Kleinian Group Orbit Trap

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Kleinian Group Orbit Trap Examples

Kleinian Marble II Kleinian Group Orbit Trap 01	Polished Rock Design Kleinian Group Orbit Trap 02	Marble Swan Kleinian Group Orbit Trap 03
Binding Attraction Kleinian Group Orbit Trap 04	Seahorse Parade Kleinian Group Orbit Trap 05	Medusa Kleinian Group Orbit Trap 06
Kleinian Marble Frieze Kleinian Group Orbit Trap 07	Kleinian Marble Inlay I Kleinian Group Orbit Trap 08	Kleinian Marble Inlay II Kleinian Group Orbit Trap 09
Kleinian Marble Inlay III Kleinian Group Orbit Trap 10	Kleinian Marble Inlay IV Kleinian Group Orbit Trap 11	Kleinian Cross Kleinian Group Orbit Trap 12
Kleinian Coat of Arms Kleinian Group Orbit Trap 13	Kleinian Marble III Kleinian Group Orbit Trap 14	Kleinian Marble Frieze II Kleinian Group Orbit Trap 15

The Kleinian Group Orbit Trap examples are based on an Orbit Trap called Kleinian Group. The Kleinian Group is a stand-alone fractal. It is implemented as an Orbit Trap so it can take advantage of Orbit Trap related features in the Fractal Science Kit, but it does not need the normal orbit processing; i.e., I set the Max Dwell property found in the Orbit Trap Orbit Generation section on the Mandelbrot / Julia / Newton page, to 1, eliminating the normal orbit processing.

Kleinian Group fractals, 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.

Performance

The Kleinian Group Orbit Trap examples are based on an Orbit Trap called Kleinian Group. There are a few properties associated with the trap that are related to quality and also affect performance. When you are exploring, you can improve performance by adjusting these properties on the trap's properties page.

Open the orbit trap's properties page:

General
    Mandelbrot / Julia / Newton
        Orbit Trap
            Orbit Trap Map
                Instructions: Kleinian Group
                    Properties

Depth, Radius Cutoff, and Min Radius, control the number of circles that are generated to define the fractal. Depth is the depth of recursion used in the algorithm. Radius Cutoff is the minimum radius of circles placed on the processing stack and is used to terminate the recursion loop early on selected branches. Min Radius is the minimum radius required for a circle to be displayed.

You can increase Depth and/or decrease Radius Cutoff or Min Radius to fill in the gaps between circles but these changes can cause dramatic increases in processing time. If you make a change that seems to be taking forever, you should click the Cancel Display command on the Tools menu of the Fractal Window to terminate the processing. Then try adjusting these properties to reduce the processing (i.e., decrease Depth and/or increase Radius Cutoff or Min Radius).

When you are exploring, you should set Depth to 32, Radius Cutoff to 0.004, and Min Radius to 0.004. This will greatly improve performance at the expense of quality. You can reset them later when you find an image you wish to save.

Change the Kleinian Group Orbit Trap Example Property

The Kleinian Group Orbit Trap examples can be divided into 2 groups: Packed and Jorden Curve.

You can change the Kleinian Group Orbit Trap Example property, but the set of acceptable values is determined by which group the example belongs to.

The Packed examples are:

Kleinian Group Orbit Trap 01
Kleinian Group Orbit Trap 02
Kleinian Group Orbit Trap 08
Kleinian Group Orbit Trap 09
Kleinian Group Orbit Trap 10
Kleinian Group Orbit Trap 11
Kleinian Group Orbit Trap 12
Kleinian Group Orbit Trap 14
Kleinian Group Orbit Trap 15

The Jorden Curve examples are:

Kleinian Group Orbit Trap 03
Kleinian Group Orbit Trap 04
Kleinian Group Orbit Trap 05
Kleinian Group Orbit Trap 06
Kleinian Group Orbit Trap 13

Note that the Kleinian Group Orbit Trap 07 example is not in either group because you cannot change the Example property for that example.

Open the Kleinian Group Orbit Trap properties page:

General
    Mandelbrot / Julia / Newton
        Orbit Trap
            Orbit Trap Map
                Instructions: Kleinian Group
                    Properties

For the the Packed examples, you can change the Example property to any of the following values:

Apollonian Gasket
1/10 Double Cusp
1/9 Double Cusp
1/8 Double Cusp
1/7 Double Cusp
1/6 Double Cusp
1/5 Double Cusp
2/9 Double Cusp
1/4 Double Cusp
2/7 Double Cusp
3/10 Double Cusp
1/3 Double Cusp
3/8 Double Cusp
2/5 Double Cusp
3/7 Double Cusp
4/9 Double Cusp
1/2 Double Cusp
5/9 Double Cusp
4/7 Double Cusp
3/5 Double Cusp
5/8 Double Cusp
2/3 Double Cusp
7/10 Double Cusp
5/7 Double Cusp
3/4 Double Cusp
7/9 Double Cusp
4/5 Double Cusp
5/6 Double Cusp
6/7 Double Cusp
7/8 Double Cusp
8/9 Double Cusp
9/10 Double Cusp
Circle Packing: 6 x 6
Circle Packing: 6 x 9
Circle Packing: 9 x 6
Circle Packing: 9 x 9
Circle Packing: 12 x 4
Circle Packing: 12 x 5
Circle Packing: 12 x 6
Circle Packing: 12 x 9
Circle Packing: 4 x 12
Circle Packing: 5 x 12
Circle Packing: 6 x 12
Circle Packing: 9 x 12

For the the Jorden Curve examples, you can change the Example property to any of the following values:

Jordan Curve 1
Jordan Curve 2
Jordan Curve 3
Jordan Curve 4
Jordan Curve 5
Jordan Curve 6
Jordan Curve 7

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.

The following examples apply a transformation to the fractal:

Kleinian Group Orbit Trap 02
Kleinian Group Orbit Trap 08
Kleinian Group Orbit Trap 09
Kleinian Group Orbit Trap 10
Kleinian Group Orbit Trap 11
Kleinian Group Orbit Trap 12

To change the transformation applied to the fractal, select the transformation's properties page:

General
    Mandelbrot / Julia / Newton
        Transformation
            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.

You can also use a different transformation altogether. Select the Composite Function page, and 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.

For those examples that do not apply a transformation to the fractal, you can apply a transformation of your own.

To apply a transformation to the fractal, select the Identity transformation:

General
    Mandelbrot / Julia / Newton
        Transformation
            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.

Textures

You can change the textures mapped to the circles in the following examples:

Kleinian Group Orbit Trap 13
Kleinian Group Orbit Trap 14
Kleinian Group Orbit Trap 15

To do this, select Pattern Map - Texture:

General
    Mandelbrot / Julia / Newton
        Orbit Trap
            Controllers
                Pattern Map - Texture

Use the Texture List Control to change the textures mapped to the circles. For these examples, using 2 or 4 textures gives the best results. The Texture List Control allows you to add, remove, and reorder the textures in the list.