Quadratic Attractor

The Quadratic Attractor example generates a wallpaper group tiling by passing the results of the attractor through a symmetry transformation that implements a plane symmetry group square lattice.

Experiment 1

Change the attractor. To do this, select the equation's Properties page:

General
    Orbital / IFS / Strange Attractor
        Orbital Equation: Quadratic Attractors
            Properties

Change the Attractor property to select the 1 of the attractors. Do not change the coefficients C01 through C12 since it is unlikely your change will produce a fractal. Most of the attractors in this list were found using the program Quadratic Attractors (Search) which generates sets of coefficients that meet a set of user-defined criteria. Try that program out later when you have time.

Change the Rotate By property to rotate the attractor image. Since the rotation is processed prior to applying the symmetry transformation, this results in significant changes to the resulting image.

Experiment 2

Change the type of symmetry transformation. To do this, select the symmetry transformation's Properties page:

General
    Orbital / IFS / Strange Attractor
        Symmetry Transformation: Plane Symmetry Groups - Square Lattice
            Properties

Change the Symmetry property to 1 of the 12 square lattice based symmetry groups.

The Subtype property does not change the shape of the resulting image but can be used to produce 2-color symmetry patterns. In order to view these patterns, you will need to change the data mapped to the image in the color controller. To do this, select the controller's Properties page:

General
    Orbital / IFS / Strange Attractor
        Controllers
            Gradient Map - Value
                Properties

Change the Value property from Attractor Index to Symmetry Index.

Now you can return to the transformation's Properties page and change the Subtype property to produce 2-color symmetry patterns. Each Symmetry property setting has a different set of Subtype property settings for the 2-color symmetry patterns associated with the Symmetry setting. Note that some attractors look better than others with respect to 2-color symmetry pattern coloring.

Experiment 3

Try out the 5 plane symmetry group hexagonal lattice patterns. To do this, select the Symmetry Transformation: Plane Symmetry Groups - Square Lattice page:

General
    Orbital / IFS / Strange Attractor
        Symmetry Transformation: Plane Symmetry Groups - Square Lattice

Change the Based On property to Symmetry Transformation: Plane Symmetry Groups - Hexagonal Lattice.

Next, select the transformation's Properties page:

General
    Orbital / IFS / Strange Attractor
        Symmetry Transformation: Plane Symmetry Groups - Hexagonal Lattice
            Properties

Set Rows to 2 and Columns to 2 and change the Symmetry property to 1 of the 5 hexagonal lattice based symmetry groups.

Experiment 4

Change the transformation applied to the base fractal before the symmetry transformation. To do this, select the transformation's Properties page:

General
    Orbital / IFS / Strange Attractor
        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.

Since this transformation is applied before the symmetry transformation (i.e., the output from this transformation is passed to the symmetry transformation), the resulting image will maintain the symmetry defined by the symmetry transformation.