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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
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
Two-dimensional packing problem
packing in a circle Sphere packing in a cube Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1991). Unsolved Problems in Geometry. New York:
Circle_packing_in_a_square
Solid with six equal square faces
Euclidean space Sphere packing in a cube, on three-dimensional sphere packing problem in a cube Cubing the cube, analogue to the two-dimensional problem
Cube
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
Set of points equidistant from a center
A sphere (from Ancient Greek σφαῖρα (sphaîra) 'ball') is a surface analogous to the circle, a curve. In solid geometry, a sphere is the set of points that
Sphere
Problems which attempt to find the most efficient way to pack objects into containers
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
Generalized sphere of dimension n (mathematics)
as a vanishingly small fraction of the unit cube is contained in the sphere. In ten dimensions, less than 2% of the cube is filled by the sphere, so
N-sphere
Sphere tangent to every edge of a polyhedron
In geometry, the midsphere or intersphere of a convex polyhedron is a sphere which is tangent to every edge of the polyhedron. Not every polyhedron has
Midsphere
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
Rubik's Cube Speedcubing Pocket Cube Rubik's Magic Rubik's Revenge Rush Hour (puzzle) Situation puzzle Sliding puzzle Snake cube Sokoban Soma cube Sphere packing
List_of_puzzle_topics
Video projection technique
pitch (longitude and latitude) of a sphere linearly to a rectangular image. It produces a signature curved look. In addition, the distribution of pixel
360_video_projection
Polyhedron with 6 rhombic and 6 trapezoidal faces
polyhedron that can tile a space with its copy, representing the three-dimensional Voronoi cell of a sphere in a hexagonal close packing; this and the face-centered
Trapezo-rhombic_dodecahedron
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
Dupin cyclide (inversion of a torus) Whitney umbrella Right conoid (a ruled surface) Apollonian gasket Apollonian sphere packing Blancmange curve Cantor dust
List_of_mathematical_shapes
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
Graph-theoretic description of polyhedra
angles between pairs of circles in a circle packing whose corresponding polyhedron has the desired relation to its sphere. In any dimension higher than three
Steinitz's_theorem
Branch of geometry that studies combinatorial properties and constructive methods
circles, spheres, or tiles) in a regular way on a surface or manifold. A sphere packing is an arrangement of non-overlapping spheres within a containing
Discrete_geometry
Solid with four equal triangular faces
{2}}}\right)} A regular tetrahedron can be embedded inside a cube in two ways such that each vertex is a vertex of the cube, and each edge is a diagonal of
Regular_tetrahedron
Hungarian mathematician (1915–2005)
Euclidean plane (a generalization of Thue's theorem, a 2-dimensional analog of the Kepler conjecture). He also investigated the sphere packing problem. He
László_Fejes_Tóth
Natural number
which form the binary tetrahedral group. The optimal sphere packing problem has been solved in dimension 24, one of the only dimensions where this has
24_(number)
Concept in euclidean geometry
corresponds to a sphere packing of edge-length-diameter spheres centered on each vertex, or (dually) inscribed in each cell instead. In the hypercubic
Tesseractic_honeycomb
Polyhedron with four faces
of each at each cube face), and cubes can fill space, so the characteristic 3-orthoscheme of the cube is a space-filling tetrahedron in this sense. (The
Tetrahedron
Geometric space with five dimensions
Alexander (1999). Sphere Packings, Lattices and Groups (3rd ed.). p. 19. ISBN 978-0-387-98585-5. Zwiebach, Barton (2004). A First Course in String Theory
Five-dimensional_space
the 3-spheres in the densest known packing of equal spheres in 4-space; its kissing number is 24, which is also the same as the kissing number in R4, as
16-cell_honeycomb
Regular tessellation in 4D Euclidean space
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
Empty space between atoms in a crystal lattice
In crystallography, interstitial sites, holes or voids are the empty space that exists between the packing of atoms (spheres) in the crystal structure
Interstitial_site
Crystallographic system where the unit cell is in the shape of a cube
of a central atom in the structure. Each sphere in a cP lattice has coordination number 6, in a cI lattice 8, and in a cF lattice 12. Atomic packing factor
Cubic_crystal_system
Regular tiling of a two-dimensional space
in many crystals. In three dimensions, the face-centered cubic and hexagonal close packing are common crystal structures. They are the densest sphere
Hexagonal_tiling
cubes along three positive axis-aligned rays with a shared apex. Several problems of tiling and packing tripods and related shapes were formulated in
Tripod_packing
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
Geometric pattern used in art
also be seen as a projection of an n-unit cube of spheres in 3-dimensional space, viewed on the diagonal axis. There are more spheres than circles because
Overlapping_circles_grid
Polyhedron with 8 triangles and 6 squares
has a dual tessellation; the cell centers in a tessellation are cell vertices in its dual tessellation. The densest known regular sphere-packing in two
Cuboctahedron
Polyhedral compromise map projection
from his work on close-packing of spheres. This involves connecting the sphere centers from cubic closest-packed spheres into a corresponding convex hull
Waterman_butterfly_projection
Tiling puzzle
original on 2007-10-22. Retrieved 2007-08-12. Gardner, Martin (2009). Sphere Packing, Lewis Caroll and Reversi. Cambridge University Press. MacMahon, Percy
Edge-matching_puzzle
Quasiregular space-filling tesselation
lattice in crystallography and is also referred to as the cubic close packed lattice as its vertices are the centers of a close-packing with equal spheres that
Tetrahedral-octahedral honeycomb
Tetrahedral-octahedral_honeycomb
Archimedean solid with 14 faces
truncated octahedron is a 6-zonohedron. It is also the Goldberg polyhedron GIV(1,1), containing square and hexagonal faces. Like the cube, it can tessellate
Truncated_octahedron
(Chapters 16–17: Geometries on Three-manifolds I, II) George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, JOURNAL OF ALGEBRA 79,78-97 (1982)
Order-4-3 pentagonal honeycomb
Order-4-3_pentagonal_honeycomb
Puzzles solved by mechanical manipulation
colours together" or "all numbers in order". The most famous of these puzzles is the original Rubik's Cube, a cubic puzzle in which each of the six faces can
Combination_puzzle
Archimedean solid with 26 faces
turned are projected onto a sphere, they are topologically identical to a rhombicuboctahedron's edges. Variants using the Rubik's Cube mechanism have been produced
Rhombicuboctahedron
Two tetrahedra crossing each other
Both tetrahedra can be inscribed in a cube, and each one shares four vertices with the cube. The two tetrahedra share a common midsphere, making the compound
Stellated_octahedron
Ordered arrangement of atoms, ions, or molecules in a crystalline material
efficient way of packing together equal-sized spheres and stacking close-packed atomic planes in three dimensions. For example, if plane A lies beneath plane
Crystal_structure
Type of crystal structure
diamond structure. The atomic packing factor of the diamond cubic structure (the proportion of space that would be filled by spheres that are centered on the
Diamond_cubic
Shape with four equal sides and angles
is the face of a spherical cube with four 120° angles, covering one sixth of the sphere's surface. Another is a hemisphere, the face of a spherical square
Square
Natural number
MR 2542837. Conway, John H.; Sloane, N. J. A. (1988). "Algebraic Constructions for Lattices". Sphere Packings, Lattices and Groups. New York, NY: Springer
8
Number of bits that differ between two strings
count) in a XOR b. The metric space of length-n binary strings, with the Hamming distance, is known as the Hamming cube; it is equivalent as a metric
Hamming_distance
Number of close-packed spheres in an octahedron
octahedral packing of spheres may be partitioned into two square pyramids, one upside-down underneath the other, by splitting it along a square cross-section
Octahedral_number
algebraic geometry Ernest Vinberg (1937–2020) J. H. Conway (1937–2020) – sphere packing, recreational geometry Robin Hartshorne (1938–) – geometry, algebraic
List_of_geometers
a NaCl-like packing of chromium cubes and cuboctahedra. The space group of this structure is called Fm3m (in Hermann–Mauguin notation) or "225" (in the
Cr23C6_crystal_structure
Natural number
12 vertices. The cubic close packing and hexagonal close packing, which are the two densest possible sphere packings in three-dimensional space (the Kepler
12_(number)
Arrangement of points on a sphere
the surface of a unit sphere that repel each other with a force given by Coulomb's law. The physicist J. J. Thomson posed the problem in 1904 after proposing
Thomson_problem
Empirical study of systems in transformation
to cast his findings in their most general philosophical context. For example, his sphere packing studies led him to generalize a formula for polyhedral
Synergetics_(Fuller)
Mathematical foam of equal-volume bubbles
Cube', for the 2008 Summer Olympics. The Pursuit of Perfect Packing, a book by Weaire on this and related problems Weaire, D.; Phelan, R. (1994), "A counter-example
Weaire–Phelan_structure
Mathematical model of the physical space
status in mathematics, it is impractical to give more than a representative sampling of applications here. A surveyor uses a level Sphere packing applies
Euclidean_geometry
Process in materials science
Cooper, Douglas W. (1989). "Random sequential packing simulations in three dimensions for aligned cubes". J. Appl. Probab. 26 (3): 664–670. doi:10.2307/3214426
Random_sequential_adsorption
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
Uniform 7-Honeycomb
Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). Springer. ISBN 0-387-98585-9. "The Lattice D7". Sphere packings, lattices, and groups
7-demicubic_honeycomb
honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 6-space. It is constructed as an alternation of the regular 6-cube honeycomb
6-demicubic_honeycomb
rectified 7-orthoplex's 84 vertices represent the kissing number of a sphere-packing constructed from this honeycomb. or rectified heptacross rectified
Rectified_7-orthoplexes
Polyhedron that tiles space by translation
parallelohedron in 1885 in his studies of crystallographic systems. They are the cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron, and truncated
Parallelohedron
Geometry problem about finding touching circles
sphère, par la geométrie". Nouvelles Annales de Mathématiques (in French). XIV: 55–71. Kasner E, Supnick F (December 1943). "The Apollonian Packing of
Problem_of_Apollonius
Mechanically-interlinked pieces to be manipulated
have been in use by humanity as early as the 3rd century BC, one of the most well-known mechanical puzzles of modern day is the Rubik's Cube, invented
Mechanical_puzzle
Polyhedral number representing a tetrahedron
tetrahedra built from Ten spheres are used as a unit, it can be shown that a space tiling with such units can achieve a densest sphere packing as long as n ≤ 4
Tetrahedral_number
Flat-sided three-dimensional shape
vertices, and the convex hull of a finite set of points is a polyhedron. Many common families of convex polyhedra, with cubes and pyramids are familiar examples
Polyhedron
States of matter for water as a solid
is hexagonal in nature, and the pores are helical channels with a diameter of about 6.10 Å (6.10×10−10 m; 2.40×10−8 in). It was reported in 2020 that cubic
Phases_of_ice
Strong permanent magnet made from alloys of rare-earth elements
magnet spheres including Zen Magnets have resumed the sale of small neodymium magnet spheres following a successful appeal by Zen Magnets in the Tenth
Rare-earth_magnet
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
Four-dimensional analog of the dodecahedron
tetrahedra can be inscribed in a cube, and five cubes can be inscribed in a dodecahedron, ten tetrahedra in five cubes can be inscribed in a dodecahedron: two opposing
120-cell
Carter, Ithiel; Rodin, Burt (December 1992). "An Inverse Problem for Circle Packing and Conformal Mapping". Transactions of the American Mathematical Society
List_of_spirals
Number of atoms, molecules or ions bonded to a molecule or crystal
structure. Regular hexagonal close packing of spheres would predict that each atom has 12 nearest neighbours and a triangular orthobicupola (also called
Coordination_number
Ordered chemical structure with no repeating pattern
a new principle for packing of atoms and molecules," stated the Nobel Committee and pointed that "this led to a paradigm shift within chemistry." In 2014
Quasicrystal
Field of knowledge
complexity—play a major role in discrete mathematics. The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the
Mathematics
Russian mathematician (1935–2017)
Peredachi Informacii, 13 (1): 3–18 G.A. Kabatiansky; V.I. Levenshtein (1978), "On Bounds for Packings on a Sphere and in Space", Problemy Peredachi Informatsii
Vladimir_Levenshtein
each vertex. 8-cubic honeycomb Uniform polytope "The Lattice D8". Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander
8-demicubic_honeycomb
Crystalline structure for solid elements
structures of many metals can be described as a nearly mathematical close-packing of equal spheres. A simple model for both of these is to assume that
Periodic table (crystal structure)
Periodic_table_(crystal_structure)
Theorem in geometry
disc obtained by packing the volume of the slice as close to the origin as possible; in the case when K(x) is not a disc, the example of a hypercube shows
Brunn–Minkowski_theorem
Hecatonicosachoron) Polytope Viewer Polytopes and optimal packing of p points in n dimensional spheres An atlas of small regular polytopes Regular polyhedra
List_of_regular_polytopes
Shape with three equal sides
Willem Einthoven. In the Thomson problem, concerning the minimum-energy configuration of n {\displaystyle n} charged particles on a sphere, and for the Tammes
Equilateral_triangle
Spatial tiling of convex uniform polyhedra
in the extended symmetry diagrams. The alternated cubic honeycomb is of special importance since its vertices form a cubic close-packing of spheres.
Convex_uniform_honeycomb
Covering by shapes without overlaps or gaps
possible: in 1887, Lord Kelvin proposed a packing using only one solid, the bitruncated cubic honeycomb with very slightly curved faces. In 1993, Denis
Tessellation
Core of an atom composed of nucleons
approximately the same total size result as packing hard spheres of a constant size (like marbles) into a tight spherical or almost spherical bag (some
Atomic_nucleus
Mathematical proof at least partially generated by computer
conjecture, 1996 Kepler conjecture, 1998 – the problem of optimal sphere packing in a box Lorenz attractor, 2002 – 14th of Smale's problems proved by Warwick
Computer-assisted_proof
Graph operation
0 ) {\displaystyle GC_{1,1}(G_{0})} is constructed on the skeleton of a cube. In the last two graphs, blue lines are edges of G 0 {\displaystyle G_{0}}
Goldberg–Coxeter_construction
History of chemical crystallography to 1895
crystals as the result of a close packing of spheres. In 1591 Thomas Harriot studied the close packing of cannonballs (spheres). In 1597 Andreas Libavius
Chemical crystallography before X-rays
Chemical_crystallography_before_X-rays
Physics theorem of interacting particles
r]^{p}\cap \bigcup _{B\in {\mathcal {P}}}B\right)}{(2r)^{p}}}} exists. Hausdorff dimension Geometric measure theory Sphere packing Riemann zeta function
Poppy-seed_bagel_theorem
Concept in molecular modelling
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
Property of being an even or odd number
by Jarvis, Josephine, New York: A Lovell & Company, pp. 240 Conway, J. H.; Sloane, N. J. A. (1999), Sphere packings, lattices and groups, Grundlehren
Parity_(mathematics)
History of geometrical crystallography to 1895
as quartz. In 1550 Gerolamo Cardano made an early attempt to explain the shape of crystals as the result of a close packing of spheres. In 1591 Thomas
Geometrical crystallography before X-rays
Geometrical_crystallography_before_X-rays
Coordinate system
from any given sphere to its 12 surrounding neighbors in the cubic close packing (CCP), equivalently the IVM (isotropic vector matrix) in Synergetics. Therefore
Quadray_coordinates
American reality television series episodes
The series is filmed in Las Vegas, Nevada, where it chronicles the activities at the World Famous Gold & Silver Pawn Shop, a 24-hour family business
List_of_Pawn_Stars_episodes
this polytope corresponds to the center of a 6-sphere in a moderately dense sphere packing, in which each sphere is tangent to 70 others; the best known
1_33_honeycomb
Seeker is a NASA CubeSat intended to demonstrate ultra-low cost in-space inspection capability. Taken from design to delivery from late 2017 to early 2019
Seeker_(spacecraft)
Invariant measure of fractal dimension
of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape
Hausdorff_dimension
Property of objects which are scaled or mirrored versions of each other
equal to the cube of the ratio of corresponding lengths of those figures (for example, when the edge of a cube or the radius of a sphere is multiplied
Similarity_(geometry)
ISBN 978-019853199-9. Conway, John H.; Sloane, Neil J. A. (December 2010). Sphere Packings, Lattices and Groups. Grundlehren der mathematischen Wissenschaften
Higman–Sims_graph
"[cover]" in the table with a hyperlink to the cover. Gardner wrote 5 other articles for Scientific American. His flexagon article in December 1956 was in all
List of Martin Gardner Mathematical Games columns
List_of_Martin_Gardner_Mathematical_Games_columns
English mathematician (1937–2020)
Richard A. Parker, and Robert Arnott Wilson). Clarendon Press, New York, Oxford University Press, 1985, ISBN 0198531990. 1988 – Sphere Packings, Lattices
John_Horton_Conway
Geometric system with a finite number of points
PG(3,2), known as packings. A spread of a projective space is a partition of its points into disjoint lines, and a packing is a partition of the lines
Finite_geometry
dimension of the apollonian sphere packing Archived 6 May 2016 at the Wayback Machine Baird, Eric (2014). "The Koch curve in three dimensions" – via ResearchGate
List of fractals by Hausdorff dimension
List_of_fractals_by_Hausdorff_dimension
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