Chained Marble

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Chained Marble Examples

Fractal: Chained Marble

Chained Marble I
Chained Marble 01

Fractal: Chained Marble

Chained Marble II
Chained Marble 02

Fractal: Chained Marble

Chained Marble III
Chained Marble 03

Fractal: Chained Marble

Chained Marble IV
Chained Marble 04

Fractal: Chained Marble

Angel of Light
Chained Marble 10

Fractal: Chained Marble

Sun and Stars
Chained Marble 11

Fractal: Chained Marble

Dot to Dot
Chained Marble 12

Fractal: Chained Marble

Solid Foundation
Chained Marble 13

Fractal: Chained Marble

Bottleneck
Chained Marble 14

Fractal: Chained Marble

Crisscross
Chained Marble 15

Fractal: Chained Marble

Twisted Reflection
Chained Marble 16

Fractal: Chained Marble

Joining Together
Chained Marble 17

Fractal: Chained Marble

Stone Totem
Chained Marble 18

Fractal: Chained Marble

Expanding Universe
Chained Marble 19

Fractal: Chained Marble

Fountain of Life
Chained Marble 20

Fractal: Chained Marble

Binocular Disparity
Chained Marble 21

   

The Chained Marble examples apply a Circle Orbit Trap to a Newton Fractal based on various Fractal Equations. The coloring is achieved by mapping the trap's dwell (i.e., orbit index value) to 1 of 4 gradients and then mapping the trapped point's magnitude into a gradient to select the color. Then the color is blended with a Perlin noise based pattern to give it a stone-like appearance.

Note the following:

Example Fractal Type Fractal Equation
Chained Marble 01 Julia Fractal Newton 2
Chained Marble 02 Julia Fractal Newton 7
Chained Marble 03 Mandelbrot Fractal Newton 1
Chained Marble 04 Julia Fractal Newton 2
Chained Marble 10 Julia Fractal Newton 2
Chained Marble 11 Julia Fractal Newton 2
Chained Marble 12 Julia Fractal Newton 2
Chained Marble 13 Julia Fractal Newton 2
Chained Marble 14 Julia Fractal Newton 2
Chained Marble 15 Julia Fractal Newton 2
Chained Marble 16 Julia Fractal Newton 2
Chained Marble 17 Julia Fractal Newton 2
Chained Marble 18 Julia Fractal Newton 2
Chained Marble 19 Julia Fractal Newton 2
Chained Marble 20 Julia Fractal Newton 2
Chained Marble 21 Julia Fractal Newton 2

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 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.

You can also try out the different optimized orbit traps. To do this, 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.

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 add additional transformations, select Transformation:

General
    Mandelbrot / Julia / Newton
        Transformation

Click the New toolbar button to add a new Identity transformation to the bottom of the list, and then click the Move Up toolbar button to move the new transformation to the desired position in the list. Normally, I move the new transformation to the top of the list, but it can be placed anywhere. See Transformation Array for details.

Then select the Identity transformation:

General
    Mandelbrot / Julia / Newton
        Transformation
            Identity

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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