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Arrangement of spheres within a space
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical
Sphere_packing
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
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
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
Mathematical theory
finite sphere packing concerns the question of how a finite number of equally-sized spheres can be most efficiently packed. The question of packing finitely
Finite_sphere_packing
Packing problem
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
Sphere_packing_in_a_cube
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 that
Apollonian_sphere_packing
Field of geometry closely arranging circles on a plane
this is called sphere packing, which usually deals only with identical spheres. The branch of mathematics generally known as "circle packing" is concerned
Circle_packing
Ukrainian mathematician (born 1984)
2 December 1984) is a Ukrainian mathematician known for her work in sphere packing. She is a full professor and Chair of Number Theory at the Institute
Maryna_Viazovska
Problems which attempt to find the most efficient way to pack objects into containers
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'
Packing_problems
On lattices and sphere packing in Euclidean space
anisohedral tiling in three-dimensional Euclidean space, and the densest sphere packing in Kepler conjecture. Respectively, these questions were answered affirmatively
Hilbert's_eighteenth_problem
Packing method for objects
Random close packing (RCP) of spheres is an empirical parameter used to characterize the maximum volume fraction of solid objects obtained when they are
Random_close_pack
Limit on the parameters of a block code
block 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
Hamming_bound
On tangency patterns of circles
of circle packings to certain packings of infinitely many circles on a sphere or open disk. His uniqueness theorem applies to circle packings in which
Circle_packing_theorem
Math theorem about sphere packing
mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of equally sized spheres filling space has a greater
Kepler_conjecture
Lattice in 8-dimensional space with special properties
n-dimensional spheres of a fixed radius in Rn so that no two spheres overlap. Lattice packings are special types of sphere packings where the spheres are centered
E8_lattice
Geometry hypothesis
three-dimensional convex body with lower packing density than the sphere? More unsolved problems in mathematics Ulam's packing conjecture, named for Stanisław
Ulam's_packing_conjecture
Two-dimensional packing problem
investigations. Square packing in a circle Circle packing in a circle Sphere packing in a cube Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1991)
Circle_packing_in_a_square
Rational number equal to an integer plus 1/2
is an integer. The densest lattice packing of unit spheres in four dimensions (called the D4 lattice) places a sphere at every point whose coordinates are
Half-integer
American mathematician
Levi L. Conant Prize for his article “A Conceptual Breakthrough in Sphere Packing,” published in 2017 in the Notices of the AMS. In 2003, with Chris Umans
Henry_Cohn
Geometric concept
unit spheres (i.e., of radius 1) that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing (arrangement
Kissing_number
Generalized sphere of dimension n (mathematics)
projective line O P 1 {\displaystyle \mathbf {OP} ^{1}} . 23-sphere A highly dense sphere-packing is possible in 24 {\displaystyle 24} -dimensional space
N-sphere
Basic noise model used in information theory
spheres therefore must not intersect, we are faced with the problem of sphere packing. How many distinct codewords can we pack into our n {\displaystyle n}
Additive_white_Gaussian_noise
Set of points equidistant from a center
Sphere Napkin ring problem Orb (optics) Pseudosphere Riemann sphere Solid angle Sphere packing Spherical coordinates Spherical cow Spherical helix, tangent
Sphere
Equation in Fourier analysis
on the density of sphere packings using the Poisson summation formula, which subsequently led to a proof of optimal sphere packings in dimension 8 and
Poisson_summation_formula
Fraction of a space filled by objects packed into that space
defines the translative packing constant of that body. Atomic packing factor Sphere packing List of shapes with known packing constant Groemer, H. (1986)
Packing_density
Natural number
Hurwitz quaternions, which form the binary tetrahedral group. The optimal sphere packing problem has been solved in dimension 24, one of the only dimensions
24_(number)
Concept in euclidean geometry
honeycombs, the tesseractic honeycomb corresponds to a sphere packing of edge-length-diameter spheres centered on each vertex, or (dually) inscribed in each
Tesseractic_honeycomb
British-American mathematician (born 1939)
contributions are in the fields of combinatorics, error-correcting codes, and sphere packing. Sloane is best known for being the creator and maintainer of the On-Line
Neil_Sloane
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
24-dimensional repeating pattern of points
Sphere packing E8 lattice Conways groups – Four finite groups derived from the Leech lattice Conway, J.H.; Sloane, N.J.A. (1999), Sphere packings, lattices
Leech_lattice
Family of error-correcting codes that encode data in blocks
\right)\right)+o\left(1\right)} Block codes are tied to the sphere packing problem which has received some attention over the years. In two dimensions
Block_code
Circle-packing on the surface of a sphere
geometry, the Tammes problem is a problem in packing a given number of points on the surface of a sphere such that the minimum distance between points
Tammes_problem
Field of knowledge
major role in discrete mathematics. The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second
Mathematics
Sphere tangent to every face of a polyhedron
the 'inspheres' of their polyhedra. Circumscribed sphere Inscribed circle Midsphere Sphere packing Coxeter, H.S.M. Regular Polytopes 3rd Edn. Dover (1973)
Inscribed_sphere
Two joined triangular cupolae
125.3°. The packing of congruent spheres can be arranged densely into a triangular orthobicupola. The twelve vertices represent the spheres, or ligancy
Triangular_orthobicupola
American mathematician and information theorist (1915–1998)
use of a Hamming matrix), the Hamming window, Hamming numbers, the sphere-packing or Hamming bound, Hamming graph concepts, and the Hamming distance.
Richard_Hamming
Type of mathematical set
contact graph of a sphere packing (a graph where vertices are the centers of spheres and edges exist if the corresponding packing elements touch each
Simplicial_complex
Danish-American mathematician
the representation theory of finite groups, the geometry of numbers, sphere packing, and quadratic forms. He is the namesake of Blichfeldt's theorem. Blichfeldt
Hans_Frederick_Blichfeldt
Erica (March 30, 2016), "Sphere Packing Solved in Higher Dimensions", Quanta Magazine Viazovska, Maryna (2016). "The sphere packing problem in dimension 8"
List of shapes with known packing constant
List_of_shapes_with_known_packing_constant
Analytic function on the upper half-plane with a certain behavior under the modular group
Modular forms also appear in other areas, such as algebraic topology, sphere packing, and string theory. More precisely, a modular form is a holomorphic
Modular_form
Topics referred to by the same term
Close-packing of equal spheres, the arrangement of ions in a crystal Packing problems, a family of optimization problems in mathematics Packing (firestopping)
Packing
ratios larger than one can pack denser than spheres. Packing problems Sphere packing Tetrahedron packing Donev, Aleksandar; Stillinger, Frank H.; Chaikin
Ellipsoid_packing
Natural number
26-dimensional Lorentzian unimodular lattice II25,1 plays a significant role in sphere packing problems and the classification of finite simple groups. 26 is the gematric
26_(number)
Model particles in statistical mechanics
statistical mechanics, hard spheres are widely used as model particles in fluids and solids. They are defined simply as impenetrable spheres that cannot overlap
Hard_spheres
Crystallography concept
In crystallography, atomic packing factor (APF), packing efficiency, or packing fraction is the fraction of volume in a crystal structure that is occupied
Atomic_packing_factor
Mathematician
Romik published a paper simplifying Maryna Viazovska's solution to the sphere packing problem in dimension 8. Viazovska's original solution relied on computer
Dan_Romik
Branch of geometry that studies combinatorial properties and constructive methods
However, sphere packing problems can be generalised to consider unequal spheres, n-dimensional Euclidean space (where the problem becomes circle packing in
Discrete_geometry
Webcomic
SMBC Spheres Part 4 (April 9, 2026), part of a series with Dr. Terence Tao explaining the mathematical problem of sphere packing.
Saturday Morning Breakfast Cereal
Saturday_Morning_Breakfast_Cereal
Topics referred to by the same term
Spherical packing may refer to: Sphere packing Spherical code This disambiguation page lists articles associated with the title Spherical packing. If an
Spherical_packing
Physical process
a random sphere packing of frictionless soft spheres that are jammed together upon applying an external hydrostatic pressure to the packing. Right at
Jamming_(physics)
23 mathematical problems stated in 1900
Resolved. Result: Yes (by Karl Reinhardt). 1928 (c) What is the densest sphere packing? Resolved, by computer-assisted proof (by Thomas Callister Hales) and
Hilbert's_problems
Canadian mathematician
Michelen, Marcus; Sahasrabudhe, Julian (2023). "A new lower bound for sphere packing". Submitted. arXiv:2312.10026. An exponential improvement for diagonal
Julian_Sahasrabudhe
vertices of this lattice are the centers of the 3-spheres in the densest known packing of equal spheres in 4-space; its kissing number is 24, which is also
16-cell_honeycomb
lowest maximum packing density of all centrally-symmetric convex plane sets Sphere packing problems, including the density of the densest packing in dimensions
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
algebraic geometry Ernest Vinberg (1937–2020) J. H. Conway (1937–2020) – sphere packing, recreational geometry Robin Hartshorne (1938–) – geometry, algebraic
List_of_geometers
Solid with six equal square faces
lines in three-dimensional Euclidean space Sphere packing in a cube, on three-dimensional sphere packing problem in a cube Cubing the cube, analogue
Cube
Mathematics award
two-dimensional random structures." "In recognition of her groundbreaking work on sphere-packing problems in eight and twenty-four dimensions." 2016 Mark Gross and Bernd
Clay_Research_Award
Hungarian mathematician (1915–2005)
2-dimensional analog of the Kepler conjecture). He also investigated the sphere packing problem. He was the first to show, in 1953, that proof of the Kepler
László_Fejes_Tóth
(1983), "The Coxeter–Todd lattice, the Mitchell group, and related sphere packings", Mathematical Proceedings of the Cambridge Philosophical Society,
Coxeter–Todd_lattice
periodically. If a 3-sphere is inscribed in each hypercell of this tessellation, the resulting arrangement is the densest known regular sphere packing in four dimensions
24-cell_honeycomb
Polyhedron related to sphere packing
packing spheres according to the cubic close(st) packing (CCP), also known as the face-centered cubic (fcc) packing, then sweeping away the spheres that
Waterman_polyhedron
American mathematician
discrete geometry, he settled the Kepler conjecture on the density of sphere packings, the honeycomb conjecture, and the dodecahedral conjecture. In 2014
Thomas_Callister_Hales
Mathematical model of the physical space
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
American theoretical scientist
conjecture for the densest packings of nonspherical particles, and providing strong theoretical evidence that the densest sphere packings in high dimensions (a
Salvatore_Torquato
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
Natural number
; Sloane, N. J. A. (1988). "Algebraic Constructions for Lattices". Sphere Packings, Lattices and Groups. New York, NY: Springer. doi:10.1007/978-1-4757-2016-7
8
Empirical study of systems in transformation
findings in their most general philosophical context. For example, his sphere packing studies led him to generalize a formula for polyhedral numbers: 2 P
Synergetics_(Fuller)
Mathematics award
– "For remarkable application of the theory of modular forms to the sphere packing problem in special dimensions." Aaron Naber – "For work in geometric
Breakthrough Prize in Mathematics
Breakthrough_Prize_in_Mathematics
Concept in three-dimensional geometry
hard, regular tetrahedra that packed more densely than spheres, demonstrating numerically a packing fraction of 77.86%. A further improvement was made in
Tetrahedron_packing
Law of sediment aggradation
random close packing. An upper bound for close-packed spherical grains is 0.74048 (see sphere packing for more details); this degree of packing is extremely
Exner_equation
Arrangement of leaves on the stem of a plant
Physical models of phyllotaxis date back to Airy's experiment of packing hard spheres. Gerrit van Iterson diagrammed grids imagined on a cylinder (rhombic
Phyllotaxis
Academic journal
Ferguson in 2006 on the Kepler conjecture on optimal three-dimensional sphere packing, earned their authors the Fulkerson Prize. Kalai, Gil (1992). "Upper
Discrete & Computational Geometry
Discrete_&_Computational_Geometry
(puzzle) Situation puzzle Sliding puzzle Snake cube Sokoban Soma cube Sphere packing Stick puzzle Sudoku Tangram Three-cottage problem Three cups problem
List_of_puzzle_topics
Theorem on the minimal volume of cells in the Voronoi decomposition of packed spheres
to sphere packing. László Fejes Tóth, a 20th-century Hungarian geometer, considered the Voronoi decomposition of any given packing of unit spheres. He
Dodecahedral_conjecture
Triangulation method
Voronoi insertion Gabriel graph Gradient pattern analysis Hamming bound – sphere-packing bound Linde–Buzo–Gray algorithm Lloyd's algorithm – Voronoi iteration
Delaunay_triangulation
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
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
Geometry
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 (1975). Applewhite
Boerdijk–Coxeter_helix
Ratio of cation radius to anion radius
can be treated as incompressible spheres, meaning the crystal structure can be seen as a kind of unequal sphere packing. The allowed size of the cation
Cation-anion_radius_ratio
Study of the properties of codes and their fitness
Perfect codes Locally recoverable code Block codes are tied to the sphere packing problem, which has received some attention over the years. In two dimensions
Coding_theory
States of matter for water as a solid
of seven- and eight-membered rings, a 4-connected net (4-coordinate sphere packing)—the densest possible arrangement without hydrogen bond interpenetration
Phases_of_ice
Overview of and topical guide to geometry
Hyperplane Lattice Ehrhart polynomial Leech lattice Minkowski's theorem Packing Sphere packing Kepler conjecture Kissing number problem Honeycomb Andreini tessellation
Outline_of_geometry
isomorphism theorem. 1998 Thomas Callister Hales Kepler conjecture sphere packing 1998 Thomas Callister Hales and Sean McLaughlin dodecahedral conjecture
List_of_conjectures
Public university in Lausanne, Switzerland
Sciences EPFL) Maryna Viazovska (Professor, Mathematician, solved the Sphere packing problem in dimension 8 and 24, awarded a Fields Medal in 2022) Mathias
École Polytechnique Fédérale de Lausanne
École_Polytechnique_Fédérale_de_Lausanne
Geometric space with five dimensions
rwth-aachen.de. Conway, John Horton; Sloane, Neil James Alexander (1999). Sphere Packings, Lattices and Groups (3rd ed.). p. 19. ISBN 978-0-387-98585-5. Zwiebach
Five-dimensional_space
Type of uniform space-filling tessellation
5-demicubic honeycomb is the D5 lattice which is the densest known sphere packing in 5 dimensions. The 40 vertices of the rectified 5-orthoplex vertex
5-demicubic_honeycomb
Capability of a computer graphic to allow whatever is "behind" it to be visible
GIF animation of an Apollonian sphere packing with transparent background
Transparency_(graphic)
Hypersurface in hyperbolic space
to his teacher Gauss. Noting that in Euclidean geometry the limit of a sphere as its radius tends to infinity is a plane, Wachter affirmed that even if
Horosphere
Dimension Surfaces". ResearchGate. The Fractal dimension of the apollonian sphere packing Archived 6 May 2016 at the Wayback Machine Baird, Eric (2014). "The
List of fractals by Hausdorff dimension
List_of_fractals_by_Hausdorff_dimension
Size of a mathematical ball
{1}{p_{n}}}+1{\bigr )}}{\Gamma {\bigl (}{\tfrac {n}{p}}+1{\bigr )}}}R^{n}.} n-sphere Sphere packing Hamming bound Equation 5.19.4, NIST Digital Library of Mathematical
Volume_of_an_n-ball
Sphere tangent to every edge of a polyhedron
said to be midscribed about this sphere. When a polyhedron has a midsphere, one can form two perpendicular circle packings on the midsphere, one corresponding
Midsphere
Mathematics prize
spaces". 2018: Henry Cohn for his article "A conceptual breakthrough in sphere packing". 2017: David H. Bailey, Jonathan Borwein, Andrew Mattingly, and Glenn
Levi_L._Conant_Prize
Theory about lossy data compression
within one or more signals Rate–distortion optimization Sphere packing – Arrangement of spheres within a space White noise – Type of signal in signal processing
Rate–distortion_theory
Particular class of intermetallic phases
Kasper, J. S. (1958-03-10). "Complex alloy structures regarded as sphere packings. I. Definitions and basic principles". Acta Crystallographica. 11 (3)
Frank–Kasper_phases
umbrella Right conoid (a ruled surface) Apollonian gasket Apollonian sphere packing Blancmange curve Cantor dust Cantor set Cantor tesseract[citation needed]
List_of_mathematical_shapes
Mathematical proof at least partially generated by computer
Robbins conjecture, 1996 Kepler conjecture, 1998 – the problem of optimal sphere packing in a box Lorenz attractor, 2002 – 14th of Smale's problems proved by
Computer-assisted_proof
Physical model for representing molecules
snowflakes and the close packing of spherical objects such as fruit. The symmetrical arrangement of closely packed spheres informed theories of molecular
Molecular_model
Physics theorem of interacting particles
between the constant C s , p {\displaystyle C_{s,p}} and the problem of sphere packing is known: lim s → ∞ ( C s , p ) 1 / s = 1 s ( α p Δ p ) 1 / p , {\displaystyle
Poppy-seed_bagel_theorem
dimensions. Generally, optimal sphere packing lattices are ideal for sampling smooth stochastic processes while optimal sphere covering lattices are ideal
Multidimensional_sampling
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