Apollonian gasket

In mathematics, an Apollonian gasket or Apollonian net is a fractal generated from triples of circles, where any circle is tangent to two others. It is named after Greek mathematician Apollonius of Perga.

Construction

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 non-intersecting 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" (see Descartes' theorem). 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 Apollonian gasket has a Hausdorff dimension of about 1.3057 [http://abel.math.harvard.edu/~ctm/papers/home/text/papers/dimIII/dimIII.pdf] .

Variations

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 circles that are tangent to one of the two straight lines form a family of Ford circles.

The three-dimensional equivalent of the Apollonian gasket is the Apollonian sphere packing.

ymmetries

If two of the original generating circles have the same radius and the third circle has a radius that is two-thirds 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 D₂.

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 D₃.

Links with hyperbolic geometry

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, which can be thought of as a tessellation of the hyperbolic plane by circles and hyperbolic triangles.

The Apollonian gasket is the limit set of a group of Möbius transformations known as a Kleinian group.

Integral Apollonian circle packings

If any four mutually tangent circles in an Apollonian gasket all have integer curvature (where the curvature of a circle is defined to be the inverse of its radius) then all circles in the gasket will have integer curvature. [ [http://citeseer.ist.psu.edu/cache/papers/cs/15837/http:zSzzSzwww.math.tamu.eduzSz~catherine.yanzSz.zSzFileszSzPart4_10.pdf/apollonian-circle-packings-number.pdf Ronald L. Graham, Jeffrey C. Lagarias, Colin M. Mallows, Alan R. Wilks; "Apollonian Circle Packings: Number Theory"] ] 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; a negative curvature indicates that all other circles are tangent to the interior of that circle (that is, it is the bounding circle), while a positive curvature indicates that all other circles are tangent to the exterior of that circle (these are the interior circles). 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.

equential curvatures

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.

Footnotes

References

*
* Alexander Bogomolny, " [http://www.cut-the-knot.org/Curriculum/Geometry/ApollonianGasket.shtml Apollonian Gasket] ", cut-the-knot
* Benoit B. Mandelbrot: "The Fractal Geometry of Nature", W H Freeman, 1982, ISBN 0-7167-1186-9
* Paul D. Bourke: " [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.
* David Mumford, Caroline Series, David Wright: "", Cambridge University Press, 2002, ISBN 0-521-35253-3
* 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, ( [http://arxiv.org/pdf/math.MG/0101066 arXiv:math.MG/0101066 v1 9 Jan 2001] )

External links

* [http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=15987&objectType=FILE A Matlab script to plot 2D Apollonian gasket with n identical circles]
* " [http://demonstrations.wolfram.com/ApollonianGasket/ Apollonian Gasket] " by Michael Screiber, The Wolfram Demonstrations Project.