A circle packing is an arrangement of disks with pairwise disjoint interiors, possibly confined to a specified
boundary. The generalization to spheres
is called a sphere packing. Tessellations
of regular polygons correspond to particular circle
packings (Williams 1979, pp. 35-41). There is a well-developed theory of circle
packing in the context of discrete conformal mapping
(Stephenson).
Doyle spirals provide discrete analogs of the exponential function. The local tangency conditions
used to describe these arrangements permit overlaps between nonneighboring circles,
so a locally defined pattern need not be a globally nonoverlapping circle packing
(Bobenko and Hoffmann 2001, pp. 142 and 146-147).
(OEIS A093766; Wells 1986, p. 30). Gauss proved that the hexagonal lattice is the densest
planelattice packing, and in 1940, L. Fejes
Tóth proved that the hexagonal lattice
is indeed the densest of all possible plane packings.
Surprisingly, the circular disk is not the least economical convex region for packing the plane.
The "worst" such shape is not known, but among centrally symmetric regions
the conjectured candidate is the smoothed octagon.
Wells (1991, pp. 30-31) considers the maximum size possible for identical circles packed on the
surface of a unit sphere.
Solutions for the smallest diametercircles into which
unit-diameter circles can be packed have been proved optimal
for
through 12 (Fodor 2000). The best known results are summarized in the following table,
and the first few cases are illustrated above (Friedman).
exact
approx.
1
1
1.00000
2
2
2.00000
3
2.15470...
4
2.41421...
5
2.70130...
6
3
3.00000
7
3
3.00000
8
3.30476...
9
3.61312...
10
3.813...
11
3.923...
12
4.029...
The exact value for
can be obtained from the contact configuration of the optimal packing, which has
eight boundary circles and two interior circles
with bilateral symmetry (Graham et al. 1998,
pp. 140-141 and 152). Here is the unique root in of
(8)
The exact value for
follows from the boundary-ring construction with nine boundary circles
(Graham et al. 1998, pp. 148-149).
Fodor (2000, p. 402) gives the corresponding minimum radius for a circle
containing 12 points with pairwise distances at least 1.
Adding
to this radius and doubling gives the enclosing diameter
for unit-diameter circles used in the table.
The following table gives the diameters of circles giving the densest known
packings of
equal circles packed inside a unit
square, the first few of which are illustrated above (Friedman). All to 20 solutions have been proved optimal (Friedman). Peikert
(1994) uses a normalization in which the centers of circles of diameter are packed into a square
of side length 1. Friedman lets the circles have unit
radius and gives the smallest square
side length . A tabulation of analytic and diagrams for to 25 circles is given by Friedman.
Coordinates for optimal packings are given by Nurmela and Östergård (1997).
1
1
1.000000
2
0.585786
1.414214
3
0.508666
1.035276
4
0.500000
1
1.000000
5
0.414214
0.707107
6
0.375361
0.600925
7
0.348915
0.535898
8
0.341081
0.517638
9
0.333333
0.500000
10
0.296408
0.421280
Wehr and Ferriera (2021) give a packing of two unit circles, one cut into two pieces by a chord, in a square
of side
(10)
They report a computer-assisted proof that this packing
is optimal.
The best known packings of circles into an equilateral
triangle are shown above for the first few cases (Friedman).
identical circles packed on the
surface of a unit sphere.
for the circle packings corresponding
to the regular tessellations and semiregular
tessellations of the plane (Williams 1979, p. 49).
exact
approx.
unit-diameter circles can be packed have been proved optimal
for 
through 12 (Fodor 2000). The best known results are summarized in the following table,
and the first few cases are illustrated above (Friedman).
exact
approx.
can be obtained from the contact configuration of the optimal packing, which has
eight boundary circles and two interior circles
with bilateral symmetry (Graham et al. 1998,
pp. 140-141 and 152). Here 
is the unique root in 
of
![(1+sqrt(3))t^3+4(3t-4)(t-2)[t-1-(2+sqrt(3))sqrt(t-1)]=0.](/images/equations/CirclePacking/NumberedEquation5.svg)
follows from the boundary-ring construction with nine boundary circles
(Graham et al. 1998, pp. 148-149).
in the 
row is the smallest positive root of
for a circle
containing 12 points with pairwise distances at least 1.
Adding 
to this radius and doubling gives the enclosing diameter
for unit-diameter circles used in the table.
of circles giving the densest known
packings of 
equal circles packed inside a unit
square, the first few of which are illustrated above (Friedman). All 
to 20 solutions have been proved optimal (Friedman). Peikert
(1994) uses a normalization in which the centers of 
circles of diameter 
are packed into a square
of side length 1. Friedman lets the circles have unit
radius and gives the smallest square
side length 
. A tabulation of analytic 
and diagrams for 
to 25 circles is given by Friedman.
Coordinates for optimal packings are given by Nurmela and Östergård (1997).
