Apollonian gasket

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.

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.

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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.

Related Topics:
Hyperbolic plane - Tessellation

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

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