Apollonian Gasket Variations

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Apollonian Gasket Variations Examples

Fractal: Apollonian Gasket Attractor

Apollonian Gasket I
Apollonian Gasket Variations 01

Fractal: Apollonian Gasket Attractor

Apollonian Gasket II
Apollonian Gasket Variations 02

Fractal: Apollonian Gasket Attractor

Apollonian Gasket III
Apollonian Gasket Variations 03

Fractal: Apollonian Gasket Attractor

Apollonian Gasket IV
Apollonian Gasket Variations 04

The Apollonian Gasket Variations examples use an attractor based on a variation of the algorithm used to generate the Apollonian Gasket.

The Apollonian Gasket, and the methods used to produce them, are described in the excellent book Indra's Pearls - The Vision of Felix Klein by David Mumford, Caroline Series, and David Wright. For additional details, see David Wright's Indra's Pearls site.

Change the Variation

You can change the variation.

Select the equation's properties page:

General
    Orbital / IFS / Strange Attractor
        Orbital Equation: Apollonian Gasket (Variations)
            Properties

Change the Variation property to one of the predefined variations or create your own by defining a Mobius transformation.

Change the Invert property to conjugate the results with the complex inversion transformation.

I do not recommend changing the Mobius transformations defined for Attractor 1 or Attractor 2 unless you understand the algorithm on which this attractor is based.

Change the Transformation

You can apply a transformation to the fractal.

Execute the Home command on the View menu of the Fractal Window to reset the fractal to the default position/magnification before you adjust the transformation. Then change the transformation and Zoom In to interesting areas of the transformed image.

Note the following:

  • Apollonian Gasket Variations 01 applies the Identity transformation to the fractal.
  • Apollonian Gasket Variations 02 applies the transformation Disc Automorphism - Hyperbolic to the fractal.
  • Apollonian Gasket Variations 03 applies the Identity transformation to the fractal.
  • Apollonian Gasket Variations 04 applies the transformation Clip to the fractal.

The Identity transformation does not alter the image.

In the remaining sections, when I refer to the transformation, I will use Identity, but you should use the transformation for the example you are working with.

To apply a transformation to the fractal, select the Identity transformation's page:

General
    Orbital / IFS / Strange Attractor
        Transformation 1
            Identity

Set the Based On property to one of the available transformations, select the transformation's properties page (found under the transformation in the page hierarchy), and play with the properties found there.

Mobius transformations work especially well in these examples because they preserve circles; i.e., circles are mapped to circles.

To use a Mobius transformation, set the Based On property to one of the following transformations:

  • Mobius Transformation
  • Mobius Transformation - Elliptic
  • Mobius Transformation - Hyperbolic
  • Mobius Transformation - Loxodromic
  • Mobius Transformation - Parabolic
  • Disc Automorphism - Elliptic
  • Disc Automorphism - Hyperbolic
  • Disc Automorphism - Parabolic
  • Disc Automorphism - General
  • Half-Plane to Disk
  • Circle To Circle
  • Poincare Disk
  • Mobius Group

Note that the default for most of these transformations is the identity transformation (which does nothing) so you will need to select the properties page found under the transformation and change the properties in the section labeled Transformation Control.

To add additional transformations, select Transformation 1:

General
    Orbital / IFS / Strange Attractor
        Transformation 1

Click the New toolbar button to add a new Identity transformation to the bottom of the list. See Transformation Array for details.

Then select the Identity transformation:

General
    Orbital / IFS / Strange Attractor
        Transformation 1
            Identity

Change the Based On property to select a transformation and then open the transformation's properties page (found under the transformation in the page hierarchy), and play with the transformation's properties. See Transformation Support for details.

 

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