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

The Newton fractal examples are:

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

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 Angle Relief adds a 3D relief based on the smoothed angle at each point. Newton Classic colors each point on the associated Julia fractal based on the root to which the point converges. 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. Newton Trap Relief adds a 3D relief based on a shape 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 Orbit Trap example, you can change the transformation applied to the orbit point prior to applying the trap. To do this, select the transformation's Properties page:

General
    Mandelbrot / Julia / Newton
        Orbit Trap
            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 6

For the Newton Shift example, you can change the transformation applied to the orbit point within the symmetry transformation. To do this, select the transformation's Properties page:

General
    Mandelbrot / Julia / Newton
        Orbit Trap
            Symmetry Transformation: Transformation Shift - Rotate
                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 7

For the Newton Shift example, you can change the properties that control the symmetry transformation. To do this, select the transformation's Properties page:

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

Change any of these properties to define the symmetry transformation.

Experiment 8

For the Newton Shift example, you can try a different symmetry transformation. To do this, select the Symmetry Transformation: Transformation Shift - Rotate page:

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

Change the Based On property to one of the following symmetry transformations:

• Transformation Shift - Rotate
• Transformation Shift - Linear Path
• Transformation Shift - Circular Path
• Transformation Shift - Reflected Sectors
• Transformation Shift - Mobius Interpolation

After selecting one of the symmetry transformations, select the transformation's Properties page and change the properties found there.