Newton's Lattice

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Newton's Lattice Examples

Fractal: Newton's Lattice

Newton's Lattice I
Newton's Lattice 01

Fractal: Newton's Lattice

Newton's Lattice II
Newton's Lattice 02

Fractal: Newton's Lattice

Newton's Lattice III
Newton's Lattice 03

Fractal: Newton's Lattice

Newton's Lattice IV
Newton's Lattice 04

The Newton's Lattice examples apply an Isogonal Polygon Orbit Trap to a Newton Fractal. The coloring uses a Perlin noise based pattern to give it a brushed metal quality.

Note the following:

Example Fractal Type Fractal Equation
Newton's Lattice 01 Julia Fractal Newton 2
Newton's Lattice 02 Julia Fractal Newton 7
Newton's Lattice 03 Julia Fractal Newton 7
Newton's Lattice 04 Mandelbrot Fractal Carlson - Newton 3

In the remaining sections, when I refer to the equation, I will use Newton 2, but you should use the equation for the example you are working with.

Zoom In/Out

Zoom In or Zoom Out to examine different parts of the fractal.

Execute the Home command on the View menu of the Fractal Window to reset the fractal to the default position/magnification, and then Zoom In to other areas.

Remember that as you Zoom In, you may need to increase the Max Dwell property found in the Orbit Trap Orbit Generation section of the General page.

Play with the Fractal Equation's Properties

You can change the equation's properties for more variations.

Select the equation's properties page:

General
    Mandelbrot / Julia / Newton
        Fractal Equation: Newton 2
            Properties

Play with the equation's properties.

Explore the Julia Fractals

For the Mandelbrot Fractal examples, you can use the Preview Julia command to explore the Mandelbrot's many different Julia Fractals. This is a common technique that can be used to generate lots of different Julia fractals from a single Mandelbrot image.

Execute the Home command on the View menu of the Fractal Window to reset the Mandelbrot fractal to the default position/magnification, and use the Preview Julia command to explore the Mandelbrot's many different Julia Fractals. See Working with Julia Fractals for details.

Change the Julia Constant

For the Julia Fractal examples, you can generate other Julia Fractals based on the same equation.

Select the Fractal Equation:

General
    Mandelbrot / Julia / Newton
        Fractal Equation: Newton 2

Uncheck the Julia checkbox, execute the Home command on the View menu of the Fractal Window to reset the Mandelbrot fractal to the default position/magnification, and use the Preview Julia command to explore the Mandelbrot's many different Julia Fractals. See Working with Julia Fractals for details.

Alternatively, you can change the Julia Constant property on the Fractal Equation page, and then click the Preview Fractal toolbar button on the Properties Window to generate a preview of your change in the Preview Window.

Change the Fractal Equation

You can change the Fractal Equation used to generate the fractal.

Select the Fractal Equation:

General
    Mandelbrot / Julia / Newton
        Fractal Equation: Newton 2

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

Then execute the Home command on the View menu of the Fractal Window to reset the Mandelbrot fractal to the default position/magnification, and use the Preview Julia command to explore the Mandelbrot's many different Julia Fractals. See Preview Julia Support for details.

Remember to navigate to the properties page for the equation (found under the equation in the page hierarchy) and play with the different properties found there. Many of the equations support properties that can be used to generate lots of different variations.

Change the Orbit Trap

You can try out different Orbit Traps.

Select Orbit Trap Map:

General
    Mandelbrot / Julia / Newton
        Orbit Trap
            Orbit Trap Map

Change the Type property to one of the following:

Each of these orbit traps have properties (on the page found under the Orbit Trap Map page) to manipulate the trap and thereby change the resulting fractal.

You can also try out the other built-in Orbit Trap programs.

Set Type to Instructions and then select Instructions: Circle:

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
  • General Polygon
  • Shape
  • Sphere
  • Patterned Sphere
  • Solid Polygon
  • Solid General Polygon
  • Patterned Polygon
  • Patterned General Polygon
  • Epicycloid Polygon
  • Hypocycloid Polygon
  • Cross
  • Crossed Lines
  • Image
  • Cassinian Curve
  • Limacon
  • Lemniscate
  • Folium
  • Cycloid of Ceva
  • Circular Vine
  • Celtic Knot
  • Borromean Rings
  • Tangent Circles
  • Steiner Chain
  • Rep-4 Tile
  • Rep-4 Tile Patterned Polygon
  • Rep-4 Tile Patterned General Polygon
  • Rep-9 Tile
  • Rep-9 Tile Patterned Polygon
  • Rep-9 Tile Patterned General Polygon
  • Polygon Motif
  • Star of David
  • String of Beads
  • Wheel
  • Ornament
  • Sectors
  • Triangles
  • Star Polygon
  • Faceted Polygon
  • Composite Shape
  • Spiral
  • Daisy
  • Swirl
  • Flower
  • Rose
  • Super Ellipse
  • 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 or are too complex to display in this context.

Change the Transformation

You can apply a transformation to the initial orbit point, or to each orbit point prior to passing it to the orbit trap.

Execute the Home command on the View menu of the Fractal Window to reset the fractal to the default position/magnification before you adjust the transformation. Then change the transformation and Zoom In to interesting areas of the transformed image.

To change the transformation applied to the initial orbit point, 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.

You can also use a different transformation altogether. Select the Composite Function page, and change the Based On property to select a transformation and then open the transformation's properties page (found under the transformation in the page hierarchy), and play with the transformation's properties. See Transformation Support for details.

To change the transformation applied to each orbit point prior to passing it to the orbit trap, 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.

You can also use a different transformation altogether. Select the Composite Function page, and change the Based On property to select a transformation and then open the transformation's properties page (found under the transformation in the page hierarchy), and play with the transformation's properties. See Transformation Support for details.

Play with Color

To play with color, select the color controller's properties page:

General
    Mandelbrot / Julia / Newton
        Orbit Trap
            Controllers
                Gradient Map - Dwell/Index
                    Properties

Change the Power, Factor, and Offset properties in the Value Map section to control how colors are mapped onto the image.

 

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