en>Smyth: Not discussed in the public domain source from which this text was copied, but could be discussed here

2014-12-28T12:05:59Z

Not discussed in the public domain source from which this text was copied, but could be discussed here

en>ChrisGualtieri: /* References */Remove stub template(s). Page is start class or higher. Also check for and do General Fixes + Checkwiki fixes using AWB

2013-12-20T00:30:02Z

References: Remove stub template(s). Page is start class or higher. Also check for and do General Fixes + Checkwiki fixes using AWB

← Older revision		Revision as of 02:30, 20 December 2013
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	~~Friends call him Tad~~. ~~His day job~~ is an ~~auditing officer~~. ~~Researching fashion~~ is ~~one~~ of the ~~things he loves most~~. ~~South Dakota~~ has ~~always been~~ his ~~home~~. He is ~~running~~ and ~~maintaining~~ a ~~blog here~~: http://~~www~~.~~serpis~~.~~com~~/~~avisolegal~~.~~html~~<br><br>~~My site~~ ... ~~boxer Calvin Klein;~~ [http://www.~~serpis~~.com/~~avisolegal~~.html ~~www~~.~~serpis~~.~~com~~],		:''This article describes the packing of circles on surfaces. For the related article on circle packing with a prescribed [[intersection graph]], please see the [[circle packing theorem]].''
			[[Image:Citrus fruits.jpg\|thumb\|The most efficient way to pack different-sized circles together is not obvious.]]

			In [[geometry]], '''circle packing''' is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that all circles touch another. The associated "packing density", ''η'', of an arrangement is the proportion of the surface covered by the circles. Generalisations can be made to higher dimensions – this is called [[sphere packing]], which usually deals only with identical spheres.

			While the circle has a relatively low maximum packing density of 0.9069 on the [[Euclidean plane]], it does not have the lowest possible. The "worst" shape to pack onto a plane is not known, but the [[smoothed octagon]] has a packing density of about 0.902414, which is the lowest maximum packing density known of any centrally-symmetric convex shape.<ref>{{MathWorld\|urlname=SmoothedOctagon \|title=Smoothed Octagon }}</ref>
			Packing densities of concave shapes such as [[star polygon]]s can be arbitrarily small.

			The branch of mathematics generally known as "circle packing" is concerned 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.

			==Packings in the plane==
			[[File:Circle packing (hexagonal).svg\|160px\|thumb\|right\|Identical circles in a ''hexagonal packing'' arrangement, the densest packing possible.]]

			In two dimensional Euclidean space, [[Joseph Louis Lagrange]] proved in 1773 that the lattice arrangement of circles with the highest density is the [[hexagon]]al packing arrangement,<ref name="ChangWang"/> in which the centres of the circles are arranged in a [[hexagonal lattice]] (staggered rows, like a [[honeycomb]]), and each circle is surrounded by 6 other circles. The density of this arrangement is

			::<math>\eta_h = \frac{\pi}{2\sqrt{3}} \approx 0.9069.</math>

			[[Axel Thue]] provided the first proof that this was optimal in 1890, showing that the hexagonal lattice is the densest of all possible circle packings, both regular and irregular. However, his proof was considered by some to be incomplete. The first rigorous proof is attributed to [[László Fejes Tóth]] in 1940.<ref name="ChangWang">{{cite arXiv \|last1=Chang\|first1=Hai-Chau \|last2=Wang\|first2=Lih-Chung \|authorlink= \|eprint=1009.4322v1 \|title=A Simple Proof of Thue's Theorem on Circle Packing \|class=math.MG \|year=2010 \|accessdate=2011-05-11 }}</ref>

			At the other extreme, very low density arrangements of rigidly packed circles have been identified.

			=== Uniform packings ===
			There are 11 circle packings based on the 11 [[Uniform_tiling#Uniform_tilings_of_the_Euclidean_plane\|uniform tiling]]s of the plane.<ref>{{The Geometrical Foundation of Natural Structure (book)\|page=35-39}}</ref> In these packings, every circle can be mapped to every other circle by reflections and rotations. The [[hexagon]]al gaps can be filled by one circle and the [[dodecagon]]al gaps can be filled with 7 circles, creating 3-uniform packings. The [[truncated trihexagonal tiling]] with both types of gaps can be filled as a 4-unform packing. The [[snub hexagonal tiling]] has two mirror-image forms.
			{\| class=wikitable
			\|+ 1-uniform packings based on uniform tilings
			\|- align=center
			\|[[File:triangular tiling circle packing.png\|180px]]<BR>[[Triangular tiling\|Triangular]]
			\|[[File:Trihexagonal tiling circle packing.png\|180px]]<BR>[[Trihexagonal tiling\|Trihexagonal]]
			\|[[File:Square tiling circle packing.png\|180px]]<BR>[[Square tiling\|Square]]
			\|[[File:Elongated triangular tiling circle packing.png\|180px]]<BR>[[Elongated triangular tiling\|Elongated triangular]]
			\|- align=center
			\|[[File:Hexagonal tiling circle packing.png\|180px]]<BR>[[Hexagonal tiling\|Hexagonal]]
			\|[[File:Truncated square tiling circle packing.png\|180px]]<BR>[[Truncated square tiling\|Truncated square]]
			\|[[File:Truncated rhombitrihexagonal tiling circle packing.png\|180px]]<BR>[[Truncated trihexagonal tiling\|Truncated trihexagonal]]
			\|[[File:Truncated hexagonal tiling circle packing.png\|180px]]<BR>[[Truncated hexagonal tiling\|Truncated hexagonal]]
			\|- align=center
			\|[[File:Snub square tiling circle packing.png\|180px]]<BR>[[Snub square tiling\|Snub square]]
			\|[[File:Rhombitrihexagonal tiling circle packing.png\|180px]]<BR>[[Rhombitrihexagonal tiling\|Rhombitrihexagonal]]
			\|[[File:Snub hexagonal tiling circle packing.png\|180px]]<BR>[[Snub hexagonal tiling\|Snub hexagonal]]
			\|[[File:Snub hexagonal tiling mirror circle packing.png\|180px]]<BR>Snub hexagonal (mirrored)
			\|}

			==Packings on the sphere==
			A related problem is to determine the lowest-energy arrangement of identically interacting points that are constrained to lie within a given surface. The [[Thomson problem]] deals with the lowest energy distribution of identical electric charges on the surface of a sphere. The [[Tammes problem]] is a generalisation of this, dealing with maximising the minimum distance between circles on sphere. This is analogous to distributing non-point charges on a sphere.

			==Packings in bounded areas==
			[[File:Circles packed in square 15.svg\|thumb\|right\|Fifteen equal circles [[Circle packing in a square\|packed within the smallest possible square]]. Only four equilateral triangles are formed by adjacent circles.]]
			{{main\|Circle packing in a circle}}
			{{main\|Circle packing in a square}}
			{{main\|Circle packing in an equilateral triangle}}
			{{main\|Circle packing in an isosceles right triangle}}
			[[Packing_problem#Packing_circles\|Packing circles]] in simple bounded shapes is a common type of problem in [[recreational mathematics]]. The influence of the container walls is important, and hexagonal packing is generally not optimal for small numbers of circles.

			==Unequal circles==
			[[File:2-d dense packing r1.svg\|thumb\|left\|A compact binary circle packing with the most similarly sized circles possible.<ref name=Kennedy/> It is also the densest possible packing of discs with this size ratio.<ref name=Heppes>{{cite journal\|last=Heppes\|first=Aladár\|title=Some Densest Two-Size Disc Packings in the Plane\|journal=Discrete and Computational Geometry\|date=1 August 2003\|volume=30\|issue=2\|pages=241–262\|doi=10.1007/s00454-003-0007-6}}</ref>]]
			There are also a range of problems which permit the sizes of the circles to be non-uniform. One such extension is to find the maximum possible density of a system with two specific sizes of circle (a ''binary'' system). Only nine particular radius ratios permit ''compact packing'', which is when every pair of circles in contact is in mutual contact with two other circles (when line segments are drawn from contacting circle-center to circle-center, they triangulate the surface).<ref name=Kennedy>{{cite journal\|author1=Tom Kennedy\|title=Compact packings of the plane with two sizes of discs\|year=2006\|pages=255–267\|volume=35\|journal=Discrete and Computational Geometry\|arxiv=math/0407145v2\|doi=10.1007/s00454-005-1172-4\|issue=2}}</ref> For seven of these radius ratios a compact packing is known that achieves the maximum possible packing fraction (above that of uniformly-sized discs) for mixtures of discs with that radius ratio.<ref name=Heppes/><ref>{{cite web\|url=http://arxiv.org/abs/math/0412418\|first=Tom\|last=Kennedy\|title=A densest compact planar packing with two sizes of discs\|date=21 Dec 2004\|accessdate=11 December 2013}}</ref>

			It is also known that if the radius ratio is above 0.742, a binary mixture cannot pack better than uniformly-sized discs.<ref name=Heppes/><!--citing separate original results by Fejes Toth and Blind--> Upper bounds for the density that can be obtained in such binary packings at smaller ratios have also been obtained.<ref>{{cite web\|last=de Laat\|first=David\|title=Upper bounds for packings of spheres of several radii\|url=http://arxiv.org/abs/1206.2608\|accessdate=11 December 2013\|coauthors=de Oliveira Filho, Fernando Mario; Vallentin, Frank\|date=12 June 2012}}</ref>
			{{clear}}

			==Applications of circle packing==
			[[Quadrature amplitude modulation]] is based on packing circles into circles within a [[phase-amplitude space]]. A [[modem]] transmits data as a series of points in a 2-dimensional phase-amplitude plane. The spacing between the points determines the noise tolerance of the transmission, while the circumscribing circle diameter determines the transmitter power required. Performance is maximized when the [[Constellation diagram\|constellation]] of code points are at the centres of an efficient circle packing. In practice, suboptimal rectangular packings are often used to simplify decoding.

			Circle packing has become an essential tool in [[origami]] design, as each appendage on an origami figure requires a circle of paper.<ref>TED.com lecture on modern origami "[http://www.ted.com/index.php/talks/robert_lang_folds_way_new_origami.html Robert Lang on TED]."</ref> [[Robert J. Lang]] has used the mathematics of circle packing to develop computer programs that aid in the design of complex origami figures.

			==See also==
			*[[Circle packing in a square]]
			*[[Apollonian gasket]]
			*[[Kepler conjecture]]
			*[[Malfatti circles]]
			*[[Packing problem]]

			==Bibliography==
			* {{cite book \| author = Wells D \| year = 1991 \| title = The Penguin Dictionary of Curious and Interesting Geometry \| publisher = Penguin Books \| location = New York \| isbn = 0-14-011813-6 \| pages = 30–31, 167}}

			==References==
			{{Reflist}}<!--added above External links/Sources by script-assisted edit-->

			{{Packing problem}}

			[[Category:Circle packing]]

en>Andreas4965: updated bibliography & removed PARCOR abbreviation (not used outside Wikipedia)

2011-11-17T23:56:04Z

updated bibliography & removed PARCOR abbreviation (not used outside Wikipedia)

New page

Friends call him Tad. His day job is an auditing officer. Researching fashion is one of the things he loves most. South Dakota has always been his home. He is running and maintaining a blog here: http://www.serpis.com/avisolegal.html<br><br>My site ... boxer Calvin Klein; [http://www.serpis.com/avisolegal.html www.serpis.com],

Partial autocorrelation function - Revision history

en>Smyth: Not discussed in the public domain source from which this text was copied, but could be discussed here

en>ChrisGualtieri: /* References */Remove stub template(s). Page is start class or higher. Also check for and do General Fixes + Checkwiki fixes using AWB

en>Andreas4965: updated bibliography & removed PARCOR abbreviation (not used outside Wikipedia)