 6) + the diminutive suffix -in.
Surname or Lastname
English (Staffordshire)
English (Staffordshire) : from the Welsh personal name Pasgen, a derivative of Latin Pascentius.
Boy/Male
American, Anglo, Australian, British, English
Little Rock; Little Peter
Surname or Lastname
English
English : from a pet form of Paul.Altered form, in the New Netherland Dutch community, of Paling. Compare Paulding.
Surname or Lastname
English
English : from Old English LÄ“ofecing, a patronymic from LÄ“ofeca (see Levick 2), or possibly, as Reaney suggests, a late derivative of Lovekin (see Lucken).
Surname or Lastname
English
English : from a diminutive of Middle English cok ‘cock’ (see Cocke).
Boy/Male
Arabic, Muslim, Sindhi
Walking
Boy/Male
American, British, English
Son of Parkin
Surname or Lastname
English (mainly Yorkshire)
English (mainly Yorkshire) : from the Middle English personal name Perkin, Parkin, a pet form of Peter with the diminutive suffix -kin. (The change from -er- to -ar- was a characteristic phonetic development in Old French and Middle English.)
PACKING DIMENSION
PACKING DIMENSION
Male
Arthurian
, lofty and wise (?).
Boy/Male
Gujarati, Hindu, Indian, Sanskrit, Telugu
Love; Affection; Friendship; Respect
Boy/Male
Indian
The elevated one
Boy/Male
Scottish
True and bold. Also 'bald'. Introduced from England and Germany during the Norman conquest, the...
Boy/Male
Arabic, Muslim
Earner; Aquirer
Biblical
the salvation of the Lord (same as Isaiah)
Boy/Male
Hindu, Indian
Unstoppable
Boy/Male
Bengali, Indian
God of Kartikeya
Girl/Female
Assamese, Bengali, Gujarati, Hindu, Indian, Kannada, Marathi, Telugu
A Devotee of Lord Krishna
Girl/Female
Hindu
PACKING DIMENSION
PACKING DIMENSION
PACKING DIMENSION
PACKING DIMENSION
PACKING DIMENSION
n.
Any material used to pack, fill up, or make close.
n.
The act of one who, or that which, marks; the mark or marks made; arrangement or disposition of marks or coloring; as, the marking of a bird's plumage.
n.
A coarse woolen fabric, used for floor cloths, to cover carpets, etc.; -- so called from the town of Bocking, in England, where it was first made.
a.
Done or made as with a pointed tool; as, a picking sound.
n.
A thin layer, or sheet, of yielding or elastic material inserted between the surfaces of a flange joint.
n.
A trick; collusion.
v. t.
Small coal produced in making the nicking.
n.
A substance or piece used to make a joint impervious
n.
Same as Filling.
p. pr. & vb. n.
of Pack
n.pl.
Packing of hemp.
n.
A union of securities given at different times, all of which must be redeemed before an intermediate purchaser can interpose his claim.
n.
A yielding ring, as of metal, which surrounds a piston and maintains a tight fit, as inside a cylinder, etc.
p. pr. & vb. n.
of Tack
a.
Distressing; worrying; perplexing; corroding; as, carking cares.
n.
The act or process of one who packs.
p. pr. & vb. n.
of Sack
n.
The substance in a stuffing box, through which a piston rod slides.
n.
Stout, coarse cloth of which sacks, bags, etc., are made.
n.
Spun yarn used in racking ropes.