Borromean Rings

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Borromean Rings Examples

Fractal: Borromean Rings Orbit Trap

Mandelbrot's Pretzel
Borromean Rings 01

Fractal: Borromean Rings Orbit Trap

Gordian Knot
Borromean Rings 02

Fractal: Borromean Rings Orbit Trap

Bound Together
Borromean Rings 03

Fractal: Borromean Rings Orbit Trap

Chaotic Vines II
Borromean Rings 04

Fractal: Borromean Rings Orbit Trap

Decisions
Borromean Rings 05

Fractal: Borromean Rings Orbit Trap

Ceramic Beadwork
Borromean Rings 06

Fractal: Borromean Rings Orbit Trap

Points of Attraction
Borromean Rings 07

Fractal: Borromean Rings Orbit Trap

Hypnotic Convergence
Borromean Rings 08

Fractal: Borromean Rings Orbit Trap

Chain of Events
Borromean Rings 09

The Borromean Rings examples apply a Borromean Rings Orbit Trap to fractals based on various Fractal Equations.

Note the following:

Example Fractal Type Fractal Equation
Borromean Rings 01 Mandelbrot Fractal Mandelbrot
Borromean Rings 02 Julia Fractal Mandelbrot
Borromean Rings 03 Julia Fractal Mandelbrot
Borromean Rings 04 Mandelbrot Fractal Mandelbrot
Borromean Rings 05 Julia Fractal Phoenix
Borromean Rings 06 Julia Fractal Newton 7
Borromean Rings 07 Julia Fractal Newton 7
Borromean Rings 08 Julia Fractal Newton 7
Borromean Rings 09 Julia Fractal Newton 7

In the remaining sections, when I refer to the equation, I will use Newton 7, 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

Note: This section applies only to the examples based on the equation Newton 7.

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

Select the equation's properties page:

General
    Mandelbrot / Julia / Newton
        Fractal Equation: Newton 7
            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 7

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.

For the examples based on the Mandelbrot equation, select the Fractal Equation:

General
    Mandelbrot / Julia / Newton
        Fractal Equation: Mandelbrot

Change the equation's Based On property to any of the other equations.

For the examples based on the Newton 7 equation, select the Fractal Equation:

General
    Mandelbrot / Julia / Newton
        Fractal Equation: Newton 7

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: Borromean Rings:

General
    Mandelbrot / Julia / Newton
        Orbit Trap
            Orbit Trap Map
                Instructions: Borromean Rings

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.

Note the following:

  • Borromean Rings 03 applies the Composite Function transformation.
  • Borromean Rings 04 applies the Composite Function transformation.
  • All of the reset of the examples apply the Identity transformation.

In the following discussion, when I refer to the transformation, I will use Identity, but you should use the transformation for the example you are working with.

To apply a transformation to the initial orbit point, 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 apply a transformation to each orbit point prior to passing it to the orbit trap, select the Identity transformation:

General
    Mandelbrot / Julia / Newton
        Orbit Trap
            Transformation 1
                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.

 

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