Fractal Examples |
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Fractal Science Kit ExamplesA small set of example fractal properties files can be downloaded to get you started. As time permits I will add to this set so check back periodically for new examples. Use the link below to download the Fractal Science Kit examples to your computer: Fractal Science Kit Examples 1.2 The following instructions are written from the perspective of Microsoft Windows XP but Windows Vista users should have no trouble following along. To install the Fractal Science Kit examples, do the following:
Fractal Science Kit Sample ImagesHere are some sample images generated from the example fractal properties files. Each image was created from the named example by performing 1 or more of the experiments given in the example's description, resizing the image to 800x600, and enabling high-quality image processing. I have included in the download both the properties files for the examples and the properties files used to generate the images. The files used to produce the images are in a folder named Experiments in the folder containing the examples. I would recommend that you start with the base examples for your exploration. The base examples are configured for exploration rather than for generating high-quality images and are much faster to generate. Fractal Science Kit Sample Properties FilesFor those of you interested in programming, I have included below the properties files used to generate the sample images above. Fractal properties files are saved as XML files, and the Fractal Programs used to generate the fractal are embedded within the file. Normally, you would use the interactive development environment to view/edit the programs but I have included the raw XML files below so you can get an idea of the programming possibilities. Of course, you will need to wade through the XML to find the embedded programs but they are easy to spot.
The examples only use the Built-in Programs included with the product but you can write your own programs too. Fractal Science Kit Examples OverviewFractal properties files are saved as XML files. In the following discussions, when referring to the example files, the .xml suffix is omitted for brevity. Once the examples are in place, you can open them by executing the Open File command on the File menu of the Fractal Window.
When the Open XML File dialog is displayed, open the folder containing the example files, select one of the examples, and click Open to load the file. A message box is displayed to ask if you want to replace the properties in the existing window or open a new window. Click Yes to replace the properties in the existing window. Finally, click Display Fractal on the Tools menu to display the example. The following sections describe the examples and give suggestions for things to try with each. The descriptions are not as detailed as those in the Tutorials and I recommend that you work through the Tutorials first so that you have a basic understanding of the application windows and the property page hierarchy. At a minimum, you should read the 1st page of the Tutorials which contains concepts required in the following sections. Each of the experiments given in the following sections provide instructions for changing 1 or more properties. After making changes to 1 or more Properties Pages, you can view the resulting fractal image by executing the Display Fractal command on the Tools menu.
Typically, you will open the Properties Window, make changes to 1 or more Properties Pages, and generate a new fractal image by executing the Display Fractal command on the Tools menu of the Fractal Window. Then you make additional changes to the Properties Pages, generate a new fractal image, and so on. Preview Julia SupportSeveral of the descriptions for the Mandelbrot examples will instruct you to use the Preview Julia command to explore the Mandelbrot's associated Julia fractals. This is accomplished by selecting the Preview Julia item on the Tools menu of the Fractal Window.
When you select this item (or press the Preview Julia toolbar button), the Fractal Science Kit changes the cursor to a cross and places the Fractal Window into a state where clicking on the Mandelbrot fractal generates a Julia fractal preview in the Preview Window. The point on the Mandelbrot fractal where you click is used as the Julia Constant for the preview. There is a single Preview Window shared by all windows and each click on the Mandelbrot image generates a new Julia fractal preview, replacing any image currently in the Preview Window. If you wish to cancel the operation, simply select the menu item or toolbar button again. While you are in this state, the Preview Julia menu item has a check next to it and the Preview Julia toolbar button is depressed. Try clicking at various points on the Mandelbrot fractal and view the resulting Julia fractals. The Julia fractal will be quite different depending on the characteristics of the point you click on. If you like the Julia preview, you can click on the image in the Preview Window and a new Fractal Window will open with a full-sized version of the preview. Transformation SupportSeveral of the descriptions below include experiments to apply a transformation to the fractal. Most of these examples use the Composite Function transformation since it is flexible, efficient, and easy to use. This transformation allows you to define a composite function from 2 complex functions. When selecting a complex function on the program's Properties page, you should skip over the first several functions in the list and start experimenting with the Pow2 function and above. The earlier functions tend to be uninteresting. You can change some of the other properties on this page for more variations. There are lots of other transformations available that you can use as well but for the sake of brevity I have not described them here but fell free to try them out. Several of the transformations are specialized programs useful only in certain situations. This is discussed in the comment section in the program's instructions. You can read the program comments or simply try to use the transformation and if it doesn't produce good results, try another one! Generating High-Quality ImagesThe examples are configured for exploration rather than for generating high-quality images. High-quality images take longer to generate than do lower quality images, so it is best to wait until you find an image you wish to save and then change the properties that affect the quality just before generating your final image so that you only incur the additional cost when necessary. The most important property with respect to quality is Anti-Aliasing. Anti-Aliasing is a method used to improve the quality of the fractal image by oversampling the fractal and then averaging the results. The Mandelbrot examples have Oversampling set to <None> and you should increase Oversampling to 2x2 Oversampling or 3x3 Oversampling when you produce the final image.
This improves the quality of the resulting image but dramatically increases the space required for sample data and the time required to compute it. The Orbital examples have Oversampling set to 2x2 Oversampling or 3x3 Oversampling since anti-aliasing does not result in as severe a time penalty as with Mandelbrot fractals so you will not need to change this property in these cases. Another way to improve the quality of the fractal is to modify the properties that control the orbit generation. For Mandelbrot fractals, the Orbit Generation properties (and Orbit Trap Orbit Generation properties for orbit trap based fractals) control the orbit generation process. Normally, the property settings in the examples are fine unless you zoom way into the fractal. In that case, you may need to increase Max Dwell to improve the quality near the Mandelbrot set boundary. For Orbital fractals, the Orbital / IFS / Strange Attractor properties control the orbit generation process. Increase Max Count to increase the density of the samples if the Orbital fractal image appears too sparse, too dark, or too grainy. Fractal Science Kit Example DescriptionsThe examples are described in the following sections:
Each section contains a description of 1 or more examples and experiments that you can use to generate additional fractals based on the given examples. Newton Fractal ExamplesThe Newton fractal examples are:
These examples illustrate a few of the ways you can visualize Newton fractals. Each example displays a Mandelbrot fractal based on Newton's method of finding the roots of an equation. Newton Classic colors each point on the associated Julia fractal based on the root to which the point converges. Newton Angle Relief adds a 3D relief based on the smoothed angle at each point. Newton Trap Relief adds a 3D relief based on a shape transformation. Newton Orbit Trap displays the fractal using a circle orbit trap. Newton Shift generates a single dwell of a circle orbit trap passed through a symmetry transformation. Experiment 1Use the Preview Julia command to explore the Julia fractals for each example. Experiment 2Change the root-finding method used to generate the fractal. To do this, select the Properties page associated with the Fractal Equation:
General Change the Method property to select the root-finding method. The supported methods are: Newton's method The Argument property is used to specify an argument to 2 of the root-finding methods. For the Chebyshev-Halley family, the argument is the family parameter Theta and for the C-Iterative family, the argument is the family parameter C. The Chebyshev-Halley family parameter Theta is related to several of the other methods as follows: Theta = 0 ->
Super-Newton method Try other values for Theta too. Small negative values (-0.05) are interesting, for example. For the C-Iterative family, set C to values between -2.0 and 2.0 for the best results. Experiment 3Change the Fractal Equation to one of the other Newton formulas. To do this, select the Fractal Equation: Newton 1 properties page:
General Change the Based On property to one of the following Fractal Equations: Newton 1 These programs generate Newton fractals based on various equations. Each of these programs have properties (on the Properties page found under the equation) to set the root-finding method and to manipulate the equation and thereby change the resulting fractal. Experiment 4Change the transformation applied to the base fractal. To do this, select the transformation's Properties page:
General 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. Experiment 5For the Newton Shift example, you can change the transformation associated with the transformation shift. To do this, select the transformation's Properties page:
General 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. Convergent Fractal ExamplesThe Convergent fractal examples are:
These examples illustrate a few of the ways you can visualize Convergent fractals. Each example displays a Mandelbrot fractal based on a convergent fractal equation. Convergent Atan 2 colors the fractal using the angle associated with each sample point. Convergent Angle Relief adds a 3D relief based on the smoothed angle at each point. Convergent Orbit Trap displays the fractal using a circle orbit trap. Experiment 1Use the Preview Julia command to explore the Julia fractals for each example. Experiment 2Change the Fractal Equation to one of the other Convergent formulas. To do this, select the Fractal Equation: Convergent Map 1 properties page:
General Note that Convergent Angle Relief is based on Fractal Equation: Convergent Map 14. Change the Based On property to one of the following Fractal Equations: Convergent Map 1 These programs generate convergent fractals based on various equations. Each of these programs have properties (on the Properties page found under the equation) to manipulate the equation and thereby change the resulting fractal. Experiment 3For the Convergent Atan 2 example, you can increase the complexity of the resulting image by setting the Repetitions property. To do this, select the color controller's Properties page:
General Change the Repetitions property to 2. Experiment with the other properties on this page too. These changes can be applied to the Mandelbrot or to any of the Julia fractals you create by clicking on the image in the Preview Window. A common approach is to click on the Mandelbrot until you get a Julia you like, then click on the Preview Window to open a full-sized version of the preview, open the Properties Window for the Julia, and then use the above techniques to adjust the Julia fractal's coloring. Experiment 4For the Convergent Angle Relief example, you can change the coloring applied to the resulting image by setting the controller's properties. To do this, select the color controller's Properties page:
General Change the Color Scheme to use a different gradient. Change the Value property to Angle. This looks best if Factor is left at the default value of 1. Change the Factor property to 2 or 3. This results in cycling through the color gradient 2 or 3 times, adding to the color complexity of the image. You can change the Power and/or Offset too for more variations. These changes can be applied to the Mandelbrot or to any of the Julia fractals you create by clicking on the image in the Preview Window. A common approach is to click on the Mandelbrot until you get a Julia you like, then click on the Preview Window to open a full-sized version of the preview, open the Properties Window for the Julia, and then use the above techniques to adjust the Julia fractal's coloring. Experiment 5For the Convergent Orbit Trap example, you can change the secondary coloring applied to the resulting image by setting the Triangle Metric property. To do this, select the Triangle Metric page: Change the Triangle Metric property in the p1 section. The other properties on this page also affect the resulting triangle metric point. When you have time, read the Triangle Metric page documentation for details. For now, ignore these other properties. A second way to affect the secondary coloring applied to the resulting image is by setting the Factor property. To do this, select the color controller's Properties page:
General Change the Factor property in the Value 2 section. Either of these methods can be applied to the Mandelbrot or to any of the Julia fractals you create by clicking on the image in the Preview Window. A common approach is to click on the Mandelbrot until you get a Julia you like, then click on the Preview Window to open a full-sized version of the preview, open the Properties Window for the Julia, and then use either of the above techniques to adjust the Julia fractal's coloring. Experiment 6Change the transformation applied to the base fractal. To do this, select the transformation's Properties page:
General 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. Julia Map ExampleThe Julia Map fractal example is called Julia Map. This example illustrates 1 of the ways you can visualize Julia Map fractals. Each example displays a Mandelbrot fractal based on a Julia Map fractal equation. The Mandelbrot fractals are not interesting but the associated Julia fractals can produce very nice results. Experiment 1Use the Preview Julia command to explore the Julia fractals for this example. Experiment 2Change the Fractal Equation to one of the other Julia Map formulas. To do this, select the Fractal Equation: Julia Map 1 properties page:
General Change the Based On property to one of the following Fractal Equations: Julia Map 1 These programs produce a Julia Map fractal. Each of these programs has a Magnitude property (on the Properties page found under the equation) that can be used to change the resulting fractal image. Experiment 3Change the transformation applied to the base fractal. To do this, select the transformation's Properties page:
General 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. Phoenix ExampleThe Phoenix example displays a phoenix fractal. Zoom into this fractal to see interesting designs. Angle Relief ExamplesThe Angle Relief fractal examples are:
These examples color the fractal using a 3D relief based on the angle at each point. Each example sets the Magnitude property in the Orbit Generation section of the Mandelbrot / Julia / Newton page and uses a color controller appropriate for the property setting. Angle Relief x2 sets Magnitude to x^2 and Angle Relief y2 sets Magnitude to y^2. Experiment 1Use the Preview Julia command to explore the Julia fractals for this example. Experiment 2For the Angle Relief x2 example, you can change the Fractal Equation to one of several different equations. To do this, select the Fractal Equation: Exp 5 properties page:
General Change the Based On property to one of the following Fractal Equations: NanoGeometry 1 Important: Many of these equations have a Bailout property on the Properties page found under the equation. If Bailout is present, it must be set to a number less than or equal to 128 for this example. Experiment 3For the Angle Relief y2 example, you can change the Fractal Equation to one of several different equations. To do this, select the Fractal Equation: Trig 13 properties page:
General Change the Based On property to one of the following Fractal Equations: NanoGeometry 2 Experiment 4Change how coloring is applied to the resulting image. To do this for the Angle Relief x2 example, select the color controller's Properties page:
General To do this for the Angle Relief y2 example, select the color controller's Properties page:
General Change any of the properties on this page to affect how the data is mapped to the gradient. These changes can be applied to the Mandelbrot or to any of the Julia fractals you create by clicking on the image in the Preview Window. A common approach is to click on the Mandelbrot until you get a Julia you like, then click on the Preview Window to open a full-sized version of the preview, open the Properties Window for the Julia, and then use the above techniques to adjust the Julia fractal's coloring. Experiment 5Change the transformation applied to the base fractal. To do this, select the transformation's Properties page:
General 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. Direct Color ExampleThe Direct Color example uses a direct color controller to color the fractal. Zoom into this fractal to see interesting designs. Experiment 1Use the Preview Julia command to explore the Julia fractals for this example. Experiment 2Change the transformation applied to the base fractal. To do this, select the transformation's Properties page:
General 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. Orbit Trap ExampleThe Orbit Trap example displays a Circle orbit trap. Zoom into this fractal to see interesting designs. Experiment 1Use the Preview Julia command to explore the Julia fractals for this example. Experiment 2Try out different Fractal Equations. To do this, select the Fractal Equation: Mandelbrot properties page:
General Change the Based On property to any of the different Fractal Equations. Note that a few of the equations do not lend themselves to an orbit trap representation (e.g., Barnsley, Julia Map). If the image does not look good, simply select another equation. Most equations look great when used with orbit traps. Experiment 3Try out different Orbit Traps. To do this, select the Instructions: Circle properties page:
General Change the Based On property to one of the following Orbit Traps: Circle Each of these programs have properties (on the Properties page found under the orbit trap) to manipulate the trap and thereby change the resulting fractal. There are several orbit traps not given in the above list since they are stand-alone fractals (e.g., Kleinian Group) or are too complex to display in this context (e.g., Complex Grid). You can also try out the different optimized orbit traps. To do this, select the Orbit Trap Map properties page:
General Change the Type property to one of the following: Circle Each of these orbit traps have properties (on the Properties page found under the Orbit Trap Map page) to manipulate the trap and thereby change the resulting fractal. Experiment 4Change the transformation applied to the base fractal. To do this, select the transformation's Properties page:
General 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. Ornament ExampleThe Ornament example displays a fractal that looks like an ornament. L-System ExamplesThe L-System fractal examples are:
These examples display a Hilbert curve using L-System instructions. L-System Classic displays the curve using the L-System subsystem while the others display the curve as an orbit trap. The experiments below apply only to the orbit trap based L-System examples (L-System Orbit Trap and L-System Shift). Experiment 1Change the color of the fractal. To do this, select the color controller's Properties page:
General Change the Gradient Offset property to the 0-based index of the gradient you want to use. View the gradients on the Pattern Map - Perlin Noise properties page. Experiment 2Change the transformation applied to the base fractal. To do this, select the transformation's Properties page:
General 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. Experiment 3For the L-System Shift example, you can change the transformation associated with the transformation shift. To do this, select the transformation's Properties page:
General 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. Kleinian Group ExampleThe Kleinian Group example generates a Kleinian group orbit trap. Experiment 1Change the Kleinian group properties. To do this, select the Kleinian group's Properties page:
General Change the Example property to any of the different examples. Some of the examples (e.g., Double Cusp examples) may require changes in several advanced properties (Depth, Radius Cutoff, Min Radius) to fill in the gaps between circles but many work well with the default settings. 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. Also, try changing the Recipe, Conjugate By, and/or UCG Transform properties. Using different combinations of these properties, you can generates many different designs. Complex Analysis ExampleThe Complex Analysis example is not a fractal but illustrates how the Fractal Science Kit can be used for visualizing complex transformations. The example shown is the Bipole() function which is defined as: Bipole(z) = 1/(z+1) + 1/(z-1) To change the transformation applied to the base image, select the transformation's Properties page:
General Set the F(z) property to one of the complex functions in the list. If you wan to test a transformation not in the list, change the Based On property from Composite Function to the Identity transformation (the 1st entry in the list) and assign the transformation expression to the variable z. For example, to use the same transformation as above, use: z = 1/(z+1) + 1/(z-1) Sierpinski Variations ExampleThe Sierpinski Variations example generates a set of attractors based on a variation of the algorithm used to generate the classic Sierpinski Triangle. A set of points is defined along with a transformation per point. A point is randomly selected and the transformation is used to alter the location of the current orbit point. Experiment 1Change the attractor. To do this, select the equation's Point Information properties page:
General Change the Example property to 1 of the 74 predefined examples or define your own set of points/transformations using the properties on this page. (The following is an advanced feature and should be skipped the first time you try this example.) The Weight property controls the relative probability of selecting the associated point during orbit processing and can add depth to the resulting image in some situations. Try setting the Weight values for all the points equal to 1 and then set the Weight for one of the points to 4. Very different results are obtained depending on which point you choose. Experiment 2Change the number of vertices in the base polygon. To do this, select the equation's General properties page:
General Change the Vertices property to try higher order polygons. You can also change the Reflect, Rotate, and Scale properties which affect the transformations applied to each of the points. Experiment 3Change the non-linear transformation applied to each point. To do this, select the transformation's Properties page:
General Set the Shape property to 1 of the 167 different shape types. When you change the Shape (or any of the other properties associated with the other experiments for this example), if the image is too dense, try reducing the Radius property. If the image is too sparse, try increasing Radius. Small changes to Radius can result in large changes in the image. Experiment 4Change the transformation applied to the base fractal. To do this, select the transformation's Properties page:
General 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. Sierpinski Cycloid ExampleThe Sierpinski Cycloid example injects a cycloid variation into the classic Sierpinski N-gon. Experiment 1Change the attractor. To do this, select the equation's Properties page:
General Change any of the property values. It works best to set Vertices relative to the Cusps property found on the cycloid variation's properties page (see Experiment 2). Set Cusps as a multiple of Vertices or vice versa, or set both with a common multiple (e.g., Vertices = 6 and Cusps = 9). Experiment 2Change the cycloid properties. To do this, select the variation's Properties page:
General Change any of the property values. Experiment 3Change the transformation applied to the base fractal. To do this, select the transformation's Properties page:
General 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. Apollonian Gasket Variations ExampleThe Apollonian Gasket Variations example is an attractor based on a variation of the algorithm used to generate the Apollonian Gasket. Options are provided to select several different variants of the attractor. Experiment 1Select the properties to define the attractor. To do this, select the equation's Properties page:
General Change the Variation property to 1 of the predefined variations or define your own by defining a Mobius transformation. Change the Invert property to conjugate the results with the complex inversion transformation. Kleinian Group Attractor ExampleThe Kleinian Group Attractor example is an attractor used to generate a Kleinian Group fractal. Options are provided to select different variations of the attractor. Experiment 1Select the properties to define the attractor. To do this, select the equation's Properties page:
General Change the Recipe, Example, Conjugate By, and/or Transform properties to define the attractor. Rep-9 Tile ExampleThe Rep-9 Tile example displays a Rep-Tile based on up to 9 affine transformations. A symmetry transformation is used to generate 3-way dihedral symmetry from the base tile. The default property settings completely fills the base tile. Options are provided to omit 1 or more of the affine transformations to generate more interesting images. Experiment 1Omit 1 or more of the Rep-9 Tile affine transformations. To do this, select the equation's Properties page:
General Uncheck 1 or more of the 9 checkboxes labeled Include T1 through Include T9. Unchecking 1-3 checkboxes works best. Experiment 2Change the base Rep-9 Tile. To do this, select the equation's Properties page:
General Change the Type property to any of the types found there; i.e., Isosceles Trapezium This changes the base shape used to generate the tile. Note that since 45 Degree Wedge Trapezium and L-Triomino do not have a 60 degree angle at the origin, the symmetry transformation as currently defined, either has gaps (45 Degree Wedge Trapezium) or overlaps parts of the base tile (L-Triomino), but it still looks good! You can change the symmetry transformation to prevent the gaps/overlap in these cases, if required. See the program comments for details. Experiment 3Change the transformation applied to the base fractal before the symmetry transformation. To do this, select the transformation's Properties page:
General 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. Since this transformation is applied before the symmetry transformation (i.e., the output from this transformation is passed to the symmetry transformation), the resulting image will maintain the 3-way dihedral symmetry defined by the symmetry transformation. Experiment 4Change the transformation applied to the base fractal after the symmetry transformation. To do this, select the transformation's Properties page:
General 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. Since this transformation is applied after the symmetry transformation (i.e., the output from the symmetry transformation is passed to this transformation), the resulting image will not necessarily maintain the 3-way dihedral symmetry defined by the symmetry transformation. Quadratic Attractor ExampleThe Quadratic Attractor example generates a wallpaper group tiling by passing the results of the attractor through a symmetry transformation that implements a plane symmetry group square lattice. Experiment 1Change the attractor. To do this, select the equation's Properties page:
General Change the Attractor property to select the 1 of the attractors. Do not change the coefficients C01 through C12 since it is unlikely your change will produce a fractal. Most of the attractors in this list were found using the program Quadratic Attractors (Search) which generates sets of coefficients that meet a set of user-defined criteria. Try that program out later when you have time. Change the Rotate By property to rotate the attractor image. Since the rotation is processed prior to applying the symmetry transformation, this results in significant changes to the resulting image. Experiment 2Change the type of symmetry transformation. To do this, select the symmetry transformation's Properties page:
General Change the Symmetry property to 1 of the 12 square lattice based symmetry groups. The Subtype property does not change the shape of the resulting image but can be used to produce 2-color symmetry patterns. In order to view these patterns, you will need to change the data mapped to the image in the color controller. To do this, select the controller's Properties page:
General Change the Value property from Attractor Index to Symmetry Index. Now you can return to the transformation's Properties page and change the Subtype property to produce 2-color symmetry patterns. Each Symmetry property setting has a different set of Subtype property settings for the 2-color symmetry patterns associated with the Symmetry setting. Note that some attractors look better than others with respect to 2-color symmetry pattern coloring. Experiment 3Change the transformation applied to the base fractal before the symmetry transformation. To do this, select the transformation's Properties page:
General 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. Since this transformation is applied before the symmetry transformation (i.e., the output from this transformation is passed to the symmetry transformation), the resulting image will maintain the symmetry defined by the symmetry transformation. Symmetric Icon ExampleThe Symmetric Icon example displays a symmetric icon but rather than tracking the orbit point as is normally the case, we track a triangle metric point instead. That is, as the orbit progresses, we 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. For any given symmetric icon attractor, we can produce many different variations simply by varying the triangle metric calculation (see Experiment 2). Experiment 1Change the base attractor that defines the symmetric icon. To do this, select the equation's Properties page:
General Change the Icon property to select 1 of the attractors. Do not change the remaining properties on this page since it is unlikely your change will produce a fractal. All of the attractors in this list were found using the program Symmetric Icon - Standard Formula (Search) which generates sets of values that meet a set of user-defined criteria. Try that program out later when you have time. Experiment 2Change the triangle metric point used to define the point into which we accumulate statistics. To do this, select the Triangle Metric properties page: Change the Triangle Metric property in the p1 section. The other properties on this page also affect the resulting triangle metric point. When you have time, read the Triangle Metric page documentation for details. For now, ignore these other properties. Experiment 3Change the transformation applied to the base fractal before the symmetry transformation. To do this, select the transformation's Properties page:
General 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. Experiment 4Change the color blend used to color the image. To do this, select the color controller's Properties page:
General Change the Blend Type to any of the supported color blends. |
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