Apollonian gasket
In mathematics, an Apollonian gasket or Apollonian net is a fractal generated starting from 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]}
Contents
ConstructionEdit
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 can be any size, as long as they have common tangents). 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]}
CurvatureEdit
The curvature of a circle (bend) is defined to be the inverse of its radius.
 Negative curvature indicates that all other circles are internally tangent to that circle. This is bounding circle.
 Zero curvature gives a line (circle with infinite radius).
 Positive curvature indicates that all other circles are externally tangent to that circle. This circle is in the interior of circle with negative curvature.
VariationsEdit
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.
SymmetriesEdit
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}.
Links with hyperbolic geometryEdit
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 packingsEdit
Integral Apollonian circle packing defined by circle curvatures of (−1, 2, 2, 3)
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.

Symmetry of integral Apollonian circle packingsEdit
No symmetryEdit
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.
D_{1} symmetryEdit
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.
D_{2} symmetryEdit
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.
D_{3} symmetryEdit
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.
AlmostD_{3} symmetryEdit
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 
Sequential curvaturesEdit
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.
See alsoEdit
 Descartes' theorem, for curvatures of mutually tangent circles
 Ford circle, the special case of integral Apollonian gasket (0,0,1,1)
 Sierpiński triangle
 Apollonian network, a graph derived from finite subsets of the Apollonian gasket
NotesEdit
 ^ Satija, I. I., The Butterfly in the Quantum World: The story of the most fascinating quantum fractal (Bristol: IOP Publishing, 2016), p. 5.
 ^ McMullen, Curtis T. (3 October 1997). "Hausdorff dimension and conformal dynamics III: Computation of dimension", Abel.Math.Harvard.edu. Accessed: 27 October 2018.
 ^ Counting circles and Ergodic theory of Kleinian groups by Hee Oh Brown. University Dec 2009
 ^ Ronald L. Graham, Jeffrey C. Lagarias, Colin M. Mallows, Alan R. Wilks, and Catherine H. Yan; "Apollonian Circle Packings: Number Theory" J. Number Theory, 100 (2003), 145
ReferencesEdit
 Benoit B. Mandelbrot: The Fractal Geometry of Nature, W H Freeman, 1982, ISBN 0716711869
 Paul D. Bourke: "An Introduction to the Apollony Fractal". Computers and Graphics, Vol 30, Issue 1, January 2006, pages 134–136.
 David Mumford, Caroline Series, David Wright: Indra's Pearls: The Vision of Felix Klein, Cambridge University Press, 2002, ISBN 0521352533
 Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks: Beyond the Descartes Circle Theorem, The American Mathematical Monthly, Vol. 109, No. 4 (Apr., 2002), pp. 338–361, (arXiv:math.MG/0101066 v1 9 Jan 2001)
External linksEdit
The Wikibook Fractals has a page on the topic of: Apollonian fractals 
 Weisstein, Eric W. "Apollonian Gasket". MathWorld.
 Alexander Bogomolny, Apollonian Gasket, cuttheknot
 An interactive Apollonian gasket running on pure HTML5 (the link is dead)
 (in English) A Matlab script to plot 2D Apollonian gasket with n identical circles using circle inversion
 Online experiments with JSXGraph
 Apollonian Gasket by Michael Screiber, The Wolfram Demonstrations Project.
 Interactive Apollonian Gasket Demonstration of an Apollonian gasket running on Java
 Dana Mackenzie. A Tisket, a Tasket, an Apollonian Gasket. American Scientist, January/February 2010.
 "Sand drawing the world's largest single artwork", The Telegraph, 16 Dec 2009. Newspaper story about an artwork in the form of a partial Apollonian gasket, with an outer circumference of nine miles.