Kleinian Group Orbit Trap

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Kleinian Group Orbit Trap Examples

Fractal: Kleinian Group Orbit Trap

Kleinian Marble II
Kleinian Group Orbit Trap 01

Fractal: Kleinian Group Orbit Trap

Polished Rock Design
Kleinian Group Orbit Trap 02

Fractal: Kleinian Group Orbit Trap

Marble Swan
Kleinian Group Orbit Trap 03

Fractal: Kleinian Group Orbit Trap

Binding Attraction
Kleinian Group Orbit Trap 04

Fractal: Kleinian Group Orbit Trap

Seahorse Parade
Kleinian Group Orbit Trap 05

Fractal: Kleinian Group Orbit Trap

Medusa
Kleinian Group Orbit Trap 06

Fractal: Kleinian Group Orbit Trap

Kleinian Marble Frieze
Kleinian Group Orbit Trap 07

Fractal: Kleinian Group Orbit Trap

Kleinian Marble Inlay I
Kleinian Group Orbit Trap 08

Fractal: Kleinian Group Orbit Trap

Kleinian Marble Inlay II
Kleinian Group Orbit Trap 09

Fractal: Kleinian Group Orbit Trap

Kleinian Marble Inlay III
Kleinian Group Orbit Trap 10

Fractal: Kleinian Group Orbit Trap

Kleinian Marble Inlay IV
Kleinian Group Orbit Trap 11

Fractal: Kleinian Group Orbit Trap

Kleinian Cross
Kleinian Group Orbit Trap 12

Fractal: Kleinian Group Orbit Trap

Kleinian Coat of Arms
Kleinian Group Orbit Trap 13

Fractal: Kleinian Group Orbit Trap

Kleinian Marble III
Kleinian Group Orbit Trap 14

Fractal: Kleinian Group Orbit Trap

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.

 

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