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Topic: Sphere packing

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List of open problems in computer science

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	Sphere packing - Wikipedia, the free encyclopedia
		A typical sphere packing problem is to find an arrangement in which the spheres fill as large a proportion of the space as possible.
		The branch of mathematics generally known as "circle packing", however, is not concerned with dense packing of equal-sized circles but with the geometry and combinatorics of packings of arbitrarily-sized circles; these give rise to discrete analogs of conformal mapping, Riemann surfaces and the like.
		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.
	en.wikipedia.org /wiki/Sphere_packing (1111 words)

	Sympathetic Vibratory Physics - John W. Keely's Sacred Science.
		Mathematicians have not yet reached consensus on a proof that a Barlow packing, including the face-centered cubic (fcc) and hexagonal (hcp) is actually the densest possible, although Gauss proved the fcc's density of approximately 0.74 optimal for a lattice (any denser arrangement would have to be more random).
		Tri-ville Packing (or Pool Ball) 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.
		Below, the focus is on the number of spheres added as a shape grows in size, versus the number of spheres exposed on a surface.
	www.svpvril.com /svpnotes/SPHERE-PACKING_116354.html (475 words)

	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.
		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.
		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.
	www.grunch.net /synergetics/sphpack.html (926 words)

	Sphere Packing (Site not responding. Last check: 2007-11-07)
		Two possible ways to pack circles together on a plane are shown; hexagonal packing, which is the most efficient, and the less efficient method of putting the circles in a square array.
		Let us take this packing, and look at it in terms of the tesselation of cubes in which it is embedded, but let us shrink the cubes to half of their size, and view the packing in terms of multiple layers, stacked one above the other.
		Naturally, the spheres have to be made into four-dimensional spheres as well, or the problem of packing them in four dimensions would be trivial, and the equivalent applies in higher dimensions.
	www.quadibloc.com /math/pakint.htm (2129 words)

	CELL AGGREGATION AND SPHERE PACKING
		Spheres in the next layer are placed in the crevices of the underlying layer.
		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.
	www.tiem.utk.edu /~gross/bioed/webmodules/spherepacking.htm (1665 words)

	Spheres and Lattices
		The sphere density would then be the volume of the enclosure times 0.7405 divided by the number of balls.
		Notice the hexagon surrounding each sphere in the packing (which is what suggests the name of the packing).
		The above sphere packing has been shown to have the highest density in 2 dimensions (the density is 0.9069).
	www.sonic.net /~surdules/articles/sphere_packings/index.html (1574 words)

	Bin Packing
		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.
		Sphere packing is an important and well-studied special case of bin packing, with applications to error-correcting codes.
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK5/NODE192.HTM (1114 words)

	Mapping the Hidden Patterns of Sphere Packing
		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.
		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.
	www.verbchu.com /crystals/patterns.htm (2305 words)

	[No title]
		The SPHERE of radius r in R^n is the set of points whose coordinates satisfy x1^2+...+xn^2=r^2; the unit sphere is the sphere of radius 1.
		The difficulty is that a sphere packing implies a distribution of arbitrarily many points on the sphere, and gives these distributions in a coherent way, whereas an individual distribution is done with just one fixed number of points.
		Certainly the easiest sphere packings to understand and use are the ones with great regularity: one places a sphere centered at each point in a crystallographic lattice.
	www.math.niu.edu /~rusin/known-math/95/sphere.faq (4423 words)

	Oddballs: Science News Online, Oct. 2, 2004 (Site not responding. Last check: 2007-11-07)
		Sphere packings, however, are intimately connected to what are called error-correcting codes.
		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.
	www.sciencenews.org /articles/20041002/bob9.asp (2461 words)

	pack.html: JEHS sphere packing page
		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.
		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.
	www.ewartshaw.co.uk /pack.html (305 words)

	Introduction to Sphere Packing
		A denser packing is the triangular packing shown in Figure 9.
		C.A Rogers showed that the density of a packing cannot exceed the fraction of the n-simplex which is interior to the spheres.
		This packing is formed by arranging the spheres the way we would arrange oranges in a pile; each layer has the form of a square packing in E
	www.mdstud.chalmers.se /~md7sharo/coding/main/node37.html (326 words)

	Sphere Packing
		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.
		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.
	www.enm.bris.ac.uk /teaching/projects/2002_03/jb8355/sphere_packing.htm (3020 words)

	The Music of the Spheres
		In another scenario, the outer spheres are suspended off the surface but stuck to the central sphere (Model II).
		Close-up of a cluster formed by spheres with diameters of 269 nm and 548 nm, respectively.
		The parameter n is the number of outer spheres with a radius r that can neatly pack around a central sphere of radius R. The ratio B in the above expression describes how the size of the spheres should relate to each other to form good packing.
	invsee.asu.edu /srinivas/spheresmod/nonpacking.html (231 words)

	The Geometry Junkyard: Sphere Packing
		The Kepler Conjecture on dense packing of spheres.
		Packing circles in the hyperbolic plane, Java animation by Kevin Pilgrim illustrating the effects of changing radii in the hyperbolic plane.
		Packing pennies in the plane, an illustrated proof of Kepler's conjecture in 2D by Bill Casselman.
	www.ics.uci.edu /~eppstein/junkyard/spherepack.html (790 words)

	Packing problem - Wikipedia, the free encyclopedia
		Packing problems are one area where mathematics meets puzzles (recreational mathematics).
		Usually the packing must be without gaps or overlaps, but in some packing problems the overlapping (of goods with each other and/or with the boundary of the container) is allowed but should be minimised.
		A classic problem is the sphere packing problem, where one must determine how many spherical objects of given diameter d can be packed into a box of size a × b × c.
	en.wikipedia.org /wiki/Packing_problem (575 words)

	Orbits of Orbs: Sphere Packing Meets Penrose Tilings American Mathematical Monthly, The - Find Articles (Site not responding. Last check: 2007-11-07)
		The density of such a packing P in a bounded container is intuitively the sum of the volumes of the bodies (or portions thereof) in P divided by the volume of the container.
		By a (densest) packing problem we mean the following: given a finite collection B of bodies in E^sup d^, to find the densest packings of E^sup d^ by congruent copies of these bodies.
		Although densest packing problems are at the heart of our discussion, we will not be directly concerned with the actual solution of such problems, but with broader questions such as the existence, and especially the qualitative features (symmetries) of their solutions.
	www.findarticles.com /p/articles/mi_qa3742/is_200402/ai_n9383703 (720 words)

	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.
		Having studied the way that sailors stack cannonballs, and the way that particles of water stack together to form snowflakes, Kepler eventually settled on an arrangment known as the face-centred cubic, which also happens to be the way that greengrocers stack oranges.
		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.com /Sphere_Packing.html (642 words)

	Sphere packing (Site not responding. Last check: 2007-11-07)
		Sphere Packing -- Hsiang's proof by Heidi Burgiel on 03/24/95.
		Re: Sphere Packing -- Hsiang's proof by Walter Whiteley on 03/26/95.
		Re: Sphere packing by John Conway on 06/01/95.
	mathforum.org /~sarah/HTMLthreads/articletocs/sphere.packing.html (91 words)

	Kepler's Sphere Packing Problem Solved (Site not responding. Last check: 2007-11-07)
		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.
		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.
		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 (1088 words)

	Introduction
		Quasicrystalline dense sphere packings are of interest in physics (see for example [14, 1]).
		For the packings, we consider here, the subshift Y has a unique shift-invariant measure and it implies that the density is well defined and can explicitly be determined.
		In section 3, we consider a finite dimensional class of strictly ergodic sphere packings and compute the density of the packings.
	www.math.harvard.edu /~knill/oldinterests/kepler/node1.html (707 words)

	Biting: Advancing Front Meets Sphere Packing (Site not responding. Last check: 2007-11-07)
		Each time when a new sphere is added to the interior, a larger protection sphere is removed (bitten away) from the domain so that no future sphere will overlap with this one.
		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.
		Although biting method uses the advancing front technique to bite the spheres from the domain, it is not necessary to always bite from the boundary of the remaining domain.
	www.andrew.cmu.edu /user/sowen/abstracts/Li621.html (866 words)

	3D Packing of Sphere-Polyhedra 2.2
		The packing of 100 particles or less is considered a demonstration only and is free.
		The packing of more than 100 particles is considered a commercial task, which will require opening an account with our company.
		The simplest case of sphere-polyhedron is a sphere; sphere-polyhedron is a usual polyhedron when r=0.
	smartimtech.com /packingmink/appletpackingmink.html (732 words)

	ARCC Workshop: Sphere Packings, Lattices, and Infinite Dimensional Algebra
		This workshop, sponsored by AIM and the NSF, will focus on sphere packings and lattice packings, with particular attention to dimensions 8 and 24 and the connection with automorphic forms and Moonshine.
		A {\it lattice packing} in ${\Bbb R}^{n}$ is then a sphere packing where the centers of spheres are placed at the points of a lattice $L\subset {\Bbb R}^{n}$, the radius of each sphere being half the length of the shortest non-zero vectors in $L$.
		The study of sphere packings and lattice packings fits naturally into a broader class of packing problems, including error-correcting codes in information transmission.
	www.aimath.org /ARCC/workshops/spherepacking.html (617 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		>3-dimensional hexagonal packing (hcp, and its counterpart, fcc, for >>the crystallographs :) is indeed the densest spherical packing for >>dimension 3.
		It therefore simultaneously establishes the densest packing, the thinnest covering, and the smallest circumscribed n-sided polyhedron for these four values of n.
		In particular, it establishes that the smallest dodecahedron containing the sphere is regular, which is closely related to the still-open conjecture that the regular dodecahedron is also the smallest possible Voronoi region in a sphere packing.
	www.math.niu.edu /~rusin/known-math/95/spherepack (515 words)

	HexDome - Close Packing
		The problems associated with packing small particles around a sphere so that they are more-or-less evenly spaced has been well studied.
		As far as sphere packing goes - it is well known that struts in geodesic domes tend not to be all the same length - and so modelling using identical spheres does not seem appropriate.
		You can build spheres and smash them into walls - but that stuff is hard work - a simpler geometric problem is needed if we want to be able to actually solve it in reasonable time.
	hexdome.com /essays/close_packing/index.php (972 words)

	sphere-packing database (home) (Site not responding. Last check: 2007-11-07)
		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.
		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.
	www.home.unix-ag.org /scholl/packungen/index.html (234 words)

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