Mandelbrot Fractals - Part 2

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In Part 2 of the tutorial, we are going to learn how to transform a fractal to create wildly different images from the same Fractal Equation.

Execute the Reset to Defaults command on the File menu of the Fractal Window. The Reset to Defaults command resets the Fractal Equation to the built-in equation Mandelbrot. All the images in this part of the tutorial are different views of this Fractal Equation.

Select the Transformation properties page:

General
    Mandelbrot / Julia / Newton
        Transformation

This page allows you to maintain a list of Transformations used to transform the pixel prior to starting each orbit. The transformations are applied in series to the pixel value; i.e., the point resulting from each transformation is passed to the next transformation in the list. As you will see, this can have a dramatic effect on the resulting image. By default, there is a single transformation, Identity, and the Identity transformation does not alter the input point.

Select the Identity properties page:

General
    Mandelbrot / Julia / Newton
        Transformation
            Identity

This page is a Program Editor for a Transformation.

We want to use the Composite Function transformation so we set the Based On property to that.

Read the comments in the program's Instructions at the bottom of the page and then select the program's Properties page just below the transformation in the hierarchy:

General
    Mandelbrot / Julia / Newton
        Transformation
            Composite Function
                Properties

This displays the default properties for the Composite Function transformation. The Composite Function transformation is defined by a composite function composed of 2 functions. Each function is defined via a base function and a conjugating map. For now, we will only set the F(z) and/or G(z) properties and leave the remaining properties set to their default value.

Try setting F(z) and/or G(z) to different functions and then execute the Display Fractal command on the Tools menu of the Fractal Window.

A few examples are shown below:

F(z): Airfoil

F(z): Bipole

F(z): Bipole, G(z): Tan

G(z) is set to Ident where not specified in the above examples.

Try additional functions and set some of the other properties on the page. Move up to the Composite Function transformation entry in the hierarchy and change the Based On property to try out different transformations altogether. For each transformation, select its properties page (usually called Properties) and play with the various settings that control the transformation.

When you are ready to move on, set the transformation to Stereographic Projection. This transformation is used to map the complex plane onto a sphere.

Select the program's Properties page just below the transformation in the hierarchy:

General
    Mandelbrot / Julia / Newton
        Transformation
            Stereographic Projection
                Properties

This displays the default properties for the Stereographic Projection transformation:

Change the Radius to 1.8.

Next, select the Transformation properties page:

General
    Mandelbrot / Julia / Newton
        Transformation

Add a new Identity transformation to the list by clicking the New toolbar button (the left-most button above the list):

Select the Identity properties page (in the page hierarchy not the list):

General
    Mandelbrot / Julia / Newton
        Transformation
            Identity

Set the Based On property to Kaleidoscope - Triangles.

This transformation simulates a kaleidoscope by reflecting a triangular area of the complex plane about the sides of the triangle, over and over again, filling the complex plane.

Select the program's Properties page just below the transformation in the hierarchy:

General
    Mandelbrot / Julia / Newton
        Transformation
            Kaleidoscope - Triangles
                Properties

This displays the default properties for the Kaleidoscope - Triangles transformation.

The properties define the location of a triangle on the complex plane. The area of the fractal inside the base 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.

Change the Central Vertex to -1, the Angle to 180, and the Scale to 0.5.

Execute the Display Fractal command on the Tools menu of the Fractal Window to generate the fractal image.

Mandelbrot Fractal (Stereographic Projection)

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