McWorter's Pentigree

Description

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).

The six red line segments in the figure give the basic motif (first iteration) for this IFS. Each subsequent iteration is done by replacing each line segment with a scaled version of the basic motif.

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 above we see that we must have

cos 36°
details

\[\begin{array}{l} r + 2r\cos {36^ \circ } = 1 \\ \Rightarrow r = \dfrac{1}{{1 + 2\cos {{36}^ \circ }}} = \dfrac{1}{{1 + 2\left( {\frac{{1 + \sqrt 5 }}{4}} \right)}} = \dfrac{{3 - \sqrt 5 }}{2} = 0.381966 \\ \end{array}\]

Iterated
Function
System

IFS
Animation

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 $( \cos36^\circ, r \sin36^\circ)$ after a rotation by 108° and a scaling by a factor of $r$ = 0.381966. Similar calculations with the other segments yields the following iterated function system [IFS Details]:

$ f_1(\mathbf{x})= \left[ \begin{array}{cc} 0.309 & -0.255\\ 0.255 & 0.309 \end{array} \right]\mathbf{x} $

scale by $r$, rotate by 36°

$ f_2(\mathbf{x})= \left[ \begin{array}{cc} -0.118 & -0.363\\ 0.363 & -0.118 \end{array} \right]\mathbf{x} + \left[ \begin{array}{c} 0.309\\ 0.225 \end{array} \right] $

scale by $r$, rotate by 108°

$ f_3(\mathbf{x})= \left[ \begin{array}{cc} 0.309 & 0.225\\ -0.225 & 0.309 \end{array} \right]\mathbf{x} + \left[ \begin{array}{c} 0.191\\ 0.588 \end{array} \right] $

scale by $r$, rotate by −36°

$ f_4(\mathbf{x})= \left[ \begin{array}{cc} -0.118 & 0.363\\ -0.363 & -0.118 \end{array} \right]\mathbf{x} + \left[ \begin{array}{c} 0.500\\ 0.363 \end{array} \right] $

scale by $r$, rotate by −108°

$ f_5(\mathbf{x})= \left[ \begin{array}{cc} 0.309 & 0.225\\ -0.225 & 0.309 \end{array} \right]\mathbf{x} + \left[ \begin{array}{c} 0.382\\ 0 \end{array} \right] $

scale by $r$, rotate by −36°

$ f_6(\mathbf{x})= \left[ \begin{array}{cc} 0.309 & -0.225\\ 0.225 & 0.309 \end{array} \right]\mathbf{x} + \left[ \begin{array}{c} 0.691\\ -0.225 \end{array} \right] $

scale by $r$, rotate by 36°

[Enlarge]

L-System

L-system
Animation

Angle 36
Axiom F
F —> +F++F−−−−F−−F++F++F−

Four iterations

Similarity
Dimension

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

\[\sum\limits_{k = 1}^6 {{r^d}} = 1 \quad \Rightarrow \quad d = \frac{{\log (1/6)}}{{\log (r )}} = 1.86172\]

Special
Properties

Five copies of the pentigree fit together to form a set with five-fold rotational symmetry as illustrated in the following figure. Edgar calls this the "2nd form of McWorter's pentigree". [2nd Form Details]

[Alternate version]

Variations

Pentadentrite

The pentadentrite is a variation of McWorter's pentigree based on a different initial angle and scaling factor.

Pentadentrite displayed as 6 self-similar copies in different colors

William A. McWorter was born on July 15, 1932 and died in Delaware, Ohio on October 22, 2009. He received his Ph.D in mathematics from The Ohio State University in 1963 with a dissertation in an area of group theory, and eventually became an associate professor of mathematics at Ohio State. The pentigree was discovered by accident 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.
Photo Source: OSU Department of Mathematics Newsletter, Summer 2021

Classic Iterated Function Systems