Circle packing in a square

Circle packing in a square is a packing problem in recreational mathematics, where the aim is to pack n unit circles into the smallest possible square; or, equivalently, to arrange n points in a unit square for the greatest minimal separation, d_n, between points.^[1] To convert between these two formulations of the problem, the square side for unit circles will be $L=2+\frac{2}{d_n}$.

Optimal solutions have been proven for N≤30. Solutions up to N=20 are shown below.^[2]:

Number of circles	Square size	dn[1]	Figure
1	2
2	$2+\sqrt{2}$ ≈ 3.414...	$\sqrt{2}$ ≈ 1.414...
3	$2+\frac{\sqrt{2}}{2}+\frac{\sqrt{6}}{2}+$ ≈ 3.931...	$\sqrt{6} - \sqrt{2}$ ≈ 1.035...
4	4	1
5	$2+2\sqrt{2}$ ≈ 4.828...	$\frac{1}{2} \sqrt{2}$ ≈ 0.707...
6	$2 + \frac{12}{\sqrt{13}}$ ≈ 5.328...	$\frac{1}{6} \sqrt{13}$ ≈ 0.601...
7	$4+ \sqrt{3}$ ≈ 5.732...	$4- 2\sqrt{3}$ ≈ 0.536...
8	$2 + \sqrt{2} + \sqrt{6}$ ≈ 5.863...	$\frac{1}{2}(\sqrt{6} - \sqrt{2})$ ≈ 0.518...
9	6	0.5
10	6.747...	0.421...
11	7.022...	0.398...
12	$2 + 15\sqrt{\frac{2}{17}}$ ≈ 7.144...	0.389...
13	7.463...	0.366...
14	$6 + \sqrt{3}$ ≈ 7.732...	0.348...
15	$4 + \sqrt{2} + \sqrt{6}$ ≈ 7.863...	0.341...
16	8	0.333...
17	8.532...	0.306...
18	$2 + \frac{24}{\sqrt{13}}$ ≈ 8.656...	0.300...
19	8.907...	0.290...
20	$\frac{130}{17} + \frac{16}{17} \sqrt{2}$ ≈ 8.978...	0.287...

References

1. ^ ^{a b} Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1991). Unsolved Problems in Geometry. New York: Springer-Verlag. pp. 108–110. ISBN 0-387-97506-3. 
2. ^ Eckard Specht (20 May 2010). "The best known packings of equal circles in a square". http://hydra.nat.uni-magdeburg.de/packing/csq/csq.html. Retrieved 25 May 2010. 

This geometry-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Mathematical problems Geometry stubs Wikimedia Foundation. 2010. Игры ⚽ Нужна курсовая? Circle packing in a circle Circle packing in an equilateral triangle Look at other dictionaries: Circle packing — 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. The most efficient way to pack different sized circles together is… …   Wikipedia Packing problem — Part of a series on Puzzles …   Wikipedia List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… …   Wikipedia Origami — Paper folding redirects here. For other uses, see Paper folding (disambiguation). For other uses of Origami, see Origami (disambiguation). Origami cranes The foldin …   Wikipedia Открытые математические проблемы — Открытые (нерешённые) математические проблемы  проблемы, которые рассматривались математиками, но до сих пор не решены. Часто имеют форму гипотез, которые предположительно верны, но нуждаются в доказательстве. В научном мире популярна… …   Википедия Wire — For other uses, see Wire (disambiguation). Wires overhead A wire is a single, usually cylindrical, flexible strand or rod of metal. Wires are used to bear mechanical loads and to carry electricity and telecommunications signals. Wire is commonly… …   Wikipedia Problem of Apollonius — In Euclidean plane geometry, Apollonius problem is to construct circles that are tangent to three given circles in a plane (Figure 1); two circles are tangent if they touch at a single point. Apollonius of Perga (ca. 262 BC ndash; ca. 190 BC)… …   Wikipedia Euclidean geometry — A Greek mathematician performing a geometric construction with a compass, from The School of Athens by Raphael. Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his… …   Wikipedia List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia Perth, Western Australia — Perth, Australia redirects here. For the town in Tasmania, see Perth, Tasmania. For other uses, see Perth (disambiguation) and Perth, Western Australia (disambiguation). Perth Western Australia …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”