Kleinian Group Attractor

The Kleinian Group Attractor example is an attractor used to generate a Kleinian Group fractal.

Experiment 1

Select the properties to define the attractor. To do this, select the equation's Properties page:

General
    Orbital / IFS / Strange Attractor
        Orbital Equation: Kleinian Group - Examples
            Properties

Change the Recipe, Example, Conjugate By, and/or Transform properties to define the attractor.

Experiment 2

Change the transformation applied to the base fractal. To do this, select the transformation's Properties page:

General
    Orbital / IFS / Strange Attractor
        Transformation 1
            Composite Function
                Properties

Set the F(z) property to one of the complex functions in the list. You can change some of the other properties on this page for more variations.

Experiment 3

You can also try changing the transformation to one of the built-in Mobius transformations. Mobius transformations work especially well in this example because they preserve circles; circles are mapped to circles. Note that lines are considered circles with infinite radius.

To change the transformation to one of the built-in Mobius transformations, select the Composite Function page:

General
    Orbital / IFS / Strange Attractor
        Transformation 1
            Composite Function

Set the Based On property to one of the following transformations:

• 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

Note that the default for most of these transformations is the identity transformation so you will need to select the Properties page found under the transformation and change the properties in the section labeled Transformation Control.