Unit Circle Group

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Unit Circle Group Examples

Metal Mesh Unit Circle Group 01	Fireworks Unit Circle Group 02	First Snow Unit Circle Group 03
Ice Crystal Unit Circle Group 04	Organic Web Unit Circle Group 05	Atomic Sprinkles Unit Circle Group 06
Autumn Morning Unit Circle Group 07	Quantum Energy Unit Circle Group 08	Atomic Structure Unit Circle Group 09
Molecular Logic Unit Circle Group 10	Implosion Unit Circle Group 11	Crystal Vortex Unit Circle Group 12
Organic Subspace Unit Circle Group 13	Magnetic Flux Unit Circle Group 14	Metallic Void Unit Circle Group 15
Nanotechnology Unit Circle Group 16

The Unit Circle Group examples are based on an Orbit Trap called the Unit Circle Group. The Unit Circle 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.

Unit Circle Group fractals are described in a paper in the Journal of Mathematics and the Arts called Iterated Mobius transformations by Anne M. Burns, a Professor of Mathematics at the C. W. Post Campus of Long Island University.

Also, see pages 88-89 in the excellent book Indra's Pearls - The Vision of Felix Klein by David Mumford, Caroline Series, and David Wright.

Change the Unit Circle Group Properties

Change the Unit Circle Group properties.

Select the Unit Circle Group's properties page:

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

Play with the properties in the U/V Controls section to control the Mobius Transformation that positions the circles. The properties in the Circle Controls section controls the number of circles and the depth of recursion. The Level Options sections control which circles at which level of recursion are displayed. Play with checking different circles at different levels to see how it affects the image.

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:

Unit Circle Group 01 - 10 apply the Identity transformation.
Unit Circle Group 11 - 16 apply the Circle Inversion transformation.

In the following discussion, 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
    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.

To add additional transformations, select Transformation:

General
Mandelbrot / Julia / Newton
Transformation

Click the New toolbar button to add a new Identity transformation to the bottom of the list, and then click the Move Up toolbar button to move the new transformation to the desired position in the list. Normally, I move the new transformation to the top of the list, but it can be placed anywhere. See Transformation Array for details.

Then select the Identity transformation:

General
    Mandelbrot / Julia / Newton
        Transformation
            Identity

Play with Color

To play with color, select the color controller's properties page:

General
    Mandelbrot / Julia / Newton
        Orbit Trap
            Controllers
                Color Map - Dwell/Index
                    Properties

Change the Color Map properties Count, Colors, and Offset to control how colors are mapped onto the circles.