Agnes Scott College
Larry Riddle, Agnes Scott College

Pentadentrite


Description

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).

construction

The six red line segments in the figure give the first iteration 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 \(r = \sqrt {\frac{{6 - \sqrt 5 }}{{31}}} = 0.34845\) and A = 11.81858573°.

Iterated
Function
System

 
 


IFS
Animation
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 scaled by r, rotated through an angle of A + 72°, then translated to (r sin A, r sin A). Similar calculations with the other segments led to the following IFS [Details]:
 
\({f_1}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {0.341} & { - 0.071} \\ {0.071} & {0.341} \\ \end{array}} \right]{\bf{x}}\)
 
   scale by r, rotate by A = 11.819°
 
\({f_2}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} { 0.038} & { - 0.346} \\ {0.346} & { 0.038} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {0.341} \\ {0.071} \\ \end{array}} \right]\)
 
   scale by r, rotate by A+72° = 83.819°
 
\({f_3}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {0.341} & {-0.071} \\ { 0.071} & {0.341} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {0.379} \\ {0.418} \\ \end{array}} \right]\)
 
   scale by r, rotate by A = 11.819°
 
\({f_4}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} { - 0.234} & {0.258} \\ { - 0.258} & { - 0.234} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {0.720} \\ {0.489} \\ \end{array}} \right]\)
 
   scale by r, rotate by A−144° = −132.181°
 
\({f_5}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {0.173} & {0.302} \\ { - 0.302} & {0.173} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {0.486} \\ {0.231} \\ \end{array}} \right]\)
 
   scale by r, rotate by A−72° = −60.181
 
\({f_6}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {0.341} & { - 0.071} \\ {0.071} & {0.341} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {0.659} \\ { - 0.071} \\ \end{array}} \right]\)
 
   scale by r, rotate by A = 11.819°
 

IFS image
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L-System

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


First three iterations of the L-system

Similarity
Dimension

The pentadentrite is self-similar with 6 non-overlapping copies of itself, each scaled by the factor r < 1. Therefore the similarity dimension, d, of the attractor of the IFS is the solution to

\[\sum\limits_{k = 1}^6 {{r^d}} = 1 \quad \Rightarrow \quad d = \frac{{\log (1/6)}}{{\log (r )}} = 1.6995\]

Special
Properties

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

Five copies
[Enlarge]

 

Reference

  1. Edgar, Gerald A. Measure, Topology, and Fractal Geometry, Springer-Verlag, 1990.