Kaleidoscope 02	Kaleidoscope 07	Kaleidoscope 09

The Kaleidoscope fractal example displays a Mandelbrot fractal based on a Julia Map fractal equation.

To produce a kaleidoscope image using this example you should do the following:

• Display the Kaleidoscope fractal example.
• Execute the Preview Julia command and then click on the Mandelbrot until you get a Julia you like displayed in the preview window, then click on the Preview Window to open a full-sized version of the preview.
• Open the Properties Window for the Julia fractal.
• Activate the kaleidoscope transformation on the Julia fractal and display the results.

To activate the kaleidoscope transformation on the Julia fractal, select the transformation's Properties page:

General
    Mandelbrot / Julia / Newton
        Transformation
            Kaleidoscope - Triangles
                Properties

Check the Enabled property to enable the transformation. You can change the other properties on this page for more variations. The properties define a triangle on the complex plane. The area of the fractal inside the triangle is replicated over the entire complex plane by first reflecting the triangle about its sides, and then reflecting each of the new triangles about their sides, and so on. It is best to examine the original image for an interesting triangular region and then set the transformation's properties to position the triangle over that region.

How you set the properties to position the triangle depends on the Type property. Type can be set to one of the following values:

• 60, 60, 60
• 30, 60, 90
• 45, 45, 90

The Type property names the triangle based on the triangle's angles. The triangles 60, 60, 60 and 30, 60, 90 are contained within a hexagon with its center at the Central Vertex and 2 of its vertices on the X axis. The height of the hexagon is given by Grid Size. The hexagon is divided into 6 60, 60, 60 triangles or 12 30, 60, 90 triangles, respectively, and the triangle in the first quadrant adjacent to the X axis is rotated by Angle degrees. The fractal image under this triangle is used to generate the kaleidoscope image.

The triangle 45, 45, 90 is contained within a square with its center at the Central Vertex. The height of the square is given by Grid Size. The square is divided into 8 45, 45, 90 triangles, and the triangle in the first quadrant adjacent to the X axis is rotated by Angle degrees. The fractal image under this triangle is used to generate the kaleidoscope image.

You can also try out the other kaleidoscope transformations. To do this, select the Kaleidoscope - Triangles page:

General
    Mandelbrot / Julia / Newton
        Transformation
            Kaleidoscope - Triangles

Set the Based On property to Kaleidoscope - Slices or Kaleidoscope - Squares and then select the transformation's Properties page and change the properties as required. These transformations are similar to the Kaleidoscope - Triangles transformation except that they use a pie shaped sector and a square, respectively, to tile the plane rather than a triangle.

To change the color of the fractal image, select the Gradient Map - Value properties page:

General
    Mandelbrot / Julia / Newton
        Classic
            Controllers
                Gradient Map - Value
                    Properties

Set the Color Scheme to one of the following:

• Custom - Red and Gold
• Custom - Green and Gold
• Custom - Blue and Gold

Note the associated gradient does not reflect the colors given in the name! This discrepancy is due to the color blend we apply later in the processing.

Set Power, Factor, and Offset to create different effects. By changing these properties you can produce many different images from a single Julia fractal.

Experiment 1

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 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.