# 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 and Steven Dutch's Rep-Tiles 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:

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-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:

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:

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.