Apollonian gasket
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In mathematics, an Apollonian gasket or Apollonian net is a fractal generated from three circles, any two of which are tangent to one another. It is named after Greek mathematician Apollonius of Perga.
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[edit] Construction
An Apollonian gasket can be constructed as follows. Start with three circles C1, C2 and C3, 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, C4 and C5, 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 C4. It is tangent to C1 and C2, so the triplet of circles C4, C1 and C2 has its own two Apollonian circles. We already know one of these - it is C3 - but the other is a new circle C6.
In a similar way we can construct another new circle C7 that is tangent to C4, C2 and C3, and another circle C8 from C4, C3 and C1. This gives us 3 new circles. We can construct another three new circles from C5, giving six new circles altogether. Together with the circles C1 to C5, this gives a total of 11 circles.
Continuing the construction stage by stage in this way, we can add 2·3n new circles at stage n, giving a total of 3n+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 [1].
[edit] 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.
[edit] Symmetries
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.
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.
[edit] 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 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.
[edit] References
- Weisstein, Eric W., Apollonian Gasket at MathWorld.
- Alexander Bogomolny, 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: "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 0-521-35253-3
