Angle Relief x2	Angle Relief y2	Apollonian Gasket Variations
Complex Analysis	Convergent Angle Relief	Convergent Atan 2
Convergent Orbit Trap	Direct Color	Julia Map
Kleinian Group Attractor	Kleinian Group	L-System Classic
L-System Orbit Trap	L-System Shift	Newton Angle Relief
Newton Classic	Newton Orbit Trap	Newton Shift
Newton Trap Relief	Orbit Trap	Ornament
Phoenix	Quadratic Attractors	Rep-9 Tile
Sierpinski Cycloid	Sierpinski Variations	Symmetric Icon

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 Files

For 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.

• Angle Relief x2
• Angle Relief y2
• Apollonian Gasket Variations
• Complex Analysis
• Convergent Angle Relief
• Convergent Atan 2
• Convergent Orbit Trap
• Direct Color
• Julia Map
• Kleinian Group Attractor
• Kleinian Group
• L-System Classic
• L-System Orbit Trap
• L-System Shift
• Newton Angle Relief
• Newton Classic
• Newton Orbit Trap
• Newton Shift
• Newton Trap Relief
• Orbit Trap
• Ornament
• Phoenix
• Quadratic Attractors
• Rep-9 Tile
• Sierpinski Cycloid
• Sierpinski Variations
• Symmetric Icon

The examples only use the Built-in Programs included with the product but you can write your own programs too.

Fractal Science Kit Examples Overview

Fractal 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 1^st 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 Support

Several 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 Support

Several 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 Images

The 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 Descriptions

The examples are described in the following sections:

• Newton Fractal Examples
• Convergent Fractal Examples
• Julia Map Example
• Phoenix Example
• Angle Relief Examples
• Direct Color Example
• Orbit Trap Example
• Ornament Example
• L-System Examples
• Kleinian Group Example
• Complex Analysis Example
• Sierpinski Variations Example
• Sierpinski Cycloid Example
• Apollonian Gasket Variations Example
• Kleinian Group Attractor Example
• Rep-9 Tile Example
• Quadratic Attractor Example
• Symmetric Icon Example

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 Examples

The Newton fractal examples are:

• Newton Classic
• Newton Angle Relief
• Newton Trap Relief
• Newton Orbit Trap
• Newton Shift

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 1

Use the Preview Julia command to explore the Julia fractals for each example.

Experiment 2

Change the root-finding method used to generate the fractal. To do this, select the Properties page associated with the Fractal Equation:

General
    Mandelbrot / Julia / Newton
        Fractal Equation: Newton 1
            Properties

Change the Method property to select the root-finding method. The supported methods are:

Newton's method
Schroder's method
Halley's method
Whittaker's method
Cauchy's method
Super-Newton method
Super-Halley method
Chebyshev-Halley family
C-Iterative family

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
Theta = 0.5      -> Halley's method
Theta = 1        -> Super-Halley method
Theta = Infinity -> Newton's method (use 1e8 to approximate Infinity)

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 3

Change the Fractal Equation to one of the other Newton formulas. To do this, select the Fractal Equation: Newton 1 properties page:

General
    Mandelbrot / Julia / Newton
        Fractal Equation: Newton 1

Change the Based On property to one of the following Fractal Equations:

Newton 1
Newton 2
Newton 3
Newton 4
Newton 5
Newton 6
Newton 7
Newton 8
Newton 9
Newton 10
Newton 11
Newton 12
Carlson - Newton 1
Carlson - Newton 2
Carlson - Newton 3
Newton Composite Function
Newton Poly 4a
Newton Poly 4b
Newton Poly 4c
Newton Poly 5a
Newton Poly 5b
Newton Poly 5c
Newton Poly 5d
Newton Poly 6a
Newton Poly 6b
Newton Poly 6c
Newton Poly 7a
Newton Poly 7b
Newton Poly 7c

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 4

Change the transformation applied to the base fractal. To do this, 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.

Experiment 5

For the Newton Shift example, you can change the transformation associated with the transformation shift. To do this, select the transformation's Properties page:

General
    Mandelbrot / Julia / Newton
        Orbit Trap
            Symmetry Transformation: Transformation Shift - Rotate
                Transformation Array
                    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.

Convergent Fractal Examples

The Convergent fractal examples are:

• Convergent Atan 2
• Convergent Angle Relief
• Convergent Orbit Trap

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 1

Use the Preview Julia command to explore the Julia fractals for each example.

Experiment 2

Change the Fractal Equation to one of the other Convergent formulas. To do this, select the Fractal Equation: Convergent Map 1 properties page:

General
    Mandelbrot / Julia / Newton
        Fractal Equation: Convergent Map 1

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
Convergent Map 2
Convergent Map 3
Convergent Map 4
Convergent Map 5
Convergent Map 6
Convergent Map 7
Convergent Map 8
Convergent Map 9
Convergent Map 10
Convergent Map 11
Convergent Map 12
Convergent Map 13
Convergent Map 14

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 3

For 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
    Mandelbrot / Julia / Newton
        Classic
            Controllers
                Gradient Map - Atan 2
                    Properties

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 4

For 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
    Mandelbrot / Julia / Newton
        Classic
            Controllers
                Gradient Map - Angle Relief (Smooth Angle)
                    Properties

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 5

For 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:

General
    Triangle Metric

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
    Mandelbrot / Julia / Newton
        Orbit Trap
            Controllers
                Gradient Map - Value Sum
                    Properties

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 6

Change the transformation applied to the base fractal. To do this, 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.

Julia Map Example

The 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 1

Use the Preview Julia command to explore the Julia fractals for this example.

Experiment 2

Change the Fractal Equation to one of the other Julia Map formulas. To do this, select the Fractal Equation: Julia Map 1 properties page:

General
    Mandelbrot / Julia / Newton
        Fractal Equation: Julia Map 1

Change the Based On property to one of the following Fractal Equations:

Julia Map 1
Julia Map 2
Julia Map 3
Julia Map 4
Julia Map 5
Julia Map 6
Julia Map 7
Julia Map 8
Barnsley 1
Barnsley 2
Barnsley 3

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 3

Change the transformation applied to the base fractal. To do this, 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.

Phoenix Example

The Phoenix example displays a phoenix fractal. Zoom into this fractal to see interesting designs.

Angle Relief Examples

The Angle Relief fractal examples are:

• Angle Relief x2
• Angle Relief y2

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 1

Use the Preview Julia command to explore the Julia fractals for this example.

Experiment 2

For 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
    Mandelbrot / Julia / Newton
        Fractal Equation: Exp 5

Change the Based On property to one of the following Fractal Equations:

NanoGeometry 1
NanoGeometry 4
NanoGeometry 5
Trig 2
Trig 11
Trig 17
Exp 1
Exp 2
Exp 3
Exp 4
Exp 5
Exp 6
Exp 7
Exp 8
Exp 9
Exp 10
Exp 11
Exp 12
Exp 13
Exp 14
Exp 15
Exp 16
Exp 17
Z Power 1
Z Power 2
Z Power 3

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 3

For 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
    Mandelbrot / Julia / Newton
        Fractal Equation: Trig 13

Change the Based On property to one of the following Fractal Equations:

NanoGeometry 2
NanoGeometry 3
NanoGeometry 6
NanoGeometry 7
Pinwheels
Trig 1
Trig 10
Trig 12
Trig 13
Trig 14
Airfoil 2
Airfoil 3

Experiment 4

Change 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
    Mandelbrot / Julia / Newton
        Classic
            Controllers
                Gradient Map - Angle Relief (x^2)
                    Properties

To do this for the Angle Relief y2 example, select the color controller's Properties page:

General
    Mandelbrot / Julia / Newton
        Classic
            Controllers
                Gradient Map - Angle Relief (y^2)
                    Properties

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 5

Change the transformation applied to the base fractal. To do this, 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.

Direct Color Example

The Direct Color example uses a direct color controller to color the fractal. Zoom into this fractal to see interesting designs.

Experiment 1

Use the Preview Julia command to explore the Julia fractals for this example.

Experiment 2

Change the transformation applied to the base fractal. To do this, 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.

Orbit Trap Example

The Orbit Trap example displays a Circle orbit trap. Zoom into this fractal to see interesting designs.

Experiment 1

Use the Preview Julia command to explore the Julia fractals for this example.

Experiment 2

Try out different Fractal Equations. To do this, select the Fractal Equation: Mandelbrot properties page:

General
    Mandelbrot / Julia / Newton
        Fractal Equation: Mandelbrot

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 3

Try out different Orbit Traps. To do this, select the Instructions: Circle properties page:

General
    Mandelbrot / Julia / Newton
        Orbit Trap
            Orbit Trap Map
                Instructions: Circle

Change the Based On property to one of the following Orbit Traps:

Circle
Line
Polygon
Cross
Crossed Lines
Cassinian Curve
Circular Vine
Borromean Rings
Tangent Circles
Ornament
Sectors
Triangles
Star Polygon
Flower
Rose
Super Ellipse
Sound Ornament
Shapes
Squares

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
    Mandelbrot / Julia / Newton
        Orbit Trap
            Orbit Trap Map

Change the Type property to one of the following:

Circle
Cross
Epicycloid
Fractal Gasket
Gear
Hypocycloid
Isogonal Polygon
Koch Triangle
L-System
Oscillator
Penrose Kite
Polygon
Rectangle
Rose
Shape
Sierpinski Triangle
Spirolateral
Star Polygon
Triangle

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 4

Change the transformation applied to the base fractal. To do this, 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.

Ornament Example

The Ornament example displays a fractal that looks like an ornament.

L-System Examples

The L-System fractal examples are:

• L-System Classic
• L-System Orbit Trap
• L-System Shift

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 1

Change the color of the fractal. To do this, select the color controller's Properties page:

General
    Mandelbrot / Julia / Newton
        Orbit Trap
            Controllers
                Pattern Map - Perlin Noise
                    Properties

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 2

Change the transformation applied to the base fractal. To do this, 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.

Experiment 3

For 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
    Mandelbrot / Julia / Newton
        Orbit Trap
            Symmetry Transformation: Transformation Shift - Linear Path
                Transformation Array
                    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.

Kleinian Group Example

The Kleinian Group example generates a Kleinian group orbit trap.

Experiment 1

Change the Kleinian group properties. To do this, select the Kleinian group's Properties page:

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

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 Example

The 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
    Mandelbrot / Julia / Newton
        Transformation
            Composite Function
                Properties

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 1^st 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 Example

The 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 1

Change the attractor. To do this, select the equation's Point Information properties page:

General
    Orbital / IFS / Strange Attractor
        Orbital Equation: Sierpinski N-gons (Variations)
            Point Information

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 2

Change the number of vertices in the base polygon. To do this, select the equation's General properties page:

General
    Orbital / IFS / Strange Attractor
        Orbital Equation: Sierpinski N-gons (Variations)
            General Options

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 3

Change the non-linear transformation applied to each point. To do this, select the transformation's Properties page:

General
    Orbital / IFS / Strange Attractor
        Orbital Equation: Sierpinski N-gons (Variations)
            Transformation Array
                Shape Value
                    Properties

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 4

Change the transformation applied to the base fractal. To do this, select the transformation's Properties page:

General
    Orbital / IFS / Strange Attractor
        Transformation 1
            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.

Sierpinski Cycloid Example

The Sierpinski Cycloid example injects a cycloid variation into the classic Sierpinski N-gon.

Experiment 1

Change the attractor. To do this, select the equation's Properties page:

General
    Orbital / IFS / Strange Attractor
        Orbital Equation: Sierpinski N-gons
            Properties

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 2

Change the cycloid properties. To do this, select the variation's Properties page:

General
    Orbital / IFS / Strange Attractor
        Orbit Variation
            Cycloid
                Properties

Change any of the property values.

Experiment 3

Change the transformation applied to the base fractal. To do this, select the transformation's Properties page:

General
    Orbital / IFS / Strange Attractor
        Transformation 1
            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.

Apollonian Gasket Variations Example

The 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 1

Select the properties to define the attractor. To do this, select the equation's Properties page:

General
    Orbital / IFS / Strange Attractor
        Orbital Equation: Apollonian Gasket (Variations)
            Properties

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 Example

The 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 1

Select the properties to define the attractor. To do this, select the equation's Properties page:

General
    Orbital / IFS / Strange Attractor
        Orbital Equation: Kleinian Group - Examples
            Properties

Change the Recipe, Example, Conjugate By, and/or Transform properties to define the attractor.

Rep-9 Tile Example

The 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 1

Omit 1 or more of the Rep-9 Tile affine transformations. To do this, select the equation's Properties page:

General
    Orbital / IFS / Strange Attractor
        Orbital Equation: Rep-9 Tile
            Properties

Uncheck 1 or more of the 9 checkboxes labeled Include T1 through Include T9. Unchecking 1-3 checkboxes works best.

Experiment 2

Change the base Rep-9 Tile. To do this, select the equation's Properties page:

General
    Orbital / IFS / Strange Attractor
        Orbital Equation: Rep-9 Tile
            Properties

Change the Type property to any of the types found there; i.e.,

Isosceles Trapezium
45 Degree Wedge Trapezium
60 Degree Wedge Trapezium
L-Triomino
Sphinx
Stellated Fish
Stellated Bird
Stellated Ampersand

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 3

Change the transformation applied to the base fractal before the symmetry transformation. To do this, select the transformation's Properties page:

General
    Orbital / IFS / Strange Attractor
        Transformation 1
            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.

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 4

Change the transformation applied to the base fractal after the symmetry transformation. To do this, select the transformation's Properties page:

General
    Orbital / IFS / Strange Attractor
        Transformation 2
            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.

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 Example

The 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 1

Change the attractor. To do this, select the equation's Properties page:

General
    Orbital / IFS / Strange Attractor
        Orbital Equation: Quadratic Attractors
            Properties

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 2

Change the type of symmetry transformation. To do this, select the symmetry transformation's Properties page:

General
    Orbital / IFS / Strange Attractor
        Symmetry Transformation: Plane Symmetry Groups - Square Lattice
            Properties

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
    Orbital / IFS / Strange Attractor
        Controllers
            Gradient Map - Value
                Properties

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 3

Change the transformation applied to the base fractal before the symmetry transformation. To do this, select the transformation's Properties page:

General
    Orbital / IFS / Strange Attractor
        Transformation 1
            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.

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 Example

The 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 1

Change the base attractor that defines the symmetric icon. To do this, select the equation's Properties page:

General
    Orbital / IFS / Strange Attractor
        Orbital Equation: Symmetric Icon - Standard Formula
            Properties

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 2

Change the triangle metric point used to define the point into which we accumulate statistics. To do this, select the Triangle Metric properties page:

General
    Triangle Metric

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 3

Change the transformation applied to the base fractal before the symmetry transformation. To do this, select the transformation's Properties page:

General
    Orbital / IFS / Strange Attractor
        Transformation 1
            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.

Experiment 4

Change the color blend used to color the image. To do this, select the color controller's Properties page:

General
    Orbital / IFS / Strange Attractor
        Controllers
            Direct Color Map - RGB Blend
                Properties

Change the Blend Type to any of the supported color blends.