Rep-9 Tile

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Rep-9 Tile Examples

Ferris Wheel Rep-9 Tile 01	Optical Illusion Rep-9 Tile 02	Planes in Perspective Rep-9 Tile 03
Cubic Snowflake Rep-9 Tile 04	Exploded Diagram Rep-9 Tile 05	Navajo Tapestry Rep-9 Tile 06
Chaotic Floorplan Rep-9 Tile 07	Persian Tile Rep-9 Tile 08	King's Cross Rep-9 Tile 09
Rubies and Gold Rep-9 Tile 10	Fractal Embroidery Rep-9 Tile 11	Royal Cross Rep-9 Tile 12
Byzantine Tile Rep-9 Tile 13	Square Quilt Rep-9 Tile 14	Shadowbox Rep-9 Tile 15
Spinning Star Rep-9 Tile 16	Borg Cube Rep-9 Tile 17	Starflake Rep-9 Tile 18
Hex Quilt Rep-9 Tile 19	Hex Tile Rep-9 Tile 20	Mandala I Rep-9 Tile 21
Mandala II Rep-9 Tile 22	Mandala III Rep-9 Tile 23	Star Quilt I Rep-9 Tile 24
Mayan Sun I Rep-9 Tile 25	Mayan Sun II Rep-9 Tile 26	Autumn Sun I Rep-9 Tile 27
Mandala IV Rep-9 Tile 28	Star Quilt II Rep-9 Tile 29	Autumn Sun II Rep-9 Tile 30
Gear Quilt Rep-9 Tile 31	Flower Quilt Rep-9 Tile 32

The Rep-9 Tile examples display a Rep-Tile based on a set of Affine Transformations.

The Rep-9 Tile examples are based on Rep-N Tile attractors. The term Rep-Tile (replicating figures on the plane) was coined by mathematician Solomon W. Golomb in 1962. See Rep-Tile for a brief description and Stewart R. Hinsley's site for additional details on Rep-Tiles. I learned about Rep-Tiles from Rep-Tiles: Replicating Figures on the Plane, chapter 19 of the book The Unexpected Hanging and Other Mathematical Diversions by Martin Gardner.

By omitting 1 or more of the Affine Transformations used to define the Rep-Tile, you can create holes in the tile, which makes the resulting fractal more interesting. Finally, the Rep-Tile is rotated about the origin to generate the image. This is accomplished using the Rosette Symmetry Group symmetry transformation.

All of the examples are based on the Orbital Equation Rep-9 Tile with the exception of the following, which are based on Rep-9 Triangle:

Rep-9 Tile 16
Rep-9 Tile 17
Rep-9 Tile 18
Rep-9 Tile 19
Rep-9 Tile 20
Rep-9 Tile 22
Rep-9 Tile 27
Rep-9 Tile 28
Rep-9 Tile 29
Rep-9 Tile 30

In the remaining sections, when I refer to the equation, I will use Rep-9 Tile, but you should use the equation for the example you are working with.

Play with the Orbital Equation's Properties

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

Select the equation's properties page:

General
    Orbital / IFS / Strange Attractor
        Orbital Equation: Rep-9 Tile
            Properties

The properties in the first section control the shape/position of the base tile. The properties in the Transformations section let you select which of the 9 Affine Transformations that define the tile are included in the attractor. Excluding a transformation creates a hole in the resulting tile. Most of the examples exclude 2 of the transformations but that is not required. Excluding 1, 2, or 3 transformations works best.

Play with the equation's properties.

Note:

The properties in the first section, control the shape/position of the Rep-Tile and you may need to adjust the Order property on the Rosette Symmetry Group symmetry transformation to match. The value you should use for Order is discussed in the comment section of the Orbital Equation so you should select the equation's page and read the comments at the top of the program. Then select the properties for the symmetry transformation:

General
    Orbital / IFS / Strange Attractor
        Symmetry Transformation: Rosette Symmetry Group
            Properties

Change the Order property to match the Rep-9 Tile you are using.

For example, consider the following table:

Type	Order	Dihedral
Isosceles Trapezium	3	checked
45 Degree Wedge Trapezium	4	checked
60 Degree Wedge Trapezium	3	checked
L-Triomino	2	checked
Sphinx	3	checked
Stellated Fish	3	checked
Stellated Bird	3	checked
Stellated Ampersand	3	checked

If the Rep-9 Tile has Rotation set to None, Reflection is unchecked, and Type is set as given in the table above, the Rosette Symmetry Group symmetry transformation should use the Order and Dihedral shown in the table. If Rotation is set to a value other than None, and/or Reflection is checked, you should set Order to 2 and check Dihedral.

Change the Orbital Equation

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

Select the Orbital Equation:

General
Orbital / IFS / Strange Attractor
Orbital Equation: Rep-9 Tile

Change the Based On property to one of the following Orbital Equations:

Rep-4 Tile
Rep-4 Equilateral Triangle
Rep-4 Triangle
Rep-4 Parallelogram
Rep-9 Tile
Rep-9 Equilateral Triangle
Rep-9 Triangle
Rep-9 Parallelogram

Select the properties page for the equation (found under the equation in the page hierarchy) and play with the different properties found there.

The properties in the first section control the shape/position of the base tile. The properties in the Transformations section let you select which of the Affine Transformations then define the tile are included in the attractor. Excluding a transformation creates a hole in the resulting tile.

Note:

General
    Orbital / IFS / Strange Attractor
        Symmetry Transformation: Rosette Symmetry Group
            Properties

Change the Order property to match the Rep-Tile you are using.

Change Transformation 1

You can change the Shape Value transformation applied to the orbit point prior to applying the symmetry transformation.

Note: The examples Rep-9 Tile 01 - 20 do not apply a transformation to the fractal, so before you can adjust the shape you need to add the Shape Value transformation. To do that, select the Identity transformation:

General
    Orbital / IFS / Strange Attractor
        Transformation 1
            Identity

Change the Based On property to Shape Value.

To change the Shape Value, select the transformation's properties page:

General
    Orbital / IFS / Strange Attractor
        Transformation 1
            Shape Value
                Properties

Change the Shape property.

Change Transformation 2

You can apply a transformation to the orbit point after applying the symmetry transformation.

To apply a transformation to the orbit point after applying the symmetry transformation, select the Identity transformation's page:

General
    Orbital / IFS / Strange Attractor
        Transformation 2
            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.

Play with Color

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

General
    Orbital / IFS / Strange Attractor
        Controllers
            Color Map - Index
                Properties

Change the Count, Colors, and Offset properties. Count should match the number of transformations you included (checked) on the Orbital Equation properties page described above. Each of the selected transformations is assigned a color using the Colors property. Click on the color boxes to set the colors. See Color Selection Dialog for details on setting the colors.