Sphere-packing - Factbites

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Topic: Sphere-packing

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	Sphere packing - Wikipedia, the free encyclopedia
		Sphere packing on the corners of a hypercube (with the spheres defined by Hamming distance) corresponds to designing error-correcting codes: if the spheres have radius d, then their centers are codewords of a d -error-correcting code.
		However, sphere packing problems can be generalised to two dimensional space (where the "spheres" are circles), to n -dimensional space (where the "spheres" are hyperspheres) and to non-Euclidean spaces such as hyperbolic space.
		In mathematics, sphere packing problems are problems concerning arrangements of non-overlapping identical spheres which fill a space.
	en.wikipedia.org /wiki/Sphere_packing

	Sphere Packing
		Kepler studied sphere packing pretty intensely and knew that you get the same fcc packing if you start with a layer of spheres packed in a square arrangement and nest the next layer in the valleys so formed.
		Spheres packed in an icosahedral conformation have the same number of spheres as in a shell of the cuboctahedron of the same frequency, although not cumulatively.
		Again, the number of spheres added per frequency is expressible in terms of a mathematical formula, as well as the cumulative total number of spheres in the growing cuboctahedron.
	www.grunch.net /synergetics/sphpack.html

	pack.html: JEHS sphere packing page
		In particular, the densest 8-dimensional sphere packing yields a 240-node degree-7 integration rule, the densest known 16-dimensional sphere packing yields a 4320-node degree-7 rule, and the densest 24-dimensional sphere packing yields a 196560-node degree-11 rule (and also a 4600-node degree-7 rule for integrating over the surface of a sphere in 23 dimensions, see e.g.
		A packing of 840 balls of radius 9°0'19" on the 3-sphere.
		Similarly in higher dimensional sphere packings, the centres of all spheres touching a given one yield suitable nodes for numerical integration over a spherical surface.
	www.ewartshaw.co.uk /pack.html

	The Geometry Junkyard: Sphere Packing
		The Kepler Conjecture on dense packing of spheres.
		Densest packings of equal spheres in a cube, Hugo Pfoertner.
		See also his list of sphere-packing and lattice theory publications.
	www.ics.uci.edu /~eppstein/junkyard/spherepack.html

	Introducton to packing circles on a sphere
		Two packings are considered the same, if one can be transformed into the other by a rigid motion of the sphere, or by an inversion in the sphere's center followed by a rigid motion.
		Eventually no further expansion is possible and we obtain a packing of the n circles on the sphere.
		Packings are local maxima in the sense that all sufficently small perturbations of the circle centers break the packing; that is, cause the circles to overlap unless they are made smaller.
	home.earthlink.net /~jbuddenh/pack/sphere

	Sphere Packing
		The so-called sphere packing problem was born in 1611, when the German astronomer Johannes Kepler asked himself which is the most efficient way to pack spheres leaving as few gaps as possible.
		According to Professor John Conway, co-author of the standard text on sphere packing,
		Traditionally mathematicians would simply alter the variables to maximise the packing efficiency for the equation, and then see which arrangement is associated with the variables, however the equation is immensely complicated, which puts the maximisation process beyond paper and pencil calculations, and even challenges the limits of computers.
	www.simonsingh.net /Sphere_Packing.html

	Kepler's Sphere Packing Problem Solved
		But the whole topic of the efficient packing of spheres is a crucial part of the mathematics that lies behind the error-detecting and error-correcting codes that are widely used to store information on compact disks and to compress information for efficient transmission around the world.
		The general problem as considered by Kepler and subsequent mathematicians is formulated not in terms of the number of spheres that can be packed together but the density of the packing, i.e., the total volume of the spheres divided by the total volume of the container into which they are packed.
		Devlin describes Kepler's sphere packing problem in his book Mathematics: The Science of Patterns, published in paperback by W. Freeman in 1996.
	www.maa.org /devlin/devlin_9_98.html

	Bin Packing
		Sphere packing is an important and well-studied special case of bin packing, with applications to error-correcting codes.
		Packing pieces of a standard jigsaw puzzle is a different problem than packing squares into a rectangular box.
		Packing boxes is much easier than packing arbitrary geometric shapes, enough so that a reasonable approach for general shapes is to pack each part into its own box and then pack the boxes.
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK5/NODE192.HTM

	Mapping the Hidden Patterns of Sphere Packing
		We can build the closest packing of spheres starting from one sphere and packing 12 spheres around the one, and that would be the first shell in the shape of a cuboctahedron.
		Chemistry has used the close packing of spheres as a model for some of the simplest structures formed by elements, that of cubic close packing (ccp) and hexagonal close packing (hcp).
		It is hard to imagine that such unique and complex relationships exist in such a simple pattern as the closest packing of spheres or stacking of oranges in the shape of pyramids.
	www.verbchu.com /crystals/patterns.htm

	Polymorf - Knowhwere- FCC packing
		Therefore the FCC sphere packing and lattice is demonstrated to be
		packed spheres that correspond to the four (111) planes of cubic symmetry.
		spheres in all three layers the arrangement is referred to as ABC.
	www.polymorf.net /matter6.htm

	CELL AGGREGATION AND SPHERE PACKING
		Since randomly packed spheres have a packing density of approximately 64%, this suggests there is some order in yeast cell aggregation to reduce the amount of space between cells or increase cell-to-cell contact.
		Sphere packing theory is also useful in testing the effects of different medications on platelet aggregation.
		We can then calculate the packing density as the volume of a single sphere divided by the volume of a single cube.
	www.tiem.utk.edu /~gross/bioed/webmodules/spherepacking.htm

	sp.html
		New Sphere Packings in Dimensions 9-15, J. Leech and N. Sloane, Bull.
		Sphere Packings Constructed from BCH and Justesen Codes, N. Sloane, Mathematika, 19 (1972), pp.
		Recent Bounds for Sphere Packings and Related Problems Obtained by Linear Programming and Other Methods, N. Sloane, Papers in Analysis, Algebra and Statistics, Contemporary Math., Vol.
	www.research.att.com /~njas/doc/sp.html

	Oddballs: Science News Online, Oct. 2, 2004
		Since the largest error a code can correct corresponds to the radius of the spheres, and the efficiency of a code is closely related to the density of the sphere packing, Delsarte's method immediately became a powerful tool for studying sphere packings.
		Since antiquity, mathematicians have considered this arrangement to be the densest-possible two-dimensional sphere packing, but their intuition was mathematically proved only in the early 20th century.
		Sphere packings, however, are intimately connected to what are called error-correcting codes.
	www.sciencenews.org /articles/20041002/bob9.asp

	Sphere Packing
		When the porosity values of random sphere packing are compared to these figures, significant discrepancies are observed; the type of snow associated with such a high volume density is firn snow – snow which has been through a summer melt season and is being compacted into glacial ice.
		It is therefore more appropriate to model the ice grains as spheres and the wet snowpack as a collection of these spherical ice grains, joined together in an appropriate manner, with liquid water present throughout the sphere packing.
		Only the inner 2000 spheres, plus their neighbours, were used in subsequent analysis (3367 in total) to restrict effects present at the outer boundary, and the volume density of this pack was found to be 0.6366.
	www.enm.bris.ac.uk /teaching/projects/2002_03/jb8355/sphere_packing.htm

	PENTAGON PACKING IN A CIRCLE
		The problems of the densest packing of equal circles in the plane, in a circle and on a sphere are well known.
		Circle packings in the plane, in a circle and on a sphere occur as motifs also in art.
		(2001) Packing of equal regular pentagons on a sphere, Proceedings of the Royal Society of London, A 457, 1043-1058.
	members.tripod.com /vismath7/proceedings/tarnai.htm

	Albert Philipse
		Stackings or networks of randomly oriented, rigid fibres seem to have little in common with random sphere packings.
		[1] E.A.J.F. Peters, M. Kollmann, Th.M.A.O.M. Barenbrug and A.P. Philipse, Caging of a d-dimensional sphere and its relevance for the random dense sphere packing, Phys.
		[5] A.P. Philipse, The random contact equation and its implications for (colloidal) rods in packings, supensions and anisotropic powders, Langmuir 12 (1996) 5971.
	www.ill.fr /SAFIN2001/abstracts/philipse.html

	sphere-packing database (home)
		I'm especially interested in packings of which you think, that it is useful to calculate their parametric density and to compare them to those in the database.
		Especially we have considered packings in the face-centered-cubic lattice (fcc) with densest and second densest facets (Groemer-packings).
		If you click on the ID of the packing, you will be lead to further information on this special packing and some pictures.
	www.home.unix-ag.org /scholl/packungen

	Biting: Advancing Front Meets Sphere Packing
		By doing this, it builds the sphere packing by adding spheres one at a time, or a layer at a time, in the same spirit as the standard advancing method; our new method uses advancing front to construct a sphere packing instead of the mesh elements themselves.
		At a high level, this new advancing front based packing algorithm first finds a sphere packing of the boundary of the domain and then grows the packing towards the interior of the domain.
		In this paper, we show that the advancing front method can be used to efficiently construct a quality sphere-packing.
	www.andrew.cmu.edu /user/sowen/abstracts/Li621.html

	kepler.html
		The Voronoi Cell of a sphere in the packing is the set of points closer to it than to any other sphere in the packing - they are therefore polyhedra (in 3-dim) or polygons (in 2-dim) containing unit spheres/circles.
		of the density of the packing in the sphere with radius r centred at the origin.
		The density of a Voronoi Cell is the ratio of the volume of the unit sphere to the volume of the cell.
	www.uz.ac.zw /science/maths/zimaths/71/kepler.html

	Maximizing the Packing Density on a Class of Almost Periodic Sphere Packings (ResearchIndex)
		Every sphere packing in R d defines a dynamical system with time R d.
		Abstract: We consider the variational problem of maximizing the packing density on some finite dimensional set of almost periodic sphere packings.
		We show that the maximal density on this manifold is obtained by periodic packings.
	citeseer.ist.psu.edu /knill95maximizing.html

	Electron - Wikipedia, the free encyclopedia
		Based on the classical electron radius and assuming a dense sphere packing, it can be calculated that the number of electrons that would fit in the observable universe is on the order of 10
		This number amounts to a density of about one electron per cubic metre of space.
		Of course, this number is even less meaningful than the classical electron radius itself.
	en.wikipedia.org /wiki/Electron

	Sphere Packing in Curved 3D Space
		The densest possible packing of four spheres in flat 3D space is given by the tetrahedral arrangement, but this arrangement can't be used to fill space because tetrahedrons don't quite "fit together".
		For a related discussion, see Packing Universes In Spacetime.
		This is because if five tetrahedrons share a common edge, they take up only 97.96% of the total 2pi arc about that axis, leaving a small gap.
	www.mathpages.com /home/kmath349.htm

	Computer Assisted Sphere Packing in Higher Dimensions (ResearchIndex)
		2 the density of sphere packings in E3: Kepler's conjecture an..
		The configuration of 24 spheres touching a central sphere in this packing is shown to be rigid, unlike the analogue in 3-space, in which the spheres can slide past each other.
		Abstract: A computer was used to help study the packing of equal spheres in dimension four and higher.
	citeseer.lcs.mit.edu /524642.html

	Hexagonal Closest Packing
		In the 2-dimensional case we created tangent hexagons from the packing (tiling) of 2-spheres (circles) on the
		Without going into details (see "Sphere Packings, Lattices and Groups" by
		This consists of layers of spheres packed in a hexagonal arrangement, each layer fitting as snuggly as
	www.7stones.com /Homepage/Publisher/rrRotate.html

	Sphere packing [Matlab]
		sphere pack reaches the brim of the cylinder.
		> sphere pack reaches the brim of the cylinder.
		compatible with spheres whose radii must all be positive.
	www.adras.com /Sphere-packing.t7236-80.html

	Contents
		The sphere packing problem is discussed in general terms, with a particular account of its connection with coding theory.
		This paper covers the basics of coding theory and its connection to the theory of sphere packing.
		The paper deals with simple error-detecting codes such as repetition and block-repetition codes as well as with the more sophisticated Hamming codes.
	www.mdstud.chalmers.se /~md7sharo/coding/main/node1.html

	Interview
		This was connected with a certain packing of spheres in twenty-four dimensions.
		A: Yes, I read his two long papers on sphere packings, and it was obvious that what he was really doing was taking binary codes and lifting them to form lattices.
		In the simplest case, which I immediately called "Construction A", the centers of the spheres are all the points which reduce to codewords when read mod 2.
	www.research.att.com /~njas/doc/interview.html

	Volume of Sphere
		Re: 3d sphere space fill with volume adjusted probability in 4 transforms...
		Experimental determination of the volume of a crystalline silicon sphere using s...
		Volume of a sphere with a hole drilled through its centre.
	www.scienceoxygen.com /math/200.html

	Joseph Malkevitch: Tidbit: Geometric Packing
		The conjecture, which goes back to Kepler is that the densest way to pack spheres in 3-space is to pack them in the way the grocers typically pack grape fruits: the fruits are laid out on a plane and then the next layer is filled in and so one.
		means that we are allowed to have the packing shapes overlap one another only on their boundaries (perimeter) and that no part of the packing shape can fall outside the shape Y into which we are trying to pack it.
		This problem can be thought of as a geometric packing problem as follows.
	www.york.cuny.edu /~malk/tidbits/tidbit-geometric-packing.html

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In the News (Fri 25 Dec 09)