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{{Short description|Fractal composed of tangent circles}}
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[[Image:Apollonian gasket.svg|thumb|An example of an Apollonian gasket]]
In [[mathematics]], an '''Apollonian gasket''',  '''Apollonian net''', or '''Apollonian circle packing''' is a [[fractal]] generated by starting with a triple of circles, each tangent to the other two, and successively filling in more circles, each [[tangent]] to another three. It is named after Greek mathematician [[Apollonius of Perga]].<ref>Satija, I. I., ''The Butterfly in the Iglesias Waseas World: The story of the most fascinating quantum fractal'' ([[Bristol]]: [[IOP Publishing]], 2016), [https://books.google.com/books?id=toIRDQAAQBAJ&pg=PR63 p. 5].</ref>

==Construction==
[[File:Descartes Circles.svg|thumb|Mutually tangent circles. Given three mutually tangent circles ({{color|black|'''black'''}}), there are in general two other circles mutually tangent to them ({{color|red|'''red'''}}).]]
The construction of the Apollonian gasket starts with three circles <math>C_1</math>, <math>C_2</math>, and <math>C_3</math> (black in the figure), that are each tangent to the other two, but that do not have a single point of triple tangency. These circles may be of different sizes to each other, and it is allowed for two to be inside the third, or for all three to be outside each other. As Apollonius discovered, there exist two more circles <math>C_4</math> and <math>C_5</math> (red) that are tangent to all three of the original circles&nbsp;– these are called ''Apollonian circles''. These five circles are separated from each other by six curved triangular regions, each bounded by the arcs from three pairwise-tangent circles. The construction continues by adding six more circles, one in each of these six curved triangles, tangent to its three sides. These in turn create 18 more curved triangles, and the construction continues by again filling these with tangent circles, ad infinitum.

Continued stage by stage in this way, the construction adds <math>2\cdot 3^n</math> new circles at stage <math>n</math>, giving a total of <math>3^{n+1}+2</math> circles after <math>n</math> stages. In the limit, this set of circles is an Apollonian gasket. In it, each pair of tangent circles has an infinite [[Pappus chain]] of circles tangent to both circles in the pair.

[[Image:ApollonianGasket-0_0_1_1-Labels.png|thumb|In the limiting case (0,0,1,1), the two largest circles are replaced by parallel straight lines.  This produces a family of [[Ford circle]]s.]]
The size of each new circle is determined by [[Descartes' theorem]], which states that, for any four mutually tangent circles, the radii <math>r_i</math> of the circles obeys the equation
<math display=block>\left(\frac1{r_1}+\frac1{r_2}+\frac1{r_3}+\frac1{r_4}\right)^2=2\left(\frac1{r_1^2}+\frac1{r_2^2}+\frac1{r_3^2}+\frac1{r_4^2}\right).</math>
This equation may have a solution with a negative radius; this means that one of the circles (the one with negative radius) surrounds the other three.
One or two of the initial circles of this construction, or the circles resulting from this construction, can [[Degeneracy (mathematics)|degenerate]] to a straight line, which can be thought of as a circle with infinite radius. When there are two lines, they must be parallel, and are considered to be tangent at a [[point at infinity]]. When the gasket includes two lines on the <math>x</math>-axis and one unit above it, and a circle of unit diameter tangent to both lines centered on the <math>y</math>-axis, then the circles that are tangent to the <math>x</math>-axis are the [[Ford circle]]s, important in [[number theory]].

The Apollonian gasket has a [[Hausdorff dimension]] of about 1.3056867, which has been extended to at least 128 decimal places.<ref>{{citation
 | last = Boyd | first = David W.| author-link=David William Boyd 
 | doi = 10.1112/S0025579300004745
 | journal = [[Mathematika]]
 | mr = 493763
 | pages = 170–174
 | title = The residual set dimension of the Apollonian packing
 | volume = 20
 | year = 1973| issue = 2}}</ref><ref>{{citation
 | last = McMullen | first = Curtis T. | author-link = Curtis T. McMullen
 | doi = 10.1353/ajm.1998.0031
 | issue = 4
 | journal = American Journal of Mathematics
 | mr = 1637951
 | pages = 691–721
 | title = Hausdorff dimension and conformal dynamics, III: Computation of dimension
 | url = https://abel.math.harvard.edu/~ctm/papers/home/text/papers/dimIII/dimIII.pdf
 | volume = 120
 | year = 1998| s2cid = 15928775 }}</ref><ref>{{citation
 | last = Caroline L. Wormell | first = Polina L. Vytnova | author-link = 
 | doi = 10.1007/s00222-024-01311-y
 | journal = Inventiones Mathematicae
 | pages = 909–946
 | title = Hausdorff dimension of the Apollonian gasket
 | url = https://link.springer.com/article/10.1007/s00222-024-01311-y
 | volume = 239
 | year = 2025| issue = 3 | arxiv = 2406.04922
 | bibcode = 2025InMat.239..909V }}</ref> Because it has a well-defined fractional dimension, even though it is not precisely [[Self-similarity|self-similar]], it can be thought of as a [[fractal]].

==Symmetries==
The [[Möbius transformation]]s of the plane preserve the shapes and tangencies of circles, and therefore preserve the structure of an Apollonian gasket.
Any two triples of mutually tangent circles in an Apollonian gasket may be mapped into each other by a Möbius transformation, and any two Apollonian gaskets may be mapped into each other by a Möbius transformation. In particular, for any two tangent circles in any Apollonian gasket, an [[inversion in a circle]] centered at the point of tangency (a special case of a Möbius transformation) will transform these two circles into two parallel lines, and transform the rest of the gasket into the special form of a gasket between two parallel lines. Compositions of these inversions can be used to transform any two points of tangency into each other. Möbius transformations are also isometries of the [[Hyperbolic geometry|hyperbolic plane]], so in hyperbolic geometry all Apollonian gaskets are congruent. In a sense, there is therefore only one Apollonian gasket, up to (hyperbolic) isometry.

The Apollonian gasket is the limit set of a group of Möbius transformations known as a [[Kleinian group]].<ref>[https://www.math.brown.edu/~heeoh/AMSKMS.pdf Counting circles and Ergodic theory of Kleinian groups by  Hee Oh Brown.  University Dec 2009]</ref>

For Euclidean symmetry transformations rather than Möbius transformations, in general, the Apollonian gasket will inherit the symmetries of its generating set of three circles. However, some triples of circles can generate Apollonian gaskets with higher symmetry than the initial triple; this happens when the same gasket has a different and more-symmetric set of generating circles. Particularly symmetric cases include the Apollonian gasket between two parallel lines (with infinite [[dihedral group|dihedral symmetry]]), the Apollonian gasket generated by three congruent circles in an equilateral triangle (with the symmetry of the triangle), and the Apollonian gasket generated by two circles of radius 1 surrounded by a circle of radius 2 (with two lines of reflective symmetry).

==Integral Apollonian circle packings==
<gallery>
Image:ApollonianGasket-1_2_2_3-Labels.png|Integral Apollonian circle packing defined by circle [[curvature]]s of (−1,&nbsp;2,&nbsp;2,&nbsp;3)
Image:ApollonianGasket-3_5_8_8-Labels.png|Integral Apollonian circle packing defined by circle curvatures of (−3,&nbsp;5,&nbsp;8,&nbsp;8)
Image:ApollonianGasket-12_25_25_28-Labels.png|Integral Apollonian circle packing defined by circle curvatures of (−12,&nbsp;25,&nbsp;25,&nbsp;28)
Image:ApollonianGasket-6_10_15_19-Labels.png|Integral Apollonian circle packing defined by circle curvatures of (−6,&nbsp;10,&nbsp;15,&nbsp;19)
Image:ApollonianGasket-10_18_23_27-Labels.png|Integral Apollonian circle packing defined by circle curvatures of (−10,&nbsp;18,&nbsp;23,&nbsp;27)
</gallery>
If any four mutually tangent circles in an Apollonian gasket all have integer [[curvature]] (the inverse of their radius) then all circles in the gasket will have integer curvature.{{sfn|Soddy|1937}}{{sfn|Graham et al. 2003}}
Since the equation relating curvatures in an Apollonian gasket, integral or not, is
<math display=block>a^2 + b^2 + c^2 + d^2 = 2ab + 2 a c + 2 a d + 2 bc+2bd+2cd,\,</math>
it follows that one may move from one quadruple of curvatures to another by [[Vieta jumping]], just as when finding a new   [[Markov number]].
The first few of these integral Apollonian gaskets are listed in the following table. The table lists the curvatures of the largest circles in the gasket. Only the first three curvatures (of the five displayed in the table) are needed to completely describe each gasket&nbsp;– all other curvatures can be derived from these three.

{{Col-begin}}
{{Col-break}}
{| class="wikitable"
|+ '''Integral Apollonian gaskets'''
! Beginning curvatures
! Symmetry
|-
| −1, 2, 2, 3, 3 || align="center"| ''D''<sub>2</sub>
|-
| −2, 3, 6, 7, 7 || align="center"| ''D''<sub>1</sub>
|-
| −3, 4, 12, 13, 13 || align="center"| ''D''<sub>1</sub>
|-
| −3, 5, 8, 8, 12 || align="center"| ''D''<sub>1</sub>
|-
| −4, 5, 20, 21, 21 || align="center"| ''D''<sub>1</sub>
|-
| −4, 8, 9, 9, 17 || align="center"| ''D''<sub>1</sub>
|-
| −5, 6, 30, 31, 31 || align="center"| ''D''<sub>1</sub>
|-
| −5, 7, 18, 18, 22 || align="center"| ''D''<sub>1</sub>
|-
| −6, 7, 42, 43, 43 || align="center"| ''D''<sub>1</sub>
|-
| −6, 10, 15, 19, 19 || align="center"| ''D''<sub>1</sub>
|-
| −6, 11, 14, 15, 23 || align="center"| ''C''<sub>1</sub>
|-
| −7, 8, 56, 57, 57 || align="center"| ''D''<sub>1</sub>
|-
| −7, 9, 32, 32, 36 || align="center"| ''D''<sub>1</sub>
|-
| −7, 12, 17, 20, 24 || align="center"| ''C''<sub>1</sub>
|-
| −8, 9, 72, 73, 73 || align="center"| ''D''<sub>1</sub>
|-
| −8, 12, 25, 25, 33 || align="center"| ''D''<sub>1</sub>
|-
| −8, 13, 21, 24, 28 || align="center"| ''C''<sub>1</sub>
|-
| −9, 10, 90, 91, 91 || align="center"| ''D''<sub>1</sub>
|-
| −9, 11, 50, 50, 54 || align="center"| ''D''<sub>1</sub>
|-
| −9, 14, 26, 27, 35 || align="center"| ''C''<sub>1</sub>
|-
| −9, 18, 19, 22, 34 || align="center"| ''C''<sub>1</sub>
|-
| −10, 11, 110, 111, 111 || align="center"| ''D''<sub>1</sub>
|-
| −10, 14, 35, 39, 39 || align="center"| ''D''<sub>1</sub>
|-
| −10, 18, 23, 27, 35 || align="center"| ''C''<sub>1</sub>
|-
| −11, 12, 132, 133, 133 || align="center"| ''D''<sub>1</sub>
|-
| −11, 13, 72, 72, 76 || align="center"| ''D''<sub>1</sub>
|-
| −11, 16, 36, 37, 45 || align="center"| ''C''<sub>1</sub>
|-
| −11, 21, 24, 28, 40 || align="center"| ''C''<sub>1</sub>
|-
| −12, 13, 156, 157, 157 || align="center"| ''D''<sub>1</sub>
|-
| −12, 16, 49, 49, 57 || align="center"| ''D''<sub>1</sub>
|-
| −12, 17, 41, 44, 48 || align="center"| ''C''<sub>1</sub>
|-
| −12, 21, 28, 37, 37 || align="center"| ''D''<sub>1</sub>
|-
| −12, 21, 29, 32, 44 || align="center"| ''C''<sub>1</sub>
|-
| −12, 25, 25, 28, 48 || align="center"| ''D''<sub>1</sub>
|-
| −13, 14, 182, 183, 183 || align="center"| ''D''<sub>1</sub>
|-
| −13, 15, 98, 98, 102 || align="center"| ''D''<sub>1</sub>
|-
| −13, 18, 47, 50, 54 || align="center"| ''C''<sub>1</sub>
|-
| −13, 23, 30, 38, 42 || align="center"| ''C''<sub>1</sub>
|-
| −14, 15, 210, 211, 211 || align="center"| ''D''<sub>1</sub>
|-
| −14, 18, 63, 67, 67 || align="center"| ''D''<sub>1</sub>
|-
| −14, 19, 54, 55, 63 || align="center"| ''C''<sub>1</sub>
|-
| −14, 22, 39, 43, 51 || align="center"| ''C''<sub>1</sub>
|-
| −14, 27, 31, 34, 54 || align="center"| ''C''<sub>1</sub>
|-
| −15, 16, 240, 241, 241 || align="center"| ''D''<sub>1</sub>
|-
| −15, 17, 128, 128, 132 || align="center"| ''D''<sub>1</sub>
|-
| −15, 24, 40, 49, 49 || align="center"| ''D''<sub>1</sub>
|-
| −15, 24, 41, 44, 56 || align="center"| ''C''<sub>1</sub>
|-
| −15, 28, 33, 40, 52 || align="center"| ''C''<sub>1</sub>
|-
| −15, 32, 32, 33, 65 || align="center"| ''D''<sub>1</sub>
|}
{{col-end}}

===Enumerating integral Apollonian circle packings===
The curvatures <math>(a, b, c, d)</math> are a root quadruple (the smallest in some integral circle packing) if <math>a < 0 \leq b \leq c \leq d</math>. They are primitive when <math>\gcd(a, b, c, d)=1</math>. Defining a new set of variables <math>(x, d_1, d_2, m)</math> by the matrix equation
<math display=block>
\begin{bmatrix}
a \\ b \\ c \\ d
\end{bmatrix}
=
\begin{bmatrix}
 1 & 0 & 0 & 0\\
-1 & 1 & 0 & 0\\
-1 & 0 & 1 & 0\\
-1 & 1 & 1 &-2
\end{bmatrix}
\begin{bmatrix}
x \\ d_1 \\ d_2 \\ m
\end{bmatrix}
</math>
gives a system where <math>(a, b, c, d)</math> satisfies the Descartes equation precisely when <math>x^2+m^2=d_1 d_2</math>. Furthermore, <math>(a, b, c, d)</math> is primitive precisely when <math>\gcd(x, d_1, d_2)=1</math>, and <math>(a, b, c, d)</math> is a root quadruple precisely when <math>x<0\leq 2m\leq d_1\leq d_2</math>.{{sfn|Graham et al. 2003}}

This relationship can be used to find all the primitive root quadruples with a given negative bend <math>x</math>. It follows from <math>2m\leq d_1</math> and <math>2m\leq d_2</math> that <math>4m^2\leq d_1d_2</math>, and hence that <math>3m^2\leq d_1d_2-m^2=x^2</math>. Therefore, any root quadruple will satisfy <math>0\leq m \leq |x|/\sqrt{3}</math>. By iterating over all the possible values of <math>m</math>, <math>d_1</math>, and <math>d_2</math> one can find all the primitive root quadruples.<ref>{{citation |last1=Bradford |first1=Alden |title=Revisiting Apollonian Gaskets |url=https://aldenbradford.com/revisiting-apollonian-gaskets.html |access-date=7 August 2022}}</ref> The following [[Python (programming language)|Python]] code demonstrates this algorithm, producing the primitive root quadruples listed above.

<syntaxhighlight lang="python">
import math

def get_primitive_bends(n: int) -> tuple[int, int, int, int]:
    if n == 0:
        yield 0, 0, 1, 1
        return
    for m in range(math.ceil(n / math.sqrt(3))):
        s = m**2 + n**2
        for d1 in range(max(2 * m, 1), math.floor(math.sqrt(s)) + 1):
            d2, remainder = divmod(s, d1)
            if remainder == 0 and math.gcd(n, d1, d2) == 1:
                yield -n, d1 + n, d2 + n, d1 + d2 + n - 2 * m

for n in range(15):
    for bends in get_primitive_bends(n):
        print(bends)
</syntaxhighlight>

====The Local-Global Conjecture====

The curvatures appearing in a primitive integral Apollonian circle packing must belong to a set of six or eight possible residues classes modulo 24, and theoretical results and numerical evidence supported that any sufficiently large integer from these residue classes would also be present as a curvature within the packing.{{sfn|Graham et al. 2003}}{{sfn|Fuchs|Sanden|2011}}{{sfn|Bourgain|Kontorovich|2014}} This conjecture, known as the local-global conjecture, was proved to be false in 2023.{{sfn|Haag et al. 2024}}<ref>{{citation |last=Levy |first=Max G. |date=August 10, 2023 |title=Two Students Unravel a Widely Believed Math Conjecture |publisher=[[Quanta Magazine]] |url=https://www.quantamagazine.org/two-students-unravel-a-widely-believed-math-conjecture-20230810 |access-date=August 14, 2023}}</ref>

===Symmetry of integral Apollonian circle packings===
There are multiple types of [[Dihedral symmetry in three dimensions#Types|dihedral symmetry]] that can occur with a gasket depending on the curvature of the circles.

====No symmetry====
If none of the curvatures are repeated within the first five, the gasket contains no symmetry, which is represented by symmetry group ''C''<sub>1</sub>; the gasket described by curvatures (−10,&nbsp;18,&nbsp;23,&nbsp;27) is an example.

====''D''<sub>1</sub> symmetry====
Whenever two of the largest five circles in the gasket have the same curvature, that gasket will have ''D''<sub>1</sub> symmetry, which corresponds to a reflection along a diameter of the bounding circle, with no rotational symmetry.

====''D''<sub>2</sub> symmetry====
If two different curvatures are repeated within the first five, the gasket will have D<sub>2</sub> symmetry; such a symmetry consists of two reflections (perpendicular to each other) along diameters of the bounding circle, with a two-fold rotational symmetry of 180°. The gasket described by curvatures (−1,&nbsp;2,&nbsp;2,&nbsp;3) is the only Apollonian gasket (up to a scaling factor) to possess D<sub>2</sub> symmetry.

====''D''<sub>3</sub> symmetry====
There are no integer gaskets with ''D''<sub>3</sub> symmetry.

If the three circles with smallest positive curvature have the same curvature, the gasket will have ''D''<sub>3</sub> symmetry, which corresponds to three reflections along diameters of the bounding circle (spaced 120° apart), along with three-fold rotational symmetry of 120°. In this case the ratio of the curvature of the bounding circle to the three inner circles is 2{{sqrt|3}}&nbsp;&minus;&nbsp;3. As this ratio is not rational, no integral Apollonian circle packings possess this ''D''<sub>3</sub> symmetry, although many packings come close.

====Almost-''D''<sub>3</sub> symmetry====
[[Image:ApollonianGasket-15_32_32_33.svg|thumb|left|(−15,&nbsp;32,&nbsp;32,&nbsp;33)]]
[[Image:ApollonianGasket-15_32_32_33-Labels.png|thumb|right|(−15,&nbsp;32,&nbsp;32,&nbsp;33)]]
The figure at left is an integral Apollonian gasket that appears to have ''D''<sub>3</sub> symmetry. The same figure is displayed at right, with labels indicating the curvatures of the interior circles, illustrating that the gasket actually possesses only the ''D''<sub>1</sub> symmetry common to many other integral Apollonian gaskets.

The following table lists more of these ''almost''-''D''<sub>3</sub> integral Apollonian gaskets. The sequence has some interesting properties, and the table lists a factorization of the curvatures, along with the multiplier needed to go from the previous set to the current one. The absolute values of the curvatures of the "a" disks obey the [[recurrence relation]] {{math|1=''a''(''n'') = 4''a''(''n'' − 1) − ''a''(''n'' − 2)}} {{OEIS|A001353}}, from which it follows that the multiplier converges to {{sqrt|3}}&nbsp;+&nbsp;2 ≈ 3.732050807.

{| class="wikitable"
|+ '''Integral Apollonian gaskets with near-''D''<sub>3</sub> symmetry'''
! colspan="4"| Curvature
! rowspan="2"|
! colspan="3"| Factors
! rowspan="2"|
! colspan="4"| Multiplier
|-
! ''a'' !! ''b'' !! ''c'' !! ''d'' !! ''a'' !! ''b'' !! ''d'' !! ''a'' !! ''b'' !! ''c'' !! ''d''
|-
| −1 || 2 || 2 || 3 ||rowspan=9|  || 1×1 || 1×2 || 1×3 ||rowspan=9|  || {{N/A}} || {{N/A}} || {{N/A}} || {{N/A}}
|-
| −4 || 8 || 9 || 9 ||  2×2 || 2×4 || 3×3 ||  4.000000000 || 4.000000000 || 4.500000000 || 3.000000000
|-
| −15 || 32 || 32 || 33 ||  3×5 || 4×8 || 3×11 ||  3.750000000 || 4.000000000 || 3.555555556 || 3.666666667
|-
| −56 || 120 || 121 || 121 ||  8×7 || 8×15 || 11×11 ||  3.733333333 || 3.750000000 || 3.781250000 || 3.666666667
|-
| −209 || 450 || 450 || 451 ||  11×19 || 15×30 || 11×41 ||  3.732142857 || 3.750000000 || 3.719008264 || 3.727272727
|-
| −780 || 1680 || 1681 || 1681 ||  30×26 || 30×56 || 41×41 ||  3.732057416 || 3.733333333 || 3.735555556 || 3.727272727
|-
| −2911 || 6272 || 6272 || 6273 ||  41×71 || 56×112 || 41×153 ||  3.732051282 || 3.733333333 || 3.731112433 || 3.731707317
|-
| −10864 || 23408 || 23409 || 23409 ||  112×97 || 112×209 || 153×153 ||  3.732050842 || 3.732142857 || 3.732302296 || 3.731707317
|-
| −40545 || 87362 || 87362 || 87363 ||  153×265 || 209×418 || 153×571 ||  3.732050810 || 3.732142857 || 3.731983425 || 3.732026144
|}

===Sequential curvatures===
[[Image: ApollianGasketNested_2-20.svg|thumb|Nested Apollonian gaskets]]
For any integer ''n''&nbsp;>&nbsp;0, there exists an Apollonian gasket defined by the following curvatures: <br />(−''n'',&nbsp;''n''&nbsp;+&nbsp;1,&nbsp;''n''(''n''&nbsp;+&nbsp;1),&nbsp;''n''(''n''&nbsp;+&nbsp;1)&nbsp;+&nbsp;1).  <br />For example, the gaskets defined by (−2,&nbsp;3,&nbsp;6,&nbsp;7), (−3,&nbsp;4,&nbsp;12,&nbsp;13), (−8,&nbsp;9,&nbsp;72,&nbsp;73), and (−9,&nbsp;10,&nbsp;90,&nbsp;91) all follow this pattern.  Because every interior circle that is defined by ''n''&nbsp;+&nbsp;1 can become the bounding circle (defined by&nbsp;−''n'') in another gasket, these gaskets can be [[Hereditarily finite set|nested]].  This is demonstrated in the figure at right, which contains these sequential gaskets with ''n'' running from 2 through 20.

==History==
Although the Apollonian gasket is named for Apollonius of Perga, because of the dependence of its construction on the solution to the [[problem of Apollonius]], the earliest description of the gasket is from 1706 by [[Gottfried Wilhelm Leibniz|Leibniz]] in a letter to [[Bartholomew Des Bosses|Des Bosses]].<ref>
Leibniz to Des Bosses, Hannover 11-17 March 1706, translated by Dr. Osvaldo Ottaviani https://humanities.technion.ac.il/en/leibniz-to-des-bosses-hannover-11-17-march-1706/</ref>
The first modern definition of the Apollonian gasket is given by [[Edward Kasner|Kasner]] and Supnick.{{sfn|Kasner|Supnick|1943}}

==See also==
[[Image:Apollonian spheres.jpg|thumb|[[Apollonian sphere packing]]]]
*[[Apollonian network]], a graph derived from finite subsets of the Apollonian gasket
*[[Apollonian sphere packing]], a three-dimensional generalization of the Apollonian gasket
*[[Sierpiński triangle]], a self-similar fractal with a similar combinatorial structure

==Notes==
<references/>

==References==
* [[Benoit B. Mandelbrot]]: ''[[The Fractal Geometry of Nature]]'', W H Freeman, 1982, {{ISBN|0-7167-1186-9}}
* {{citation |last1=Bourgain |first1=Jean | author-link1=Jean Bourgain |last2=Kontorovich |first2=Alex |author-link2=Alex Kontorovich |year=2014 |title=On the Local-Global Conjecture for integral Apollonian gaskets |journal=Inventiones Mathematicae |volume=196 |number=196  |pages=589–650|doi=10.1007/s00222-013-0475-y |arxiv=1205.4416 |bibcode=2014InMat.196..589B }}
* Paul D. Bourke: "[https://web.archive.org/web/20060914030236/http://local.wasp.uwa.edu.au/~pbourke/papers/apollony/ An Introduction to the Apollony Fractal]". Computers and Graphics, Vol 30, Issue 1, January 2006, pages 134–136.
* {{citation |last1=Fuchs |first1=Elena |last2=Sanden |first2=Katherine |date=2011-11-28 |title=Some Experiments with Integral Apollonian Circle Packings |journal=Experimental Mathematics |language=en |volume=20 |issue=4 |pages=380–399 |doi=10.1080/10586458.2011.565255 |issn=1058-6458|arxiv=1001.1406 }}
* {{citation | first1=Ronald L. | last1=Graham | author-link1=Ronald Graham | first2=Jeffrey C. | last2=Lagarias | author-link2=Jeffrey Lagarias| first3=Colin L. | last3=Mallows | author-link3=Colin Lingwood Mallows | first4=Alan R. | last4=Wilks | first5= Catherine H. | last5=Yan |author-link5=Catherine Yan | title=Apollonian Circle Packings: Number Theory | journal=J. Number Theory | volume=100 | number=1 | year=2003 | pages=1–45 | doi=10.1016/S0022-314X(03)00015-5 | url=https://www.sciencedirect.com/science/article/pii/S0022314X03000155 | ref={{harvid|Graham et al. 2003}} | arxiv=math/0009113 }}
* {{citation | arxiv=2307.02749 | author=Summer Haag | author2=Clyde Kertzer | author3=James Rickards | author4=Katherine E. Stange | author-link4=Katherine E. Stange | title=The local-global conjecture for Apollonian circle packings is false | journal=Annals of Mathematics | year=2024 | volume=200 | issue=2 | doi=10.4007/annals.2024.200.2.6 | ref={{harvid|Haag et al. 2024}}}}
* {{citation |last1=Kasner |first1=Edward |author-link=Edward Kasner|last2=Supnick|first2=Fred|title=The Apollonian packing of circles|journal=Proceedings of the National Academy of Sciences|year=1943|volume=29|issue=11 |pages=378–384 |doi=10.1073/pnas.29.11.378 |doi-access=free |pmid=16588629 |pmc=1078636 |bibcode=1943PNAS...29..378K }}
* [[Alexandre Kirillov|A.A. Kirillov]]: ''A Tale of Two Fractals'', Birkhauser, 2013.
* {{citation | arxiv=1208.5460 | last=Kontorovich | first=Alex | author-link=Alex Kontorovich | title=From Apollonius to Zaremba: Local-global phenomena in thin orbits | journal=Bull. Amer. Math. Soc.| date=2013 | volume=50 | issue=2 | pages=187–228 | doi=10.1090/S0273-0979-2013-01402-2 }}
* {{citation
 | last1 = Lagarias | first1 = Jeffrey C.
 | last2 = Mallows | first2 = Colin L.
 | last3 = Wilks | first3 = Allan R.
 | arxiv = math/0101066
 | doi = 10.2307/2695498
 | issue = 4
 | journal = American Mathematical Monthly
 | mr = 1903421
 | pages = 338–361
 | title = Beyond the Descartes circle theorem
 | volume = 109
 | year = 2002}}
* [[David Mumford]], [[Caroline Series]], David Wright: ''[[Indra's Pearls (book)|Indra's Pearls: The Vision of Felix Klein]]'', Cambridge University Press, 2002, {{ISBN|0-521-35253-3}}
* {{citation| last=Soddy | first=Frederick | author-link=Frederick Soddy | year=1937 | title=The bowl of integers and the hexlet | journal=[[Nature (journal)|Nature]] | place=London | pages=77–79 | volume=139| doi = 10.1038/139077a0 | issue=3506| doi-access = free | bibcode=1937Natur.139...77S }}.

==External links==
{{Wikibooks|Fractals|Apollonian fractals  }}
* {{mathworld|urlname=ApollonianGasket |title=Apollonian Gasket}}
* [[Alexander Bogomolny]], ''[http://www.cut-the-knot.org/Curriculum/Geometry/ApollonianGasket.shtml Apollonian Gasket]'', cut-the-knot
* {{Webarchive|url=https://web.archive.org/web/20110502081052/http://closet.zao.se/emilk/circles.html|title=An interactive Apollonian gasket running on pure HTML5}}
* [http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=15987&objectType=FILE A Matlab script to plot 2D Apollonian gasket with n identical circles] {{Webarchive|url=https://web.archive.org/web/20081007104804/http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=15987&objectType=FILE |date=2008-10-07 }} using [[circle inversion]]
* [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Apollonian_circle_packing Online experiments with JSXGraph]
* ''[http://demonstrations.wolfram.com/ApollonianGasket/ Apollonian Gasket]'' by Michael Screiber, [[The Wolfram Demonstrations Project]].
* ''[https://code.google.com/p/fract-iag/ Interactive Apollonian Gasket]'' Demonstration of an Apollonian gasket running on Java
* Dana Mackenzie. [https://www.jstor.org/stable/27859441 Computing Science: A Tisket, a Tasket, an Apollonian Gasket]. American Scientist, January/February 2010.
*{{citation|url=https://www.telegraph.co.uk/culture/art/art-news/6824326/Sand-drawing-the-worlds-largest-single-artwork.html|title=Sand drawing the world's largest single artwork|journal=The Telegraph|date=16 Dec 2009}}. Newspaper story about an artwork in the form of a partial Apollonian gasket, with an outer circumference of nine miles.
* ''[http://www.pietrocola.eu/maecla/tartapelago/frattali/apollonio2/ Dynamic apollonian gaskets]'', Tartapelago by Giorgio Pietrocola, 2014. {{in lang|it}}

{{Packing problem}}
{{Ancient Greek mathematics}}

{{Fractals}}

[[Category:Fractals]]
[[Category:Hyperbolic geometry]]
[[Category:Circle packing]]