Iterated Function
Systems
Larry Riddle
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Pentadentrite

The pentadentrite is a variation of the McWorter's pentigree. Starting from the origin, draw a line segment of length r at an angle A. Turn left by 72° and draw another line segment of length r. Next turn right by 72° 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 144° followed by two left turns of 72°. The end of the last line segment will be at the point (1,0).


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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 A1 for this construction. The scaling factor r and the angle A must be chosen so that the endpoint of the last segment is at the point (1,0).

The following calculations show that and A = 11.8186°.

It is now possible to determine how each of the six rotated segments must be translated. For example, the first segment must be scaled by r and rotated through the angle A. The second segment is rotated through an angle of A + 72°, scaled by r, then translated to (r sin A°, r sin A°). Similar calculations with the other segments led to the following IFS:

Angle 72
Axiom F
Generator F --> F+F-F- -F+F+F
(this will be rotated from the IFS constructed curve)


First three iterations of the L-system

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

This is called a pentadentrite because five copies of the limit set fit perfectly together to form the following shape:

 
  1. Edgar, Gerald A. Measure, Topology, and Fractal Geometry, Springer-Verlag, 1990.
Home Sierpinski Gasket Sierpinski Carpet Sierpinski Pentagon Heighway Dragon
McWorter Pentigree Pentadentrite Koch Curve Koch Snowflake Levy Dragon