Borromean Rings Examples
Mandelbrot's Pretzel |
Gordian Knot |
Bound Together |
Chaotic Vines II |
Decisions |
Ceramic Beadwork |
Points of Attraction |
Hypnotic Convergence |
Chain of Events |
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:
- Circle
- Cross
- Epicycloid
- Fractal Gasket
- Gear
- Hypocycloid
- Isogonal Polygon
- Koch Triangle
- L-System
- Oscillator
- Penrose Kite
- Polygon
- Rectangle
- Rose
- Shape
- Sierpinski Triangle
- Spirolateral
- Star Polygon
- Triangle
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.