In mathematics, an Apollonian gasket or Apollonian net 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.^{[1]}
An Apollonian gasket can be constructed as follows. Start with three circles C_{1}, C_{2} and C_{3}, each one of which is tangent to the other two (in the general construction, these three circles may be different sizes, and they must have a common tangent). Apollonius discovered that there are two other nonintersecting circles, C_{4} and C_{5}, which have the property that they are tangent to all three of the original circles – these are called Apollonian circles. Adding the two Apollonian circles to the original three, we now have five circles.
Take one of the two Apollonian circles – say C_{4}. It is tangent to C_{1} and C_{2}, so the triplet of circles C_{4}, C_{1} and C_{2} has its own two Apollonian circles. We already know one of these – it is C_{3} – but the other is a new circle C_{6}.
In a similar way we can construct another new circle C_{7} that is tangent to C_{4}, C_{2} and C_{3}, and another circle C_{8} from C_{4}, C_{3} and C_{1}. This gives us 3 new circles. We can construct another three new circles from C_{5}, giving six new circles altogether. Together with the circles C_{1} to C_{5}, this gives a total of 11 circles.
Continuing the construction stage by stage in this way, we can add 2·3^{n} new circles at stage n, giving a total of 3^{n+1} + 2 circles after n stages. In the limit, this set of circles is an Apollonian gasket.
The sizes of the new circles are determined by Descartes' theorem. Let k_{i} (for i = 1, ..., 4) denote the curvatures of four mutually tangent circles. Then Descartes' Theorem states

(1) 
The Apollonian gasket has a Hausdorff dimension of about 1.3057.^{[2]}
The curvature of a circle (bend) is defined to be the reciprocal of its radius.
An Apollonian gasket can also be constructed by replacing one of the generating circles by a straight line, which can be regarded as a circle passing through the point at infinity.
Alternatively, two of the generating circles may be replaced by parallel straight lines, which can be regarded as being tangent to one another at infinity. In this construction, the additional circles form a family of Ford circles.
The threedimensional equivalent of the Apollonian gasket is the Apollonian sphere packing.
If two of the original generating circles have the same radius and the third circle has a radius that is twothirds of this, then the Apollonian gasket has two lines of reflective symmetry; one line is the line joining the centres of the equal circles; the other is their mutual tangent, which passes through the centre of the third circle. These lines are perpendicular to one another, so the Apollonian gasket also has rotational symmetry of degree 2; the symmetry group of this gasket is D_{2}.
If all three of the original generating circles have the same radius then the Apollonian gasket has three lines of reflective symmetry; these lines are the mutual tangents of each pair of circles. Each mutual tangent also passes through the centre of the third circle and the common centre of the first two Apollonian circles. These lines of symmetry are at angles of 60 degrees to one another, so the Apollonian gasket also has rotational symmetry of degree 3; the symmetry group of this gasket is D_{3}.
The three generating circles, and hence the entire construction, are determined by the location of the three points where they are tangent to one another. Since there is a Möbius transformation which maps any three given points in the plane to any other three points, and since Möbius transformations preserve circles, then there is a Möbius transformation which maps any two Apollonian gaskets to one another.
Möbius transformations are also isometries of the 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.^{[3]}
Integral Apollonian circle packing defined by circle curvatures of (−1, 2, 2, 3)
Integral Apollonian circle packing defined by circle curvatures of (−3, 5, 8, 8)
Integral Apollonian circle packing defined by circle curvatures of (−12, 25, 25, 28)
Integral Apollonian circle packing defined by circle curvatures of (−6, 10, 15, 19)
Integral Apollonian circle packing defined by circle curvatures of (−10, 18, 23, 27)
If any four mutually tangent circles in an Apollonian gasket all have integer curvature then all circles in the gasket will have integer curvature.^{[4]} Since the equation relating curvatures in an Apollonian gasket, integral or not, is
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 – all other curvatures can be derived from these three.

There are multiple types of dihedral symmetry that can occur with a gasket depending on the curvature of the circles.
If none of the curvatures are repeated within the first five, the gasket contains no symmetry, which is represented by symmetry group C_{1}; the gasket described by curvatures (−10, 18, 23, 27) is an example.
Whenever two of the largest five circles in the gasket have the same curvature, that gasket will have D_{1} symmetry, which corresponds to a reflection along a diameter of the bounding circle, with no rotational symmetry.
If two different curvatures are repeated within the first five, the gasket will have D_{2} symmetry; such a symmetry consists of two reflections (perpendicular to each other) along diameters of the bounding circle, with a twofold rotational symmetry of 180°. The gasket described by curvatures (−1, 2, 2, 3) is the only Apollonian gasket (up to a scaling factor) to possess D_{2} symmetry.
There are no integer gaskets with D_{3} symmetry.
If the three circles with smallest positive curvature have the same curvature, the gasket will have D_{3} symmetry, which corresponds to three reflections along diameters of the bounding circle (spaced 120° apart), along with threefold rotational symmetry of 120°. In this case the ratio of the curvature of the bounding circle to the three inner circles is 2√3 − 3. As this ratio is not rational, no integral Apollonian circle packings possess this D_{3} symmetry, although many packings come close.
The figure at left is an integral Apollonian gasket that appears to have D_{3} 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_{1} symmetry common to many other integral Apollonian gaskets.
The following table lists more of these almostD_{3} 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 a(n) = 4a(n − 1) − a(n − 2) (sequence A001353 in the OEIS), from which it follows that the multiplier converges to √3 + 2 ≈ 3.732050807.
Curvature  Factors  Multiplier  

a  b  c  d  a  b  d  a  b  c  d  
−1  2  2  3  1×1  1×2  1×3  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 
For any integer n > 0, there exists an Apollonian gasket defined by the following curvatures:
(−n, n + 1, n(n + 1), n(n + 1) + 1).
For example, the gaskets defined by (−2, 3, 6, 7), (−3, 4, 12, 13), (−8, 9, 72, 73), and (−9, 10, 90, 91) all follow this pattern. Because every interior circle that is defined by n + 1 can become the bounding circle (defined by −n) in another gasket, these gaskets can be nested. This is demonstrated in the figure at right, which contains these sequential gaskets with n running from 2 through 20.
The Wikibook Fractals has a page on the topic of: Apollonian fractals 