Iterated Function
Systems
Larry Riddle
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McWorter's Pentigree









Construction
Animation
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 Starting from the origin, draw a line segment of length r at an angle of 36°. Turn left by 72° and draw another line segment of length r. Next turn right by 144° and draw a third line segment of length r. The remaining three line segments, each of length r, are formed by making a right turn of 72° followed by two left turns of 72°. The end of the last line segment will be at the point (1,0).

If A0 is the line segment from the origin to the point (1,0), then the six red line segments in the figure give the first iteration for this IFS.

These six transformations are given by rotations of 36°, 108°, -36°, -108°, -36°, and 36° respectively. Thus the endpoints of the fourth segment and the last segment lie on the x-axis. The value of r must be chosen so that the endpoint of the last segment is at the point (1,0). From the figure we see that we must have

Sin 54°
Details

Iterated Function System It is now possible to determine how each of the six rotated segments must be translated. For example, the second segment must be translated to (r cos36°, r sin36°) after a rotation by 108° and a scaling by a factor of r=0.381966. Similar calculations with the other segments led to the following IFS:

Angle 36
Axiom F
Generator F --> +F++F- - - -F- -F++F++F-



First three iterations of the L-system

Similarity Dimension We have a hyperbolic IFS with each map being a similitude of ratio r < 1. Therefore the similarity dimension, d, of the unique invariant set of the IFS is the solution to

Special Properties Five copies of the pentigree fit together to form a set with five-fold rotational symmetry as illustrated in the following figure.

The pentigree was discovered by accident by William A. McWorter, Jr. during his hunts for "dragons", a category of fractals and space-filling curves that McWorter described as "organism[s] of cells arranged according to a genetic code." The most complete analysis of the pentigree can be found in the text by Gerald Edgar.

 
  1. Edgar, Gerald A. Measre, Topology, and Fractal Geometry, Springer-Verlag, 1990.
  2. McWorter Jr., William A. and Jane Morrill Tazelaar. "Creating Fractals," Byte, August 1987, 123-132.
Home Sierpinski Gasket Sierpinski Carpet Sierpinski Pentagon Heighway Dragon
McWorter Pentigree Pentadentrite Koch Curve Koch Snowflake Levy Dragon