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:''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]] | |||
Revision as of 02:30, 20 December 2013
- 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.
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.[1] Packing densities of concave shapes such as star polygons 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
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In two dimensional Euclidean space, Joseph Louis Lagrange proved in 1773 that the lattice arrangement of circles with the highest density is the hexagonal packing arrangement,[2] 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
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.[2]
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 tilings of the plane.[3] In these packings, every circle can be mapped to every other circle by reflections and rotations. The hexagonal gaps can be filled by one circle and the dodecagonal 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.
| File:Triangular tiling circle packing.png Triangular |
 Trihexagonal |
 Square |
 Elongated triangular |
 Hexagonal |
File:Truncated square tiling circle packing.png Truncated square |
 Truncated trihexagonal |
 Truncated hexagonal |
 Snub square |
 Rhombitrihexagonal |
 Snub hexagonal |
 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
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. 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
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).[4] 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.[5][6]
It is also known that if the radius ratio is above 0.742, a binary mixture cannot pack better than uniformly-sized discs.[5] Upper bounds for the density that can be obtained in such binary packings at smaller ratios have also been obtained.[7] 50 year old Petroleum Engineer Kull from Dawson Creek, spends time with interests such as house brewing, property developers in singapore condo launch and camping. Discovers the beauty in planing a trip to places around the entire world, recently only coming back from .
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 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.[8] 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
Bibliography
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
- ↑
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- ↑ Template:The Geometrical Foundation of Natural Structure (book)
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Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ 5.0 5.1 5.2 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ Template:Cite web
- ↑ Template:Cite web
- ↑ TED.com lecture on modern origami "Robert Lang on TED."