Symmetric Icon 

Symmetric Icon ExamplesThe Symmetric Icon examples are based on the Symmetric Icon Orbital Equation. Symmetric Icons and the methods used to produce them are described in the book Symmetry in Chaos by Michael Field and Martin Golumitsky. For additional details, see Images of Chaos and Symmetry. Note the following:
In the remaining sections, when I refer to the equation, I will use Symmetric Icon  Variations, but you should use the equation for the example you are working with. Add a Symmetry Transformation for PerformanceYou can add a Symmetry Transformation to improve the performance. Symmetric Icons are by definition symmetric. They exhibit rotational symmetry of order N about the origin. The order N is evident by looking at the image but is given here for completeness:
A Rosette Symmetry Group Symmetry Transformation injects rotational symmetry of order N into a fractal, so applying a Rosette Symmetry Group with the same value for the order N as given by the Symmetric Icon example, would not change the image in any way. However, in most cases, the Symmetry Transformation processing is much faster than the Orbital Equation processing, so you can improve performance by applying a Symmetry Transformation. For example, Symmetric Icon 01 takes 2.73 minutes to generate the data without the Symmetry Transformation but only 41 seconds if you add a Rosette Symmetry Group Symmetry Transformation with Order set to 6. Of course, these are the times required to produce a highquality image. If you follow the recommendations outlined in Step 2 of Fractal Science Kit Examples Overview, both these times will be significantly lower. To add the Rosette Symmetry Group Symmetry Transformation, select the Identity Symmetry Transformation:
General Change the Based On property to Rosette Symmetry Group. Next, select the symmetry transformation's properties:
General Change the Order property to match the rotational symmetry of the Symmetric Icon you are working with. I did not do this in these examples because some of the experiments given on this page, change or eliminate the symmetry in the image, and in those cases, the symmetry transformation would significantly alter the image. For example, selecting a different design, as discussed in the next section, will change the symmetry to match the new design, and you would need to change the symmetry transformation's Order to match. 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 Icon property to select a different design. Change the Color Offset property to alter how the color is mapped to the design. The remaining properties should not be changed. Change the Orbital EquationYou can change the Orbital Equation used to generate the fractal. Select the Orbital Equation:
General Change the Based On property to one of the following Orbital Equations:
Then navigate to the properties page for the equation (found under the equation in the page hierarchy) and change the Icon property to select an icon. Search for Additional Symmetric IconsYou can search for new Symmetric Icon designs. Select the Orbital Equation:
General Change the Based On property to one of the following Orbital Equations:
Then navigate to the properties page for the equation (found under the equation in the page hierarchy) and change the properties found there to control the search. Note that the default settings work, so you do not need to change anything, however, if you are working with Symmetric Icon  Variations (Search), I recommend that you change F0 and F1. Now, every time you execute the Display Fractal command on the associated Fractal Window, the program will search for a new icon and display the results. If you don't like the image, execute the Display Fractal command again to search for another icon. If you like the image and want to save the corresponding parameters, you must open the Error/Debug Window, and copy/paste the parameters found at the end of the message window into your favorite text editor. To use these parameters, you must copy them into the companion program without the (Search) in the name. To do that, open the companion program, find the line that begins with Ex01, "Example 01", and replace the remainder of the line with the parameters that you copied from the Error/Debug Window. That will replace the Example 01 icon with the one you just found. Then save the fractal with a different name. In fact, this is how I found all the icons shown here! Change the Point to TrackNormally, an Orbital Fractal collects statistics during the orbit of a fractal formula and uses these to create the fractal image. Rather than tracking the orbit point as is normally the case, you can track a triangle metric point instead. That is, as the orbit progresses, you use the last 3 orbit points to define a triangle, compute a triangle metric point based on the triangle, and accumulate statistics into the triangle metric point rather than the original orbit point as is normally done. Starting with any of the Symmetric Icon examples, you can produce many different variations simply by varying the triangle metric calculation. To do this, select Orbital / IFS / Strange Attractor:
General Set the Point to Track property to Triangle Metric Point. Note that Symmetric Icon 01 and Symmetric Icon 02 do this already. Next, select the Triangle Metric:
General Change the Triangle Metric property in the p1 section. You can change the other properties on this page too. See Triangle Metric for details. Change the TransformationYou can apply a transformation to the fractal. To apply a transformation to the fractal, 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. Turn on EmbossingYou can apply embossing to the image to give it a 3D relief look. To turn on embossing, select the General page: In the Embossing section, check the Emboss Image checkbox. The other properties in this section can be used to control the strength and direction of the embossing. Play with ColorTo play with color, select the color controller's properties page:
General Change the Color Scheme, Power, Factor, and Offset properties to control the way data is mapped to color. 
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