Mobius Patterns (page 1) 

Mobius Patterns ExamplesThe Mobius Patterns examples are based on an IFS formed from a set of Mobius Transformations using the Orbital Equation Mobius Patterns. The Mobius Patterns examples are based on information found 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. Symmetry TransformationsSome of these examples apply a Symmetry Transformation to improve the fractal generation performance. For example, if a fractal design exhibits reflective symmetry about the X axis (i.e., the fractal image below the X axis is a reflection of the image above the X axis), you can add a Symmetry Transformation that reflects points about the X axis to improve the performance of the fractal processing without altering the fractal image. The reason you need to be aware of this is because you might change something in the example that alters or eliminates the symmetry in the fractal, and if you do not change or remove the Symmetry Transformation, it will continue to inject the symmetry of the original example into the image. Sometimes this can result in an interesting image, but it can also result in a mess, so my rule of thumb is to check if there is a Symmetry Transformation applied to the image, and if so, reset it to the Identity transformation if your experiments seem to be adversely affected. To check if a Symmetry Transformation is in effect, look for the Identity transformation:
General If you find the Identity transformation is selected as shown above, then no Symmetry Transformation is in effect. Otherwise, select the properties page for the transformation that is in effect to view the details. To reset the the symmetry transformation to the Identity transformation, select the existing symmetry transformation and change the Based On property to Identity. Finally, when you apply a transformation to a fractal, you can apply the transformation before applying the symmetry transformation or after applying the symmetry transformation. If no Symmetry Transformation is in effect, then the distinction is irrelevant. However, if a Symmetry Transformation is in effect, the distinction is important. Zoom In/OutZoom In or Zoom Out to examine different parts of the fractal. Execute the Home command on the View menu of the Fractal Window to reset the fractal to the default position/magnification, and then Zoom In to other areas. Remember that as you Zoom In, you may need to increase the Max Count property found in the Orbital / IFS / Strange Attractor section of the Orbital / IFS / Strange Attractor page to strengthen the image. Play with the Orbital Equation's PropertiesYou can change the equation's properties for more variations. Select the equation's properties page:
General Change the Pattern property to select a design. The remaining properties control the weights and index values assigned to the individual transformations that make up the IFS. You probably do not need to change these values, but they can be of help when you are trying to bring a design into focus. If you decide to experiment with the weights, I recommend that you make a small change to a single weight and view the effect. That way you can develop an appreciation for how the weights effect the image. Additional information is given in the comment section of the Orbital Equation so select the equation's page and read the comments at the top of the program. Change Transformation 1You can apply a transformation to the orbit point before applying the symmetry transformation. 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. 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 orbit point before applying the symmetry transformation, select the Identity transformation's page:
General 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 1:
General 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 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. Change Transformation 2You can apply a transformation to the orbit point after applying the symmetry transformation. 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. To apply a transformation to the orbit point after applying the symmetry transformation, select the Identity transformation's page:
General 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 2:
General 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 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. Change the Symmetry TransformationYou can apply a Symmetry Transformation to inject symmetry into the image. Typically, a Symmetry Transformation is used to improve performance by adding a transformation that mirrors the symmetry inherent in the fractal image already. However, in some cases, you can add a Symmetry Transformation to inject symmetry into an image where none existed before. In either case, the method is the same. Select the symmetry transformation:
General Change the Based On property to select a Symmetry Transformation and then open the transformation's properties page (found under the transformation in the page hierarchy), and set the transformation's properties. 
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